Anomalous Thermodynamics of Nonideal Gas Physisorption on Nanostructured Carbons - CaltechTHESIS

Anomalous Thermodynamics of Nonideal Gas Physisorption on Nanostructured Carbons - CaltechTHESIS
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Anomalous Thermodynamics of Nonideal Gas Physisorption on Nanostructured Carbons
Citation
Murialdo, Maxwell Robert
(2017)
Anomalous Thermodynamics of Nonideal Gas Physisorption on Nanostructured Carbons.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z9GH9FXM.
Abstract
Mesoporous and microporous adsorbents play critical roles in gas storage and separation applications. This thesis describes previously unexplored anomalous thermodynamics in the field of gas physisorption and their impact on energy relevant gases including methane, ethane, krypton and carbon dioxide. Physisorption occurs when an adsorbent induces gas molecules to form a locally densified layer at its surface due to physical interactions. This increases gas storage capacity over pure compression and its efficacy is dependent on the surface area of the adsorbent and the isosteric heat of adsorption. The isosteric heat of adsorption is the molar change in the enthalpy of the adsorptive species upon adsorption and serves as a measure of adsorbent-adsorbate binding strength.
Unlike conventional adsorbate-adsorbent systems, which have isosteric heats of adsorption that decrease with surface loading, zeolite-templated carbon is shown to have isosteric heats of methane, ethane and krypton adsorption that increase with surface loading. This is a largely beneficial effect that can enhance gas storage and separation. The unique nanostructure and uniform pore periodicity of the zeolite-templated carbon promote lateral interactions among the adsorbed molecules that cause the isosteric heats of adsorption to increase with loading. These results have been tested and corroborated by developing robust fitting techniques and thermodynamics analyses. The anomalous thermodynamics are shown to result from cooperative adsorbate-adsorbate interactions among the nonideal species and are modeled with an Ising-type model.
As a second theme of this thesis, the study of nonideal gas adsorption has enabled the development of a Generalized Law of Corresponding States for Physisorption. A predictive understanding of high-pressure physisorption on a variety of adsorbents would facilitate the further development of tailored adsorbents and adsorption analysis. Prior attempts at developing a predictive understanding, however, have been hindered by nonideal gas effects.
By approaching physisorption from both empirical and fundamental perspectives, a Generalized Law of Corresponding States for Physisorption was established that accounts for a number of nonideal effects. This new Law of Corresponding States allows one to predict adsorption isotherms for a variety of classical gases from data measured with a single gas. In brief: "At corresponding conditions on the same adsorbent, classical gases physisorb to the same fractional occupancy."
Corresponding conditions are met when the reduced variables of each nonideal gas are equivalent, and fractional occupancy gives the fraction of occupied adsorption sites. This Law of Corresponding States for Physisorption is determined using monolayer, BET and Dubinin-Polanyi adsorption theories along with measured adsorption isotherms across a number of conditions and adsorbents. Furthermore, the anomalous cooperative adsorbate-adsorbate interactions discussed in this thesis are shown to be consistent with the Generalized Law of Corresponding States for Physisorption.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
adsorption, physisorption, thermodynamics, carbon materials, methane storage, gas separation
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Fultz, Brent T.
Thesis Committee:
Fultz, Brent T. (chair)
Bernardi, Marco
Faber, Katherine T.
Goddard, William A., III
Johnson, William Lewis
Defense Date:
28 July 2016
Funders:
Funding Agency
Grant Number
EFree (Energy Frontier Research in Extreme Environments Center)
DE-SC0001057
Record Number:
CaltechTHESIS:08292016-233907648
Persistent URL:
DOI:
10.7907/Z9GH9FXM
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Article Adapted for Ch. 5
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Article Adapted for Ch. 8 and Ch. 9
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
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Collection:
CaltechTHESIS
Deposited By:
Maxwell Murialdo
Deposited On:
23 Sep 2016 22:50
Last Modified:
25 May 2021 23:04
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Chapter 1

An Introduction to Adsorption
1. Overview
Recently, after I finished a talk on the thermodynamics of adsorption, an audience
member stood up with a critical comment. “But adsorption has been studied for over a
hundred years.” Yes, exactly!
Serious studies of adsorption have been ongoing for over a hundred years now.
Qualitative knowledge of adsorption goes back thousands of years. This is not to the detriment
of the field. Rather it is a testament to the broad utility of adsorption and the complexity of
trying to understand such a diverse array of phenomena.
Adsorption is the densification of a fluid at an interface. The nature of the interface
may be solid-liquid, gas-liquid, liquid-liquid or solid-gas. In this work we focus on adsorption
between a solid “adsorbent” and a gaseous “adsorptive species” which is denoted the
“adsorbate” in the adsorbed phase. Adsorption is strictly an interfacial phenomenon, unlike
“absorption” wherein the absorptive species penetrates the absorbent. Each molecule in the
adsorbed phase actually has less entropy than in the bulk gas phase; however it also has a
reduced enthalpy owing to attractive interactions at the interface. This sets up the basic
equilibrium of adsorption: a reduction in entropy compared to the gas phase is balanced by a
comparable reduction in enthalpy. Adsorption is thus an exothermic process. The favorable
interactions at the interface, which enable all adsorption, may be of chemical or physical
nature. When chemical bonds occur between the adsorbent and the adsorbate the
phenomenon is called chemisorption. When only physical interactions are present (e.g. van der

Waals forces), it is called physisorption. “Sorption” is a more general term used for the
ambiguous case. In general, chemisorption results in much stronger adsorbent-adsorbate
interactions and is effective at higher temperatures, but requires specific adsorbate-adsorbent
systems and is less reversible. In this work we focus on physisorptive systems.

2. Pores
The uses of physisorption are extremely diverse, the most primary of which is the
characterization of the adsorbents themselves. Solid-gas physisorption may occur on any solid,
but as an interfacial phenomenon, physisorption is strictly a surface effect. High surface area
materials, especially porous materials, therefore lend themselves to adsorptive studies. Often
porous materials can only be easily and effectively characterized by physisorption. The pores
of a porous material may be broadly lumped into one of three size categories: macropores,
mesopores and micropores. Macropores are greater than 50 nanometers in width. Mesopores
have widths between 2 and 50 nanometers. Micropores have widths of less than 2 nanometers.
Both micropores and mesopores are considered “nanopores”. Pores are typically modeled as
either cylinders with unique pore diameters, or slits with unique pore widths. Macropores,
mesopores and micropores each physisorb gases in distinct ways and require different
modeling.
In macropores the pore walls are sufficiently well separated to have non-overlapping
effects. This is because physical interactions like van der Waals forces fall off rapidly with
distance. Macropores can thus be treated in the same way as a nonporous material, where only
a single surface is considered at a time. In macropores, layer-by-layer adsorption is expected
and can be effectively modeled using Brunauer, Emmett and Teller (BET) Theory. BET
Theory has proven highly successful in determining the surface area of macroporous

adsorbents. Using subcritical adsorption isotherm measurements of nitrogen, argon, carbon
dioxide or krypton (for low surface area adsorbents), up to near-saturation pressures, the
monolayer coverage can be solved from BET Theory. This quantitative monolayer coverage is
in turn correlated with a specific surface area based on the size of the adsorbate molecule. One
caveat is that different results may be obtained with different gases (probe molecules) as
different pores and topologies are accessible to different size probe molecules.
Mesopores may be treated in much the same way as macropores, with the additional
complication of capillary condensation. In appropriately sized mesopores, surface tension
(through capillary action) can cause the adsorptive species to condense into a liquid phase in
the mesopores at pressures below the bulk saturation pressure. This phenomenon is unique to
mesopores as macropores are too large and micropores are too small. Capillary condensation
results in hysteresis between the adsorption and desorption isotherms. However, the presence
of capillary condensation can be used to obtain information about the pore-size distribution
via the Kelvin Equation and variants.
Micropores are of similar dimensions to the gases adsorbed. Within micropores,
opposite pore walls often exhibit overlapping potentials. Thus micropores are more accurately
treated with a pore-filling model than a layer-by-layer model for adsorption. In general the
adsorptive species will holistically fill the micropore volume, making metrics of specific
micropore volume more important than specific surface area. The micropore volume may be
determined empirically with the Dubinin-Radushkevich equation and variants. Moreover, of
critical importance, the micropore-size distribution may be obtained semi-empirically via
nonlocal density functional theory (NLDFT). As such, physisorption may be used to
characterize pores varying in size from 0.35 nanometers to greater than 100 nanometers.

3. Applications
Adsorbents are also widely used in industrial scale processes like catalysis and gas
separation. Catalysis is an enabling factor in ~90% of chemical and materials manufacturing
worldwide and may employ homogeneous or heterogeneous catalysts.1 Unlike homogeneous
catalysts, heterogeneous catalysts are of a different phase than the underlying reaction. Often
porous solids are used as heterogeneous catalysts in conjunction with a liquid or gas phase.
The fluid phase is first adsorbed onto the catalyst, followed by dissociation of the fluid, surface
diffusion, a surface reaction and finally desorption of the product. Adsorption is thus a
prerequisite for most heterogeneous catalysis and a fundamental understanding of the
adsorption process is vital. Moreover, as adsorbents typically have large accessible surface
areas, they may be used as structural supports that keep catalysts well dispersed to maximize
efficacy.
Gas separation and purification is a second key industrial use of adsorption. Separating
chemically inert gases or otherwise removing gaseous impurities is often done by cryogenic
distillation. For gases with similar boiling points this process can be both expensive and energy
intensive. Gas separation using physisorption offers an efficient alternative, especially when
high levels of purity are not required. In physisorptive separation, gases are flowed through an
adsorbent bed where one gas preferentially adsorbs over another, ideally with great selectivity.
The adsorbent bed may then be regenerated by reducing the pressure or increasing the
temperature to desorb the adsorbate. These cyclic processes are referred to as Pressure Swing
Adsorption (PSA) and Temperature Swing Adsorption (TSA), respectively. The selectivity is
typically due to differences in the adsorbent-adsorbate physical interactions between each gas
and the adsorbent. In some cases steric effects are used to enhance selectivity when the gases

to be separated are of dissimilar size or shape and only small and correctly shaped molecules
can penetrate a well-defined pore structure. This is commonly used to dry steam from the
cracking process or to dry natural gas using zeolites with well-defined micropore structures.
Kinetic separation mechanisms may also be employed in molecular sieves where non-uniform
pore size distributions allow different molecules to diffuse at different rates. One common
example is the separation of nitrogen from air using molecular sieves. Overall, adsorption
offers an efficient means of gas separation, purification and in some cases solvent recovery
from both industrial and vented sources.

4. Krypton
In this work we study krypton adsorption in detail as a step towards improving
krypton separation from other inert gases. Krypton, the fourth noble gas, is an unreactive
monatomic gas that otherwise bears many similarities to methane. The two gases share a
similar size (Kr: 3.9 Å, CH4: 4.0 Å)2 and approximately spherical symmetry, as well as similar
boiling points (120 K and 112 K, respectively)3 and critical temperatures (209 K and 190 K,
respectively). Conveniently, monatomic krypton allows for very simple calculations of
thermodynamic properties such as entropy, since rotational and internal vibrational modes do
not exist. Krypton has applications in the photography, lighting4 and medical industries5,6, and
is commonly used as an adsorbate for characterizing low-surface-area materials7,8. There is also
significant active interest in finding a cost effective and efficient means of separating krypton
from xenon derived from nuclear waste9,10.
Nuclear power plants supply over 10% of the world’s electricity and are a valuable
source of energy in the United States.11 Unfortunately, the United States is already fraught with
over 70,000 tons of nuclear waste and no good storage options12,13. One avenue towards

diminishing the amount of nuclear waste generated in the future is to reprocess spent
nuclear fuel. Reprocessing entails chopping up and dissolving spent nuclear fuel to recover
fissionable remains. These remains can be used to generate additional electricity often
exceeding 25% of the initial generation.14 While the US does not currently reprocess nuclear
fuel, reprocessing is typical in Europe, Russia and Japan, and as a member of the International
Framework for Nuclear Energy Cooperation, the US has partnered with other countries to
improve and develop closed nuclear fuel cycles with reprocessing. During reprocessing,
radioactive krypton-85 and nonradioactive xenon are off-gassed. As a radioactive mixture,
these gases should be stored as radioactive waste, often in inefficient mole fraction of ~90%
nonradioactive xenon to ~10% radioactive krypton-85.15 While cryogenic distillation is an
energy intensive means of separating krypton from xenon, adsorbents offer a potentially
efficient alternative. With properly tuned adsorptive separation, less nonradioactive xenon
would need to be stored, putting less stress on current nuclear waste storage options.

5. Natural Gas
Another large and quickly growing application of physisorption is for the densified
storage of natural gas, particularly within the transportation sector. This is a major focus of this
thesis.
Natural gas powers 22% of the world and 33% of the US, and yet its importance is
projected to grow.16 From 2010 to 2013 worldwide natural gas consumption has grown at a
rate of 2% annually from 113,858 to 121,357 billion cubic feet.16 Over the same time, the
proven reserves have grown at a rate of 1% annually, up to 6,972.518 trillion cubic feet.16

Assuming a 2% annual increase in consumption, but no increase in the proven reserves, our
current proven reserves will last into the 23rd century, although estimates vary.16
Natural gas predominantly originates from two naturally occurring processes. Biogenic
methane results when methanogenic archaea break organic matter into simple hydrocarbons
like methane. These microorganism live in oxygen depleted regions of the earth’s crust and in
the intestines of most animals. Collecting the gases emanating from manure and landfills has
proven a clean and renewable source of natural gas. Nonetheless, to date these operations are
small scale and pale in comparison to the scope of thermogenic natural gas collection.
Thermogenic methane is synthesized from ancient organic matter under the high temperatures
and pressures found deep in the earth’s crust. This process takes millions of years and is thus
not considered renewable on a human timescale. These natural gas fossil fuels are often
discovered alongside oil, although deeper deposits synthesized at higher temperatures tip the
balance towards a higher fraction of natural gas synthesis.
Thermogenic methane is also found alongside coal (coalbed gas) or as a methane
hydrate (clathrate). Methane hydrates form under temperatures below 15 °C and pressures
greater than 19 bar. These conditions are met in offshore continental shelves and the
permafrost of Siberia. Recently Japan Oil, Gas and Metals National Cooperation has recovered
commercially viable quantities of natural gas from oceanic methane clathrates.17 This presents a
particularly exciting advancement given estimates of vast quantities of oceanic methane
hydrates (estimated at 5x1015 cubic meters of methane hydrate).16
Other unconventional sources of natural gas have been studied and commercialized
within the past decade. These sources have historically been economically prohibitive, but this
is changing with recent technological advancements. The collection of deep gas, tight gas, and

shale gas are on the rise. Shale gas, in particular has boomed from less than 1% of the US
natural gas production in 2000 to 39% in 2012.16 It is estimated that there may be more than a
quadrillion cubic feet of unconventional natural gas reserves in the US alone.16
At present, 15 countries account for 84% of worldwide natural gas production, with
Russia, Iran, Qatar, Turkmenistan and the US at the top of the list.16 Before transportation or
use, the natural gas must be purified. Impurities like water, sand and other gases are separated
out. Some of the purified gas byproducts like propane, butane and hydrogen sulfide are sold
on secondary markets. Landfill methane is prone to have large quantities of carbon dioxide
and hydrogen sulfide, which must be removed before transportation to prevent pipeline
corrosion. The gas may be further purified to achieve a high quality gas comprised of almost
pure methane and known as “dry” natural gas. In the presence of significant quantities of
other hydrocarbons, natural gas is deemed “wet”. The main constituent in natural gas is always
methane, followed by ethane. The composition varies significantly but is generally in
accordance with Table 1.
Table 1. Typical Composition of Natural Gas18
gas
methane
ethane
propane
butane
alkanes #C > 4
nitrogen
carbon dioxide
oxygen
hydrogen

mole fraction (%)
87-97
1.5-7
0.1-1.5
0.02-0.6
trace
0.2-5.5
0.1-1.0
0.01-0.1
trace

The transportation and utilization of natural gas has been employed since 500 B.C.
when Chinese in the Ziliujing district of Sichuan developed crude natural gas pipelines from
bamboo.19 This natural gas was harnessed to boil seawater. It wasn’t until 1785, however, that
natural gas first found widespread commercialization as a fuel for streetlamps and homes in
England.19 This natural gas emanated from fissures above naturally occurring pockets. In 1821
William A. Hart drilled the first intentional natural gas well, in Fredonia, New York.19 Since
then the industry has exploded in the US and abroad. An emerging natural gas industry set to
explode is for use in the transportation sector. Worldwide tens of millions of natural gas
vehicles dot the roads. Ordinary internal combustions engines can be converted to run on
natural gas for less than $10,000 and commercial vehicles designed specifically to run on
natural gas are gaining traction. Starting in 2008 with the Honda Civic natural gas, major auto
manufacturers have churned out a number of compressed natural gas commercial vehicles
including the Chevrolet Silverado 2500, Dodge Ram 2500, Ford F-250 and Chevrolet Savana.
This surge in natural gas cars proffers environmental benefits. For the same amount of
energy, thermogenic methane emits 16% less carbon dioxide than diesel.16 Landfill biogas can
emit a net 88% less carbon dioxide than diesel.16 In power generation, methane power plants
emit almost 50% less carbon dioxide than coal-powered plants.16 The carbon dioxide released
per million BTU, for a variety of common fuels, is listed in Table 2. Methane is the cleanest
burning hydrocarbon. Additionally, natural gas emits less trace pollutants including carbon
monoxide, nitrogen oxides, sulfur oxides and particular matter than other fuels.

10
16

Table 2. Pounds of CO2 Emissions Per MBTU for Common Fuels
fuel
Coal (anthracite)
Coal (bituminous)
Coal (lignite)
Coal (subbituminous)
Diesel fuel & heating oil
Gasoline
Propane
Natural Gas

Lbs. CO2/MBTU
228.6
205.7
215.4
214.3
161.3
157.2
139
117

The main hindrance to wide implementation of natural gas in the transportation sector
stems from the onboard storage problem. Natural gas has a high gravimetric energy density of
-1

56 MJ kg , competitive with other fuel sources, but an abysmally low volumetric energy
-3

density of 37 MJ m at standard conditions.3 This is a common problem for gaseous fuels
with critical temperatures significantly below room temperature. Three potential solutions have
been separately implemented in commercial applications: cryogenic liquefaction, high-pressure
compression, and compression in the presence of adsorbent materials.
Cryogenic liquefaction requires cooling natural gas to below the critical temperature of
methane, which is 190K. This is typically achieved with liquid nitrogen as a coolant in a
cryogenic setup. While cryogenic liquefaction can achieve volumetric energy densities as high
-3

as 22 GJ m , it is an energy intensive process and requires expensive equipment and
monitoring3. These drawbacks have thus far prevented the widespread adoption of cryogenic
liquefaction as a means for natural gas storage on privately owned vehicles.

11
At high pressures of 700 bar, natural gas has a volumetric energy density of ~17 GJ
-3

m .3 However, high-pressure compression requires specialized storage tanks. As pressure
requirements are increased, the requisite class of storage tank shifts from Type I (all metal,
which is the cheapest and currently makes up 93% of the onboard natural gas storage market)
to Type IV, all composite with high associated costs.20 Moreover, high pressures limit potential
tank designs (as a necessity of eliminating weak points) and potentially pose a significant threat
if ruptured intentionally or unintentionally.
The use of adsorbents and moderate compression allows for significant volumetric
energy density improvements over pure compression at moderate pressures and temperatures.
At low and moderate pressures the favorable interactions between the adsorbent and the
adsorbate densify natural gas under equilibrium conditions. Improved natural gas storage is a
significant subject of inquiry in this thesis.
Adsorption also serves a number of other niche purposes in areas such as heat pumps
and spacecraft environmental controls.21 All told, adsorption spans chemistry, biology, physics
and engineering and is an integral part of our world. It demands further fundamental scientific
inquiry and expertise in adsorbent engineering.

12

References:
1.$Chorkendorff,$ I.;$ Niemantsverdriet,$ J.$ W.$ Concepts$ of$ Modern$ Catalysis$ and$
Kinetics,$2nd$ed.;$Wiley:$New$York,$2007.$
2.$Cuadros,$ F.;$ Cachadina,$ I.;$ Ahumada,$ W.$ Determination$ of$ LennardKJones$
Interaction$Parameters$Using$a$New$Procedure.$Mol.%Eng.%1996,$6,$319K325.$
3.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$version$8.0;$
National$Institute$of$Standards$and$Technology:$Gaithersburg,$MD,$2007;$CDKROM.$
4.$Hwang,$ H.$ S.;$ Baik,$ H.$ K.;$ Park,$ K.$ W.;$ Song,$ K.$ M.;$ Lee,$ S.$ J.$ Excitation$ Energy$
Transfer$of$Metastable$Krypton$Atoms$in$KrKHeKXe$Low$Pressure$Glow$Discharge$for$
MercuryKFree$Lighting.$Jpn.%J.%Appl.%Phys.%2010,$49,$1K3.$
5.$Chon,$D.;$Beck,$K.$C.;$Simon,$B.$A.;$Shikata,$H.;$Saba,$O.$I.;$Hoffman,$E.$A.$Effect$of$
LowKXenon$ and$ Krypton$ Supplementation$ on$ Signal/Noise$ of$ Regional$ CTKbased$
Ventilation$Measurements.$J.%Appl.%Physiol.%2007,$102,$1535K1544.$
6.$Pavlovskaya,$ G.$ E.;$ Cleveland,$ Z.$ I.;$ Stupic,$ K.$ F.;$ Basaraba,$ R.$ J.;$ Meersmann,$ T.$
Hyperpolarized$ KryptonK83$ as$ a$ Contrast$ Agent$ for$ Magnetic$ Resonance$ Imaging.$
Proc.%Natl.%Acad.%Sci.%U.S.A.%2005,$102,$18275K18279.$
7.$Takei,$ T.;$ Chikazawa,$ M.$ Measurement$ of$ Pore$ Size$ Distribution$ of$ LowKSurfaceK
Area$ Materials$ by$ Krypton$ Gas$ Adsorption$ Method.$ J.% Ceram.% Soc.% Jpn.% 1998,$ 106,$
353K357.$
8.$Youssef,$A.$M.;$Bishay,$A.$F.;$Hammad,$F.$H.$Determination$of$Small$SurfaceKAreas$
by$Krypton$Adsorption.$Surf.%Technol.%1979,$9,$365K370.$
9.$Ryan,$ P.;$ Farha,$ O.$ K.;$ Broadbelt,$ L.$ J.;$ Snurr,$ R.$ Q.$ Computational$ Screening$ of$
MetalKOrganic$Frameworks$for$Xenon/Krypton$Separation.$AIChE%J.%2011,$57,$1759K
1766.$
10.$Bae,$ Y.KS.;$ Hauser,$ B.$ G.;$ Colon,$ Y.$ J.;$ Hupp,$ J.$ T.;$ Farha,$ O.$ K.;$ Snurr,$ R.$ Q.$ High$
Xenon/Krypton$ Selectivity$ in$ a$ MetalKOrganic$ Framework$ with$ Small$ Pores$ and$
Strong$Adsorption$Sites.$Microporous%Mesoporous%Mater.%2013,$169,$176K179.$
11.$World$ Statistics:$ Nuclear$ Energy$ Around$ the$ World.$ Nuclear% Energy% Institute.$
(accessed$2016).$
12.$Stop$Dithering$on$Nuclear$WasteKThree$Decades$After$Chernobyl,$the$US$Needs$
to$Tackle$its$Own$Ominous$Nuclear$Safety$Problem.$Sci%Am.%2016,$314,$10.$
13.$Stuckless,$ J.$ S.;$ Levich,$ R.$ A.$ The$ Road$ to$ Yucca$ MountainKEvolution$ of$ Nuclear$
Waste$Disposal$in$the$United$States.$Environ.%Eng.%Geosci.%2016,$22,$1K25.$
14.$Processing$ of$ Used$ Nuclear$ Fuel.$ http://www.worldKnuclear.org/informationK
library/nuclearKfuelKcycle/fuelKrecycling/processingKofKusedKnuclearKfuel.aspx$
(accessed$2016).$
15.$Banerjee,$ D.;$ Cairns,$ A.$ J.;$ Liu,$ J.;$ Motkuri,$ R.$ K.;$ Nune,$ S.$ K.;$ Fernandez,$ C.$ A.;$
Krishna,$R.;$Strachan,$D.$M.;$Thallapally,$P.$K.$Potential$of$MetalKOrganic$Frameworks$
for$Separation$of$Xenon$and$Krypton.$Acc.%Chem.%Res.%2015,$48,$211K219.$
16.$Independent$ Statistics$ and$ Analysis.$ U.S.% Energy% Information% Administration.$

13
17.$Tabuchi,$H.$An$Energy$Coup$for$Japan:$'Flammable$Ice'.$New%York%Times,%%Mar.$
12,$2013,$B1.$
18.$Chemical$ Composition$ of$ Natural$ Gas.$ Union% Gas,$ https://www.uniongas.com$
(accessed$2016).$
19.$A$ Brief$ History$ of$ Natural$ Gas.$ American% Public% Gas% Association.$
20.$LeGault,$ M.,$ Pressure$ Vessel$ Tank$ Types.$ Composites% World.$
21.$Dabrowski,$ A.$ Adsorption$ K$ From$ Theory$ to$ Practice.$ Adv.% Colloid% Interface% Sci.%
2001,$93,$135K224.$

14
Chapter 2

Fundamentals of Physisorption
1. Background
Early work into determining a universal equation of state for all gases led to the
development of the ideal gas law:
!" = !"#

(1)

This law, derived by Emile Clapeyron in 1834, brings together 3 linear gas relationships:
Boyle’s Law, Charles’s Law and Avogadro’s Law. The ideal gas law is simple, functional and
fairly accurate at dilute conditions. However, at high pressures or low temperatures, the
assumptions of the ideal gas law, namely that gases are composed of non-interacting point
particles, break down. This shortcoming was addressed later in the 19th century by introducing
nonlinear gas equations of state such as the van der Waals equation:
!! !

! + !!

! − !" = !"#

(2)

Introduced by Johannes Diderick van der Waals in 1873, this equation of state
incorporates two gas-dependent parameters, a, and b, which account for the attractive
interactions between gas molecules, and the finite volume of real gas molecules, respectively.
Johannes van der Waals adamantly believed that gases collided as hard spheres and did not
possess any other repulsive interactions. This is now known to be false.
Real gases exhibit both attractive and repulsive interactions that are strongly correlated
with intermolecular spacing. For electrically neutral molecules, these forces (e.g. van der Waals)
typically fall into one of 4 categories: Keesom Forces, Debye Forces, London Dispersion
forces and Pauli repulsive forces. The first three result from some combination of permanent

15
and/or induced multipole interactions while the Pauli repulsive force is a purely quantum
effect.
The sum of the attractive and repulsive intermolecular interactions forms an
“interaction potential.” While early on it was generally agreed upon that van der Waals
interactions fall off quickly with increasing intermolecular spacing, the precise function was
unknown. A number of pair potentials were proposed including the commonly used LennardJones Potential, VLJ:
!!" = 4!

! !"

! !

(3)

where r is the intermolecular separation and σ and ε are the Lennard-Jones parameters, specific
to each gas. The Lennard-Jones potential balances the longer-range attractive interactions (that
-6

fall as r ) with the very short-range Pauli repulsion forces (arbitrarily modeled as falling with r
12

-6

-12

). While the r dependence derives from the London dispersion force, the r

repulsive

term has no physical basis. Thus while a useful heuristic, the Lennard-Jones potential is not
rigorously accurate in describing potentials between two molecules. In physisorption, gas
molecules interact with an adsorbent surface, which is generally considered to be much wider
than the molecule itself. Patchwise, these interactions may be modeled as between a flat
crystalline material and a small molecule, by generalizing the ideas of the Lennard-Jones
potential into a new form called the Steele potential1:

! ! = 2!!!" !! ! ! Δ

! ! !"

! !

!!
!! !!!.!"! !

(4)

16
where εsf is the solid-fluid well-depth given by Berthelot mixing rules2, ρs is the density of the
solid, Δ is interplanar spacing of the crystalline material, and σ is a Lennard-Jones type distance
parameter determined by Lorentz mixing rules2.
In a dynamic view, gases may collide with a solid interface, either elastically or
inelastically. Occasionally a gas molecule that collides inelastically will undergo an interaction
with the surface, wherein it is briefly localized by the surface potential. This is the essence of
physisorption at the microscale. The surface potential may vary over the surface due to
impurities, defects, or overall structural features. In microporous materials, the surface
potential heterogeneity is largely dictated by the pore-size distribution. Physisorptive systems
typically have only shallow (weak) surface potentials that allow adsorbed molecules to explore
multiple sites on the two-dimensional potential surface before reentering the gas phase. This is
known as a mobile adsorption, as opposed to localized adsorption, which is typically associated
with deeper potential wells and chemisorption.
From a fundamental thermodynamics perspective, adsorption compresses a 3dimensional gas phase into a 2-dimensional adsorbed phase. This presents a significant drop in
the molar entropy of the adsorptive species. The difference in molar entropy between the
adsorbed and gas phases (at constant coverage) is called the isosteric entropy of adsorption
(ΔSads). The gas-phase entropy (Sg) may be read from data tables. The adsorbed-phase entropy
(Sa) depends on a number of factors including coverage.
In order to establish equilibrium, the isosteric entropy of adsorption must be offset by
a comparable decrease in molar enthalpy upon adsorption. The difference between the
adsorbed-phase enthalpy (Ha) and the gas-phase enthalpy (Hg) (at constant coverage) is called

17
the isosteric enthalpy of adsorption (ΔHads). Physisorption is always exothermic, yielding a
negative isosteric enthalpy of adsorption. By convention, the isosteric heat of adsorption (qst) is
defined as a positive quantity as follows
!!" ≡ − !! − !!

(5)

The isosteric heat of adsorption may be thought of as a proxy metric of the binding energy
between the adsorbent and adsorbate that results from the interaction potential.
Over time an adsorbed molecule may explore many adsorption sites, but due to
energetic constraints, the most favorable sites (with the largest isosteric heats) will have the
highest average occupation. For conventional adsorbents, the highest isosteric heat values are
observed at the lowest coverage. As coverage is increased, the most favorable adsorption sites
become saturated. This leads to a decreasing isosteric heat (an average quantity) with increasing
coverage.

2. History of Adsorption
Simple adsorptive applications have been employed since at least 1550 BC, when
records indicate that the Egyptians made use of charcoals to adsorb putrid gases expelled
during human dissection.3 Scientific adsorptive experiments are more recent. Scheele in 1773,
followed by Fontana in 1777, were the first scientists to measure the uptake of gases by porous
solids.3 Saussure built upon this work, and in 1814 determined that adsorption was exothermic
in nature.3 A theoretical understanding of adsorption followed far behind experiments. It
wasn’t until 1888 that Bemmelen made the first known attempt at fitting adsorption data,
introducing a fitting equation now known as the “Freundlich Equation.”3 More precise

18
terminology soon followed when Bois-Reymond and Kayser introduced the term
“adsorption” into standard scientific lexiconography.3
By the 20th century, the field of adsorption was full steam ahead. 1903 saw the
discovery of selective adsorption (Twsett)4. In 1909 McBain introduced the term “absorption”
to differentiate bulk uptake from the surface phenomenon of “adsorption.”5 Within the next
few decades physisorption had been cast into a number of rigorous theoretical frameworks,
namely Eucken-Polanyi Theory (1914)6, the Langmuir Isotherm (1918)7, BET Theory (19351939)8,9,10 and Dubinin Theory (1946)11.

3. Adsorption Theory
Apart from limited calorimetric work, the “adsorption isotherm” forms the
fundamental basis of adsorption measurement and theory. Constant-temperature isotherms
may be measured in a number a ways but ultimately yield the same information: Gibbs excess
adsorption as a function of pressure. A simple model for adsorption may be drawn up as
follows (Figure 1).

Figure 1. Cartoon depiction of adsorption. The gray rectangle represents the adsorbent

19
surface. All blue molecules represent absolute adsorption molecules. The dark blue circles
represent excess adsorption molecules. The red line indicates a dividing surface. The green
circles represent the gas-phase molecules.
Here section 1 represents the solid adsorbent surface, section 2 represents the
adsorbed phase (densified molecules near the interface), and section 3 represents the molecules
that remain unaffected by the adsorbent and remain free in the gas phase. The quantity of
absolute adsorption (na) comprises all of the molecules in the adsorbed phase. The volume of
the adsorbed phase (Vads), however, is not rigorously established. Thus absolute adsorption
cannot be directly measured via experiment. Instead Gibbs worked around this problem by
defining excess adsorption (ne) as follows:
!! = !! − !!"# !(!, !)

(6)

Gibbs excess adsorption differs from absolute adsorption by an amount equal to the volume
of the adsorption layer multiplied by the density of the free gas phase (ρ(P,T)). Here the
volume of the adsorbed phase is defined as the volume between the adsorbent surface and a
dividing surface. The quantity of Gibbs excess adsorption measures the amount of adsorbate
in the vicinity of the adsorbent surface that is in excess of the free gas phase density. For this
reason, the Gibbs excess adsorption is directly measurable by volumetric or gravimetric
methods without need for assumptions about the volume of the adsorbed layer. Absolute
adsorption, however, cannot be easily and directly measured. Instead it is often crudely
assumed that absolute adsorption equals excess adsorption. This assumption is only valid
when the gas phase is dilute, and quickly breaks down at high pressures and low temperatures.
Thus in this thesis, we instead use a robust fitting method (presented in Chapter 3) to back out

20
reasonable values of the absolute adsorption.
The first attempts to fit physisorption isotherms came as early as 1888 when
Bemmelen introduced what is now known as the “Freundlich Equation”12,3:

= !!!

(7)

where n is the uptake, m is the mass of the adsorbent, P is equilibrium pressure, and K and η
are adsorbent-specific parameters. This equation is only empirical and does not purport to
capture or contain the physics of adsorption. Nonetheless, it displays key behaviors that are
common to fit functions for type 1 isotherms. At low pressures, uptake increases linearly with
pressure per Henry’s Law:
! = !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(8)!
where KH is the Henry’s Law constant. Henry’s Law was first formulated by William Henry in
1803 and can be derived from ideal gas assumptions. If we assume that the adsorbed phase
takes the form of a two-dimensional ideal gas following
!! !! = !!"

(9)

where Pa and Aa are the spreading pressure and surface area, respectively, and also assume that
the equilibrium spreading pressure is proportional to the equilibrium bulk gas pressure, then
! !

!"!

! = !!"! = !"! = !! !

(10)

Accordingly, the low-pressure regime of an isotherm where uptake is proportional to applied
gas pressure is commonly referred to as the Henry’s Law regime. At higher pressures, the
adsorption isotherm saturates as all of the available adsorption sites are filled. This correlation,
however, is not exact as unlike absolute adsorption, experimentally measured excess
adsorption data behaves in a non-monotonic fashion at high pressures. Nonetheless, most type

21
1 adsorption fit functions subsume both the Henry’s Law regime behavior and the saturation
regime behavior.
The Langmuir isotherm, derived by Irving Langmuir in 19187, provides a fundamental
model for gas adsorption that incorporates both the Henry’s Law regime the saturation regime
behavior. Here the adsorbed phase is assumed to exist as a monolayer of adsorbate directly
above the adsorbent surface. Langmuir’s model makes a number of simplifying assumptions as
follows:
1. The adsorbent surface is perfectly flat.
2. The gas adsorbs into an immobile state.
3. There is a finite number of adsorption sites that can each by filled by no more than
one adsorbate molecule.
4. All adsorption sites are energetically identical.
5. Adsorbate molecules do not interact.

With these simplifying assumptions in hand, the Langmuir adsorption isotherm may be
derived from kinetic theory, statistical mechanics, or from a phenomological perspective, and
takes the form:
!"

! = !!!"

(11)

where θ is fractional occupancy and K is an equilibrium constant given by an Arrhenius-type
equation (Equation 12):
!! =

!!

!!!

! !"

where Ai is a prefactor and Ei is an energy of the ith isotherm.

(12)

22
Unfortunately, the five simplifying assumptions above are almost never entirely
satisfied and the Langmuir isotherm cannot be applied over broad ranges of conditions. Many
of the drawbacks of the Langmuir isotherm may be overcome by fitting excess adsorption data
with a weighted superposition of Langmuir isotherms (see Chapter 3).
In particular, the Langmuir model breaks down when multilayer adsorption is possible,
as found in larger micropores, mesopores, and macropores. In 1938 Stephen Brunauer, Paul
Emmett, and Edward Teller extended the Langmuir model to consider multilayer adsorption.8
They realized that in multilayer adsorption, molecules do not successively fill one complete
monolayer after another. Rather, fragments of multilayer stacks of varying sizes dot the
adsorbent surface. Each layer is in dynamic equilibrium with the layers above and below it,
much in the same way that the Langmuir model assumes a dynamic equilibrium between the
adsorbed monolayer and the gas phase above it. The Brunauer, Emmett, Teller, or BET
method has been elaborated on in detail in literature13,14 and will not be rederived here. Rather,
the results and key insights are elucidated from the BET equation:

!!
!!

=!

!!!

!"# ! !!

+!

!"# !

(13)

where n is uptake, P is equilibrium pressure, Po is saturation pressure, nmax is maximum possible
uptake, and C is the BET constant. BET Theory assumes that infinite layers may be adsorbed
successively on a surface. Moreover, these layers do not interact with one another and each
follow the Langmuir model. Two additional assumptions are made:
1. The E1 parameter is the isosteric heat between the adsorbent and first adsorbed layer.
2. All higher layers have an EL parameter equal to the heat of liquefaction of the
adsorbate.

23
BET theory has proven particularly useful at measuring the specific surface areas of
porous carbons. For high quality surface area determinations, nitrogen, argon, carbon dioxide,
and krypton have been used. In particular, a plot of
line in the relative pressure range of 0.05<

!!

!!
!!

vs

!!

should yield a straight

<0.3. Using linear regression, the slope and y-

intercept of this line are determined. The parameters nmax and C are determined by:

!!"# = !"#$%!!"#$%&$'#
!=

!"#$%
!"#$%&$'#

+1

(14)
(15)

The surface area may then be determined from nmax, using the established cross-sectional area
of the probe molecule.
While the Langmuir isotherm rapidly gained popularity and contributed to Irving
Langmuir’s 1932 Nobel Prize, it competed with Polanyi’s theory of adsorption, which has now
earned its place in annals of science history. Whereas Langmuir conceptualized adsorption as a
monolayer effect localized at the adsorbent surface, Polanyi’s approach was more amenable
holistic pore filling with longer-range effects. Polanyi reasoned that the density of adsorptive
molecules near a surface diminishes with distance from the attractive surface, much the way
the atmosphere of a planet thins out at high altitudes. For adsorption this requires a longerrange interaction potential, now called the Polanyi Adsorption Potential6.

Polanyi recognized that at equilibrium, the chemical potential (µ) of the adsorbed
phase at an arbitrary distance, x, from the interface and a corresponding pressure Px, must

24
equal the chemical potential of the gas phase at an infinite distance and corresponding bulk
pressure, P.
! !, !! = ! ∞, !

(16)

! !,!!
! !,!

(17)

!" =! ! !, !! − ! ∞, ! = 0

Moreover,
!" = −!"# + !"# + !"

(18)

where S is entropy, V is volume and U is the potential energy. Constant temperature
(isothermal) conditions yield
!" = !"# + !"

(19)

! !,!!
! !,!

(20)

!" =! ! ! !"#! + ! ! − ! ∞ = 0

where U(x) is the potential energy at a distance x from the surface and U(∞) is the potential
energy at an infinite distance, which Polanyi took to be zero:

−! ! = ! ! ! !"#

(21)

By substituting in the ideal gas law
! !"

−! ! = ! ! ! ! !"

(22)

or
! ! = !!"#$

!!

≡!

(23)

where A is the Polanyi potential. While the Polanyi potential went unappreciated for many
years, it was given new life in 1946 when Dubinin and Radushkevich introduced the “theory of
the volume filling of micropores (TVFM)”.11,15,16 In this theory the Polanyi potential is the
negative of the work done by the sorption system:

25
! = −∆!

(24)

This insight extended the Polanyi potential to broad thermodynamic analysis, codified in the
Dubinin-Radushkevich equation:
! = !!"!! !

! !
!!!

(25)

where n is the uptake, nmax is the maximum possible uptake, β is the affinity coefficient, and Eo
is the standard characteristic energy. Other modified and more generalized forms were later
introduced, such as the Dubinin-Astakhov equation17

! = !!"#! ! !!!

(26)

where χ is an adsorbent-specific heterogeneity parameter.
If uptake is plotted as a function of the Polanyi potential, the Dubinin-Radushkevich
equation15 yields a single characteristic curve for each gas-adsorbent system. In theory the
characteristic curve may be used to predict uptake over a wide range of temperatures and
pressures, and its accuracy has been generally confirmed by experiment.18,19 Moreover, plotting

ln(n) as a function of − !!

yields a linear trend, wherein the y-intercept gives the maximal

uptake (nmax) and the slope gives the characteristic energy (Eo). The parameter nmax may be
used to determine the total micropore volume of the adsorbent by multiplying by the
established molecular volume of the adsorbate. The parameter Eo may be used to estimate an
average micropore width.

26

References:
1.$Siderius,$D.$W.;$Gelb,$L.$D.$Extension$of$the$Steele$10K4K3$Potential$for$Adsorption$
Calculations$ in$ Cylindrical,$ Spherical,$ and$ Other$ Pore$ Geometries.$ J.% Chem.% Phys.%
2011,$135,$084703K1K7.$
2.$Boda,$D.;$Henderson,$D.$The$Effects$of$Deviations$from$LorentzKBerthelot$Rules$on$
the$Properties$of$a$Simple$Mixture.$Mol.%Phys.%2008,$106,$2367K2370.$
3.$Dabrowski,$ A.$ AdsorptionKFrom$ Theory$ to$ Practice.$ Adv.% Colloid% Interface% Sci.%
2001,$93,$135K224.$
4.$Berezkin,$V.G.$(Compiler).$Chromatographic$Adsorption$Analysis:$Selected$Works$
of$M.S.$Tswett.$Elis%Horwood:%New$York,$1990.$
5.$McBain,$ J.$ W.$ The$ Mechanism$ of$ the$ Adsorption$ ("Sorption")$ of$ Hydrogen$ by$
Carbon.$Philos.%Mag.%1909,$18,$916K935.$
6.$Polanyi,$M.,$Potential$Theory$of$Adsorption.$Science.%1963,$141,$1010K1013.$
7.$Langmuir,$ I.$ The$ Adsorption$ of$ Gases$ on$ Plane$ Surfaces$ of$ Glass,$ Mica$ and$
Platinum.$J.%Am.%Chem.%Soc.%1918,$40,$1361K1403.$
8.$Brunauer,$ S.;$ Emmett,$ P.$ H.;$ Teller,$ E.$ Adsorption$ of$ Gases$ in$ Multimolecular$
Layers.$J.%Am.%Chem.%Soc.%1938,$60,$309K319.$
9.$Brunauer,$ S.;$ Emmett,$ P.$ H.$ The$ Use$ of$ Low$ Temperature$ van$ der$ Waals$
Adsorption$ Isotherms$ in$ Determining$ the$ Surface$ Areas$ of$ Various$ Adsorbents.$ J.%
Am.%Chem.%Soc.%1937,$59,$2682K2689.$
10.$Brunauer,$ S.;$ Emmett,$ P.$ H.$ The$ Use$ of$ van$ der$ Waals$ Adsorption$ Isotherms$ in$
Determining$the$Surface$Area$of$Iron$Synthetic$Ammonia$Catalysts.$J.%Am.%Chem.%Soc.%
1935,$57,$1754K1755.$
11.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR%Phys.%Chem.$1947,$55,$331K337.$
12.$Skopp,$ J.$ Derivation$ of$ the$ Freundlich$ Adsorption$ Isotherm$ from$ Kinetics.$ J.%
Chem.%Educ.%%2009,$86,$1341K1343.$
13.$Gregg,$ S.$ J.;$ Jacobs,$ J.$ An$ Examination$ of$ the$ Adsorption$ Theory$ of$ Brunauer,$
Emmett,$ and$ Teller,$ and$ Brunauer,$ Deming,$ Deming$ and$ Teller.$ T.% Faraday% Soc.%
1948,$44,$574K588.$
14.$Legras,$ A.;$ Kondor,$ A.;$ Heitzmann,$ M.$ T.;$ Truss,$ R.$ W.$ Inverse$ Gas$
Chromatography$ for$ Natural$ Fibre$ Characterisation:$ Identification$ of$ the$ Critical$
Parameters$ to$ Determine$ the$ BrunauerKEmmettKTeller$ Specific$ Surface$ Area.$ J.%
Chromatogr.%A.%2015,$1425,$273K279.$
15.$Nguyen,$ C.;$ Do,$ D.$ D.$ The$ DubininKRadushkevich$ Equation$ and$ the$ Underlying$
Microscopic$Adsorption$Description.$Carbon.%2001,$39,$1327K1336.$
16.$Dubinin,$ M.$ M.$ Generalization$ of$ the$ Theory$ of$ Volume$ Filling$ of$ Micropores$ to$
Nonhomogeneous$Microporous$Structures.$Carbon.%1985,$23,$373K380.$
17.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
18.$Saeidi,$N.;$Parvini,$M.$Accuracy$of$DubininKAstakhov$and$DubininKRadushkevich$
Adsorption$ Isotherm$ Models$ in$ Evaluating$ Micropore$ Volume$ of$ Bentonite.$ Period.%
Polytech.%Chem.%2016,$60,$123K129.$

27
19.$Do,$ D.$ D.;$ Nicholson,$ D.;$ Do,$ H.$ D.$ Adsorption$ in$ Micropores$ (Nanopores):$ A$
Computer$Appraisal$of$the$Dubinin$Equations.$Mol.%Simul.%2009,$35,$122K137.$

38
Chapter 4

Materials

Three carbonaceous materials, MSC-30, CNS-201, and zeolite-templated carbon (ZTC) were
studied repeatedly with different gases and under differing conditions in this work. These
materials were obtained and characterized in varied ways described herein.

1. MSC-30
MSC-30 (Maxsorb) is a microporous superactivated carbon obtained from Kansai
Coke & Chemicals Company Ltd. (Japan) and is an “AX21-type” superactivated carbon. MSC30 is synthesized by activating petroleum coke with molten KOH in a process patented by
Standard Oil Company (later Amoco Corporation).
Nitrogen adsorption isotherms were carried out at 77K in a Micromeritics ASAP 2420.
The specific surface area was determined by applying BET theory to the data as implemented
by Micromeritics ASAP 2420 version 2.02 software. The BET surface area of MSC-30 was
determined to be 3244+ 28 m2 g-1. Using nonlocal density functional theory (NLDFT)1 and a
slit-pore model, the MSC-30 pore-size distribution was determined as shown in Figure 1.
MSC-30 has a broad range of pore sizes (from 6 to 35 Å). Over 40% of the micropore volume
is contained in pores of greater than 21 Å (width). The micropore volume was found to be
1.54 cm3 g-1 by the Dubinin-Radushkevich method2,3. The skeletal density was measured by
helium pycnometry and the skeletal density determined to be 2.1 g cm-3.

39
Cu Kα X-ray diffraction of MSC-30 on a PANalytic Pro powder diffractometer gave
a broad peak at 2θ= 34 degrees, in accordance with that reported for AX21. The elemental
composition (CHN) was determined by the Dumas method4 in triplicate combustion
experiments, indicating that 1.16 wt% of MSC-30 is hydrogen. X-Ray Photospectroscopy
(XPS) measurements were made on a Kratos AXIS Ultra DLD spectrometer and the results
are summarized in Table 1. Electron Energy Loss Spectroscopy (EELS) measurements were
made on a FEI Technai F20 with a Gatan Imaging Filter system. MSC-30 has an sp3
hybridized carbon content of 16%. Transmission electron microscope (TEM) images were
taken with a Tecnai TF30 with a LaB6 filament and 80 keV electrons (Figure 2).

Figure 1. The pore-size distribution (left) of MSC-30 (red), CNS-201 (black), and ZTC (purple)
as calculated by the NLDFT method.

40

Figure 2. TEM image of MSC-30 and an accompanying fast Fourier transform of the image.
Table 1. Summary of XPS data on MSC-30 and ZTC
peak
position
(eV)
component
ZTC
MSC-30

285.0
285.7 286.4
287.3 288.1
289.4 290.2 291.5
C-C sp C-C sp C-OR C-O-C C=O COOR
53.4
18.0
8.6
6.0
1.1
4.2
1.0
7.7
48.0
18.8
6.8
4.8
6.1
4.2
3.6
7.7

2. CNS-201
CNS-201 is a microporous activated carbon obtained from A. C. Carbone Inc.
(Canada). It is synthesized by pyrolysis of coconut shells. Nitrogen adsorption isotherms were
carried out at 77K in a Micromeritics ASAP 2420. The specific surface area was determined by
applying BET theory to the data as implemented by Micromeritics ASAP 2420 version 2.02
software. As determined by nitrogen adsorption and BET analysis, CNS-201 has a surface area
of 1095 + 8 m2 g-1. Using the nitrogen adsorption data and a slit pore model, NLDFT1 pore-

41
size analysis was conducted to determine the pore size distribution (Figure 1). CNS-201 has
a three dominant pore widths of 5.4, 8.0, and 11.8 Å, containing roughly 50%, 20%, and 15%
of the total micropore volume respectively. The micropore volume was found to be 0.45 cm3
g-1 by the Dubinin-Radushkevich method2,3. The skeletal density was measured by helium
pycnometry and the skeletal density determined to be 2.1 g cm-3. Transmission electron
microscope (TEM) images were taken with a Tecnai TF30 with a LaB6 filament and 80 keV
electrons (Figure 3).

Figure 3. TEM image of CNS-201 and an accompanying fast Fourier transform of the image.

42

3. Zeolite-Templated Carbon
Zeolite-Templated Carbon (ZTC) is a microporous templated carbon synthesized in
multi-gram quantities with the following procedure at HRL Laboratories.
3.1 ZTC Synthesis
Faujasite-type Zeolum® zeolite Nay, HSZ-320NAA (faujasite structure, Na cation,
SiO2/Al2O3 = 5.5 mol/mol) (NaY) was obtained from Tosoh Corporation. 6.0 grams of
zeolite NaY were dried under vacuum at 450 °C for 8 hours. This powder was cooled and
mixed with 12 mL of furfuryl alcohol (98% Sigma Aldrich) and stirred under Argon for 24
hours. The resulting zeolite-furfuryl alcohol mixture was separated by vacuum filtration and
rinsed four times in 100 mL aliquots of xylenes.
Next the powder was loaded in a quartz boat and placed in quartz tube-furnace/CVD
reactor. The tube furnace was purged with argon and held under argon at 80 °C for 24 hours.
The reactor was heated to 150 °C (under argon) for 8 hours to induce polymerization. Next
the temperature was ramped up at a rate of 5 degrees °C per minute to a final temperature of
700 °C whereupon the gas flow was switched to a 7% propylene/ 93% nitrogen mixture for 4
hours. After 4 hours, the reactor was purged with argon at 700 °C for 10 minutes. Next the
temperature was increased to 900 °C and held for 3 hours under argon.
This product was then cooled and transferred to a PTFE beaker. 200 milliliters of
aqueous hydrofluoric acid (48% Sigma Aldrich) were added. The solution was left for 16 hours
before collecting the ZTC by vacuum filtration and rinsing 10 times with 50 mL aliquots of
water. Finally the ZTC product was dried at 150 °C under vacuum. Careful control of the

43
inert atmosphere and thorough drying were found to be critical to obtaining high surface
area product.
3.2 ZTC Characterization
Nitrogen adsorption isotherms at 77K were measured on a BELSORP-max
volumetric instrument (BEL-Japan Inc.). This data was analyzed with BET theory to
determine a specific surface area of 3591 + 60 m2 g-1 (among the highest to date for
carbonaceous materials)5. Using nitrogen adsorption data and a slit-pore model, NLDFT1
pore-size analysis was conducted to determine the pore-size distribution (Figure 1). Other
geometrical models including a cylindrical pore model were also tried, but none fit the data
better than the slit-pore model. ZTC has a narrow pore-size distribution centered at 12 Å.
Over 90% of the micropore volume is contained in pores of widths between 8.5 and 20$ Å.
The micropore volume was found to be 1.66 cm3 g-1 by the Dubinin-Radushkevich method2,3.
The skeletal density was measured by helium pycnometry and determined to be 1.8 g cm-3.
This is lower than most carbonaceous adsorbents (2.1 g cm-3), likely owing to a higher percent
of hydrogen terminations. The elemental composition (CHN) was determined via the Dumas
method4 in triplicate combustion experiments, indicating that 2.44 wt% of ZTC is hydrogen.
This higher percentage of hydrogen terminations may result from the hydrofluoric acid
treatment during synthesis.
Cu Kα X-ray diffraction of ZTC was measured with a PANalytic X’Pert Pro powder
diffractometer and produced a single sharp peak at 2θ= 6 degrees, indicative of the template
periodicity (14 Å). No signal from the original zeolite material was registered in the final
product. The absence of other peaks suggests that the product is (as expected) amorphous

44

carbon without any remnant zeolite. Applying the Scherrer equation to the peak (with a
Scherrer constant K=0.83 for spherical particles) suggests an ordering length scale of 24 nm.
X-Ray Photospectroscopy (XPS) measurements were made on a Kratos AXIS Ultra
DLD spectrometer and the results are summarized in Table 1. No significant differences were
noted as compared to MSC-30. Electron Energy Loss Spectroscopy (EELS) measurements
were made on a FEI Technai F20 with a Gatan Imaging Filter system. Similar to MSC-30,
ZTC has an sp3-hybridized carbon content of 18%. Transmission electron microscope (TEM)
images were taken with a Tecnai TF30 with a LaB6 filament and 80 keV electrons (Figure 4).
The samples were prepared by placing the powder on an aluminum foil square and putting a
lacey carbon copper TEM grid on top to pick up some particles. Unlike MSC-30 and CNS201, ZTC shows nongraphitic crystalline order in the Fourier transform of the TEM images.
In the fast Fourier transform of the images (Figure 4), ZTC shows spots indicative of a
spacing of 1.2 to 1.3 nanometers, in agreement with other measurements of the periodicity of
the pores. The spots also suggest hexagonal symmetry in the pore arrangement.

Figure 4. TEM image of ZTC and an accompanying fast Fourier transform of the image.

45

References:
1.$Tarazona,$P.;$Marconi,$U.$M.$B.;$Evans,$R.$PhaseKEquilibria$of$Fluid$Interfaces$and$
Confined$ Fluids$ K$ Nonlocal$ Versus$ Local$ Density$ Functionals.$ Mol.% Phys.% 1987,$ 60,$
573K595.$
2.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR%Phys.%Chem.$1947,$55,$331K337.$
3.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
4.$Dumas,$J.$B.$A$Method$of$Estimating$Nitrogen$in$Organic$Material.$Ann.%Chim.%Phys.$
1833,$58,$171K173.$
5.$Yang,$ Z.$ X.;$ Xia,$ Y.$ D.;$ Mokaya,$ R.$ Enhanced$ Hydrogen$ Storage$ Capacity$ of$ High$
Surface$Area$ZeoliteKLike$Carbon$Materials.$J.%Am.%Chem.%Soc.%2007,$129,$1673K1679.$
6.$Langford,$ J.$ I.;$ Wilson,$ A.$ J.$ C.$ Scherrer$ After$ 60$ Years$ K$ Survey$ and$ Some$ New$
Results$in$Determination$of$Crystallite$Size.$J.%Appl.%Crystallogr.%1978,$11,$102K113.$

46
Chapter 5

Methane on Microporous Carbons
N.P. Stadie, M. Murialdo, C.C. Ahn, and B. Fultz, “Unusual Entropy of Adsorbed Methane on
Zeolite-Templated Carbon,” J. Phys. Chem. C, 119, 26409 (2015).
DOI: 10.1021/acs.jpcc.5b05021
N.P. Stadie, M. Murialdo, C.C. Ahn, and B. Fultz, “Anomalous Isosteric Enthalpy
of Adsorption of Methane on Zeolite-Templated Carbon,” J. Am. Chem. Soc.
135, 990 (2013).
DOI: 10.1021/ja311415m

1. Adsorptive Uptake
The ability to store the necessary quantities of natural gas in inexpensive and
moderately sized onboard tanks remains a significant challenge to the wide-spread adoption of
natural gas as a fuel in the transportation sector. By filling the onboard tanks with specially
designed physisorptive materials, the storage capacity can be significantly improved and the
volumetric energy density significantly increased. Here we study high-pressure methane
adsorption on three microporous carbons (ZTC, MSC-30, and CNS-201). The zeolitetemplated carbon (ZTC) is of particular interest due to its unique synthesis and morphology (a
narrow pore-size distribution centered at 1.2 nm).
Methane isotherms were measured at 13 temperatures from 238 to 526K and up to
pressures of ~10 MPa in a volumetric Sieverts apparatus, commission and verified for accurate
measurements up to 10 MPa1. Multiple adsorption runs were completed at each temperature

47
with research-grade methane (99.999%) and errors between cycles were less than 1%.
Adsorption/desorption cycles demonstrated full reversibility of the isotherms. The excess
adsorption isotherms are plotted in Figure 1. Unlike absolute adsorption, excess adsorption
reaches a maximum at high pressures. This maximum is a readily accessible figure of merit for
the gravimetric performance of a material at a fixed temperature. The excess maximum is
similar for ZTC-3 and MSC-30 at room temperature, but slightly higher for MSC-30 at 14.5
mmol g-1 at 8 MPa. While excess adsorption increases faster for MSC-30 at pressures between
0 and 0.8 MPa, uptake in ZTC-3 increases fastest between 0.8 and 5.7 MPa. CNS-201 has
much lower maxima due to its significantly smaller surface area. The highest measured excess
uptake of this study is for ZTC-3 at 238 K: 22.1 mmol g-1 (26.2 wt%) at 4.7 MPa, despite a
gentler initial increase at low pressure. Interestingly, the excess uptake in ZTC-3 is also greater
than for MSC-30 at high temperatures, although neither reaches a maximum between 0 and 9
MPa. At all temperatures, methane uptake in ZTC-3 is characterized by a gradual initial rise
and delayed increase at pressures between 0.2 and 2 MPa, leading to higher eventual methane
capacity than MSC-30, a material of comparable specific surface area. The measured excess
adsorption maxima (at a sample temperature of 298K) scale linearly with specific surface area
and are consistent with the reported linear trend for methane uptake at 3.5 MPa and 298 K.2

48

Figure 1. Measured methane excess adsorption as a function of temperature and pressure on
CNS-201 (top), MSC-30 (middle), and ZTC (bottom). The curves indicate the best fit obtained
with a dual-Langmuir fitting function.

49

2. Analysis
The excess adsorption data were fitted with a superposition of two Langmuir
isotherms, as detailed in Chapter 3. In general, the best-fit parameters obtained correlate with
physical properties of the materials studied. For example, the parameter indicative of the
maximum volume of the adsorbed layer, Vmax, can be independently verified through
comparison to the micropore volume of the adsorbent as measured with the DubininRadushkevich method3,4. In ZTC, if taken to be proportional to surface area, the Vmax
parameter corresponds to half of the mean pore width of the material: a thickness of 0.6 nm.
Likewise, the maximum possible adsorption quantity, given by parameter nmax, correlates well
with an estimate determined by the product of the micropore volume and the molar liquid
density of the adsorbate (methane)5.

50

Figure 2. The isosteric heats of methane adsorption on CNS-201 (top), MSC-30 (middle), and
ZTC (bottom) as a function of temperature and fractional site occupancy (θ).
The isosteric enthalpy of adsorption is the molar change in enthalpy of the adsorptive
species upon adsorption. While adsorption is an exothermic process, the isosteric heat of
adsorption is reported as a positive quantity by convention, as shown in Figure 2. These curves
were obtained by applying the Clapeyron relationship to the dual-Langmuir fits. It is necessary
to use the general form of the Clapeyron relationship for methane adsorption at high pressure

51
because of the significant nonideality of methane gas-state properties. Its derivation and
explanation with respect to the ideal-gas form of the equation are given in Chapter 3.
The Henry’s Law value of adsorption heat, -ΔH0, is calculated by extrapolating the
heat of adsorption to zero pressure. The Henry’s law values for CNS-201, MSC-30, and ZTC3 are 18.1-19.3, 14.4-15.5, and 13.5-14.2 kJ mol-1, respectively. The isosteric heats of methane
adsorption as a function of fractional occupancy, θ, in the activated carbons (CNS-201 and
MSC-30) are typical of other carbon materials, with the isosteric heats decreasing with θ. In the
range 0 < θ < 0.6, the more graphitic CNS-201 shows a more gradual decrease of isosteric
heat than MSC-30, indicative of more heterogeneous site energies in MSC-30. Surprisingly, the
isosteric heat of adsorption in ZTC increases to a maximum at θ = 0.5-0.6 at temperatures
from 238 to 273 K. This increase is anomalous compared to previous experimental reports of
methane adsorption on carbon.
This anomalous effect results from adsorbate-adsorbate intermolecular interactions, as
suggested by theoretical work.6,7,8 We have reported similar effects for ethane (Chapter 6) and
krypton (Chapter 7). Accurately assessing the contribution of intermolecular interactions to the
isosteric heat requires knowledge of the adsorption binding-site energies. A heterogeneous
distribution of site energies, as in MSC-30, is reflected in the relatively rapid decrease of the
isosteric heat with θ. This behavior is common as the most favorable sites are filled first (on
average). The material properties of ZTC, including a narrow distribution of pore width,

periodic pore spacing, and high content of sp -hybridized carbon, suggest a high degree of
homogeneity of the binding-site energies. We expect that the measured increase of 0.5 kJ mol-1
in the isosteric heat at 238K reflects most of the contribution from favorable intermolecular

52
interactions, and this increase is in good agreement with calculations of lateral interactions of
methane molecules on a surface.6,7An increasing isosteric heat, as seen with methane on ZTC,
is highly desirable as it enhances deliverable storage capacity. This effect enables a larger
fraction of the adsorption capacity at pressures above the lower bound of useful storage.
Indeed, the deliverable gravimetric methane capacities of ZTC at temperatures near ambient
are the highest of any reported carbonaceous material.
A clearer picture of adsorption thermodynamics can be achieved by evaluating the
specific enthalpy of the adsorbed phase, as shown in Figure 3. This removes the gas-phase
dependency of the isosteric heat and focuses solely on adsorbed-phase thermodynamics. This
is particularly useful because at a constant temperature, it is a reasonable approximation that
the specific properties of the adsorbed phase as a function of increasing site occupancy do not
depend on contributions from internal (intramolecular) phenomena. Here, a decreasing
enthalpy of the adsorbed phase as a function of uptake, as seen on ZTC, corresponds to an
“increasing” (or, decreasing negative) isosteric heat of adsorption.

53

Figure 3. Adsorbed-phase enthalpy of CNS-201 (top), MSC-30 (middle), and ZTC (bottom) as
a function of temperature and fractional occupancy (θ).
The adsorbed-phase entropy was determined in a similar manner that is described in
more detail in Chapter 3. Adsorbed-phase entropy isotherms are shown in Figure 4. While all

54
three adsorbed phases show qualitatively similar entropies, a notable difference is seen
between the smaller pore materials (CNS-201 and ZTC) and MSC-30, which has a significant
fraction of pores of widths >2 nm. The molar entropy of methane adsorbed on CNS- 201 and
ZTC at 238 K approaches the value of the liquid reference state rather closely (within 12 J K-1
mol-1), indicating a liquid-like character of the adsorbed layer, unlike on MSC-30 (reaching a
minimum of 22 J K-1 mol-1 at 238 K). We must note, however, the relatively arbitrary nature
of the reference state; the entropy of saturated liquid CH4 varies by ~45 J K-1 mol-1 along its
liquidus phase boundary.

55

Figure 4. Adsorbed-Phase Entropy of methane on CNS-201 (top), MSC-30 (middle), and ZTC
(bottom) as a function of temperature and fractional occupancy. Curves indicate
corresponding statistical mechanics estimates.
For comparison, statistical mechanics calculations were carried out to independently
estimate the adsorbed-phase entropy. The only experimental parameter used in the theoretical
calculations was the material’s specific surface area. A remarkable consistency between theory

56
and experiment is observed across all three adsorbents, especially in the limit of high
temperature and low pressure where the approximations in the theoretical model are most
justified9. Of particular note, the theoretical calculations very closely reproduce the measured
molar entropies of the adsorbed phase on MSC-30, where the largest errors between θ = 0-0.2
are <3.5% at all temperatures measured (see Figure 4). The agreement between experimental
data and the statistical mechanical calculations for the adsorbed phase on CNS-201 is similarly
close in the dilute limit, but strays significantly beyond θ = 0.35. On ZTC, however, the
estimated adsorbed-phase entropy exceeds measured values, especially at low temperatures and
moderate coverage. This suggests the mechanism by which enhanced adsorbate-adsorbate
interactions are promoted in the adsorbed phase on ZTC, namely clustered configurations.
For ZTC, confinement of an adsorbed phase in narrow pores is likely to lead to
clustering as a result of enhanced lateral interactions. The formation of such clusters, or
adsorbate “islands”, on an adsorbent surface due to attractive intermolecular interactions, is a
well-known feature of physisorption of strongly interacting molecular species (e.g., methanol
on indium-tin-oxide glass10) and moderately interacting molecular species (e.g., subcritical CO2
on several MOFs and zeolites11,12) and also of chemisorption of fairly weakly interacting atomic
species (e.g., oxygen on Pt(111)13). This clustering behavior results in a reduction in entropy
due to the reduced number of accessible configurations of the cluster(s) in the same total
number of sites. This is consistent with the reduced entropy measured for methane on ZTC as
compared to the statistical mechanics estimate. The topic of clustering accounting for
enhanced adsorbate-adsorbate interactions is investigated in more detail in Chapter 8.
As a general note, the entropies of methane adsorbed between 238 and 526 K on the

57
various carbon materials measured in this work are very high as compared to some historical
theoretical estimates, approaching values of bulk gaseous methane in the dilute limit. In fact,
this observation has recently been made across many materials and adsorbed molecular
species.14 The ratio of adsorbed-phase entropy to the gas-phase entropy for CH4 on MSC-30,
for example, spans from 0.3 (at 238 K and 2 MPa) to 0.8 (at 0.1 MPa and 521 K), similar to the
value reported for methane on graphite(0001) in the dilute limit: 0.76 at 55 K.14,15

58

References:
1.$Bowman,$R.$C.;$Luo,$C.$H.;$Ahn,$C.$C.;$Witham,$C.$K.;$Fultz,$B.$The$Effect$of$Tin$on$the$
Degradation$of$LaNi5KySny$MetalKHydrides$During$Thermal$Cycling.$J.%Alloys%Compd.%
1995,$217,$185K192.$
2.$Sun,$ Y.;$ Liu,$ C.$ M.;$ Su,$ W.;$ Zhou,$ Y.$ P.;$ Zhou,$ L.$ Principles$ of$ Methane$ Adsorption$
and$Natural$Gas$Storage.$Adsorption.%2009,$15,$133K137.$
3.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR%Phys.%Chem.%Sect.$1947,$55,$331K337.$
4.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
5.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$Version$8.0$
[CDKROM],$2007.$
6.$Salem,$ M.$ M.$ K.;$ Braeuer,$ P.;$ von$ Szombathely,$ M.;$ Heuchel,$ M.;$ Harting,$ P.;$
Quitzsch,$ K.;$ Jaroniec,$ M.$ Thermodynamics$ of$ HighKPressure$ Adsorption$ of$ Argon,$
Nitrogen,$ and$ Methane$ on$ Microporous$ Adsorbents.$ Langmuir.% 1998,$ 14,$ 3376K
3389.$
7.$Sillar,$ K.;$ Sauer,$ J.$ Ab$ Initio$ Prediction$ of$ Adsorption$ Isotherms$ for$ Small$
Molecules$ in$ MetalKOrganic$ Frameworks:$ The$ Effect$ of$ Lateral$ Interactions$ for$
Methane/CPOK27KMg.$J.%Am.%Chem.%Soc.%2012,$134,$18354K18365.$
8.$AlKMuhtaseb,$ S.$ A.;$ Ritter,$ J.$ A.$ Roles$ of$ Surface$ Heterogeneity$ and$ Lateral$
Interactions$on$the$Isosteric$Heat$of$Adsorption$and$Adsorbed$Phase$Heat$Capacity.$
J.%Phys.%Chem.%B.%1999,$103,$2467K2479.$
9.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Unusual$ Entropy$ of$ Adsorbed$
Methane$on$ZeoliteKTemplated$Carbon.$J.%Phys.%Chem.%C%2015,$119,$26409K26421.$
10.$Wang,$L.;$Song,$Y.$H.;$Wu,$A.$G.;$Li,$Z.;$Zhang,$B.$L.;$Wang,$E.$K.$Study$of$Methanol$
Adsorption$ on$ Mica,$ Graphite$ and$ ITO$ Glass$ by$ Using$ Tapping$ Mode$ Atomic$ Force$
Microscopy.$Appl.%Surf.%Sci.%2002,$199,$67K73.$
11.$Krishna,$R.;$van$Baten,$J.$A.$Investigating$Cluster$Formation$in$Adsorption$of$CO2,$
CH4,$and$Ar$in$Zeolites$and$Metal$Organic$Frameworks$at$Suberitical$Temperatures.$
Langmuir.%2010,$26,$3981K3992.$
12.$Krishna,$R.;$van$Baten,$J.$M.,$Highlighting$a$Variety$of$Unusual$Characteristics$of$
Adsorption$and$Diffusion$in$Microporous$Materials$Induced$by$Clustering$of$Guest$
Molecules.$Langmuir.%2010,$26,$8450K8463.$
13.$Parker,$ D.$ H.;$ Bartram,$ M.$ E.;$ Koel,$ B.$ E.$ Study$ of$ High$ Coverages$ of$ Atomic$
Oxygen$on$the$Pt(111)$Surface.$Surf.%Sci.%1989,$217,$489K510.$
14.$Campbell,$C.$T.;$Sellers,$J.$R.$V.$The$Entropies$of$Adsorbed$Molecules.$J.%Am.%Chem.%
Soc.%2012,$134,$18109K18115.$
15.$Tait,$S.$L.;$Dohnalek,$Z.;$Campbell,$C.$T.;$Kay,$B.$D.$NKAlkanes$on$Pt(111)$and$on$
C(0001)/Pt(111):$ Chain$ Length$ Dependence$ of$ Kinetic$ Desorption$ Parameters.$ J.%
Chem.%Phys.%2006,$125,$234308K1K15.$

59
Chapter 6

Observation and Investigation of Increasing Isosteric Heat of
Adsorption of Ethane on Zeolite-Templated Carbon
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, "Observation and Investigation of
Increasing Isosteric Heat of Adsorption of Ethane on Zeolite-Templated Carbon," J. Phys.
Chem. C, 119, 994 (2015).
DOI: 10.1021/jp510991y
Abstract
Ethane adsorption was measured on zeolite-templated carbon (ZTC) and compared to
superactivated carbon MSC-30. Isotherms measured at temperatures between 252 and 423 K
were fitted using a superposition of two Langmuir isotherms and thermodynamic properties
were assessed. Unlike typical carbon adsorbents, the isosteric heat of adsorption on ZTC
increases by up to 4.6 kJ mol-1 with surface coverage. This increase is attributed to strong
adsorbate-adsorbate intermolecular interactions, a hypothesis that is shown to be consistent
with fundamental estimates of intermolecular interactions. Furthermore, the molar entropy of
the adsorbed phase was measured and compared to an estimate derived from statistical
mechanics. While the measured and estimated entropies of the adsorbed phase of ethane on
MSC-30 are in agreement, they differ significantly on ZTC at high coverage, indicative of the
atypical properties of ethane adsorption on ZTC.

60

1. Introduction
Ethane is the second most abundant component in natural gas and an important
petrochemical feedstock. It is a common reactant for the synthesis of ethylene, and its
separation from natural gas has been an important process for many years. Currently, the
separation of ethane from natural gas is predominantly carried out via cryogenic distillation, an
energy intensive process.1 Physisorption materials have been proposed as a more efficient
separation solution.2,3,4 Physisorption materials also hold promise in improving the volumetric
energy density of stored ethane.5,6 An understanding of ethane adsorption is thus essential for
natural gas storage and separation processes.
Physisorption occurs when weak physical interactions between a solid adsorbent and a
gas induce the formation of a locally densified adsorbate layer at the solid surface. This
interaction depends sensitively on the surface chemistry and surface structure of the
adsorbent.7,8,9,10 The isosteric heat of adsorption, qst, is often reported as a critical figure of
merit for physisorption. This proxy measure of binding energy determines the equilibrium
uptake quantity at finite temperatures and pressures.
Microporous carbons have gained significant attention as potential adsorbents due
their light weight, low cost, and wide variability.11,12,13 However, these materials often exhibit
binding energies that are below desired optimal values.14,15 Furthermore, the isosteric heat of
adsorption typically decreases with surface coverage due to binding site heterogeneity, further
reducing the deliverable gas storage capacity in the range of practicality for applications. We
recently reported the observation of increasing isosteric heat of adsorption of methane on
zeolite-templated carbon,16 a unique and anomalous behavior with respect to methane which
typically has very weak intermolecular interactions. Recently, Yuan et al. reported the synthesis

61
of a mesoporous carbon wherein the isosteric heat of ethane adsorption increases as a
function of coverage.17 This was attributed to favorable ethane-ethane intermolecular
interactions and a relatively homogeneous adsorbent surface. While intriguing, the latter results
have limited applications for gas storage and separation due to limitations of the mesoporous
carbon in question. With an average pore width of 48 Å and a specific surface area of 599 m2
g-1, this mesoporous carbon has only a small total excess uptake capacity of ~2.5 mmol g-1 at
278 K. This effect has not been investigated or observed for ethane adsorption on a microporous
carbon with a large specific surface area, a system that would have superior potential for
advanced applications. In this work, we report that a zeolite-templated carbon (ZTC) with a
narrow distribution of pore widths centered at 12 Å and a large specific surface area of 3591
m2 g-1 exhibits an increasing isosteric heat of ethane adsorption as a function of coverage. This
material has an exceptional uptake capacity of 22.8 mmol g-1 (at 252K), owing to its very large
surface area and optimized structural properties. Furthermore, several novel analysis methods
including comparisons to methane adsorption, Lennard-Jones parameters and statistical
mechanics calculations are implemented to corroborate and assist in the understanding of the
phenomenon of increasing isosteric heat of adsorption.

2. Experimental Methods
2.1 Materials Synthesis
Two materials were chosen for comparison in this study: MSC-30 and ZTC. The
superactivated carbon “Maxsorb” MSC-30 was obtained from Kansai Coke & Chemicals
Company Ltd. (Japan). The zeolite-templated carbon (ZTC) was synthesized in a multistep
process that was optimized to achieve high template fidelity of the product.18 The faujasite-

62
type zeolite NaY (obtained from Tosoh Corp., Japan) was impregnated with furfuryl alcohol
which was subsequently polymerized at 150 °C, augmented by a propylene CVD step at 700
°C, and carbonized at 900 °C. The ZTC product was freed by dissolution of the zeolite
template in HF. ZTC was confirmed to exhibit very high fidelity with the zeolite template and
outstanding microporous periodicity by X-ray diffraction and transmission electron
microscopy, described in detail elsewhere.16
2.2 Materials Characterization
Equilibrium nitrogen adsorption isotherms were measured at 77 K using a BELSORPmax volumetric instrument from BEL-Japan Inc. The Dubinin-Radushkevich (DR) method19,20
was employed to calculate micropore volumes and the Brunauer-Emmett-Teller (BET)
method21 was used to calculate specific surface areas. Pore-size distributions were determined
by non-local density functional theory (NLDFT) analysis implemented by software provided
by Micromeritics Instrument Corp., and a carbon slit-pore model was utilized.22 Skeletal
densities of the materials were determined by helium pycnometry. Finally, the Dumas
method23 was employed to determine the elemental composition (CHN) of MSC-30 and
ZTC.24
2.3 Measurements
Equilibrium ethane adsorption isotherms were measured on ZTC and MSC-30 at 9
temperatures between 252 and 423 K. Research grade ethane (99.999%) obtained from
Matheson Tri-Gas Inc. was used in a custom Sieverts apparatus that was tested for accuracy up
to 10 MPa.25 The reactor containing the sample was held at a constant set temperature while
the remaining gas manifold always remained at room temperature. For low temperature
isotherms, the reactor was submerged in a circulated chiller bath leading to temperature

63
deviations no larger than +0.1 K. For high temperature isotherms, the reactor was encased
in a copper heat exchanger and wrapped with insulating fiberglass-heating tape. A PID
controller was used to maintain a constant temperature with fluctuations of less than +0.4 K.
In both setups, the temperature was measured by K-type thermocouples placed in direct
contact with the reactor. The manifold temperature was measured with a platinum resistance
thermometer in contact with the outer wall of the manifold.
High-pressure measurements were made with an MKS Baratron (Model 833) pressure
transducer. For degassing, vacuum pressures were ensured with a digital cold cathode pressure
sensor (I-MAG, Series 423). The Sieverts apparatus is equipped with a molecular drag pump
capable of achieving vacuum of 10-4 Pa. Each sample was degassed at 520 K under vacuum of
less than 10-3 Pa prior to measurements. Multiple adsorption/desorption isotherms were
measured to ensure reversibility, and errors between identical runs were less than 1%. Gas
densities were determined using the REFPROP Standard Reference Database.26

3. Results
3.1 Adsorbent Characterization
BET specific surface areas of ZTC and MSC-30 were found to be 3591 + 60 and 3244
+ 28 m2 g-1, respectively. Likewise, both materials have similar DR micropore volumes: 1.66
mL g-1 (ZTC) and 1.54 mL g-1 (MSC-30). The distribution of the pore sizes, however, differs
significantly between the two materials. ZTC was determined to have a uniform pore-size
distribution centered at 12 Å (see Figure 1). Over 90% of the micropore volume of ZTC is
contained in pores with widths between 8.5 and 20 Angstroms. MSC-30 exhibits a wide range
of pore sizes from 6 to 35 Å. 40% of the pore volume of MSC-30 is contained in pore widths

64
greater than 21 Angstroms. Furthermore, while MSC-30, like most activated carbons, has a
skeletal density of 2.1 g mL-1, ZTC has an unusually low skeletal density of 1.8 g mL-1 (in
agreement with other ZTCs).24 This discrepancy can be explained by the significantly higher
hydrogen content found in ZTC by elemental analysis experiments. Hydrogen was found to
account for 2.4% (by weight) of ZTC but only 1.2% of MSC-30.24

Figure 1. Pore-size distribution and relative pore volume of ZTC (orange) and MSC-30
(purple).

65
3.2 Adsorption Measurements:
Equilibrium ethane adsorption isotherms were measured at 9 temperatures between
252 and 423K and at pressures of up to 32 bar (see Figure 2). The maximum excess adsorption
quantities measured on ZTC and MSC-30 (at ~253K) were 22.8 and 26.8 mmol g-1,
respectively. At room temperature (297K), the maximum uptake quantities measured on ZTC
and MSC-30 were 19.2 and 22.1 mmol g-1, respectively. Thus at both temperatures, MSC-30
exhibits greater excess adsorption capacities, with differences between the maximums being
less than 15%. This is in contrast to methane adsorption on the same two materials, where
ZTC exhibited higher excess adsorption capacities than MSC-30 at low temperatures.16
Moreover, the differences in excess adsorption capacities between the two materials are
smaller (less than 5%) for methane adsorption.

66

Figure 2. Equilibrium excess adsorption isotherms of ethane on ZTC and MSC-30. The lines
indicate the best-fit analysis using a superposition of two Langmuir isotherms.

4. Discussion
4.1 Fitting Methodology
Thermodynamic analysis of adsorption requires interpolation of the adsorption data
points, generally with a fitting function. It is common in literature to assume that the excess

67
adsorption well approximates the absolute adsorption. This assumption, while valid at low
pressures and high temperatures, is invalid at temperatures near the critical point, particularly
in high-pressure studies. To determine the absolute adsorption quantities and avoid the welldocumented errors associated with equating excess adsorption and absolute adsorption, we
follow a method initially described by Mertens.27 We extend this method by modifications for
the nonideal gas regime.16
The Gibbs excess adsorption28 (ne) is a function of the bulk gas density (ρ):$

!! = ! !! − ! !! !

$$$$$$$$$$$$$$$$$$$$$

(1)

Determining the absolute adsorption quantity (na) is simplest when the volume of the
adsorption layer (Va) is known. We left Va as an independent fitting parameter, and assessed it
later. The measured excess adsorption quantities were fitted with a generalized (multi-site)
Langmuir isotherm:
!! (!, !) = !!"# − !!"# !(!, !)

!! !
! !! !!! !

(2)

Excess adsorption and density are functions of pressure (P) and temperature (T). The
independent fitting parameters in this fitting model are nmax, a scaling factor indicative of the
maximum absolute adsorption, αi, a weighting factor for the ith Langmuir isotherm(

! !! =

1), Va, which scales with coverage up to Vmax (the maximum volume of the adsorption layer),
and Ki , an equilibrium constant for the ith Langmuir isotherm. The parameter Ki is defined by
an Arrhenius-type equation:
!! =

!!

! !!! !"

(3)

68
Here, Ai is a prefactor and Ei is a binding energy associated with the ith Langmuir
isotherm. Using two superimposed isotherms (i=2), we obtained satisfactory results while
limiting the number of independent fitting parameters to 7: nmax, Vmax, α1, A1, A2, E1, and E2.
For ZTC and MSC-30, accurate fits were obtained with residual mean square values of 0.21
and 0.13 (mmol g-1)2. These fits are shown in Figure 2 and the best-fit values of the fitting
parameters are given in Table 1. For comparison, fitting parameters for methane adsorption on
ZTC and MSC-30 were obtained using the same fitting procedure16, and are also shown in
Table 1.
Table 1. Least-squares minimized fitting parameters of the excess adsorption isotherms of
ethane on MSC-30 and ZTC described by a two-site Langmuir isotherm.
Ethane on
ZTC
Ethane on
MSC-30
Methane on
ZTC
Methane on
MSC-30

nmax
25 mmol/g
36 mmol/g
36 mmol/g
41 mmol/g

Vmax
1.6
mL/g
2.6
mL/g
2.0
mL/g
2.3
mL/g

α1
0.82
0.71
0.46
0.70

A1
2.1E-7
K1/2/MPa
0.086
K1/2/MPa
0.059
K1/2/MPa
0.068
K1/2/MPa

A2
0.044
K1/2/MPa
0.0065
K1/2/MPa
0.00018
K1/2/MPa
0.0046
K1/2/MPa

E1
41
kJ/mol
20
kJ/mol
12
kJ/mol
13
kJ/mol

E2
18
kJ/mol
18
kJ/mol
20
kJ/mol
13
kJ/mol

Many of the independent fitting parameters in this method have physical significance.
For example, nmax represents the maximum specific absolute adsorption of the system. If the
entire micropore volume of the material is assumed to be completely filled at this condition,
then it follows that it should be approximately comparable to the value obtained by
multiplying the density of the liquid phase of the adsorbate by the total micropore volume.

69
26

Using the density of liquid ethane and liquid methane near the triple point, the maximum
possible adsorption quantities estimated in this simplified way for ethane and methane on
MSC-30 are 33 and 43 mmol g-1, respectively. These values are within 10% of the nmax values
determined through fitting. For ethane and methane adsorption on ZTC, the estimated and
fitted values of nmax both deviate more significantly, with estimated values greater by 44% and
32% respectively.
The Vmax parameter approximates the maximum volume of the adsorbed layer.
Dividing by surface area, this gives an average width for the adsorbed layer. For ethane and
methane adsorption on ZTC, this gives average adsorbed layer widths of 4.5 and 5.5 Å, both
of which are in agreement with the measured ZTC micropore half-width of 6 Å. This suggests
fairly effective filling of the ZTC micropores. Likewise, the ethane and methane Vmax
parameters on MSC-30 give adsorbed layer widths of 8 and 7 Å, which are in agreement with
the average measured micropore half-width, 7 Å. Furthermore, for ethane on ZTC, Vmax
equals 1.6 mL g-1, which is in good agreement with the micropore volume measured using the
DR method and nitrogen isotherms, 1.66 mL g-1. The Vmax values for ethane on MSC-30 and
methane on both materials deviate more significantly from measured DR micropore volumes
with discrepancies of up to 41%. This, however, is not unexpected as different adsorbates with
differing size and shape are confined and adsorbed in micropores distinctly.
4.2 Isosteric Enthalpy of Adsorption
The isosteric enthalpy of adsorption (ΔHads) is a widely used figure of merit that is
indicative of the strength of binding interactions at a fixed temperature, pressure, and
coverage. Typically it is determined using the isosteric method and reported as a positive value,

70
qst, the so-called isosteric heat (a convention that is followed herein), defined according to the
Clapeyron equation:

!!" = −∆!!"# = −!

!"
!" !!

Δ!!"#

(4)

The molar change in volume of the adsorbate upon adsorption (Δvads) is given by the
difference between the gas-phase molar volume and

!!"#
!!"#

. The fitting method used in this

work is especially convenient for thermodynamic calculations with the Clapeyron equation
because the generalized Langmuir equation with i=2 can be analytically differentiated.27 The
isosteric heats of adsorption of ethane on ZTC and MSC-30 calculated in this way are shown
in Figure 3.

71

Figure 3. Isosteric heats of ethane adsorption on ZTC and MSC-30.
It can be observed that the isosteric heat of adsorption of ethane on ZTC increases as
a function of coverage while MSC-30 (and a majority of other known systems) exhibits the
typical decreasing isosteric heat with coverage (in this case represented as the absolute uptake
quantity). This effect is most pronounced at low temperatures where the isosteric heat rises by
4.6 kJ mol-1 above its Henry’s Law value of 20.6 kJ mol-1 (at 252 K). This effect (an increasing
isosteric heat of adsorption) has also been observed for methane adsorption on the same
material (ZTC)16 and is dependent on gas properties as well as structural and surface properties

72
of the adsorbent. In particular, the effect is expected to increase with the strength of
intermolecular interactions of the adsorbate when adsorbate molecules are on nanostructured
surfaces that promote intermolecular interactions. ZTC is an ideal candidate adsorbent for
observing such a phenomenon due to its uniform pore-size distribution with pore widths
centered at 12 Å and homogeneous chemical nature24.
In the absence of intermolecular interactions and binding site heterogeneity, the
isosteric heat should be constant at all coverages and temperatures. The increase in isosteric
heat (as a function of coverage) reported in this work is hypothesized to result from attractive
intermolecular interactions between ethane molecules. Assuming random site occupancy in the
low coverage regime (e.g. less than 50% of the available sites filled), the probability of any site
being occupied is equal to the fractional site occupancy, θ. If z is the number of nearest
neighbor adsorption sites, on average an adsorbate molecule will have zθ occupied nearest
neighbor adsorption sites. By assuming that nearest neighbor interactions have a binding
energy of ε and higher order neighbors have a negligible binding energy, the average
adsorbate-adsorbate

binding

energy

per

molecule,

!!!

!= !

ξ,

is

given:

(5)

Taking the derivative of ξ with respect to θ gives the estimated slope of the isosteric heat as a
function of coverage (resulting from adsorbate-adsorbate interactions):
!(!)

!"

= !
!!

(6)

The Lennard-Jones parameter ε (which describes the well depth of the Lennard-Jones 12-6
interaction potential) is 1.7 kJ mol-1 for ethane-ethane interactions29. For adsorbed molecules
on a two-dimensional surface the number of nearest neighbor adsorption sites, z, is posited to

73
-1

be 4. This results in an estimated slope for ethane on ZTC of 3.4 kJ mol . The average slope
of the measured isosteric heat of adsorption of ethane on ZTC (Figure 3) has a similar value of
3.3 kJ mol-1 (at the lowest measured temperature 252K, up to 50% coverage). Interestingly,
these measured and estimated slopes are of a similar magnitude to the slope of the isosteric
heat of ethane adsorption on the mesoporous carbon of Yuan et al. (~4-5 kJ mol-1 at
~298K).17
This very simple model (Equation 6) also gives a reasonable estimate for the increase in the
isosteric heat of methane adsorption on ZTC. Here, ε for methane-methane interactions is 1.2
kJ mol-1,29 giving a predicted slope of 2.4 kJ mol-1. This is in agreement with the measured
slope of 2.2 kJ mol-1 (up to 50% coverage at 255K).16 It is important to note that z=4, while
intuitively reasonable, is a posited value. The true value of z is difficult, if not impossible, to
obtain, and likely varies for different gases.
A number of metrics suggest that ethane has stronger attractive intermolecular
interactions than methane by a factor of ~1.4 to 1.6. These metrics include Lennard-Jones
potential (adsorbate-adsorbate) well depth, boiling point, and critical temperature, and are
shown in Table 2.
Table 2. Gas properties of ethane and methane and their ratios.

ε (Lennard-Jones)
Boiling Point (1 atm)
Critical Temperature

ethane
1.7
kJ/mol29
184.57 K26
305.32 K26

methane
1.2
kJ/mol29
111.67 K26
190.56 K26

ratio
1.4
1.6528
1.6022

In agreement with the ratios in Table 2, the ratio of the slopes of the ethane and
methane heats of adsorption (as a function of coverage) on ZTC is 1.5. Stronger

74
intermolecular interactions within ethane correspond to an isosteric heat that increases more
steeply than methane on the same material. Furthermore, the average Henry’s Law values (zero
coverage limit) of the isosteric heat were determined to be 20 and 21 kJ mol-1 for ethane on
ZTC and MSC-30, respectively, and 14 and 15 kJ mol-1 for methane. For both ZTC and MSC30, these ratios of Henry’s law values are 1.4 for ethane and methane.
4.3 Entropy
At equilibrium, the Gibbs free energy of the adsorbed phase (Ga) equals the Gibbs free
energy of the gas phase (Gg). The isosteric entropy of adsorption (ΔSads) is
Δ!!"# = !" − !" = Δ!!"# − !Δ!!"# = 0
Δ!!"# =

!!!"#

(7)
(8)

The isosteric entropy of adsorption is the change in entropy of the adsorbate upon adsorption
and the measured values for ethane adsorption on MSC-30 and ZTC are shown in Figure 4
(reported as positive values).

75

Figure 4. The isosteric entropy of ethane adsorption on ZTC and MSC-30 between 252 and
423 K.
By adding the isosteric entropy of adsorption to the gas-phase entropy
(from Refprop26) we obtain the molar entropy of the adsorbed phase (Figure 5).

76

Figure 5. The molar adsorbed-phase entropies of ethane on ZTC and MSC-30. The curves
indicate measured data and the points are calculated values (from statistical mechanics).

4.4 Statistical Mechanics
For comparison, the adsorbed-phase molar entropies were also calculated using
statistical mechanics. Adsorbed ethane has numerous entropic contributions including
vibrational, rotational, and configurational components. Each contribution was accounted for

77
as follows using standard partition functions and established values for characteristic
frequencies. The effects of intermolecular interactions between the adsorbed molecules were
not included, and the resulting discrepancies are discussed following the analysis.
Ethane in the gas phase has 12 internal vibrational modes with well-known characteristic
vibrational frequencies and degeneracies,30 and minimal changes in these frequencies are
expected upon adsorption.

Near ambient temperature, these vibrational modes are of little

significance to the adsorbed-phase entropy, but their influence increases with temperature (see
Figure 6). The partition function for vibrational modes (qvib i) is:
!!"#!! ! =

! !!!"#!! !!
!!! !!!"#!! !

(9)

Here, Θvib i is the characteristic vibrational temperature of the ith vibrational mode.
Adsorbed gases also vibrate with respect to the adsorbent surface. While the partition
function for these vibrations is also given by Equation 9, the characteristic frequency is not
readily accessible. Instead we have estimated these characteristic harmonic frequencies by

!=

!!

(10)

Here K is the force constant and m is the molecular mass (of ethane). For simplicity (in the
absence of detailed knowledge of the adsorbent surface geometry) the adsorbate-adsorbent
potential was modeled as a Lennard-Jones potential with ε given by the average Henry’s law
value of the isosteric heat per molecule. For a Lennard-Jones potential, the force constant and
frequency are:
!"!

!=! !

(11)

78
!=

!"!

!!

!(!! )!

(12)

Here rm is the distance wherein the potential reaches its minimum. Rotationally, ethane is a
symmetric top with characteristic frequencies Θrota=Θrotb=0.953K, Θrotc=3.85K.31 For ethane,
the high temperature approximation of the partition function for the rotational modes (qrot) is:

!!

!!"# = !

!!"#$

!!"#$

(13)

The configurational entropy of the adsorbed phase was determined using the partition
function:
!!"# =

!!

!!

(14)

Here λ is specific surface area divided by specific absolute uptake, and Λ is the thermal de
Broglie wavelength. Using the partition functions above, individual entropy contributions were
calculated by taking the negative temperature derivative of the Helmholtz free energy. These
individual contributions were summed to obtain the total molar entropy of the adsorbed phase
(shown in Figure 5). The relative contributions from each component to the total entropy at a
representative fixed adsorption quantity are shown in Figure 6.

79

Figure 6. Relative contributions of each component to the total adsorbed-phase entropy of
ethane on MSC-30 at the absolute uptake value of 2 mmol g-1.

As shown in Figure 5, there is good agreement between the measured and calculated
values of the adsorbed-phase entropy of ethane on MSC-30. Errors are less than 10% (without
applying any fitting or offset) throughout the measured regime, and are especially low at high
temperatures. At high coverages the measured entropy of ethane on MSC-30 gradually levels
out, but maintains positive concavity. In contrast, the measured entropy of ethane adsorbed on
ZTC deviates significantly from calculated values. At high temperatures, the measured entropy
is in agreement with the calculation from statistical mechanics, as in MSC-30. However, at low
temperatures and high coverages, the measured entropy is well below the calculated value
(with discrepancies of up to 42%). This is associated with the increase in isosteric heat in this
regime; ethane adsorption on ZTC has an anomalously increasing isosteric heat, and likewise
an anomalously decreasing entropy in the adsorbed phase. This is expected to be caused by
enhanced adsorbate-absorbate interactions on the surface of ZTC. Stronger intermolecular
interactions correspond to stiffer vibrational modes, hindered rotational motion, and inhibited

80
molecular motion/rearrangement, all of which can lead to a decrease in the entropy of the
adsorbed phase.
The increase in isosteric heat above the Henry’s Law value is largest at low
temperatures. We expect that temperature will disrupt the adsorbate-adsorbate lateral
interactions, suppressing the increase in isosteric heat. The thermal behavior may be a
cooperative one, where the loss of lateral interactions makes other lateral interactions less
favorable. There appears to be a critical temperature around 300 K where the effect of lateral
interactions between ethane molecules is lost.

5. Conclusions
Ethane adsorption was measured on a zeolite-templated carbon material (ZTC) that
has an exceptionally high surface area and narrow and uniform microporosity. An increasing
isosteric heat of adsorption as a function of coverage was observed. The isosteric heat rises by
4.6 kJ mol-1 from a Henry’s law value of 20.6 kJ mol-1 at low coverage to a peak of 25.2 kJ
mol-1 at a coverage of 21.4 mmol g-1. By comparing ethane adsorption on ZTC to methane
adsorption on the same material, it was found that the slope of the isosteric heat of adsorption
with coverage approximately scales with the strength of the adsorbate-adsorbate
intermolecular interactions. A control material, superactivated carbon MSC-30, behaved as a
normal microporous carbon adsorbent, exhibiting a monotonically decreasing isosteric heat
with coverage. The measured adsorbed-phase entropy of ethane on MSC-30 was also
successfully estimated with a statistical mechanics based approach (without intermolecular
interactions), exhibiting discrepancies of less than 10%. The measured entropy of ethane
adsorbed on ZTC deviated significantly from this standard model prediction at high coverage

81
and low temperature, indicating atypical adsorption properties in this system. The behavior
of both the adsorbed-phase entropy and the isosteric heat of ethane on ZTC can be explained
by attractive adsorbate-adsorbate interactions promoted by the nanostructured surface of
ZTC.

82

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the$ Degradation$ of$ LANI5KYSNY$ MetalKHydrides$ During$ Thermal$ Cycling.$ J.% Alloys%
Compd.%1995,$217,$185K192.$
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111
Chapter 8

A Generalized Law of Corresponding States for the Physisorption of
Classical Gases with Cooperative Adsorbate-Adsorbate Interactions
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “A Generalized Law of Corresponding
States for the Physisorption of Classical Gases with Cooperative Adsorbate-Adsorbate
Interactions,” J. Phys. Chem. C, 120, 11847 (2016).
DOI: 10.1021/acs.jpcc.6b00289
Abstract
The Law of Corresponding States for classical gases is well established. Recent attempts at
developing an analogous Law of Corresponding States for gas physisorption, however, have
had limited success, in part due to the omission of relevant adsorption considerations such as
the adsorbate volume and cooperative adsorbate-adsorbate interactions. In this work, we
modify a prior Law of Corresponding States for gas physisorption to account for adsorbate
volume, and test it with experimental data and a generalized theoretical approach.
Furthermore, we account for the recently-reported cooperative adsorbate-adsorbate
interactions on the surface of zeolite-templated carbon (ZTC) with an Ising-type model, and in
doing so, show that the Law of Corresponding States for gas physisorption remains valid even
in the presence of atypically enhanced adsorbate-adsorbate interactions.

112

1. Introduction
Gas physisorption on microporous carbons has been extensively explored for a variety
of applications ranging from improved gas storage to efficient gas separation. At high
pressures, nonideal effects can significantly influence both the gaseous and adsorbed phases.
In bulk gases, the Law of Corresponding States generalizes the gas nonideality and provides a
simplified equation of state. According to this law, nonideal gases behave similarly and have
similar compressibility factors at corresponding conditions (i.e., when the conditions of the
gases relative to the critical point are equal). The Law of Corresponding States has been shown
to be highly accurate for bulk gases through a number of studies1,2.
In 2002 Quinn hypothesized an extension to the Law of Corresponding States for
gas physisorption, based on empirical evidence.3 Quinn posited that gases have approximately
equal excess adsorption uptake quantities (ne) at corresponding conditions on the same
adsorbent, which we call “Quinn’s hypothesis of corresponding states.” Quinn provided
evidence to support this hypothesis by comparing argon, carbon monoxide, methane,
nitrogen, and oxygen physisorption on four microporous carbons. Quinn found general
agreement among the corresponding excess adsorption uptake quantities, although
discrepancies often exceeded 50%. Hydrogen physisorption did not follow the trend, a fact
that was initially attributed to hydrogen being able to better penetrate the adsorbent
micropores, but later explained more satisfactorily as a quantum effect.4 Recently, others have
expanded Quinn’s hypothesis to include gate-opening MOFs and have noted deviations for
larger molecules,5 but to date this issue has not been resolved. In this work we present a novel
Law of Corresponding States for gas physisorption that includes adsorbate volume

113
considerations. It builds upon previous work but is more successful, especially for larger
molecules such as ethane.

2. Quinn’s Hypothesis of Corresponding States
Excess adsorption uptake of methane, krypton and ethane on three microporous
carbons are compared at corresponding conditions in Figure 1 using values from Ref. 6,7,8 along
with new measurements. These three carbons (ZTC, MSC-30 and CNS-201) have significant
differences in synthesis, specific surface area, and pore-size distribution, but all contain
predominantly micropores (see Supporting Information, S1). For each adsorbent,
experimentally measured isotherms of the three gases are compared at reduced temperatures
(TR) of 1.25 + 0.02 and 1.38 + 0.03 and at corresponding reduced pressures (PR), which are
defined as

!! = !

(1)

!! = !

(2)

!! = !

(3)

Here T, P and V are the system temperature, bulk phase pressure and bulk phase molar
volume, and Tc, Pc, and Vc are the critical temperature, pressure, and molar volume of the bulk
fluid.

114

Figure 1. Comparison of methane (squares), krypton (circles), and ethane (triangles) excess
adsorption at corresponding conditions as per Quinn’s hypothesis (at reduced temperatures of
1.25+0.02 (blue) and 1.38+0.03 (red)).

115
As seen in Figure 1, there is good agreement between the measured methane and
krypton isotherms using Quinn’s hypothesis of corresponding states, with only moderate
discrepancies (less than 25%). The ethane isotherms, however, deviate significantly from those
of the other gases, with discrepancies of ~50%.
It is notable that all of the adsorptive species used for empirical support of Quinn’s
hypothesis of corresponding states (argon, carbon monoxide, methane, nitrogen, and oxygen)
have similar molecular size. This is seen by comparing their 12-6 Lennard-Jones parameters, σ,
which serve as proxies for excluded-volume diameters (see Table 1). Argon, carbon monoxide,
methane, nitrogen, oxygen, and krypton all have σ values that are within 5% of their group
average, 0.3560 nm. Ethane, however, is a significantly larger molecule with a σ that is 26%
larger than the group average. Accordingly, ethane does not adhere to Quinn’s hypothesis of
corresponding states.
Table 1. Lennard-Jones σ Parameters of Relevant Gases

gas
argon
carbon monoxide
methane
nitrogen
oxygen
krypton
ethane

Lennard-Jones
σ (nm)
0.34059
0.36619
0.37379
0.36139
0.33829
0.36369
0.448010
$$

116

3. Law of Corresponding States Comparing Fractional Occupancy
In this work, we define fractional occupancy (θ) as the ratio of the absolute adsorbed
amount (na) to the maximum possible quantity of adsorption for the adsorbent at any
temperature or pressure (nmax):

!=! !

!"#

(4)

By comparing fractional occupancies instead of excess adsorption, we develop a Law of
Corresponding States for the physisorption of classical gases interacting through London
dispersion forces that better fits empirical data:
“At corresponding conditions on the same adsorbent, classical adsorbed gases have
the same fractional occupancy.”
Unfortunately, absolute adsorption and hence fractional occupancy are not easily accessible
through experimentation. Consequently we also define a proxy metric, the excess occupancy
(θe):

!! = ! !

!"#

(5)

At low gas densities, the excess occupancy well approximates the fractional occupancy, but
excess occupancies are more easily obtained by experiment. For this reason, here, we compare
excess occupancies instead of the more fundamental fractional occupancies, which must be
determined indirectly. For completeness, we also extrapolate fractional occupancies using a
fitting procedure and compare these values at corresponding conditions in the Supporting
Information (S3). The maximum adsorption quantity was estimated by multiplying the liquid

117
molar density of the adsorptive species at its triple point by the measured micropore volume
of the adsorbent (see Supporting Information S2).
The experimentally-derived excess occupancies of methane, krypton and ethane on
ZTC, MSC-30 and CNS-201 are compared at reduced temperatures of 1.25+0.02 and
1.38+0.03 (Figure 2, A-C). Using a robust fitting technique described in detail elsewhere11,
ethane excess occupancies have been extrapolated to higher reduced temperatures of
1.43+0.01, 1.80+0.02 and 2.07+0.05 and compared to experimentally measured methane and
krypton data at corresponding conditions (Figure 2, D-E).

118

A.$

B.$

C.$

119

D.

E.
Figure 2.

Comparison of excess occupancies at corresponding conditions for methane

(squares), krypton (circles), and ethane (triangles) adsorption. The data in A-C are at reduced
temperatures of 1.25+0.02 (blue) and 1.38+0.03 (red). The data in D-E are at reduced
temperatures of 1.43+0.01 (black), 1.80+0.02 (magenta), and 2.07+0.05 (red). The lines
indicate extrapolated ethane results, calculated from the fit parameters obtained by fitting
experimental data with a superposition of Langmuir isotherms.

120
Figure 2, which presents the experimentally-derived excess occupancies of methane,
krypton, and ethane at corresponding conditions, shows good correspondence between the
curves, with discrepancies of less than 25%, except for ethane at high temperatures on CNS201. This may result from a rotational hindrance of the ethane molecules within the very small
pores of CNS-201 (as small as ~0.54 nm in width). This comparison of excess occupancies
instead of excess uptake gives a significant improvement over Quinn’s hypothesis of
corresponding states where discrepancies of ~50% can be found with the same experimental
data. Likewise, the extrapolated excess occupancies of ethane on MSC-30 are in reasonable
agreement (discrepancies of less than 25%) with the experimentally derived excess occupancies
of methane and krypton at reduced temperatures of 1.43+0.01, 1.80+0.02, and 2.07+0.05. On
ZTC, deviations between the experimental and the extrapolated isotherms are larger and likely
derive from small fitting inaccuracies magnified over the huge extrapolation range (>220K).
The fitting and extrapolation procedure was not applied to ethane on CNS-201 due to an
insufficient number of available isotherms.

4. Law of Corresponding States for Physisorption
There are two fundamentally distinct approaches to understanding gas-solid
physisorption: the mono and multi-layer adsorption models developed by Langmuir, Brunauer,
Emmett, Teller and others12, and the pore-filling model developed by Euken, Polanyi,
Dubinin, and others12. Each model successfully treats relevant physisorption phenomenon
under differing conditions and both have widespread use. Here the Law of Corresponding
States for physisorption is justified in the context of each model. ,

121
4.1 Monolayer Adsorption Model,
Adsorbed molecules form a densified layer near the adsorbent surface, in a dynamic
equilibrium with the gas phase. The significant decrease in the molar entropy of the adsorptive
species upon adsorption is offset by a commensurate decrease in the molar enthalpy. At
equilibrium
∆!!"#

= ∆!!"#

(6)

Δ!!"# = !! − !!

(7)

Δ!!"# = !! − !!

(8)

where ΔHads is the isosteric enthalpy of adsorption and ΔSads is the isosteric entropy of
adsorption. It should be possible to predict the fractional occupancy of the adsorbed species at
a fixed temperature and gas pressure with knowledge of the gas-phase enthalpy (Hg) and
entropy (Sg), and knowledge of the constant-coverage, adsorbed-phase enthalpy (Ha) and
entropy (Sa) (and how they change with fractional occupancy at a fixed temperature).
The gas-phase molar entropy, Sg, (in reference to the boiling-point liquid molar
entropy, SL1), of monatomic gases with similar critical volumes is well approximated by a
function that depends only on reduced quantities, f(TR, VR) (Supporting Information, S4). We
assume that the molar entropy of the adsorbed phase (Sa) is given by the molar entropy of the
liquid phase (SL1) with the addition of a θ-dependent entropy of configurations of the
adsorbate molecules on the adsorbent, f(θ).13
!! = !!! + !(!)

(9)

122
The Sg (in reference to SL1) is approximated by f(TR, PR),

−Δ!!"# = ! ! ! , !! − !(!)!

(10)

For a monatomic gas, the right-hand side of Equation 6 (ΔSads) depends only on fractional
occupancy and reduced quantities. Although polyatomic gases have additional degrees of
freedom from internal vibrational and rotational modes, for many adsorbate-adsorbent
systems, these internal vibrational and rotational modes are only negligibly altered upon
physisorption13,14 and do not significantly contribute to ΔSads. The assumption that rotational
modes remain largely unchanged upon physisorption may break down in special
circumstances, particularly in pores small enough to inhibit rotational modes.
The left-hand side of Equation 6 depends on the isosteric enthalpy of adsorption, a proxy
metric of the physisorption binding-site energies. We first consider an idealized adsorbent with
completely homogeneous binding-site energies and no adsorbate-adsorbate interactions.
Under these assumptions, ΔHads is a constant for an ideal gas-adsorbent system, independent
of pressure, or fractional occupancy at a fixed reduced temperature. These assumptions are
later relaxed.
To begin, we assume that the isosteric enthalpy of adsorption is proportional to the
critical temperature of the adsorptive species,5 as detailed in the Supporting Information (S5,
S6).
!!!"# = !! !!

(11)

Here c1 is an undetermined (adsorbent specific) coefficient and the left-hand side of Equation
6 becomes

123
!!!"#

!! !!

(12)

By substituting Equation 1 into Equation 12
!!!"#

= !!!

(13)

Hence under the idealized assumptions above, the left-hand side of Equation 6 only depends
on reduced quantities. Since both sides of Equation 6 only depend on reduced quantities and
fractional occupancy, the fractional occupancy of distinct gases individually adsorbed on a
specific idealized adsorbent must be equal at corresponding conditions.

4.2 Heterogeneities and Adsorbate-Adsorbate Interactions
Real adsorbents typically exhibit a heterogeneous distribution of binding sites. Such a
distribution of binding-site energies leads to an isosteric heat (-ΔHads) that decreases as a
function of fractional occupancy as the most favorable sites are filled first. The distribution of
binding sites is unique to the adsorbent and depends on pore-size distribution, surface
structure, and chemical homogeneity. One may posit that each adsorbent has a characteristic
binding-site energy distribution that varies with fractional occupancy and is proportional to the
critical temperature of the adsorptive species, but is otherwise independent of the adsorptive
species at corresponding conditions (see Supporting Information, S5, S6). In this
approximation, c1 is no longer a constant in Equation 11. Instead c1 becomes a function of
fractional occupancy that is unique to each adsorbent, but independent of the adsorptive
species at corresponding conditions.
Furthermore, cooperative adsorbate-adsorbate interactions are important in some
physisorptive systems. In these systems, favorable interactions can lead to an isosteric heat that

124
6,7

increases as a function of fractional occupancy . The contribution of these interactions is
assumed to be proportional to the product of the critical temperature, Tc, and a function that
depends only on fractional occupancy, f(θ), at a fixed reduced temperature (see Supporting
Information, S7). The expression for ΔHads in Equation 11 is thus modified by adding a term
that is proportional to f(θ)Tc:
!!!"#

! (!)!! !!!!! ! ! !!

= !

(14)

Here c2 is an undetermined coefficient that is independent of the adsorptive species at
corresponding conditions. By substitution
!!!"#

! (!)!!! ! !

= !

!!

(15)

The left-hand side of Equation 6 remains approximately independent of the adsorptive
species, even upon taking into account binding-site heterogeneity and adsorbate-adsorbate
interactions (Equation 15). Consequently, Equation 6 only depends on reduced quantities and
the fractional occupancy, consistent with the proposed Law of Corresponding States for gas
physisorption.
4.3 BET Model
A simple extension of the Langmuir model to incorporate multiple layers of
adsorption was worked out by Brunauer, Emmett, and Teller and is known as BET Theory15.
This theory is widely used to determine the surface area of porous materials and gives the
fractional occupancy (θ) as a function of pressure (P), saturation pressure (Po), and a
parameter, cBET.

125

!=

!!"#
!!

!!

!!"# = !

!!

!!

(16)

!!!"#
!!
!!

!!!!"# !!!
!"

(17)

The parameter cBET depends on the heat of adsorption for the first layer, -ΔHads, the heat of
liquefaction of the adsorbate, Hl, and the temperature, T. The assumption that ΔHads is
proportional to Tc has been previously justified. We may similarly assume that Hl is
proportional to Tc (see Supporting Information, Section S10). Finally, we assume that for a
given gas, Po is proportional to Pc, allowing Equation 16 to be reduced as:
!!

!∝
!!! !

! !!!

!!

!!! ! !

!!
! !!!

(18)
!!

In this expression, c3 is the factor relating the heat of adsorption and the heat of liquefaction.
Equation 18 gives the fractional occupancy in terms of only reduced parameters, consistent
with the Law of Corresponding States for physisorption.
4.4 Pore-Filling Model
The similarity between the Law of Corresponding States and Dubinin-Polanyi theory
was recently noted by Sircar et al.5 Here we further develop this insight to show the
importance of fractional occupancy to the Law of Corresponding States for physisorption.

126
Specifically, we consider a pore-filling model of adsorption in the form of the DubininAstakhov equation:16

!=!

!"#$( ! )

(19)

In this equation, T is the temperature, P is the pressure, Po is the equilibrium vapor pressure, E
is the characteristic binding energy, and χ is an adsorbent-specific heterogeneity parameter.
! = !!!

!=

(20)

(21)

!!

The affinity coefficient (β) relates the characteristic binding energy of a sample adsorbate (E)
to that of the standard adsorbate (Eo) and depends on the ratio of their static polarizabilities, α
and αo, respectively. The adsorbate polarizability is assumed to be proportional to the critical
temperature of the adsorptive species (see Supporting Information, S5).
Equation 19 then reduces to:
(! !" ! )

!!

!"#( !
!)
!! !!

!!

! ! !"( !
!)
!!

(22)

where

!!! = !!

(23)

The undetermined constant, c4, is derived from the polarizability and characteristic binding
energy of the standard adsorbate. Both sides of Equation 22 are independent of the adsorptive

127
species at corresponding conditions, consistent with the Law of Corresponding States for
gas physisorption.

5. Anomalous Surface Thermodynamics
As we previously reported, both methane and krypton physisorption on ZTC yield
anomalous surface thermodynamics at supercritical temperatures.11,6,7 Methane and krypton
isotherms were fitted with a superposition of Langmuir isotherms to extract thermodynamic
quantities11,6,7. This yields analytically differentiable fits that are useful in determining the
absolute adsorption, and the isosteric enthalpy of adsorption.11,6,7 On ZTC, the isosteric heats
of methane and krypton adsorption increase with fractional occupancy, a property that is
attributed to enhanced adsorbate-adsorbate interactions within the uniquely nanostructured
pores.
From Equation 7, the molar enthalpy of the adsorbed phase, Ha, (as a function of
absolute adsorption) was determined by adding the isosteric enthalpy of adsorption (ΔHads) to
the gas-phase enthalpy values (Hg) at the same conditions (obtained from REFPROP17). On
ZTC, enhanced adsorbate-adsorbate interactions cause the low-temperature adsorbed-phase
enthalpies to decrease toward minimum, most favorable values, which are reached at moderate
fractional occupancies (Figure 3).

128

Figure 3. Adsorbed-phase enthalpies of krypton and methane on ZTC. Labels are
temperatures in K.
The adsorbed-phase enthalpies may also be plotted as a function of the average
intermolecular spacing in the adsorbed phase (xavg) (Figure 4). Here xavg was estimated by
dividing the micropore volume of ZTC (Vmic), as measured by the Dubinin-Radushkevich
method18,19), by the quantity of absolute adsorption (na) and taking the cube root.7

!!"# =

!!"# !
!!

(24)

129

Figure 4. Adsorbed-phase enthalpies of methane and krypton on ZTC, as a function of xavg.
Labels are temperatures in K.

As shown in Figure 4, the adsorbed-phase enthalpies (as a function of xavg) change
with temperature and resemble a 12-6 Lennard-Jones potential. The deepest “potential wells”
are observed at the lowest temperatures. The significant temperature dependence shows that
the enhanced adsorbate-adsorbate interactions are disrupted by thermal motion, and suggests

130
that they may be modeled with a cooperative interaction model such as the Ising model. The
heat capacity of the adsorbed phase at constant pressure (CP) is determined from the
adsorbed-phase enthalpy (Figure 5).

!! =

!!!
!" !

(25)

Figure 5. Constant pressure heat capacities of methane (squares) and krypton (circles)
adsorbed phases on ZTC (black) and MSC-30 (orange) at an example pressure of 2 MPa.

For both methane and krypton there is good agreement between adsorbed-phase heat
capacities at high temperatures for both ZTC and MSC-30. In the case of krypton, a
monatomic gas, the CP on ZTC at high temperatures and the CP on MSC-30 at all
temperatures are in good agreement with the CP for an ideal two-dimensional gas (16.6 J mol-1
K-1). At low temperatures, however, the CP of both gases are significantly larger on ZTC than
on MSC-30, with values that exceed even the liquid-phase heat capacities (methane~55.7 J

131
-1

-1

-1

-1 17

mol K and krypton~44.7 J mol K ) . These unexpectedly large adsorbed-phase heat
capacities may be attributed to a phase transition in the adsorbed phase on the surface of ZTC.

6. Model for Cooperative Interactions
As temperature is increased, the effects of adsorbate-adsorbate interactions decrease,
and disappear as the slopes in Figure 3 go to zero at a critical temperature (To) of ~300 K.
Figure 5 indicates the presence of a phase transition (in the adsorbed phase) around 270 K. We
assume that below To, two phases arise: a vacancy-rich phase with a low concentration of
occupied sites (α’), and an adsorbate-rich phase with a high concentration of occupied sites
(α’’) and more adsorbate-adsorbate interactions. We expect an unmixing phase diagram where
the concentration of occupied sites in the α’ and α’’ phases are temperature dependent. The
simplest phase diagram is symmetric, so these concentrations, cα’ and cα’’, are related by
!!! = 1 − !!!!

(26)

The average adsorbate-adsorbate interaction energy per adsorbed molecule (Uavg) should be
dominated by the interactions in the adsorbate-rich α’’ phase, with phase fraction Fα’’
!!!

!!

!!!! = !! !!!!!
!!!

(27)

For simplicity we consider the case when θ=0.5 and thus equal phase fractions of 0.5
for all temperatures. If we assume (as in the point approximation) that the arrangement of
molecules within the “clustered” α’’ phase is random and that each nearest-neighbor pair has
an interaction strength of ε (from the Lennard-Jones potential)9, then for θ=0.5

132
!"!!!! !

!!"# =

(28)

With the following definitions
! = 2!!!! − 1
!!!

=e
!!!

!!! !

(29)

(30)

the unmixing problem is transformed to the Bragg-Williams ordering problem with one order
parameter, L.20 By substitution

!!"# =

!"(

!!! !

(31)

In the context of this model, cooperative adsorbate-adsorbate interactions on ZTC at
temperatures below To are consistent with the Law of Corresponding States for physisorption
as explained in Section 4.2 and elaborated on in the Supporting Information, Section S7.
For experimental comparison, for each measured temperature, we determined the
difference between the minimum enthalpy (θ≈0.5) and the low coverage enthalpy (θ≈0) (in
Figure 3) as a proxy measure of Uavg. These “potential well” depths were then normalized by
!"

so they could be compared directly to (

!!! !

) in Figure 6, per Equation 31. In Figure 6,

we set To=300 K and assume z=4 (square-lattice).

133

Figure 6. Normalized “potential well” depths of methane and krypton on ZTC, compared to
calculated (

!!! !

) .

Figure 6 compares the measured adsorbate-adsorbate interaction enthalpies to thermal
trends from the Ising model. Qualitative similarities between the measured and modeled
temperature dependence support the hypothesis of cooperative adsorbate-adsorbate
interactions on ZTC. The observed deviations are not surprising for such a simple model and
uncertainty in the nmax, To and z parameters.

7. Conclusions
The principle that distinct gases have similar adsorptive fractional occupancies at
corresponding conditions on the same adsorbent is established as a novel Law of
Corresponding States for gas physisorption. This principle is tested empirically using
measurements of methane, krypton, and ethane physisorption on ZTC, MSC-30, and CNS201 at reduced temperatures of 1.25+0.02 and 1.38+0.03. Reasonable agreement is found

134
across different size gases with discrepancies of less than 25%. Accordingly this principle is
useful for estimating pure gas isotherms, though should not be considered a replacement for
experimental isotherm data. Further support is obtained from statistical mechanics wherein the
validity of this principle as a first approximation is established for a number of conditions. The
principle of corresponding states proves successful even in the presence of cooperative
adsorbate-adsorbate interactions, which were modeled using the Ising model in a low-level
approximation.

Acknowledgement:
This work was sponsored as a part of EFree (Energy Frontier Research in Extreme
Environments), an Energy Frontier Research Center funded by the US Department of Energy,
Office of Science, Basic Energy Sciences under Award Number DE-SC0001057.

135

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990K993.$
7.$Murialdo,$ M.;$ Stadie,$ N.$ P.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Krypton$ Adsorption$ on$ ZeoliteK
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Methane$on$ZeoliteKTemplated$Carbon.$J.%Phys.%Chem.%C%2015,$119,$26409K26421.$
12.$Dabrowski,$ A.$ Adsorption$ K$ From$ Theory$ to$ Practice.$ Adv.% Colloid% Interface% Sci.%
2001,$93,$135K224.$
13.$Campbell,$C.$T.;$Sellers,$J.$R.$V.$The$Entropies$of$Adsorbed$Molecules.$J.%Am.%Chem.%
Soc.%2012,$134,$18109K18115.$
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136
18.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
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137
Chapter 9

Supporting Information for Chapter 8
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “A Generalized Law of Corresponding
States for the Physisorption of Classical Gases with Cooperative Adsorbate-Adsorbate
Interactions,” J. Phys. Chem. C, 120, 11847 (2016).
(Supporting Information)
DOI: 10.1021/acs.jpcc.6b00289

S1. Adsorbent Materials
MSC-30
MSC-30 (Maxsorb) is a microporous superactivated carbon obtained from Kansai
Coke & Chemicals Company Ltd. (Japan). As determined by nitrogen adsorption and BET
analysis, MSC-30 has a surface area of 3244 + 28 m2 g-1. Using nitrogen adsorption data and a
slit-pore model, NLDFT pore-size analysis1 was conducted to determine the pore-size
distribution. MSC-30 has a broad range of pore sizes (from 6 to 35 Å). Over 40% of the
micropore volume is contained in pores of greater than 21 Å in width. The total micropore
volume was found to be 1.54 cm3 g-1 by the Dubinin-Radushkevich method. The skeletal
density was measured by helium pycnometry and determined to be 2.1 g cm-3. Cu Kα X-ray
diffraction of MSC-30 gave a broad peak at 2θ=34°, in accordance with that reported for AX21. The elemental composition (CHN) was determined via the Dumas method2 in combustion
experiments, indicating that 1.16 wt% of MSC-30 is hydrogen. Results from XPS experiments
on MSC-30 are summarized in Table S1.3

138
Table S1. Summary of XPS Data on MSC-30 and ZTC
peak
position
(eV)
285.0
285.7 286.4
287.3 288.1
289.4
C-C sp C-C sp C-OR C-O-C C=O COOR
component
ZTC
53.4
18.0
8.6
6.0
1.1
4.2
MSC-30
48.0
18.8
6.8
4.8
6.1
4.2

290.2
1.0
3.6

291.5
7.7
7.7

CNS-201
CNS-201 is a microporous activated carbon obtained from A. C. Carbone Inc.
(Canada). As determined by nitrogen adsorption and BET analysis, CNS-201 has a surface area
of 1095 + 8 m2 g-1. Using nitrogen adsorption data and a slit-pore model, NLDFT pore-size
analysis was conducted to determine the pore-size distribution. CNS-201 has a three dominant
pore widths of 5.4, 8.0, and 11.8 Å, containing roughly 50%, 20%, and 15% of the total
micropore volume respectively. The micropore volume was found to be 0.45 cm3 g-1 by the
Dubinin-Radushkevich method4,5. The skeletal density was measured by helium pycnometry
and determined to be 2.1 g cm-3.

Zeolite-Templated Carbon (ZTC)
ZTC is a templated carbon synthesized using zeolite NaY and a process described
elsewhere.6,7 As determined by nitrogen adsorption and BET analysis, ZTC has a surface area
of 3591 + 60 m2 g-1. Using nitrogen adsorption data and a slit-pore model, NLDFT pore-size
analysis was conducted to determine the pore-size distribution. ZTC has a narrow pore-size
distribution centered at 12 Å. Over 90% of the micropore volume is contained in pores of

139

-1

widths between 8.5 and 20$Å. The total micropore volume was found to be 1.66 cm g by
the Dubinin-Radushkevich method. The skeletal density was measured by helium pycnometry
and determined to be 1.8 g cm-3. Cu Kα X-ray diffraction of ZTC produced a sharp peak at
2θ=6°, indicative of the template periodicity of ~15 Å. The elemental composition (CHN) was
determined using the Dumas method in combustion experiments, indicating that 2.44 wt% of
MSC-30 is hydrogen. Results from XPS experiments on ZTC are summarized in Table S1.
TEM analysis has also provided evidence of the periodic structure of ZTC.3

S2. Approximation of nmax
All three materials tested are predominantly microporous. The maximum adsorption
quantity (nmax) was estimated as the product of the total micropore volume and the liquid
density of the adsorptive species at its triple point. This assumes complete micropore filling at
a maximal density given by the liquid-phase density. The micropore volumes of each adsorbent
were determined by applying the Dubinin-Radushkevich method to nitrogen adsorption
isotherms measured at 77 K.3 The liquid densities were obtained from REFPROP8. Both the
micropore volumes and the liquid densities are listed in Table S2 along with the as-determined
values for nmax.
Table S2. Liquid Molar Densities, Adsorbent Micropore Volumes,
and Estimated nmax Values
density (mol dm-3)
micropore volume (cm-3)
krypton
methane
ethane

29.2
28.1
21.7

ZTC MSC-30 CNS-201
1.66
1.54
0.45
48.5
45.0
13
46.7
43.3
13
36.0
33.4
9.8

140

S3. Comparison of Fractional Occupancies at Corresponding
Conditions
While absolute adsorption is of greater physical relevance, excess adsorption is the
experimentally-determined quantity in physisorption experiments. To obtain absolute
adsorption uptake quantities (na) from excess adsorption quantities (ne) we used Gibb’s
definition of excess adsorption
!! = !! − !!"# !!

(S1)

where ρg is the gas-phase density. We fit the unknown absolute adsorption uptake quantities
and adsorption volumes (Vads) with a superposition of Langmuir isotherms9 to obtain
!! = !!"# − !!"# !!

! !!

!! !

(S2)

!!!! !

where Ki is the equilibrium constant of the ith Langmuir isotherm, αi is the weighting factor
for the ith Langmuir isotherm ( ! !! = 1), and Vmax is the maximum adsorption volume.
While the number of Langmuir isotherms (i) may be adjusted, we found that setting i=2 gives
high quality fits with a minimum number of parameters. Moreover many of the fitting
parameters hold physical significance that may be verified by comparison to independent
estimates. For i=2, the absolute adsorption is given by
!! = !!"#

1−!

!! !
!!!! !

! !

+ ! !!!

(S3)

This fitting procedure has been described in more detail elsewhere.9
Having determined absolute adsorption, we compare the fractional occupancies of
methane, krypton and ethane on ZTC and MSC-30 at corresponding reduced temperatures of

141
1.25+0.02 and 1.38+0.03 and corresponding reduced pressures (Figure S1). CNS-201 was
not considered due to an insufficient number of experimental isotherms.

Figure S1. Comparison of krypton, methane and ethane fractional occupancies on ZTC and
MSC-30 at corresponding conditions.
As with the excess occupancies, the fractional occupancies compared at corresponding
conditions are in reasonable agreement with one another. Additional errors may have been
introduced in the fitting procedure.

142

S4. Assumption that Sg (in Reference to SL1) is Well Approximated
by f(TR, PR) for Monatomic Gases with Similar Critical Volumes
According to the Trouton-Hildebrand-Everett rule, “the entropy of vaporization for
normal liquids is the same when evaporated to the same concentration”.10 Despite notable
exceptions (e.g. due to hydrogen bonding), the Trouton-Hildebrand-Everett rule is generally
accurate.11,12 The entropy of vaporization (ΔSvap) is defined as the difference between the gasphase entropy (Sg1) and the liquid-phase entropy (SL1), where the “1” subscript indicates that
the entropies are measured at the normal boiling temperature.
Δ!!"# = !!! − !!!

(S4)

For a monatomic gas at dilute conditions, additional changes to the gas-phase enthalpy due to
increasing temperature are accounted for as

∆!!!!→! = ! !!!" !

!"

(S5)

where the subscript "!!1 → 2" indicates that the entropy change takes the gas from the
normal boiling temperature, 1, to a new temperature, 2. Noting the proportionality between
the normal boiling temperature and the critical temperature for gases with similar Vc,

∆! ∝ ! !!!" ! !

(S6)

For gases with similar Vc, the gas concentration, or equivalently the molar volume (V), is
approximately proportional to VR, and the difference between Sg2 and SL1 is well
approximated as a function that depends only on TR and VR, f(TR, VR). Figure S2 compares the
gas-phase entropies (measured in reference to SL1) of three monatomic gases with similar
critical volumes (argon, 0.096 dm3 mol-1; krypton 0.12 dm3 mol-1; xenon 0.15 dm3 mol-1) and

143

-1

one monatomic gas with a different critical volume (neon, 0.051 dm mol ) at corresponding
conditions.8 The gas-phase entropies of argon, krypton, and xenon are all in good agreement
with one another, while that of neon deviates significantly due to its smaller critical volume and
quantum effects.

Figure S2. Gas-Phase molar entropies of neon (+), argon (-), krypton (*), and xenon (x) at
reduced temperatures of 1.25 (blue) and 2.07 (red), at corresponding reduced pressures.

S5. Assumption that ΔHads is Proportional to Tc
For interactions from purely London dispersion forces, ΔHads is expected to be
proportional to the static polarizability (α) of the adsorbate following London’s theory13. This
assumption is valid for small and moderately-sized classical molecules (e.g., ethane) that have a
fairly uniform charge distribution (e.g., non-polar species without a strong quadrupole
moment).
For small and moderately-sized classical molecules that interact through

144
London dispersion forces, the critical temperature is found to be proportional to the square
root of the static polarizability14. In many cases, however, the curvature is minimal and the
trend is essentially linear. This is illustrated for noble gases in Figure S3.

Figure S3. Static polarizability of noble gases15 as a function of the gas critical temperature8. A
linear fit is shown.

Therefore, a simple approximation that the critical temperature is directly proportional to the
polarizability (α) is acceptable:
! ≈ !! !!

(S7)

Here, c5 is the proportionality constant. It then follows that:
!!!"# ≈ !! !! $
(!!!"# )
!!

≈ !!

(S8)
(S9)

The validity of this approximation is investigated in Table S3, wherein -ΔHads is derived
experimentally from adsorption measurements (at θ≈0 on ZTC, MSC-30, and CNS-201). We
find that c6 is reasonably constant across different gases on the same adsorbent. As is a

145
commonly employed assumption elsewhere, the effect of temperature on the isosteric
enthalpy of adsorption is assumed to be negligible compared to its absolute magnitude.

Table S3. Empirical Values of

(!!!!"# )
!!

for Krypton,

Methane and Ethane on MSC-30, ZTC and CNS-201

krypton
methane
ethane
mean and
spread

MSC-30
0.062
0.076
0.069

ZTC
0.058
0.072
0.066

CNS-201
0.085
0.097
0.080

0.069+10%

0.065+11%

0.087+11%

The validity of Equation S9 is also investigated by comparing

(!!!!"# )
!!

!as a function of

fractional occupancy on MSC-30 and ZTC (based on the availability of high pressure isosteric
enthalpy of adsorption data at corresponding temperatures). On MSC-30, the

(!!!!"# )
!!

!values

approximately track one another as they decrease monotonically with fractional occupancy due
to binding-site heterogeneity (Figure S4). On ZTC, methane and krypton exhibit anomalous
surface thermodynamics at temperatures below To due to cooperative adsorbate-adsorbate
interactions. This leads to

(!!!!"# )
!!

!values that initially increase with fractional occupancy, but

nonetheless approximately track one another (Figure S5) due to the considerations explained
in Sections 4.2 (Chapter 8) and S7 (Chapter 9). Additional discrepancies may result from
uncertainty in the nmax parameter.

146

Figure$ S4.$ Comparison$ of$

(!!!!"# )
!!

!$ of$ methane$ (orange),$ krypton$ (purple),$ and$

ethane$(black)$on$MSCK30$at$a$reduced$temperature$of$1.25.$

Figure S5. Comparison of

(!!!!"# )

reduced temperature of 1.25.

!!

! of methane (orange) and krypton (purple) on ZTC at a

147

S6. Generalizing the Proportionality
Between ΔHads and Tc to NonIdeal Gases
The molar enthalpy of an ideal gas depends solely on the temperature. This, along with
the assumptions in Section 4.1 of Chapter 8, allows for a constant

!!!!"#

for multiple gases

individually adsorbed on a single adsorbent at a fixed reduced temperature. The enthalpy of a
nonideal gas, however, depends on pressure and volume as well as temperature, as illustrated
by the case of a van der Waals gas (HvdW)
!! !

!!"# = ! !!! ! − ! + !"

(S10)

where N is the number of gas molecules, kB is Boltzmann’s constant, T is the temperature, P is
the pressure, V is the volume of the system, and a and b are the van der Waals parameters.
Expressed as a function of TR, PR, and VR:

!!"# =

! !!! !

!! ! ! !

!!

!"

(S11)

The van der Waals critical temperature, !! =

!!"# =

!"!! !!!

!!! !

− !! ! +

!!
!"!"!!

, can also be included giving:

!! ! ! !
!"

(S12)

The gas-phase molar enthalpy (Hg) of a van der Waals gas is therefore proportional to the
critical temperature and otherwise depends only on the reduced quantities as:
!! ∝ !! ! ! ! , !! , ! !

(S13)

148
Furthermore, we assume that the adsorbed-phase molar enthalpy (Ha) is
proportional to the critical temperature, and otherwise depends only on the fractional
occupancy and reduced quantities (as empirically supported by the data in Figure 3 of Chapter
8).
!! ∝ !! ! !, ! ! , !! , ! !

(S14)

Since the isosteric enthalpy of adsorption is the difference between the gas-phase and
adsorbed-phase enthalpies at constant coverage, Tc can be factored out of both:
Δ!!"# ∝ !! ! !! ! , !! , ! ! − ! !, ! ! , !! , ! !

(S15)

The isosteric enthalpy of adsorption of a van der Waals gas is therefore proportional to its
critical temperature, and otherwise depends only on the fractional occupancy and reduced
quantities.

S7. Assumption that the Contributions of Adsorbate-Adsorbate
Interactions to ΔHads are Proportional to f(θ)Tc
Methane and krypton adsorption on ZTC exhibit anomalous surface thermodynamics
where the isosteric heats of adsorption increase with coverage.16,17 This results from adsorbateadsorbate interactions that are enhanced by the surface nanostructure. We have previously
provided both enthalpic and entropic evidence of adsorbed-phase clustering on ZTC. 9,17 In
this work we draw a connection between the adsorbed-phase clustering and the Ising model by
analyzing the temperature dependence of the clustering as an unmixing phase transition. The
applicability of this model is supported by considerations of the adsorbed-phase heat capacity
(Figure 5 of Chapter 8) and the general temperature dependence of the effect (Figure 6 of
Chapter 8). To use the Ising model, we assume the critical temperature of the unmixing phase

149
transition (To) is proportional to the Lennard-Jones interaction potential (ε) of the adsorbate,
and hence proportional to the critical temperature of the adsorbate (Tc).18$We define a reduced
phase-transition critical temperature !!! such that

!!! =

!!

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(S16)$

By the transitive property
!!! ∝ ! !

(S17)

For !!! < 1, unmixing occurs, and the adsorbate-vacancy binary solid solution separates into a
vacancy-rich α’ phase and an adsorbate-rich α’’ phase. The adsorbate concentration in each
phase, cα’ and cα’’, respectively, depends on !!! . The relative phase fraction of the α’’ phase
(Fα’’) is given by the lever rule:
!!!

!!

!!!! = !! !!!!!

(S18)

!!!

The adsorbate-adsorbate interaction energy is predominantly from the adsorbate-rich α’’ phase.
Under the assumption of random ordering (the point approximation), the average adsorbateadsorbate energy per molecule in the α’’ phase, Uα’’, is
!!!! =

!"!!!!

(S19)

where z is the coordination number. The overall adsorbate-adsorbate energy per molecule,
Uavg, in both phases is given by a weighted average of the mean adsorbate energy in each phase
and the fraction of adsorbate molecules in each phase
! ! ! !!

!!"# = !! !! !! !!!! !!! !!!

(S20)

150
where the energy of the vacancy-rich phase, Uα’ is assumed to be zero. By substitution we
arrive at a general expression to model Uavg.
!!"# =

!"!!!!

!!!!!! !!

!!!!! !!

(S21)

For a fixed reduced temperature, cα’’ is a constant. The coordination number, z, is also
assumed to be a constant. Equation S21 is thus the product of a function proportional to ε and
a function f(θ) that only varies with θ. Notably this equation displays the expected qualitative
behaviors, increasing as a function of fractional occupancy at low coverage and leveling off at
high coverage. We may make the assumption that ε is proportional to Tc,18 leaving an equation
that only varies with f(θ)Tc.

S8. Justification for the Law of Corresponding States in Bulk Gases
The Law of Corresponding States in bulk fluids is well established and can be readily
understood by examining the van der Waals equation of state:
!"

! = !!! − ! !

(S22)

Here, P is pressure, T is temperature, V is volume, R is the gas constant, and a and b are the
van der Waals parameters (unique for each gas). By substituting in the reduced temperature,
reduced pressure and reduced volume:
!! ∗ !

!∗ !! = ! ∗ ! !!
− ! ∗!

The critical quantities in terms of the van der Waals parameters (a and b)19 are

(S23)

151
!!

!! = !"!"

(S24)

!! =
!"! !
!! = 3!

(S25)

(S26)

Substituting these critical quantities into Equation S23:
!! !

!! = !! ! !! −

!!

(S27)

Equation 11 depends only on reduced quantities, which are by definition identical for all gases
at corresponding conditions, thus providing a basis for the Law of Corresponding States in
bulk fluids.

S9. Additional Data Compared with Law of Corresponding States for
Physisorption
Using experimental physisorption data published elsewhere, the Law of Corresponding
States for physisorption is illustrated on three additional adsorbents, each representing a
unique class of materials. These examples showcase the importance of the adsorbate molecular
volume. Two plots are shown for each adsorbent, the first comparing excess adsorption at
corresponding conditions and the second comparing excess occupancies at corresponding
conditions. In each case, excess occupancies at corresponding conditions have better
agreement than excess adsorption at corresponding conditions. This is especially pronounced
for adsorbates that have very different liquid molar volumes (e.g. argon and cyclohexane).
Agot Grade Artificial Nuclear Graphite20
In Figure S6, argon and nitrogen adsorption isotherms on highly pure graphite are
compared at a reduced temperature of 0.6 + 0.01. Argon and nitrogen have similar van der
Waals molar volumes of 0.03201 and 0.0387 L mol-1 respectively. Thus switching from a

152
comparison of excess adsorption to excess occupancy only minimally improves the
agreement between the corresponding isotherms.

Figure S6. Comparison of excess adsorption (top) and excess occupancy (bottom) of argon
(circles) and nitrogen (+) adsorption on Agot grade artificial nuclear graphite at corresponding
conditions (TR= 0.6 + 0.01).20

Ni(bodc)(ted)0.5 Metal-Organic Framework21
In Figure S7, argon and cyclohexane adsorption isotherms on Ni(bodc)(ted)0.5 are
compared at a reduced temperature of 0.57+ 0.01. Argon and cyclohexane have dissimilar van
der Waals molar volumes of 0.03201 and 0.1424 L mol-1 respectively. This leads to large

153
discrepancies when comparing excess adsorption that are resolved by comparing excess
occupancy.

Figure S7. Comparison excess adsorption (top) and excess occupancy (bottom) of argon
(circles) and cyclohexane (*) adsorption on [Ni(bodc)(ted)0.5

at corresponding conditions (TR

= 0.57+ 0.01).21
Zeolite NaX22
In Figure S8, carbon dioxide, ethane, and sulfur hexafluoride adsorption isotherms on
zeolite NaX are compared at a reduced temperature of 0.94 + 0.02. The van der Waals molar
volumes of carbon dioxide, ethane, and sulfur hexafluoride are 0.04267, 0.0638, and 0.08786 L
mol-1, respectively. Comparing excess occupancies rather than excess adsorption improves the

154
agreement of the isotherms. It is notable that all three gases have similar critical temperatures
of 304, 305, and 319 K respectively, and yet the differences in the van der Waals molar
volumes yield significant discrepancies when comparing excess adsorption.

Figure S8. Comparison of excess adsorption (top) and excess occupancy (bottom) of carbon
dioxide (x), ethane (triangles), and sulfur hexafluoride (squares) adsorption on zeolite NaX at
corresponding conditions (TR = 0.94 + 0.02).22

155

S10. Heat of Liquefaction vs. Critical Temperature$

Figure S9. Plotting the heat of liquefaction as a function of critical temperature for moderatelysized classical gases shows an approximately linear correlation.8

156

References:
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Templated$Carbon$Materials$for$HighKPressure$Hydrogen$Storage.$Langmuir%2012,$
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Activated$Charcoal.$Proc.%Acad.%Sci.%USSR,%Phys.%Chem.$Sect.$1947,$55,$331K337.$
5.$Nguyen,$ C.;$ Do,$ D.$ D.$ The$ DubininKRadushkevich$ Equation$ and$ the$ Underlying$
Microscopic$Adsorption$Description.$Carbon%2001,$39,$1327K1336.$
6.$Kyotani,$ T.;$ Nagai,$ T.;$ Inoue,$ S.;$ Tomita,$ A.$ Formation$ of$ New$ Type$ of$ Porous$
Carbon$by$Carbonization$in$Zeolite$Nanochannels.$Chem.%Mater.%1997,$9,$609K615.$
7.$Nishihara,$ H.;$ Hou,$ P.KX.;$ Li,$ L.KX.;$ Ito,$ M.;$ Uchiyama,$ M.;$ Kaburagi,$ T.;$ Ikura,$ A.;$
Katamura,$ J.;$ Kawarada,$ T.;$ Mizuuchi,$ K.;$ Kyotani,$ T.$ HighKPressure$ Hydrogen$
Storage$in$ZeoliteKTemplated$Carbon.$J.%Phys.%Chem.%C%2009,$113,$3189K3196.$
8.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$version$8.0;$
National$Institute$of$Standards$and$Technology:$Gaithersburg,$MD,$2007;$CDKROM.$
9.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Unusual$ Entropy$ of$ Adsorbed$
Methane$on$ZeoliteKTemplated$Carbon.$J.%Phys.%Chem.%C%2015,$119,$26409K26421.$
10.$Hildebrand,$ J.$ H.$ The$ Entropy$ of$ Vaporization$ as$ a$ Means$ of$ Distinguishing$
Normal$Liquids.$J.%Am.%Chem.%Soc.%1915,$37,$970K978.$
11.$Hildebrand,$ J.$ H.$ Liquid$ Structure$ and$ Entropy$ of$ Vaporization.$ J.% Chem.% Phys.%
1939,$7,$233K235.$
12.$Green,$ J.$ A.;$ Irudayam,$ S.$ J.;$ Henchman,$ R.$ H.$ Molecular$ Interpretation$ of$
Trouton's$ and$ Hildebrand's$ Rules$ for$ the$ Entropy$ of$ Vaporization$ of$ a$ Liquid.$ J.%
Chem.%Thermodyn.%2011,$43,$868K872.$
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33,$8K26.$
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Prediction%of%Transport%and%Other%Physical%Properties%of%Fluids;$Pergamon$Press:$New$
York,$1971.$
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Polarizabilities$ Alpha$ and$ Gamma$ for$ the$ NobleKGases.$ J.% Chem.% Phys.% 1991,$ 94,$
4972K4979.$
16.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Anomalous$ Isosteric$ Enthalpy$ of$
Adsorption$ of$ Methane$ on$ ZeoliteKTemplated$ Carbon.$ J.% Am.% Chem.% Soc.% 2013,$ 135,$
990K993.$

157
17.$Murialdo,$M.;$Stadie,$N.$P.;$Ahn,$C.$C.;$Fultz,$B.$Krypton$Adsorption$on$ZeoliteK
Templated$ Carbon$ and$ Anomalous$ Surface$ Thermodynamics.$ Langmuir% 2015,% 31,%
7991K7998.%
18.$Evans,$ G.$ J.$ Cooperative$ Molecular$ Behavior$ and$ Field$ Effects$ on$ Liquids:$
Experimental$Considerations.$In$Dynamical%Processes%in%Condensed%Matter;$Evans,$M.$
W.,$Ed.;$John$Wiley$and$Sons:$New$York,$1985;$293K376.$
19.$Haentzschel,$ E.$ The$ Calculation$ of$ the$ A$ and$ B$ Constants$ of$ van$ der$ Waals's$
Equation$from$Critical$Values.$Ann.%Phys.%1905,$16,$565K573.$
20.$Banaresmunoz,$M.$A.;$Gonzalez,$L.$V.$F.;$Llorenta,$J.$M.$M.$AdsorptionKIsotherms$
of$Nitrogen$and$Argon$on$an$Agot$Grade$Artificial$Nuclear$Graphite$at$77KK$and$90K
K.$Carbon%1987,$25,$603K608.$
21.$Li,$K.$H.;$Lee,$J.;$Olson,$D.$H.;$Emge,$T.$J.;$Bi,$W.$H.;$Eibling,$M.$J.;$Li,$J.$Unique$Gas$
and$Hydrocarbon$Adsorption$in$a$Highly$Porous$MetalKOrganic$Framework$Made$of$
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1159.$

158
Chapter 10

A Thermodynamic Investigation of Adsorbate-Adsorbate
Interactions of Carbon Dioxide on Nanostructured Carbons
M. Murialdo, C.C. Ahn, and B. Fultz, “A Thermodynamic Investigation of AdsorbateAdsorbate Interactions of Carbon Dioxide on Nanostructured Carbons,” AIChE J.
(submitted, 2016).

Abstract
A thermodynamic study of carbon dioxide adsorption on a zeolite-templated carbon (ZTC), a
superactivated carbon (MSC-30), and an activated carbon (CNS-201) was carried out at
temperatures from 240 to 478 K and pressures up to 5.5 MPa. Excess adsorption isotherms
were fitted with generalized Langmuir-type equations, allowing the isosteric heats of
adsorption and adsorbed-phase heat capacities to be obtained as a function of absolute
adsorption. On MSC-30, a superactivated carbon, the isosteric heat of carbon dioxide
adsorption increases with occupancy from 19 to 21 kJ mol-1, before decreasing at high loading.
This increase is attributed to attractive adsorbate-adsorbate intermolecular interactions as
evidenced by the slope and magnitude of the increase in isosteric heat and the adsorbed-phase
heat capacities. An analysis of carbon dioxide adsorption on ZTC indicates a high degree of
binding-site homogeneity. A generalized Law of Corresponding States analysis indicates lower
carbon dioxide adsorption than expected.

159

1. Introduction
Anthropogenic contributions to carbon dioxide in the atmosphere have become an
ever-increasing concern, recently highlighted by the IPCC Fifth Assessment Report1. Given
the growing nature of world energy demands and our continued reliance on fossil fuels,
carbon-capture and storage (CCS) should be considered in any plan to mitigate climate
change.2,3
Physisorptive materials are promising for a variety of gas applications because they
require only low regeneration energies and are stable through thousands of cycles.4,5 In
physisorption, solid adsorbents induce a local densification of gas at the adsorbent surface by
weak physical interactions. Physisorptive materials have been widely used for gas storage
applications. 6,7 Because different gases adsorb to a given surface with differing selectivities,
physisorptive materials are also promising for gas separations8,9. A number of studies have
shown that physisorptive materials are viable candidates for separating carbon dioxide from
mixed-gas streams.10,11,12
An adsorbent’s selectivity of one gas over another is dependent on the thermodynamic
properties of the gas-adsorbent system. As gas molecules are adsorbed, their molar entropy
decreases. This is offset by a decrease in molar enthalpy from favorable interactions between
the adsorbate and the adsorbent. The differential change in molar enthalpy at constant surface
occupancy, defined as the isosteric enthalpy of adsorption (ΔHads), is an important and readily
accessible figure of merit for physisorptive processes.13 Optimizing the isosteric enthalpy of
adsorption has been the topic of active research.14,15,16 A variety of high surface area materials
including metal-organic frameworks, covalent-organic frameworks, zeolites, zeolite-templated

160
carbons, and activated carbons have been investigated as carbon dioxide adsorbents, and
-1

have exhibited isosteric heats of adsorption of up to 90 kJ!mol .17,18,19,20,21,22,23
In general the isosteric heat of adsorption (-ΔHads) decreases as a function of surface
loading due to heterogeneity of the binding sites.24 This has the effect of limiting the pressure
range over which gas separation and storage are optimal. However, recent investigations of
nonideal gases adsorbed on carbonaceous materials at high pressures have shown that, in some
systems, strong lateral intermolecular interactions between adsorbate molecules can lead to an
isosteric heat of adsorption that increases with increasing occupancy near ambient
temperatures.25,26,27,28 This effect is realized in adsorbent materials with optimal surface
structure and a narrow distribution of binding-site energies.29,30 At high pressures, carbon
dioxide gas has strong intermolecular interactions and nonideal behavior, and is therefore a
candidate for this effect. In this paper we report that the superactivated carbon MSC-30 has an
atypical, increasing isosteric heat of carbon dioxide adsorption, and we provide evidence that
this effect derives from adsorbate-adsorbate intermolecular interactions.

2. Experimental Methods
Carbon dioxide uptake was measured on three carbonaceous materials: CNS-201,
MSC-30, and ZTC. The activated carbon CNS-201 was obtained from A.C. Carbons Canada
Inc., and MSC-30, a superactivated carbon, was obtained from Kansai Coke & Chemicals
Company Ltd. ZTC was synthesized by infiltrating zeolite NaY with furfuryl alcohol followed
by polymerization, carbonization and the ultimate dissolution of the zeolite template. This
synthesis has been described in detail elsewhere31. Each material was degassed at 520 K under

161
-3

vacuum (less than 10 Pa) prior to measurements. The skeletal densities of the materials
were determined by helium pycnometry.
Equilibrium N2 adsorption isotherms were measured at 77 K with a BELSORP-max
volumetric instrument (BEL-Japan Inc.), and surface areas were calculated using the BrunauerEmmett-Teller (BET) method32. The Dubinin-Radushkevich (DR) method33,34 was used to
calculate micropore volumes. Pore-size distributions were calculated by the non-local density
functional theory (NLDFT) method35 from high-resolution data collected on a Micromeritics
ASAP 2020, using a carbon slit-pore model and software provided by Micromeritics.
Equilibrium carbon dioxide adsorption isotherms were measured at up to nine
temperatures between 240 and 478 K using a custom Sieverts apparatus that was designed and
tested for accuracy to 10 MPa36. The sample holder was submerged in an isothermal chiller
bath for low temperature isotherms, or placed inside a cylindrical copper heat exchanger and
wrapped with insulated fiberglass heating tape for high temperature isotherms. A proportionalintegral-derivative (PID) controller was used to maintain a constant temperature during
measurements; fluctuations were less than ±0.1 K at low temperatures and no higher than
±0.4 K at high temperatures.
The Sieverts apparatus is equipped with a digital cold cathode pressure sensor (I-4

MAG, Series 423) and a molecular drag pump capable of achieving a vacuum pressure of 10

Pa. High-pressure measurements were made using an MKS Baratron (Model 833) pressure
transducer. Temperature was measured on the wall of the manifold and on the outer wall of
the sample holder using K-type thermocouples and platinum resistance thermometers.
Supercritical fluid chromatography (SFC) grade carbon dioxide (99.995%) gas was obtained
from Air Liquide America Corporation, and multiple adsorption/desorption isotherms were

162
measured to ensure reversibility and reproducibility. Gas densities were determined from the
REFPROP Standard Reference Database37.

3. Results
3.1. Nitrogen Adsorption and Helium Pycnometry
The pore-size distributions calculated by nonlocal density functional methods of CNS201, MSC-30, and ZTC are shown in Figure 1. CNS-201 was found to have distinct, extremely
narrow pore sizes of < 1.5 nm in width. MSC-30, on the other hand, has a broad pore size
distribution with pore widths ranging from 0.6 to >3 nm. ZTC has a single dominant pore
width of 1.2 nm, consistent with the inverse of the zeolite structure. The BET surface areas of
MSC-30, CNS-201, and ZTC were found to be 3244+28, 1095+8, and 3591+60 m2!g-1,
respectively. ZTC is one of the highest specific surface area carbonaceous materials reported
to date. The micropore volumes of MSC-30, CNS-201, and ZTC were determined to be 1.54,
0.45, and 1.66 mL!g-1, respectively. The theoretical maximum possible carbon dioxide
adsorption (nm) was estimated as the product of the micropore volume and the density of
liquid carbon dioxide at its triple point,30 yielding estimated nm values of 41.3, 12, and 44.5
mmol!g-1 for MSC-30, CNS-201, and ZTC, respectively. MSC-30 and CNS-201 have skeletal
densities of 2.1 g!mL-1, while ZTC has a lower skeletal density of 1.8 g!mL-1 due to a higher
hydrogen content38.

163

Figure 1. The pore-size distribution of MSC-30, CNS-201, and ZTC as calculated by the
NLDFT method.

3.2. Carbon Dioxide Adsorption
Equilibrium excess adsorption isotherms of carbon dioxide on the nanostructured
carbons are shown in Figure 2. The low temperature isotherms display maxima at pressures
between 0.5-5 MPa, as expected for Gibbs excess quantities. At all measured temperatures,
MSC-30 has the highest maximum excess adsorption quantities. At room temperature (298 K),
the maximum excess uptakes on MSC-30, CNS-201, and ZTC were 22.7, 7.25, and 17.8
mmol!g-1 respectively, corresponding to 7.00, 6.62, and 4.96 mmol!(1000 m2) -1. This
correlation of BET surface area and Gibbs surface excess uptake maximum capacity is
consistent with other similar materials, which have an average “Chahine’s-type rule” for
carbon dioxide uptake at 298 K of ~7 mmol (1000 m2) -1 39.

164

Figure 2. Equilibrium excess adsorption isotherms of carbon dioxide on MSC-30 (top), ZTC
(middle), and CNS-201 (bottom). The curves show the best simultaneous fit of the measured
data for a given material, using Equation 2 below.

165
3.3. Adsorption Data Analysis
Adsorption measurements give excess adsorption (ne), not absolute adsorption (na).40
At low pressures, excess adsorption approximates absolute adsorption. At high pressures,
however, the absolute adsorption remains a monotonically increasing quantity, but excess
adsorption does not. By utilizing a fitting equation consistent with the definition of excess
adsorption, measured data can be fitted beyond the maximum of excess adsorption. Our
carbon dioxide excess adsorption isotherms were fitted using a generalized Langmuir equation,
recently described and applied to high-pressure methane adsorption.29 This fitting technique
was adapted from that implemented by Mertens41.
The fitting procedure incorporates the Gibbs definition of excess adsorption, which is
related to absolute adsorption as follows:
!! = ! !! − ! !! !(!, !)

(1)

The gas density (ρ) is a function of pressure (P) and temperature (T) and was estimated from
the modified Benedict–Webb–Rubin equation of state37. The volume of the adsorption layer,
Va, is the only unknown preventing direct calculation of absolute adsorption. Here Va, is left
as an independent fitting parameter. The Gibbs excess adsorption was fitted using the
following generalized Langmuir equation:
!! (!, !) = !!"# − !!"# !(!, !)
!! =

!!

! !!! !"

! !! = 1

!! !
! !! !!! !

(2)
(3)
(4)

The independent fitting parameters are αi (weighting factors) and Ki (equilibrium constants)
for the i Langmuir isotherms, the scaling factor nmax, and the maximum volume of the

166
adsorption layer Vmax. The equilibrium constants (Equation 3) depend on the prefactors
(Ai) and energies (Ei) of the Arrhenius-type exponentials and R, the universal gas constant. For
a good balance between number of fitting parameters and goodness of fit, the number of
superimposed Langmuir equations used was i=2. The residual mean square values of the
resulting fits of the adsorption data for MSC-30, CNS-201, and ZTC were 0.1, 0.07, and 0.7
(mmol g-1)2 respectively.
The best-fit parameters of carbon dioxide on MSC-30, CNS-201, and ZTC are
compared to the best-fit parameters of methane25, ethane26, and krypton27 on the same
materials (using the same fitting equation) in Table 1. In many cases, the fit parameters
correlate to physical properties, and may sometimes be validated by comparison to
independent estimates. For example, the adsorbent micropore volume can be determined by
the Dubinin-Radushkevich method33 and compared to Vmax. Likewise the product of the
adsorbent micropore volume and the liquid molar density of the adsorbate37 (nm) provides an
estimate of nmax. Rigorous comparisons of the fitted and independently estimated parameters
are shown in Table 2.
Table 1. Best-fit parameters from Generalized Langmuir Fits.
!!
CO2 /ZTC
CO2 /CNS-201
CO2/MSC-30
CH4/ZTC
CH4/CNS-201
CH4/MSC-30
C2H6/ZTC
C2H6/MSC-30
Kr/ZTC
Kr/CNS-201
Kr/MSC-30

nmax%
(mmol%gD1)%

45.4
12.9
81.1
35.6
9.77
41.0
25.0
36.1
39.3
10.9
57.8

α%

Vmax%
(mL%gD1)%

A1%
(k1/2%MPaD1)%

E1%
(kJ%molD1)%

A2%
(k1/2%MPaD1)%

E2%
(kJ%molD1)%

1.64E-12
0.800
0.207
0.460
0.580
0.700
0.827
0.713
0.686
0.462
0.726

12.4
3.41
10.0
2.04
0.490
2.30
1.58
2.60
2.02
0.490
2.98

0.00143
0.0456
0.000107
0.0590
0.0610
0.0680
2.14E-07
0.0865
1.81E-06
0.00590
0.112

21.6
23.0
23.6
11.6
17.2
13.4
41.0
19.8
30
15.1
11.6

0.121
0.00244
0.0635
0.000180
0.00440
0.00460
0.0444
0.00647
0.0924
0.0689
0.00306

3.94
22.0
14.3
20.4
16.4
12.9
18.5
17.8
10.0
16.3
12.8

167
Table 2. Comparison of V max and n max Parameters to Independent Estimates
!!

Vmax%(mL%gD1)%

CO2 /ZTC$
CO2 /CNS-201$
CO2/MSC-30$
CH4/ZTC$
CH4/CNS-201$
CH4/MSC-30$
C2H6/ZTC$
C2H6/MSC-30$
Kr/ZTC$
Kr/CNS-201$
Kr/MSC-30$

micropore% volume%
(mL%gD1)%

12.4$
3.41$
10.0$
2.04$
0.490$
2.30$
1.58$
2.60$
2.02$
0.490$
2.98$

1.66
0.45
1.54
1.66
0.45
1.54
1.66
1.54
1.66
0.45
1.54

nmax%
%(mmol%gD1)%

45.4$
12.9$
81.1$
35.6$
9.77$
41.0$
25.0$
36.1$
39.3$
10.9$
57.8$

nm%
%(mmol%gD1)%

44.5
12
41.3
46.6
13
43.3
36.0
33.4
48.5
13
45.0

Although more difficult to validate through independent comparison, the Ei
parameters give important insight into the adsorbent-adsorbate binding energies of the ith
isotherm, and the prefactor Ai gives insight about the relative number of adsorption sites with
energy Ei. Specifically, the product of αi, the ith isotherm weighting factor, and Ai gives an
overall “weight” of sites with energy Ei. For carbon dioxide, methane, ethane and krypton
adsorption on MSC-30 and CNS-201, the contributions of each isotherm in the fitted
superposition of isotherms are moderately well-balanced. No isotherm accounts for less than
1% of the overall “weight”, except for carbon dioxide on MSC-30. This may result from
carbon dioxide adsorbing on a more limited set of MSC-30 adsorption sites than other gases,
as noted on single-wall carbon nanotube bundles42.
For ZTC, each generalized Langmuir fit heavily favors just a single isotherm (see Table
3). This suggests that ZTC usually has a higher degree of binding-site homogeneity than MSC30 or CNS-201, consistent with the pore-size distributions of the three carbonaceous
adsorbents. Unlike MSC-30 and CNS-201, ZTC has a single sharply peaked pore width (1.2

168
nm). In microporous carbons, the predominant contribution to binding-site heterogeneity
often results from the spectrum of pore widths. ZTC eliminates much of this heterogeneity
with its micropores of approximately a constant width.

Table 3. Normalized Relative Weights of Isotherms 1 and 2 as Determined by
Multiplying the Isotherm Weighting Value (αi ) by the Isotherm Prefactor (A i )
Isotherm 1 Weight
Isotherm 2 Weight
1.94E-14
1.00E+00
9.87E-01
1.32E-02
4.40E-04
1.00E+00
9.96E-01
3.57E-03
9.50E-01
4.96E-02
9.72E-01
2.82E-02
2.30E-05
1.00E+00
9.71E-01
2.92E-02
4.28E-05
1.00E+00
6.85E-02
9.32E-01
9.90E-01
1.02E-02

CO2 /ZTC
CO2 /CNS-201
CO2/MSC-30
CH4/ZTC
CH4/CNS-201
CH4/MSC-30
C2H6/ZTC
C2H6/MSC-30
Kr/ZTC
Kr/CNS-201
Kr/MSC-30

The Clapeyron relation was used to determine the isosteric heat of adsorption:
!!" = −∆!!"# (!! ) = −!

!"
!" !!

Δ!!"#

(5)

Here the isosteric heat of adsorption (qst) is (by convention) a positive value when adsorption is
exothermic. The coverage-dependent change in enthalpy upon adsorption is ΔΗads(na). The
change in molar volume of the adsorbate upon adsorption is Δvads.

169

4. Discussion
4.1. Isosteric Heat of Carbon Dioxide Adsorption
The isosteric heats of carbon dioxide adsorption on MSC-30, CNS-201, and ZTC
derived according to Equation 5 are shown in Figure 3. The isosteric heat of carbon dioxide
adsorption on MSC-30 differs significantly from that on a conventional activated carbon like
CNS-201 in its dependence on absolute uptake. For carbon dioxide adsorption on CNS-201,
the isosteric heat displays typical behavior.$ It is a decreasing function of occupancy. On the
other hand, the isosteric heat of carbon dioxide adsorption on MSC-30 first increases as a
function of occupancy before reaching a maximum, and then decreases at high occupancy. On
ZTC, the isosteric heat of carbon dioxide adsorption decreases very gradually with loading,
-2

especially at low temperatures where the slope is ~ -25kJ!mol . This behavior is in agreement
with the high degree of binding-site homogeneity expected from the pore-size distribution and
fit parameters for ZTC. Carbon dioxide adsorption differs from methane, ethane, and krypton
adsorption where the isosteric heats on MSC-30 decrease with occupancy, while the isosteric
heats on ZTC increase with occupancy due to enhanced adsorbate-adsorbate interactions. This
suggests that carbon dioxide adsorbate-adsorbate interactions are better optimized for the
micropore distribution of MSC-30 than ZTC.

170

Figure 3. Isosteric heats of adsorption of carbon dioxide on CNS-201, MSC-30, and ZTC as a
function of absolute uptake.

171
For carbon dioxide adsorbed on MSC-30, the total increase in isosteric heat (peak
value minus low coverage value) is more pronounced at lower temperatures, reaching a
maximum measured increase of 2.1 kJ!mol-1 at 241 K. This energy is consistent with lateral
intermolecular interactions between adsorbed carbon dioxide molecules. For example, the well
depth of the Lennard-Jones potential between two carbon dioxide molecules is ε = 1.8
kJ!mol-1 43.
At low temperatures and low occupancy, the isosteric heat of adsorption of carbon
dioxide on MSC-30 increases approximately linearly with occupancy. This linear increase is
nearly identical for the lowest temperatures measured (e.g. 107 kJ!g!mol-2 at 241 K, 111
kJ!g!mol-2 at 247 K, 113 kJ!g!mol-2 at 262 K and 106 kJ!g!mol-2 at 283 K). Similar trends
hold true for methane, ethane, and krypton adsorption on ZTC.25,26,27 The gases with stronger
intermolecular interactions have larger slopes, consistent with the hypothesis that the increases
in the isosteric heats with loading result from adsorbate-adsorbate intermolecular interactions.
The slopes of the isosteric heats with respect to fractional occupancy,

!(!!!!"# )
!"

, may be

reasonably estimated with Equation 6 (see Table 4) in the low coverage regime

!(!∆!!"# )
!"

!"

= !

(6)

where the coordination number, z, is posited to be 5 and ε is the well depth of the LennardJones 12-6 interaction potential43. This simple first approximation (Equation 6) assumes that
adsorbed molecules are randomly situated and only interact with first nearest neighbors, each
interaction having strength ε.

172
Table 4. Measured and Estimated Slopes of Increasing Isosteric Heats of Methane,
Ethane, and Krypton on ZTC, and Carbon Dioxide on MSC-30 as a Function of
Fractional Occupancy (at Low Coverage and Low Temperature)
$$

Estimated$Slope$(kJ!molK1)$

Measured$Slope$(kJ!molK1)$

Methane$

3.0$

3.2$

Krypton$

3.3$

3.4$

Ethane$

4.3$

4.0$

Carbon$Dioxide$

4.5$

4.5$

A qualitatively similar increase in the isosteric heat of carbon dioxide adsorption on
Maxsorb® (from Kansai Netsu Kagaku Co.) was previously noted by Himeno et al.44, however
it is unclear to what degree those results are quantitatively accurate. Himeno et al. used the
Toth equation to fit three experimental isotherms (at 273, 298, and 323K). This fit was then
used to calculate an isosteric heat via a reduced form of the Clausius-Clapeyron Equation
(Equation 7).
∆!
!! !

!"#$
!"

(7)

The calculated “isosteric heat” (ΔH) was determined from the partial derivative of the
logarithm of the pressure with respect to temperature at constant coverage (N). In this case,
however, excess adsorption, not absolute adsorption was held constant and what was
calculated may more aptly be called an “isoexcess heat of adsorption”. At low coverage, excess
adsorption accurately approximates absolute adsorption, but deviations arise and become
significant as the equilibrium gas-phase density increases. In our work we employ a generalized
Langmuir-type fitting function that determines absolute adsorption and gives true “isosteric

173
25,26,27,29

heats of adsorption”. We have elaborated on this methodology in prior publications

Additionally, Equation 7, which is employed by Himeno et al. but not in our work, makes two
idealized assumptions that break down for nonideal gas conditions. Equation 7 assumes that
the volume of the adsorbed phase is zero, and further assumes the validity of the ideal gas law.
These assumptions lead to significant errors under nonideal gas conditions. Futhermore, our
analysis determines the temperature dependence of the isosteric heat, which is ignored by
Himeno et al. Himeno et al. report an increasing isosteric heat for carbon dioxide adsorption
-2

on Maxsorb® with a slope of ~300 (kJ!g!mol ). We report a more moderate slope (~110
kJ!g!mol-2) on the comparable superactivated carbon MSC-30 after considering isotherms
taken at eight temperatures. This more moderate increase is more consistent with the strength
of carbon dioxide intermolecular interactions as measured by the Lennard-Jones parameter,
ε=1.8 kJ/mol43, Equation 6 and the trend in Table 3.$

4.2 Adsorbed-Phase Enthalpies and Heat Capacities
The adsorbed-phase enthalpies for carbon dioxide adsorption on MSC-30, CNS-201, and ZTC
were calculated as a function of absolute adsorption by subtracting the isosteric heats in Figure
3 from gas-phase enthalpy values determined from data tables at equivalent conditions37.
Figure 4 shows that MSC-30 has different behavior than CNS-201 or ZTC in that the
adsorbed-phase enthalpy is not a monotonically increasing function of coverage. Rather the
adsorbed-phase enthalpy on MSC-30 decreases with coverage at low temperatures and low
coverages, consistent with enhanced favorable adsorbate-adsorbate interactions.

174

Figure 4. Adsorbed-phase enthalpies of carbon dioxide on MSC-30, ZTC, and CNS-201.

175
The adsorbed-phase constant pressure heat capacities of carbon dioxide on MSC-30, CNS201, and ZTC were calculated by taking the partial derivative of the adsorbed-phase enthalpies
with respect to temperature at constant pressure. Figure 5 shows the adsorbed-phase heat
capacities at a constant sample pressure of 2 MPa.

Figure 5. Adsorbed-phase molar heat capacities of carbon dioxide on MSC-30 ("), CNS-201
(+), and ZTC (•). Values are given at a constant sample pressure of 2 MPa.

Figure 5 shows that the constant pressure adsorbed-phase heat capacities of carbon
dioxide on CNS-201 and ZTC gradually increase with temperature as expected for a
polyatomic gas. On MSC-30 a different behavior is observed where the adsorbed-phase heat
capacity rises significantly around a temperature of 250K. This is suggestive of a phase
transition and indicates that the origin of the enhanced adsorbate-adsorbate interactions may
be an adsorbed-phase clustering transition as previously noted in other systems.29,30

176
4.3 Law of Corresponding States and Selectivities
It is well established that different nonideal gases behave similarly at corresponding
conditions, at equal reduced temperatures and reduced pressures.45 The reduced temperature
and reduced pressure are defined as the ratios of the system temperature to the gas critical
temperature, and the system pressure to the gas critical pressure, respectively. We have recently
reported an extension to the Law of Corresponding States that applies to physisorbed
molecules.30 Specifically:
Classical gases adsorb to the same fractional occupancy on the same adsorbent at corresponding conditions.
Fractional occupancy (θ) is defined as the ratio of the absolute adsorption (na) to
maximum possible adsorption (nm). To facilitate comparisons in this work and others,30 we
instead compare the more accessible quantity, excess occupancy (θe), defined as the ratio of
excess adsorption (ne) to maximum possible adsorption (nm). For each adsorbent, nm was
estimated by multiplying the adsorbent micropore volume by the liquid molar density of the
adsorbate at its triple point.30,37 Carbon dioxide, ethane, methane, and krypton excess
occupancy isotherms on ZTC and MSC-30 are compared at corresponding conditions in
Figure 6.

177

Figure 6. Comparison of carbon dioxide (+), ethane (∆), methane ("), and krypton (!) excess
occupancies at corresponding conditions (with reduced temperatures given in the key).
At low temperatures, there is reasonable agreement between carbon dioxide and
ethane isotherms at corresponding conditions. At higher temperatures, however, the carbon
dioxide isotherms are significantly smaller than corresponding isotherms of the other gases.
This may be due in part to the large quadrupole moment of carbon dioxide46, which is not
present in the other gases considered. The large quadrupole moment accounts for ~50% of
the cohesive energy of solid carbon dioxide.47,48 Notably the bulk critical temperature and
isosteric heats of carbon dioxide and ethane are similar in spite of carbon dioxide’s significantly

178
lower polarizability. This is also consistent with a strong quadrupole interaction of carbon
dioxide.49,50,51 It is possible that the short-range quadrupole-induced dipole interactions
between carbon dioxide and the adsorbent are disrupted with temperature, but further
investigation is needed.
The selectivities of carbon dioxide with respect to methane, ethane and krypton were
calculated as the ratio of the Henry’s Law constants. Henry’s Law constants were calculated
directly from the excess adsorption data by extrapolating to zero coverage the logarithm of
pressure divided by excess adsorption (ln(P/ne)).52 The measured room temperature Henry’s
Law constants and selectivities are given in Table 5. Methane and krypton have similar Henry’s
Law constants, as one might expect based on the similarity of the gases, particularly their
similar critical temperatures. While carbon dioxide and ethane have nearly identical critical
temperatures, ethane has significantly higher room temperature Henry’s Law constants for
each adsorbent. This is consistent with the trends noted in Figure 6.

Table 5. Henry’s Law Constants (mmol g-1 MPa-1) and Equilibrium Adsorption
Selectivities at Room Temperature
Henry’s Law Constants
C2H6
CO2
CH4
Kr
CO2/CH4
C2H6/CO2
CO2/Kr
C2H6/CH4
C2H6/Kr

ZTC
45.7
19.7
7.03
6.80
2.80
2.32
2.90
6.50
6.72

MSC-30
51.9
19.9
8.99
9.03
2.21
2.61
2.20
5.77
5.75

CNS-201
35.1
21.4
6.99
6.57
3.06
1.64
3.26
5.02
5.34

179

5. Conclusions
The excess uptakes of carbon dioxide on MSC-30, CNS-201, and ZTC were measured
volumetrically and fitted with a generalized Langmuir-type equation. The fitted data were used
in thermodynamic analyses that show how MSC-30 exhibits an atypical, increasing isosteric
heat of carbon dioxide adsorption, while CNS-201 has the thermodynamic properties of a
conventional carbon. The isosteric heat on ZTC suggests a high degree of binding-site
homogeneity. At near-ambient temperatures the isosteric heat on MSC-30 rises with uptake
from 19 to 21 kJ!mol-1. The measured adsorbed-phase enthalpies, adsorbed-phase heat
capacities, and comparisons to studies with other nonideal gases indicate that this increasing
isosteric heat results from enhanced adsorbate-adsorbate interactions within the pores of
MSC-30.

Acknowledgements
This work was supported as part of EFree, an Energy Frontier Research Center under Award
No. DE-SC0001057. A special thanks to Christopher Gardner for his contributions in the lab.

180

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and$carbon$dioxide$on$several$activated$carbons.$J$Chem$$Eng$Data.$2005;$50:$369K
376.$
45.$Guggenheim$ EA.$ The$ principle$ of$ corresponding$ states.$ J$ Chem$ Phys.$ 1945;$ 13:$
253K261.$
46.$Graham$C,$Pierrus$J,$Raab$RE.$Measurement$of$the$electric$quadrupoleKmoments$
of$CO2,$CO$and$N2.$Mol$Phys.$1989;$67:$939K955.$
47.$Slusarev$ VA,$ Freiman$ YA,$ Krupskii$ IN,$ Burakhov$ IA.$ Orientational$ Disordering$
and$ Thermodynamic$ Properties$ of$ Simple$ MolecularKCrystals.$ Phys$ Status$ Solidi$ B.$
1972;$54:$745K754.$
48.$Hirshfeld$ FL,$ Mirsky$ K.$ Electrostatic$ term$ in$ latticeKenergy$ calculations$ K$ C2H2,$
CO2,$AND$C2N2.$Acta$Crystallogr$Sect$A.$1979;$35:$366K370.$
49.$Buckingham$AD,$Disch$RL.$Quadrupole$moment$of$carbon$dioxide$molecule.$Proc$
R$Soc$A.$1963;$273:$275K289.$
50.$Do$DD,$Do$HD.$Effects$of$potential$models$on$the$adsorption$of$carbon$dioxide$on$
graphitized$ thermal$ carbon$ black:$ GCMC$ computer$ simulations.$ Colloids$ Surf$ A.$$
2006;$277:$239K248.$
51.$Terlain$ A,$ Larher$ Y.$ PhaseKdiagrams$ of$ films$ of$ linearKmolecules$ with$ large$
quadrupoleKmoments$(CO2,$N2O,$C2N2)$adsorbed$on$graphite.$Surf$Sci.$1983;$125:$
304K311.$
52.$Zhang$ SY,$ Talu$ O,$ Hayhurst$ DT.$ HighKpressure$ adsorption$ of$ methane$ in$ NaX,$
MgX,$CaX,$SrX,$and$BaX.$J$Phys$Chem.$1991;$95:$1722K1726.$

28
Chapter 3

Experimental Methods and Analysis
1. Equipment
The fundamental basis of quantitative adsorption analysis is the measurement of
excess adsorption isotherms. Each isotherm comprises a series of excess adsorption uptake
values, measured stepwise at increasing pressures and a constant temperature. At high
pressure, the excess adsorption becomes non-monotonic as a function of pressure, after
reaching an excess adsorption maximum. A desorption isotherm may be measured in reverse,
by starting with a preloaded adsorbent at high pressures and reducing the pressure stepwise.
Observed hysteresis between pairs of adsorption and desorption isotherms yields valuable
information about the adsorptive system, often indicating the presence of capillary
condensation in mesopores.
In this thesis, excess adsorption isotherms were measured by the volumetric method
also known as the Sieverts’ method using a custom Sieverts apparatus designed and tested for
accuracy up to 10 MPa (Figure 1).1 The Sieverts apparatus comprises a number of rigid,
stainless steel, and leak-proof compartments, each interconnected with controllable on-off
valves (either hand-turned or pneumatic). The volume of each compartment is known with
high precision (+ 0.01 mL). For standard adsorption measurements, only two of the
compartments are of interest: the manifold and the reactor. The manifold is equipped with a
midrange (3000 PSI) MKS Baratron (Model 833) pressure transducer for high-pressure
measurements and an MKS Baratron (Model 120AA) for low-pressure measurements of
higher resolution. The temperature of the gas in the manifold was measured with platinum

29
resistance thermometers. The temperature of the reactor was monitored with K-type
thermocouples. In a preparatory step the adsorbent sample of interest was weighed and sealed
within the reactor, which seals by the tightening of a conflat flange with a copper gasket.
Additional nickel filter gaskets with a 0.5-micron mesh size prevented the adsorbent from
escaping from the reactor. After sealing the reactor, each sample was degassed at ~520 K
-3

under a vacuum of less than 10 Pa prior to testing. The Sieverts apparatus is equipped with a
-4

molecular drag pump capable of achieving a vacuum of 10 Pa and vacuum pressures were
verified using a digital cold cathode pressure sensor (I-MAG, Series 423). To obtain low
temperature isotherms, the reactor was submerged in a circulated chiller bath or cryogenic bath
with temperature fluctuations no larger than + 0.1 K. High temperature isotherms were
obtained by encasing the reactor in a copper heat conductor wrapped with insulating fiberglass
heating tape. Using a proportional integral derivative (PID) controller, the reactor temperature
was maintained with fluctuations no larger than + 0.4 K. Prior to measurements, the entire
Sieverts was purged multiple times with the gas of interested to eliminate any impurity
residues. On each sample, multiple adsorption/desorption isotherms were taken to ensure
complete reversibility and identical measurements were found to be reproducible to within 1%
error.

30

Figure 1. Sieverts Apparatus

2. Methodology
For each isotherm data point, a predetermined pressure of research-grade gas was
introduced into and cached in the manifold. Upon reaching equilibrium, the temperature and
pressure of this gas were measured with high precision and the gas density was determined
from REFPROP data tables2. Given that the manifold volume is known, the moles of gas in
the manifold were thus determined. Next, a valve was opened to allow the gas to occupy both
the manifold and reactor volumes. The reactor housed the porous sample of known mass.
Upon opening the valve, the gas was allowed to occupy a volume given by the sum of the

31
manifold and reactor volumes minus the volume of the sample. The volume of the sample
was given by the product of the sample mass and the sample skeletal density (obtained by
helium pycnometry). Once equilibrium was reestablished, pressure and temperature
measurements were taken and a final gas-phase density determined for the step. The moles of
gas left in the gas phase were thus calculated. Any quantity of gas introduced into the manifold,
but no longer contributing to the gas-phase pressure was considered to be in the adsorbed
phase. The sample within the reactor was held at a constant temperature over the course of a
successive set of pressure measurements, resulting in an excess adsorption isotherm. Upon
completion of an isotherm, the temperature was adjusted for the next isotherm, in order to
measure a multitude of isotherms over a wide temperature range.
At equilibrium the chemical potential of the gas phase (μg) and the adsorbed phase (μa)
are equal
!! = !!

(1)

By taking the total differential of both sides of Equation 1:
−!! !" + !! !" = −!! !" + !! !"
!"
!"

! !!

= !! !!!

(2)
(3)

Thus the derivative of pressure with respect to temperature is related to the change in entropy
(upon adsorption) divided by the change in volume (upon adsorption). At constant coverage,
the difference between the entropy of the adsorbed phase (Sa) and the entropy of the gas
phase (Sg) is the isosteric entropy of adsorption (ΔSads) and is given by Equation 4.
∆!!"# = !! − !!

(4)

32
!"

∆!!"# = !! − !! !"

(5)

The isosteric entropy of adsorption is in turn related to isosteric enthalpy of adsorption (ΔHads)
by Equation 6
∆!!"#

= ∆!!"#

(6)

At constant coverage, the difference between the adsorbed and gas-phase enthalpies is thus
given by Equation 7
!"

∆!!"# = ! !! − !! !"

(7)

This is the Clapeyron equation, and is fundamental to the calculation of the isosteric enthalpy
of adsorption. A number of simplifying assumptions may be made in dealing with the
Clapeyron equation. The two most common are as follows:
1. That the volume of the gas phase is significantly larger than that of the adsorbed phase
such that the overall change in volume is well approximated as the gas-phase volume.
2. That the gas follows the ideal gas law.
Together these assumptions transform the Clapeyron equation into Equation 8
!! ! !"

∆!!"# !! = − !

!" !!

(8)

This may in turn be rearranged into the Van’t Hoff form:
∆!!"# !! = !

!"#!!

(9)
!!

By plotting ln(P) vs (1/T), a Van’t Hoff plot is formed. The slope of the Van’t Hoff plot
multiplied by R gives the isosteric heat. In common practice, adsorption measurements directly
determine excess adsorption (ne), not absolute adsorption (na). It is thus common to hold
excess adsorption, not absolute adsorption, constant in Equation 9. This results in an isoexcess

33
enthalpy of adsorption that approximates the isosteric enthalpy of adsorption, but only at
low gas densities. The previously mentioned assumptions fail when applied broadly to gas data.
Thus we must consider alternatives to these oversimplifying assumptions.
First, the ideal gas assumption may be avoided by inserting values from gas data tables
directly into the Clapeyron equation. This is the course of action followed in this thesis. Using
gas data tables enables thermodynamic calculations outside of the ideal gas regime. Second, the
simplified Clausius-Clapeyron equation assumes that the adsorbed phase has virtually zero
volume and that the net difference between the molar volume of the gas phase and the
adsorbed phase equals the molar volume of the gas phase. In reality, the adsorbed phase has a
finite molar volume that approaches that of the liquid phase of the adsorbed species. Thus the
assumption of a zero molar volume adsorbed phase may be replaced with one of two options.
Either the adsorbed-phase molar volume is assumed to be equal to that of the liquid phase
molar volume, or, fit functions are used to approximate the adsorbed-phase molar volume.
In applying the Clapeyron equation, it is necessary to take the derivative of pressure
with respect to temperature at constant coverage conditions. Where absolute adsorption is
held constant, this determines the isosteric enthalpy of adsorption and where the excess
adsorption is held constant, this determines the isoexcess enthalpy of adsorption. In practice,
however, neither absolute nor excess adsorption is an experimentally tunable variable. The
uptake quantity is never directly selected, rather the pressure is roughly selected. Thus
obtaining data points at constant coverage conditions is not simple. In some cases, with
sufficient data, constant coverage conditions across a number of temperatures may be
achieved for select data points by pure coincidence. This can be thought of as “analysis
without fitting”. Fortuitously positioned data points, however, are sparse and unreliable.

34
In general it is necessary to establish an interpolation function to interpolate between
the measured data points. Interpolation functions of a variety of forms have been employed in
literature. The simplest entails linear interpolation between data points.
While simple, this approach fails to accurately capture the adsorption behavior, leading
to scatter and errors. In this thesis research I employ a superposition of Langmuir isotherms as
the fitting function of choice. Specifically, starting from the definition of excess adsorption
(ne):
!! = !! − !! !(!, !)

(10)

I fit the absolute adsorption (na) and the volume of the adsorption layer (Va) with
superpositions of Langmuir isotherms with appropriate prefactors:
!! (!, !) = !!"#

! !!

!! (!, !) = !!"#

! !!

!! !

(11)

!!!! !
!! !

(12)

!!!! !

th

where ai is the respective weight of the i isotherm

! ∝! = 1 , P is the pressure, and Ki is an

equilibrium constant given by an Arrhenius-type equation such that:
!! =

!!

!!!

! !"

(13)

where Ai is a prefactor and Ei is an energy of the ith isotherm. Altogether the fit function
becomes:
!! (!, !) = !! − !!"# !(!, !)

! !!

!! !
!!!! !

(14)

One particular advantage of this fitting procedure is that the absolute adsorption is
easily accessible. It is one of the quantities that is directly fitted and given by Equation 11.

35
Absolute adsorption is a more fundamental quantity than excess adsorption, and its
determination is critical for in-depth analysis of the adsorption physics.
In addition to producing high quality fits, the dual-Langmuir fitting method used in
this thesis gives physically realistic fitting parameters. In particular the nmax and vmax parameters
have been found to be in reasonable agreement with independently determined physical data
(See Chapter 9). The nmax parameter gives the maximum possible absolute adsorption as
determined by fits. This is directly comparable to an estimate of the maximum possible
adsorption as obtained by multiplying the measured micropore volume by the liquid density of
the adsorptive species2. Here we assume that the entirety of the micropores is filled with
adsorbate at a density equal to the liquid density (upon maximal adsorption). Furthermore, the
vmax parameter indicates the maximum possible volume of the adsorbed phase, which may be
directly compared to the measured micropore volume of the adsorbent.
While the isosteric enthalpy of adsorption (ΔHads) is a popularly cited proxy metric of
binding site energy, a more fundamental metric exists in the adsorbed-phase molar enthalpy
(Ha) given by Equation 15. The isosteric enthalpy of adsorption is the difference between the
adsorbed-phase and gas-phase (Hg) molar enthalpies and thus retains a dependence on gasphase properties.
!! = !! + Δ!!"#

(15)

For nonideal gas conditions, this dependence can obscure interesting phenomenon occurring
strictly within the adsorbed phase. The gas-phase enthalpy is obtained from reference tables2.
The adsorbed-phase enthalpy gives critical insight into the nature of the adsorbent-adsorbate

36
binding-site energy. Moreover, the constant pressure molar heat capacity of the adsorbed
phase (CP) follows directly from the adsorbed-phase enthalpy (Equation 16):
!!

!! = !"!

(16)

The adsorbed-phase molar heat capacities provide critical qualitative insight into the adsorbedphase layer. For a monatomic gas like krypton, comparison to theoretical estimates of the heat
capacity provides further means to peer into the underpinnings of the adsorbed-phase
thermodynamics. Furthermore, the isosteric entropy of adsorption (ΔSads) is directly accessible
from the isosteric enthalpy of adsorption:
Δ!!"# =

!!!"#

(17)

By adding the isosteric entropy of adsorption to the gas-phase molar entropy (from
REFPROP2), we obtain the adsorbed-phase molar entropy (Sa).
!! = !! + ∆!!"#

(18)

37

References:
1.$Bowman,$R.$C.;$Luo,$C.$H.;$Ahn,$C.$C.;$Witham,$C.$K.;$Fultz,$B.$The$Effect$of$Tin$on$the$
Degradation$ of$ LaNi5KYsNy$ MetalKHydrides$ During$ Thermal$ Cycling.$ J.% Alloys% and%
Compd.%1995,$217,$185K192.$
2.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$version$8.0;$
National$Institute$of$Standards$and$Technology:$Gaithersburg,$MD,$2007;$CDKROM.$

84
Chapter 7

Krypton Adsorption on Zeolite-Templated Carbon and Anomalous
Surface Thermodynamics
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “Krypton Adsorption on Zeolite-Templated
Carbon and Anomalous Surface Thermodynamics,” Langmuir, 31, 7991 (2015).
DOI: 10.1021/acs.langmuir.5b01497
Abstract
Krypton adsorption was measured at 8 temperatures between 253 and 433 K on a zeolitetemplated carbon and two commercial carbons. The data were fitted using a generalized
Langmuir isotherm model and thermodynamic properties were extracted. Differing from that
on commercial carbons, krypton adsorption on the zeolite-templated carbon is accompanied
by an increasing isosteric heat of adsorption, rising by up to 1.4 kJ mol-1 as a function of
coverage. This increase is a result of enhanced adsorbate-adsorbate interactions promoted by
the ordered, nanostructured surface of the adsorbent. An assessment of the strength and
nature of these adsorbate-adsorbate interactions is made by comparing the measured isosteric
heats of adsorption (and other thermodynamic quantities) to fundamental metrics of
intermolecular interactions of krypton and other common gases.

85

1. Introduction
High surface area carbon materials have garnered interest for a variety of adsorptive
applications1,2,3,4 ranging from hydrogen storage5,6 to carbon capture7,8 and many others. The
recently emerged class of templated carbon materials9,10,11 exhibiting controlled pore-size
distributions that depend on the template, have shown exceptional performance in many
adsorptive applications owing to their uniquely ordered structure. Zeolite-templated carbon
(ZTC) is one of the highest surface area carbonaceous materials known12, and therefore
exhibits a high specific adsorptive capacity toward small molecular species including
hydrogen13,14 nitrogen15, carbon dioxide16, methane17, and water18. Further, our recent work has
shown that ZTC exhibits not only a high specific adsorptive capacity, but also hosts an
adsorbed phase with highly unusual properties; both ethane19 and methane20 exhibit isosteric
heats of adsorption on ZTC that increase with increasing surface coverage, a particularly rare
and advantageous property for deliverable storage capacity. Due to its chemical homogeneity20
and narrow pore-size distribution centered at a width of 12 Å, the surface of ZTC is optimized
for promoting lateral interactions between adsorbed molecules, even when these interactions
are exceedingly weak (e.g., as for methane).
Krypton, the fourth noble gas, is an unreactive monatomic gas that otherwise bears
many similarities to methane. The two gases share a similar size (Kr: 3.9 Å, CH4: 4.0 Å)21 and
approximately spherical symmetry, as well as similar boiling points (120 K and 112 K,
respectively)22 and critical temperatures (209 K and 190 K, respectively)22. Conveniently,
monatomic krypton allows for very simple calculations of thermodynamic properties such as
entropy, since rotational and internal vibrational modes do not exist. Krypton has applications
in the photography, lighting23, and medical industries,24,25 and is commonly used as an

86
26,27

adsorbate for characterizing low-surface-area materials

. There is also active interest in

finding adsorbent materials that can effectively separate krypton from xenon, especially the
radioactive isotope 85Kr. 28,29 These gases are off-gassed from spent nuclear fuel and their
separation is crucial to the development of “closed” nuclear fuel cycles30. Nevertheless,
krypton adsorption across a wide range of temperatures and pressures is a relatively
unexplored topic, and the results can have relevant implications for many other more complex
adsorptive systems.

2. Experimental
2.1 Materials Synthesis
Three microporous carbons were chosen for this study: MSC-30, CNS-201 and ZTC.
MSC-30 was obtained from Kansai Coke & Chemicals Company Ltd. (Japan) and CNS-201
was obtained from A. C. Carbone Inc. (Canada). ZTC is a zeolite-templated carbon that was
synthesized in a multistep process31 based on a previously reported approach designed to
achieve high template fidelity of the product13. The template used was a NaY zeolite (faujasite)
obtained from Tosoh Corp. (Japan). Briefly, the zeolite was first impregnated with furfuryl
alcohol, which was polymerized at 423 K, before undergoing a 973 K propylene chemical
vapor deposition step, and finally carbonization at 1173 K. The zeolite template was removed
by dissolution in 48% hydrofluoric acid. The synthesis is described in detail elsewhere31.

2.2 Materials Characterization
Nitrogen adsorption isotherms were measured at 77 K using a BELSORP-max
instrument (BEL-Japan Inc.). From these measurements, micropore volumes (Dubinin-

87
32,33

Radushkevich (DR) method

) and specific surface areas (Brunauer-Emmett-Teller (BET)

method34) were determined. Pore-size distributions were obtained using non-local density
functional theory (NLDFT) analysis35,36 with a carbon slit-pore model and software from
Micromeritics Instrument Corp. The skeletal density of each material was determined by
helium pycnometry.
2.3 Measurements
Equilibrium adsorption isotherms of krypton on the three carbon adsorbents were
measured at 8 temperatures between 253 and 433 K. Research-grade krypton (99.998%) was
obtained from Air Liquide America Corp. and used in a custom Sieverts apparatus designed
and tested for accuracy up to 10 MPa.37 Measurements were made up to high pressures using
an MKS Baratron (Model 833) pressure transducer. Each of the samples was degassed at 520
K under a vacuum of less than 10-9 MPa prior to testing. The Sieverts was equipped with a
molecular drag pump capable of achieving a vacuum of 10-10 MPa and vacuum pressures were
verified using a digital cold cathode pressure sensor (I-MAG, Series 423). The adsorbent was
loaded into a stainless steel reactor, sealed with a copper gasket, and held at a constant
temperature. To obtain low temperature isotherms, the reactor was submerged in a circulated
chiller bath with temperature fluctuations no larger than + 0.1 K. High temperature isotherms
were obtained by encasing the reactor in a copper heat exchanger wrapped with insulating
fiberglass heating tape. Using a PID controller, the reactor temperature was maintained with
fluctuations no larger than + 0.4 K. The temperature of the reactor was monitored with Ktype thermocouples while the temperature of the gas manifold was measured with platinum
resistance thermometers. For calculations of excess uptake, bulk phase gas densities were
obtained from the REFPROP Standard Reference Database22. Multiple adsorption/desorption

88
isotherms were taken to ensure complete reversibility and identical measurements were
found to be reproducible to within 1% error.

3. Results
3.1 Adsorbent Characterization
The BET surface areas of ZTC, MSC-30, and CNS-201 were determined to be 3591 +
60, 3244 + 28, and 1095 + 8 m2 g-1, respectively. ZTC and MSC-30 also have similar
micropore volumes of 1.66 and 1.54 cm3 g-1 while CNS-201 has a much smaller micropore
volume of 0.45 cm3 g-1. Despite their similarities, ZTC and MSC-30 have very different poresize distributions (Figure 1). Due to its templated nature, ZTC exhibits a single, sharp peak in
its pore-size distribution, corresponding to a pore width of 12 Å. This has been determined by
NLDFT pore-size analysis and further evidence is given by X-ray diffraction and transmission
electron microscopy (TEM) investigations20. Based on NLDFT pore-size analysis, over 90% of
the micropore volume of ZTC is contained in pores with widths between 8.5 and 20 Å. MSC30 on the other hand has a broad pore-size distribution with micropore widths ranging from 6
to 35 Å and over 40% of its micropore volume is in pores of widths greater than 21 Å. CNS201, has three prominent pore widths at approximately 5.4, 8.0, and 11.8 Å, which contain
roughly 50%, 20%, and 15% of the micropore volume respectively. The skeletal density of
both activated carbons (MSC-30 and CNS-201) was found to be 2.1 g cm-3, which is close to
the ideal density of graphite (~2.2 g cm-3). The templated carbon ZTC, however, was found to
have a skeletal density of 1.8 g cm-3, which is in agreement with other zeolite-templated
carbons.20,31

89

Figure 1. Pore-size distributions of the three carbon materials (CNS-201, MSC-30, and ZTC)
derived from NLDFT analysis of nitrogen adsorption measurements at 77 K.
3.2 Adsorption Measurements
Equilibrium excess adsorption isotherms of krypton on ZTC, MSC-30 and CNS-201
are presented in Figure 2. At high pressures and low temperatures, excess adsorption reaches a
maximum, a well-known phenomenon for Gibbs excess adsorption38. At 253 K, ZTC, MSC30, and CNS-201 have excess adsorption maxima of 22.6, 23.3, and 7.9 mmol g-1, respectively.
At 298 K, the excess adsorption maxima are 16.3, 17.7, and 6.6 mmol g-1, respectively. MSC-30
exhibits a greater excess adsorption maximum than ZTC at all temperatures measured. This is
in contrast to methane adsorption on the same materials where excess adsorption quantities on
ZTC exceeded those on MSC-30 at low temperatures (238-265 K). CNS-201 exhibits the
smallest excess adsorption uptake of the three materials due to its lower surface area.

90

Figure 2. Equilibrium excess adsorption isotherms of krypton on ZTC, MSC-30, and CNS201. The lines indicate the best fit as determined using a generalized (two-site) Langmuir
isotherm model.,

91

4. Data Analysis
4.1 Fitting Methodology
Thermodynamic analysis requires fitting the adsorption data points to a continuous
function. While it is common to assume that excess adsorption is equivalent to absolute
adsorption at low pressures, this assumption becomes invalid at high pressures and low
temperatures. Our method for both fitting and determining the absolute quantity of
adsorption from experimental data is based on a generalized-Langmuir model. Briefly, a
previously described method39 has been further modified to account for phenomena that are
relevant to the nonideal gas regime; the complete details of this methodology are described
elsewhere.40
Gibbs excess adsorption (ne) is a function of both the absolute adsorption (na) and the
gas density in the bulk phase (ρ):
!! = ! !! − ! !! !

(1)

If the volume of the adsorption layer (Va) is known, determining absolute adsorption is trivial
(given excess adsorption). However, as there is no generally accurate method for determining
Va, we have left it as an independent fitting parameter. Excess adsorption quantities were
fitted with the following generalized (multisite) Langmuir isotherm:
!! (!, !) = !!"# − !!"# !(!, !)

!! !
! !! !!! !

(2)

Here the independent fitting parameters are nmax, the maximum absolute adsorption which
serves as a scaling factor, Va, which scales with coverage up to the maximum volume of the
adsorption layer (Vmax), αi which weights the ith Langmuir isotherm (Σi αi=1), and Ki the

92
equilibrium constant of the ith Langmuir isotherm. Ki is given by an Arrhenius-type equation
where Ai is a prefactor and Ei is the binding energy of the ith Langmuir isotherm:
!! =

!!

! !!! !"

(3)

Pressure and temperature are denoted by P and T respectively. By setting the number of
Langmuir isotherms equal to two (i=2) we limit the number of independent fitting parameters
to seven while still obtaining highly accurate fits. The residual mean square values of the fits on
ZTC, MSC-30, and CNS-201 are 0.067, 0.070, and 0.0078 (mmol g-1)2 respectively. Individual
fitting parameters for adsorption on the three materials are given in Table 1.

Table 1. Least Squares Minimized Fitting Parameters of Krypton Excess Adsorption

ZTC
MSC-30

A1
nmax
Vmax
(K1/2
-1
3 -1
(mmol g ) (cm g ) α1
MPa-1)
39
2.0
0.31 0.092
58
3.0
0.73 0.11

A2
(K1/2
MPa-1)
1.8E-6
0.0031

CNS-201

11

0.0059

0.069

0.49

0.46

E1
(kJ mol-1)
10
12

E2
(kJ mol-1)
30
13

15

16

The optimized fit parameters were found to be in reasonable agreement with
independent estimates of physical quantities. For example, Vmax corresponds to the maximum
micropore filling volume. Dividing Vmax by the BET surface area of the adsorbent gives an
average maximum adsorption layer width. For ZTC, CNS-201, and MSC-30, the maximum
adsorption layer widths determined from Vmax are 5.6, 4.5, and 9.2 Å. These are in reasonable
agreement with the average micropore half-widths for ZTC, CNS-201, and MSC-30 as
determined by NLDFT analysis of the nitrogen adsorption uptake at 77K, which are 6, 4, and

93
7 Å, respectively. Additionally, estimates of the maximum possible absolute adsorption can
be made by multiplying the measured micropore volume by the density of liquid krypton (28.9
mmol cm-3)22. For each material, nmax was within 30% of the estimated maximum possible
absolute adsorption.

4.2 Determination of Isosteric Enthalpy of Adsorption
The isosteric enthalpy of adsorption (ΔHads) is a commonly used metric for assessing
the strength of adsorbent-adsorbate interactions at constant coverage conditions. Here it is
evaluated via the isosteric method and reported as a positive value, qst, the isosteric heat
defined by the Clapeyron equation:
!"

!!" = −∆!!"# = −! !"

!!

Δ!!"#

(4)

The molar change in volume of the adsorbate upon adsorption (Δvads) is determined by taking
the difference between the gas-phase molar volume and the average adsorbed-phase molar
volume (the average is approximated as

!!"#
!!"#

). The isosteric heats of krypton adsorption on

ZTC, MSC-30, and CNS-201 calculated in this way are shown in Figure 3.

94

Figure 3. Isosteric heats of krypton adsorption on ZTC, MSC-30, and CNS-201.

95

5. Discussion
5.1 Isosteric Heat of Adsorption
The isosteric heats of krypton adsorption on MSC-30 and CNS-201 decrease as a
function of absolute adsorption, or equivalently surface coverage, as shown in Figure 3. This is
the typical behavior of gas adsorption on a heterogeneous surface, where binding sites are
filled according to energetic favorability. CNS-201 has significantly higher isosteric heat of
adsorption Henry’s Law (zero coverage) values due to its small average pore width (8 Å).
Krypton adsorption on ZTC, however, is accompanied by an initially increasing isosteric heat
with coverage. At 253 K the isosteric heat rises to 14.6 kJ mol-1, 1.4 kJ mol-1 above its Henry’s
Law value of 13.2 kJ mol-1 (an 11% increase). This effect has also been observed in both
ethane and methane adsorption investigations on ZTC.19,20 The increasing isosteric heat is a
result of adsorbate-adsorbate interactions promoted by the nanostructured surface of ZTC, an
effect that becomes larger at low temperatures. As temperature is increased, the effect is
severly diminished. At temperatures above 300 K no increase in the isosteric heat is observed.
This suggests that the adsorbate-adsorbate interactions responsible for the increasing isosteric
heat of adsorption have cooperative behavior that can be thermally disrupted.

5.2 Slope of Increasing Isosteric Heat of Adsorption
The slope of the increasing isosteric heat as a function of fractional coverage roughly
scales with the strength of the intermolecular interactions, as determined by fundamental
metrics such as the critical temperature (CT), boiling point (BP), and the Lennard-Jones well
depth (ε). For krypton, methane, and ethane on ZTC, the average slopes of the isosteric heat

96
up to 50% surface coverage are reported alongside the CT, BP, and ε parameters for each
gas (see Table 2).$
Table 2. Slopes of Isosteric Heats of Adsorption as a Function of Fractional Coverage on ZTC
at the Lowest Measured Temperature and Gas Properties of Krypton, Methane, and Ethane.
Slope (kJ mol-1)
Krypton
Methane
Ethane

ε (kJ mol-1)
CT (K)
BP (K)
2.7
20922
12022
1.321
2.2
19022
11222
1.221
3.3
30522
18522
1.741

The ratios of the krypton/methane and krypton/ethane slopes are 1.2 and 0.82 respectively.
These ratios are similar to the krypton/methane and krypton/ethane ratios of CT, BP, and ε.
Furthermore, the slopes of the isosteric heat are in good agreement with a simplistic model
that we have previously proposed19:
!(!)

!"

= !
!!

(5)
!(!)

The left hand side of Equation 5 (

!!

) is the slope of the isosteric heat as a function of

fractional coverage while z represents the number of nearest neighbors (posited to be 4) and ε
is the Lennard-Jones potential well depth of the gas. Using Equation 5 the slopes of the
krypton, methane and ethane isosteric heats on ZTC are estimated to be 2.6, 2.4, and 3.4 kJ
mol-1 (all within 10 percent of the average measured slope for each gas).

5.3 Isosteric Heat of Adsorption Maxima
At high coverage the isosteric heat of krypton adsorption on ZTC reaches a maximum
and decreases with further coverage. In this regime the adsorbed-phase interatomic
interactions are dominated by short-range repulsion due to the high density of adsorbates. The

97
optimal density for promoting adsorbate-adsorbate interactions is the adsorbate density at
the maximum of the isosteric heat (for a given temperature). Here we label this optimal
adsorbate density “ρΔHmax” and make comparisons to the bulk gas phase via the
compressibility factor (Z). The compressibility factor provides a good metric of the nonideality
of a gas under specific conditions.
!"

! = !"#

(6)

While an ideal gas has a compressibility factor of 1, attractive intermolecular interactions
decrease Z and repulsive interactions increase Z. For a van der Waals gas, the compressibility
factor can be recast in terms of the coefficients of the van der Waals equation of state (a and
b):

!"

! = !!!" − !"#

(7)

In this representation the minimum of the compressibility factor (where attractive interactions
are most dominant) occurs at:

!=

!! !"# !
!!

(8)

As temperature is increased at a fixed volume (V), the minimum point of the compressibility
factor shifts to a lower number of particles (n) and hence to a lower density. The actual
compressibility factor of krypton22 shows similar behavior (see Figure 4).

98

Figure 4. Compressibility factor of krypton between 253-433 K, as calculated by REFPROP22.
The minima are indicated by orange circles.
The importance of the minimum in the compressibility factor is that it represents a
critical point after which repulsive interactions begin to dominate over attractive interactions in
the gas. The density at the compressibility factor minimum (ρZmin), shown in Figure 4, can
therefore be expected to correlate with the density of the adsorbed phase at the maximum in
isosteric heat of adsorption (ρΔHmax). Low temperature values (253-273 K) of ρΔHmax (for
krypton on ZTC) were determined by dividing the absolute adsorption quantity at the isosteric
heat maximum by the ZTC micropore volume (1.66 cm3 g-1). There is reasonable agreement
between ρZmin and ρΔHmax at low temperatures (less than 12% discrepancy) (see Figure 5).

99

Figure 5. Comparison of ρZmin (squares) and ρΔHmax (triangles).
5.4 Adsorbed-Phase Enthalpy
The adsorbed-phase enthalpy (Ha) of krypton on ZTC, MSC-30, and CNS-201 was
determined as a function of coverage by adding the isosteric enthalpy of adsorption to the gasphase enthalpy (Hg) (determined by REFPROP22) (see Figure 6).

!! = !! + Δ!!"#

(9)

100

Figure 6. Adsorbed-phase enthalpy of krypton on ZTC, MSC-30, and CNS-201.

101
Due to favorable adsorbate-adsorbate interactions, the adsorbed-phase enthalpy of
krypton on ZTC decreases towards a minimum (most favorable) enthalpy with coverage.
Conversely, the adsorbed-phase enthalpy of krypton on MSC-30 and CNS-201 increases with
coverage. The adsorbed-phase enthalpy may also be determined as a function of average
interatomic distance. For example, the average interatomic distance of adsorbed krypton (xavg)
at a given state of surface coverage can be determined by dividing the micropore volume, Vmic,
(1.66 cm3 g-1 for ZTC) by the quantity of absolute adsorption (na), and taking the cube root:

!!"# =

!!"# !
!!

(10)

The adsorbed-phase enthalpy at 253 K on ZTC as a function of average interatomic
distance is comparable to the 12-6 Lennard-Jones potential between two krypton atoms21 (see
Figure 7).

Figure 7. Comparison of the adsorbed-phase enthalpy of krypton on ZTC at 253 K (red) and
the 12-6 Lennard-Jones potential of krypton (dashed blue). The inset shows both curves
translated and superimposed for easier comparison.

102
At the lowest measured temperature in this work (253 K), the magnitude and form
of the adsorbed-phase enthalpy of krypton on ZTC is remarkably similar to the 12-6 LennardJones potential. Conversely, the adsorbed-phase enthalpies of krypton on MSC-30 and CNS201 display no such behavior and are monotonically increasing functions. This provides
further evidence that the anomalous isosteric heat of adsorption of krypton on ZTC results
from enhanced interatomic interactions which can be rather accurately accounted for by the
classic 12-6 interaction potential. The apparent offset in energy seen in Figure 7 is a result of
the adsorbent-adsorbate interactions and the arbitrary nature of the enthalpy reference state (in
this case the reference state is saturated liquid krypton at its normal boiling point, 120 K). The
offset in interatomic distance (~0.07 nm) between the adsorbed-phase enthalpy and the 12-6
Lennard-Jones potential may result from clustering of the krypton atoms. While some of the
krypton atoms may temporarily cluster into more optimally spaced groupings which reproduce
the 12-6 Lennard-Jones potential shape and distance, the presence of non-clustered krypton
atoms with larger interatomic distances could shift the average interatomic spacing to higher
values resulting in the offset in interatomic distance.
The Henry’s Law values of the adsorbed-phase enthalpies are also indicative of the
atypical properties of ZTC as an adsorbent for krypton. The enthalpy of a two-dimensional
ideal gas is 2RT and therefore depends linearly on temperature with a slope of 2R (16.6 J mol-1
K-1). Correspondingly, the Henry’s Law values of the adsorbed-phase enthalpy of krypton on
MSC-30 and CNS-201 also depend linearly on temperature, with slopes of 15.6 and 16.7 J mol1

K-1, respectively (see Figure 8).

103

Figure 8. Henry’s law enthalpies of adsorbed krypton on ZTC (circles), MSC-30 (triangles) and
CNS-201 (squares) as a function of temperature. Lines are to guide the eye.
The Henry’s Law values of the adsorbed-phase enthalpy of krypton on ZTC, however,
do not vary linearly with temperature until beyond 350 K. At high temperatures (>350 K) the
slope converges to approximately 2R. At low temperatures, however, deviations between the
measured Henry’s Law values and those predicted using the ideal gas slope of 2R, are
observed. These deviations likely result from the unique structure of ZTC and may in part be
due to a loss of favorable interactions with increasing temperature.

5.5 Entropy
The isosteric entropy of adsorption (ΔSads) is the change in entropy upon adsorption: the
difference between the entropy of the adsorbed phase and the entropy of the gas phase at
isosteric conditions. At equilibrium, the isosteric entropy of adsorption and the isosteric
enthalpy of adsorption (ΔHads) are related by:
Δ!!"# =

!!!"#

(13)19

104
As for the enthalpy, the entropy of the adsorbed phase can be determined by adding the
isosteric entropy of adsorption to the entropy of krypton gas (calculated using REFPROP22).
The molar entropy of adsorbed-phase krypton as a function of coverage on the three materials
in this study is shown in Figure 9. The reference state in this case is solid krypton at absolute
zero.

105

Figure 9. The entropy of adsorbed-phase krypton on ZTC, MSC-30 and CNS-201 derived by
experiment (lines) and calculated using statistical mechanics (asterisks).

106
For comparison to the experimental data, the adsorbed-phase entropy of krypton
was also calculated using statistical mechanics (shown in Figure 9). A basic statistical mechanics
model based on a two-dimensional lattice gas was used, as described elsewhere19. Since krypton
is a monatomic gas with spherical symmetry and no internal vibrational modes, only partition
functions for the surface vibrational modes and configurational modes were considered.$The
entropies corresponding to these individual contributions were determined and summed to
obtain the total entropy of the adsorbed phase (see Figure 9).
For krypton adsorbed on MSC-30 and CNS-201, agreement between the measured
and calculated adsorbed-phase entropies is good, with discrepancies of 5% and 15%
respectively. The small pores and high isosteric heat of adsorption of krypton on CNS-201
result in less accurate statistical mechanics approximations of surface vibrational modes and
hence somewhat larger deviations than for MSC-30. Moreover, the general temperature
dependence is preserved in both cases, especially at low quantities of uptake. For krypton on
ZTC, however, discrepancies are in excess of 23% despite moderate pore sizes; the
experimental adsorbed-phase entropy is much lower than estimated values due to enhanced
interatomic interactions. It is reasonable to attribute a large fraction of this discrepancy to
clustering effects (reduced configurations of the adsorbed phase due to interatomic
interactions)40, and further investigation of such phenoma is warranted.

6. Conclusions
Equilibrium excess adsorption uptake of krypton was measured on three microporous
carbon materials: ZTC, MSC-30 and CNS-201. By fitting the data using a robust generalized
Langmuir isotherm model, absolute adsorption quantities were determined along with

107
physically realistic fitting parameters and thermodynamic quantities of adsorption. While the
isosteric heat of adsorption decreases with coverage on MSC-30 and CNS-201 (the typical
case), it increases by over 10% on ZTC, reaching its maximum at a surface coverage of 19.6
mmol g-1 at 253 K. This previously unreported effect for supercritical krypton adsorption on a
high surface area carbon results from the enhancement of favorable krypton-krypton
interactions on the ZTC surface due to its uniquely ordered porous nanostructure. Moreover,
the magnitude of the increase is dependent on the strength of the interatomic interactions of
krypton, a result that is corroborated by comparisons to ethane and methane. Additional
analysis of the isosteric heat of adsorption maxima, adsorbed-phase enthalpy, and adsorbedphase entropy provide further evidence and insight into the nature of the interactions
responsible for the anomalous surface thermodynamics reported in this paper.

Acknowledgement:
This work was sponsored as a part of EFree (Energy Frontier Research in Extreme
Environments), an Energy Frontier Research Center funded by the US Department of Energy,
Office of Science, Basic Energy Sciences under Award Number DE-SC0001057.

108

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Anomalous Thermodynamics of
Nonideal Gas Physisorption on
Nanostructured Carbons

Thesis by

Maxwell Robert Murialdo

In Partial Fulfillment of the Requirements for
the degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
2017
(Defended July 28, 2016)

ii

Maxwell Murialdo

iii

ACKNOWLEDGEMENTS
I am blessed to have been surrounded by the most wonderful people who have guided,
supported and generally made this work possible. My advisor, Prof. Brent Fultz, has been a
brilliant source of guidance through my years at Caltech and I hold in high esteem his piercing
insight and penchant for navigating the obstacles of fundamental science. Dr. Nicholas Stadie
has served as a great mentor with dedicated enthusiasm and a wealth of knowledge both
hands-on and theoretical. I would like to thank the entire Fultz Group (Dr. David Abrecht,
Dr. Channing Ahn, Dr. David Boyd, Dr. Olle Hellman, Dr. Tian Lan, Dr. Jiao Lin, Dr. Lisa
Mauger, Dr. Jorge Muñoz, Dr. Hillary Smith, Dr. Hongjin Tan, Dr. Sally Tracy, Dr. Heng
Yang, Bryce Edwards, Jane Herriman, Dennis Kim, Claire Saunders, Yang Shen, Xiao Tong,
Nick Weadock, and Fred Yang along with summer students Christopher Gardner and Jun-Ren
Chen) for their help, guidance, ideas, conversations, and general support. This work would also
not have been possible without the generous support of the Energy Frontier Research Center
in Extreme Environments (EFree) and the dedication of Dr. Russell Hemley and Dr. Stephen
Gramsch. EFree has provided a number of opportunities for collaborations and I am grateful
for the contributions of my inter-institutional collaborators (Prof. John Badding, Prof. Nasim
Alem, Stephen Juhl, Prof. Kai Landskron and Yiqun Liu). A special thanks to my thesis
committee members (Prof. Marco Bernardi, Prof. Katherine Faber, Prof. William Goddard III,
and Prof. William Johnson) for their guidance both inside and outside of the classroom.
Finally I would like to thank my wonderful family and friends for their endless encouragement
and understanding.

iv

ABSTRACT
Mesoporous and microporous adsorbents play critical roles in gas storage and
separation

applications.

This

thesis

describes

previously

unexplored

anomalous

thermodynamics in the field of gas physisorption and their impact on energy relevant gases
including methane, ethane, krypton and carbon dioxide. Physisorption occurs when an
adsorbent induces gas molecules to form a locally densified layer at its surface due to physical
interactions. This increases gas storage capacity over pure compression and its efficacy is
dependent on the surface area of the adsorbent and the isosteric heat of adsorption. The
isosteric heat of adsorption is the molar change in the enthalpy of the adsorptive species upon
adsorption and serves as a measure of adsorbent-adsorbate binding strength.
Unlike conventional adsorbate-adsorbent systems, which have isosteric heats of
adsorption that decrease with surface loading, zeolite-templated carbon is shown to have
isosteric heats of methane, ethane and krypton adsorption that increase with surface loading.
This is a largely beneficial effect that can enhance gas storage and separation. The unique
nanostructure and uniform pore periodicity of the zeolite-templated carbon promote lateral
interactions among the adsorbed molecules that cause the isosteric heats of adsorption to
increase with loading. These results have been tested and corroborated by developing robust
fitting techniques and thermodynamics analyses. The anomalous thermodynamics are shown
to result from cooperative adsorbate-adsorbate interactions among the nonideal species and
are modeled with an Ising-type model.
As a second theme of this thesis, the study of nonideal gas adsorption has enabled the
development of a Generalized Law of Corresponding States for Physisorption. A predictive
understanding of high-pressure physisorption on a variety of adsorbents would facilitate the

further development of tailored adsorbents and adsorption analysis. Prior attempts at
developing a predictive understanding, however, have been hindered by nonideal gas effects.
By approaching physisorption from both empirical and fundamental perspectives, a
Generalized Law of Corresponding States for Physisorption was established that accounts for
a number of nonideal effects. This new Law of Corresponding States allows one to predict
adsorption isotherms for a variety of classical gases from data measured with a single gas. In
brief:
“At corresponding conditions on the same adsorbent, classical gases physisorb to the same
fractional occupancy.”
Corresponding conditions are met when the reduced variables of each nonideal gas are
equivalent, and fractional occupancy gives the fraction of occupied adsorption sites. This Law
of Corresponding States for Physisorption is determined using monolayer, BET and DubininPolanyi adsorption theories along with measured adsorption isotherms across a number of
conditions and adsorbents. Furthermore, the anomalous cooperative adsorbate-adsorbate
interactions discussed in this thesis are shown to be consistent with the Generalized Law of
Corresponding States for Physisorption.

vi

PUBLISHED CONTENT AND CONTRIBUTIONS
1. M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “A Generalized Law of Corresponding
States for the Physisorption of Classical Gases with Cooperative Adsorbate-Adsorbate
Interactions,” J. Phys. Chem. C, 120, 11847 (2016).
DOI: 10.1021/acs.jpcc.6b00289
M. M. participated in the data collection, analysis, conceptualization and writing of the manuscript.

2. N.P. Stadie, M. Murialdo, C.C. Ahn, and B. Fultz, “Unusual Entropy of Adsorbed Methane
on Zeolite-Templated Carbon,” J. Phys. Chem. C, 119, 26409 (2015).
DOI: 10.1021/acs.jpcc.5b05021
M. M. participated in the data collection, analysis, conceptualization and writing of the manuscript.

3. M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “Krypton Adsorption on ZeoliteTemplated Carbon and Anomalous Surface Thermodynamics,” Langmuir, 31, 7991 (2015).
DOI: 10.1021/acs.langmuir.5b01497
M. M. participated in the data collection, analysis, conceptualization and writing of the manuscript.

4. M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, "Observation and Investigation of
Increasing Isosteric Heat of Adsorption of Ethane on Zeolite-Templated Carbon," J. Phys.
Chem. C, 119, 994 (2015).
DOI: 10.1021/jp510991y
M. M. participated in the data collection, analysis, conceptualization and writing of the manuscript.

5. N.P. Stadie, M. Murialdo, C.C. Ahn, and B. Fultz, “Anomalous Isosteric Enthalpy
of Adsorption of Methane on Zeolite-Templated Carbon,” J. Am. Chem. Soc.
135, 990 (2013).
DOI: 10.1021/ja311415m
M. M. participated in the data collection, analysis, conceptualization and writing of the manuscript.

6. M. Murialdo, C.C. Ahn, and B. Fultz, “A Thermodynamic Investigation of AdsorbateAdsorbate Interactions of Carbon Dioxide on Nanostructured Carbons,” AIChE J.
(submitted, 2016).
M. M. participated in the data collection, analysis, conceptualization and writing of the manuscript.

vii

TABLE OF CONTENTS

Acknowledgements…………………………………………………………...iii
Abstract ………………………………………………………………………iv
Published Content and Contributions…………………………………….........vi
Table of Contents……………………………………………………………. vii
Nomenclature……………………………………………………………….…viii
Chapter I: An Introduction to Adsorption .......................................................... 1
Chapter II: Fundamentals of Adsorption .......................................................... 14
Chapter III: Experimental Methods and Analysis ............................................ 28
Chapter IV: Materials ....................................................................................... 38
Chapter V: Methane on Microporous Carbons ................................................ 46
Chapter VI: Observation and Investigation of Increasing Isosteric Heat ............
Of Adsorption of Ethane on Zeolite-Templated Carbon............. 59
Chapter VII: Krypton Adsorption on Zeolite-Templated Carbon
And Anomalous Surface Thermodynamics ............................... 84
Chapter VIII: A Generalized Law of Corresponding States for the
Physisorption of Classical Gases with Cooperative
Adsorbate-Adsorbate Interactions ........................................... 111
Chapter VIII: Supporting Information............................................................ 137
Chapter IX: A Thermodynamic Study of Carbon Dioxide Adsorption
On Nanostructured Carbons at High Pressures .......................... 158
Conclusions ..................................................................................................... 183

viii

NOMENCLATURE

Adsorption. The densification of molecules near an interface due to favorable interactions.
Physisorption. Adsorption resulting from physical (not chemical) interactions.
Adsorbent. Solid surface providing the adsorption interface.
Adsorbate. Molecules that have been adsorbed.
Absolute Adsorption (n a ). A measure of all of the molecules in the adsorbed phase.
Excess Adsorption (n e ). A measure of the molecules in the adsorbed phase in excess of the
gas-phase density.
Isotherm. Measured adsorption at a series of pressures and a constant temperature.
Isosteric Enthalpy of Adsorption (ΔH ads ). The difference between the adsorbed-phase and
gas-phase enthalpy at constant coverage conditions.
Isosteric Heat of Adsorption (q st ). The positive value of the isosteric enthalpy of
adsorption.
Isosteric Entropy of Adsorption (ΔS ads ). The difference between the adsorbed-phase and
gas-phase entropy at constant coverage conditions.
Adsorbed-Phase Enthalpy (H a ). The molar enthalpy of the molecules in the adsorbed
phase.
Adsorbed-Phase Entropy (S a ). The molar entropy of the molecules in the adsorbed phase.

ix
Adsorbed-Phase Heat Capacity at Constant Pressure (C P ). The constant-pressure heat
capacity of the molecules in the adsorbed phase.
Gas-Phase Enthalpy (H g ). The molar enthalpy of the molecules in the gas phase.
Gas-Phase Entropy (S g ). The molar entropy of the molecules in the gas phase.

Chapter 1

An Introduction to Adsorption
1. Overview
Recently, after I finished a talk on the thermodynamics of adsorption, an audience
member stood up with a critical comment. “But adsorption has been studied for over a
hundred years.” Yes, exactly!
Serious studies of adsorption have been ongoing for over a hundred years now.
Qualitative knowledge of adsorption goes back thousands of years. This is not to the detriment
of the field. Rather it is a testament to the broad utility of adsorption and the complexity of
trying to understand such a diverse array of phenomena.
Adsorption is the densification of a fluid at an interface. The nature of the interface
may be solid-liquid, gas-liquid, liquid-liquid or solid-gas. In this work we focus on adsorption
between a solid “adsorbent” and a gaseous “adsorptive species” which is denoted the
“adsorbate” in the adsorbed phase. Adsorption is strictly an interfacial phenomenon, unlike
“absorption” wherein the absorptive species penetrates the absorbent. Each molecule in the
adsorbed phase actually has less entropy than in the bulk gas phase; however it also has a
reduced enthalpy owing to attractive interactions at the interface. This sets up the basic
equilibrium of adsorption: a reduction in entropy compared to the gas phase is balanced by a
comparable reduction in enthalpy. Adsorption is thus an exothermic process. The favorable
interactions at the interface, which enable all adsorption, may be of chemical or physical
nature. When chemical bonds occur between the adsorbent and the adsorbate the
phenomenon is called chemisorption. When only physical interactions are present (e.g. van der

Waals forces), it is called physisorption. “Sorption” is a more general term used for the
ambiguous case. In general, chemisorption results in much stronger adsorbent-adsorbate
interactions and is effective at higher temperatures, but requires specific adsorbate-adsorbent
systems and is less reversible. In this work we focus on physisorptive systems.

2. Pores
The uses of physisorption are extremely diverse, the most primary of which is the
characterization of the adsorbents themselves. Solid-gas physisorption may occur on any solid,
but as an interfacial phenomenon, physisorption is strictly a surface effect. High surface area
materials, especially porous materials, therefore lend themselves to adsorptive studies. Often
porous materials can only be easily and effectively characterized by physisorption. The pores
of a porous material may be broadly lumped into one of three size categories: macropores,
mesopores and micropores. Macropores are greater than 50 nanometers in width. Mesopores
have widths between 2 and 50 nanometers. Micropores have widths of less than 2 nanometers.
Both micropores and mesopores are considered “nanopores”. Pores are typically modeled as
either cylinders with unique pore diameters, or slits with unique pore widths. Macropores,
mesopores and micropores each physisorb gases in distinct ways and require different
modeling.
In macropores the pore walls are sufficiently well separated to have non-overlapping
effects. This is because physical interactions like van der Waals forces fall off rapidly with
distance. Macropores can thus be treated in the same way as a nonporous material, where only
a single surface is considered at a time. In macropores, layer-by-layer adsorption is expected
and can be effectively modeled using Brunauer, Emmett and Teller (BET) Theory. BET
Theory has proven highly successful in determining the surface area of macroporous

adsorbents. Using subcritical adsorption isotherm measurements of nitrogen, argon, carbon
dioxide or krypton (for low surface area adsorbents), up to near-saturation pressures, the
monolayer coverage can be solved from BET Theory. This quantitative monolayer coverage is
in turn correlated with a specific surface area based on the size of the adsorbate molecule. One
caveat is that different results may be obtained with different gases (probe molecules) as
different pores and topologies are accessible to different size probe molecules.
Mesopores may be treated in much the same way as macropores, with the additional
complication of capillary condensation. In appropriately sized mesopores, surface tension
(through capillary action) can cause the adsorptive species to condense into a liquid phase in
the mesopores at pressures below the bulk saturation pressure. This phenomenon is unique to
mesopores as macropores are too large and micropores are too small. Capillary condensation
results in hysteresis between the adsorption and desorption isotherms. However, the presence
of capillary condensation can be used to obtain information about the pore-size distribution
via the Kelvin Equation and variants.
Micropores are of similar dimensions to the gases adsorbed. Within micropores,
opposite pore walls often exhibit overlapping potentials. Thus micropores are more accurately
treated with a pore-filling model than a layer-by-layer model for adsorption. In general the
adsorptive species will holistically fill the micropore volume, making metrics of specific
micropore volume more important than specific surface area. The micropore volume may be
determined empirically with the Dubinin-Radushkevich equation and variants. Moreover, of
critical importance, the micropore-size distribution may be obtained semi-empirically via
nonlocal density functional theory (NLDFT). As such, physisorption may be used to
characterize pores varying in size from 0.35 nanometers to greater than 100 nanometers.

3. Applications
Adsorbents are also widely used in industrial scale processes like catalysis and gas
separation. Catalysis is an enabling factor in ~90% of chemical and materials manufacturing
worldwide and may employ homogeneous or heterogeneous catalysts.1 Unlike homogeneous
catalysts, heterogeneous catalysts are of a different phase than the underlying reaction. Often
porous solids are used as heterogeneous catalysts in conjunction with a liquid or gas phase.
The fluid phase is first adsorbed onto the catalyst, followed by dissociation of the fluid, surface
diffusion, a surface reaction and finally desorption of the product. Adsorption is thus a
prerequisite for most heterogeneous catalysis and a fundamental understanding of the
adsorption process is vital. Moreover, as adsorbents typically have large accessible surface
areas, they may be used as structural supports that keep catalysts well dispersed to maximize
efficacy.
Gas separation and purification is a second key industrial use of adsorption. Separating
chemically inert gases or otherwise removing gaseous impurities is often done by cryogenic
distillation. For gases with similar boiling points this process can be both expensive and energy
intensive. Gas separation using physisorption offers an efficient alternative, especially when
high levels of purity are not required. In physisorptive separation, gases are flowed through an
adsorbent bed where one gas preferentially adsorbs over another, ideally with great selectivity.
The adsorbent bed may then be regenerated by reducing the pressure or increasing the
temperature to desorb the adsorbate. These cyclic processes are referred to as Pressure Swing
Adsorption (PSA) and Temperature Swing Adsorption (TSA), respectively. The selectivity is
typically due to differences in the adsorbent-adsorbate physical interactions between each gas
and the adsorbent. In some cases steric effects are used to enhance selectivity when the gases

to be separated are of dissimilar size or shape and only small and correctly shaped molecules
can penetrate a well-defined pore structure. This is commonly used to dry steam from the
cracking process or to dry natural gas using zeolites with well-defined micropore structures.
Kinetic separation mechanisms may also be employed in molecular sieves where non-uniform
pore size distributions allow different molecules to diffuse at different rates. One common
example is the separation of nitrogen from air using molecular sieves. Overall, adsorption
offers an efficient means of gas separation, purification and in some cases solvent recovery
from both industrial and vented sources.

4. Krypton
In this work we study krypton adsorption in detail as a step towards improving
krypton separation from other inert gases. Krypton, the fourth noble gas, is an unreactive
monatomic gas that otherwise bears many similarities to methane. The two gases share a
similar size (Kr: 3.9 Å, CH4: 4.0 Å)2 and approximately spherical symmetry, as well as similar
boiling points (120 K and 112 K, respectively)3 and critical temperatures (209 K and 190 K,
respectively). Conveniently, monatomic krypton allows for very simple calculations of
thermodynamic properties such as entropy, since rotational and internal vibrational modes do
not exist. Krypton has applications in the photography, lighting4 and medical industries5,6, and
is commonly used as an adsorbate for characterizing low-surface-area materials7,8. There is also
significant active interest in finding a cost effective and efficient means of separating krypton
from xenon derived from nuclear waste9,10.
Nuclear power plants supply over 10% of the world’s electricity and are a valuable
source of energy in the United States.11 Unfortunately, the United States is already fraught with
over 70,000 tons of nuclear waste and no good storage options12,13. One avenue towards

diminishing the amount of nuclear waste generated in the future is to reprocess spent
nuclear fuel. Reprocessing entails chopping up and dissolving spent nuclear fuel to recover
fissionable remains. These remains can be used to generate additional electricity often
exceeding 25% of the initial generation.14 While the US does not currently reprocess nuclear
fuel, reprocessing is typical in Europe, Russia and Japan, and as a member of the International
Framework for Nuclear Energy Cooperation, the US has partnered with other countries to
improve and develop closed nuclear fuel cycles with reprocessing. During reprocessing,
radioactive krypton-85 and nonradioactive xenon are off-gassed. As a radioactive mixture,
these gases should be stored as radioactive waste, often in inefficient mole fraction of ~90%
nonradioactive xenon to ~10% radioactive krypton-85.15 While cryogenic distillation is an
energy intensive means of separating krypton from xenon, adsorbents offer a potentially
efficient alternative. With properly tuned adsorptive separation, less nonradioactive xenon
would need to be stored, putting less stress on current nuclear waste storage options.

5. Natural Gas
Another large and quickly growing application of physisorption is for the densified
storage of natural gas, particularly within the transportation sector. This is a major focus of this
thesis.
Natural gas powers 22% of the world and 33% of the US, and yet its importance is
projected to grow.16 From 2010 to 2013 worldwide natural gas consumption has grown at a
rate of 2% annually from 113,858 to 121,357 billion cubic feet.16 Over the same time, the
proven reserves have grown at a rate of 1% annually, up to 6,972.518 trillion cubic feet.16

Assuming a 2% annual increase in consumption, but no increase in the proven reserves, our
current proven reserves will last into the 23rd century, although estimates vary.16
Natural gas predominantly originates from two naturally occurring processes. Biogenic
methane results when methanogenic archaea break organic matter into simple hydrocarbons
like methane. These microorganism live in oxygen depleted regions of the earth’s crust and in
the intestines of most animals. Collecting the gases emanating from manure and landfills has
proven a clean and renewable source of natural gas. Nonetheless, to date these operations are
small scale and pale in comparison to the scope of thermogenic natural gas collection.
Thermogenic methane is synthesized from ancient organic matter under the high temperatures
and pressures found deep in the earth’s crust. This process takes millions of years and is thus
not considered renewable on a human timescale. These natural gas fossil fuels are often
discovered alongside oil, although deeper deposits synthesized at higher temperatures tip the
balance towards a higher fraction of natural gas synthesis.
Thermogenic methane is also found alongside coal (coalbed gas) or as a methane
hydrate (clathrate). Methane hydrates form under temperatures below 15 °C and pressures
greater than 19 bar. These conditions are met in offshore continental shelves and the
permafrost of Siberia. Recently Japan Oil, Gas and Metals National Cooperation has recovered
commercially viable quantities of natural gas from oceanic methane clathrates.17 This presents a
particularly exciting advancement given estimates of vast quantities of oceanic methane
hydrates (estimated at 5x1015 cubic meters of methane hydrate).16
Other unconventional sources of natural gas have been studied and commercialized
within the past decade. These sources have historically been economically prohibitive, but this
is changing with recent technological advancements. The collection of deep gas, tight gas, and

shale gas are on the rise. Shale gas, in particular has boomed from less than 1% of the US
natural gas production in 2000 to 39% in 2012.16 It is estimated that there may be more than a
quadrillion cubic feet of unconventional natural gas reserves in the US alone.16
At present, 15 countries account for 84% of worldwide natural gas production, with
Russia, Iran, Qatar, Turkmenistan and the US at the top of the list.16 Before transportation or
use, the natural gas must be purified. Impurities like water, sand and other gases are separated
out. Some of the purified gas byproducts like propane, butane and hydrogen sulfide are sold
on secondary markets. Landfill methane is prone to have large quantities of carbon dioxide
and hydrogen sulfide, which must be removed before transportation to prevent pipeline
corrosion. The gas may be further purified to achieve a high quality gas comprised of almost
pure methane and known as “dry” natural gas. In the presence of significant quantities of
other hydrocarbons, natural gas is deemed “wet”. The main constituent in natural gas is always
methane, followed by ethane. The composition varies significantly but is generally in
accordance with Table 1.
Table 1. Typical Composition of Natural Gas18
gas
methane
ethane
propane
butane
alkanes #C > 4
nitrogen
carbon dioxide
oxygen
hydrogen

mole fraction (%)
87-97
1.5-7
0.1-1.5
0.02-0.6
trace
0.2-5.5
0.1-1.0
0.01-0.1
trace

The transportation and utilization of natural gas has been employed since 500 B.C.
when Chinese in the Ziliujing district of Sichuan developed crude natural gas pipelines from
bamboo.19 This natural gas was harnessed to boil seawater. It wasn’t until 1785, however, that
natural gas first found widespread commercialization as a fuel for streetlamps and homes in
England.19 This natural gas emanated from fissures above naturally occurring pockets. In 1821
William A. Hart drilled the first intentional natural gas well, in Fredonia, New York.19 Since
then the industry has exploded in the US and abroad. An emerging natural gas industry set to
explode is for use in the transportation sector. Worldwide tens of millions of natural gas
vehicles dot the roads. Ordinary internal combustions engines can be converted to run on
natural gas for less than $10,000 and commercial vehicles designed specifically to run on
natural gas are gaining traction. Starting in 2008 with the Honda Civic natural gas, major auto
manufacturers have churned out a number of compressed natural gas commercial vehicles
including the Chevrolet Silverado 2500, Dodge Ram 2500, Ford F-250 and Chevrolet Savana.
This surge in natural gas cars proffers environmental benefits. For the same amount of
energy, thermogenic methane emits 16% less carbon dioxide than diesel.16 Landfill biogas can
emit a net 88% less carbon dioxide than diesel.16 In power generation, methane power plants
emit almost 50% less carbon dioxide than coal-powered plants.16 The carbon dioxide released
per million BTU, for a variety of common fuels, is listed in Table 2. Methane is the cleanest
burning hydrocarbon. Additionally, natural gas emits less trace pollutants including carbon
monoxide, nitrogen oxides, sulfur oxides and particular matter than other fuels.

10
16

Table 2. Pounds of CO2 Emissions Per MBTU for Common Fuels
fuel
Coal (anthracite)
Coal (bituminous)
Coal (lignite)
Coal (subbituminous)
Diesel fuel & heating oil
Gasoline
Propane
Natural Gas

Lbs. CO2/MBTU
228.6
205.7
215.4
214.3
161.3
157.2
139
117

The main hindrance to wide implementation of natural gas in the transportation sector
stems from the onboard storage problem. Natural gas has a high gravimetric energy density of
-1

56 MJ kg , competitive with other fuel sources, but an abysmally low volumetric energy
-3

density of 37 MJ m at standard conditions.3 This is a common problem for gaseous fuels
with critical temperatures significantly below room temperature. Three potential solutions have
been separately implemented in commercial applications: cryogenic liquefaction, high-pressure
compression, and compression in the presence of adsorbent materials.
Cryogenic liquefaction requires cooling natural gas to below the critical temperature of
methane, which is 190K. This is typically achieved with liquid nitrogen as a coolant in a
cryogenic setup. While cryogenic liquefaction can achieve volumetric energy densities as high
-3

as 22 GJ m , it is an energy intensive process and requires expensive equipment and
monitoring3. These drawbacks have thus far prevented the widespread adoption of cryogenic
liquefaction as a means for natural gas storage on privately owned vehicles.

11
At high pressures of 700 bar, natural gas has a volumetric energy density of ~17 GJ
-3

m .3 However, high-pressure compression requires specialized storage tanks. As pressure
requirements are increased, the requisite class of storage tank shifts from Type I (all metal,
which is the cheapest and currently makes up 93% of the onboard natural gas storage market)
to Type IV, all composite with high associated costs.20 Moreover, high pressures limit potential
tank designs (as a necessity of eliminating weak points) and potentially pose a significant threat
if ruptured intentionally or unintentionally.
The use of adsorbents and moderate compression allows for significant volumetric
energy density improvements over pure compression at moderate pressures and temperatures.
At low and moderate pressures the favorable interactions between the adsorbent and the
adsorbate densify natural gas under equilibrium conditions. Improved natural gas storage is a
significant subject of inquiry in this thesis.
Adsorption also serves a number of other niche purposes in areas such as heat pumps
and spacecraft environmental controls.21 All told, adsorption spans chemistry, biology, physics
and engineering and is an integral part of our world. It demands further fundamental scientific
inquiry and expertise in adsorbent engineering.

12

References:
1.$Chorkendorff,$ I.;$ Niemantsverdriet,$ J.$ W.$ Concepts$ of$ Modern$ Catalysis$ and$
Kinetics,$2nd$ed.;$Wiley:$New$York,$2007.$
2.$Cuadros,$ F.;$ Cachadina,$ I.;$ Ahumada,$ W.$ Determination$ of$ LennardKJones$
Interaction$Parameters$Using$a$New$Procedure.$Mol.%Eng.%1996,$6,$319K325.$
3.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$version$8.0;$
National$Institute$of$Standards$and$Technology:$Gaithersburg,$MD,$2007;$CDKROM.$
4.$Hwang,$ H.$ S.;$ Baik,$ H.$ K.;$ Park,$ K.$ W.;$ Song,$ K.$ M.;$ Lee,$ S.$ J.$ Excitation$ Energy$
Transfer$of$Metastable$Krypton$Atoms$in$KrKHeKXe$Low$Pressure$Glow$Discharge$for$
MercuryKFree$Lighting.$Jpn.%J.%Appl.%Phys.%2010,$49,$1K3.$
5.$Chon,$D.;$Beck,$K.$C.;$Simon,$B.$A.;$Shikata,$H.;$Saba,$O.$I.;$Hoffman,$E.$A.$Effect$of$
LowKXenon$ and$ Krypton$ Supplementation$ on$ Signal/Noise$ of$ Regional$ CTKbased$
Ventilation$Measurements.$J.%Appl.%Physiol.%2007,$102,$1535K1544.$
6.$Pavlovskaya,$ G.$ E.;$ Cleveland,$ Z.$ I.;$ Stupic,$ K.$ F.;$ Basaraba,$ R.$ J.;$ Meersmann,$ T.$
Hyperpolarized$ KryptonK83$ as$ a$ Contrast$ Agent$ for$ Magnetic$ Resonance$ Imaging.$
Proc.%Natl.%Acad.%Sci.%U.S.A.%2005,$102,$18275K18279.$
7.$Takei,$ T.;$ Chikazawa,$ M.$ Measurement$ of$ Pore$ Size$ Distribution$ of$ LowKSurfaceK
Area$ Materials$ by$ Krypton$ Gas$ Adsorption$ Method.$ J.% Ceram.% Soc.% Jpn.% 1998,$ 106,$
353K357.$
8.$Youssef,$A.$M.;$Bishay,$A.$F.;$Hammad,$F.$H.$Determination$of$Small$SurfaceKAreas$
by$Krypton$Adsorption.$Surf.%Technol.%1979,$9,$365K370.$
9.$Ryan,$ P.;$ Farha,$ O.$ K.;$ Broadbelt,$ L.$ J.;$ Snurr,$ R.$ Q.$ Computational$ Screening$ of$
MetalKOrganic$Frameworks$for$Xenon/Krypton$Separation.$AIChE%J.%2011,$57,$1759K
1766.$
10.$Bae,$ Y.KS.;$ Hauser,$ B.$ G.;$ Colon,$ Y.$ J.;$ Hupp,$ J.$ T.;$ Farha,$ O.$ K.;$ Snurr,$ R.$ Q.$ High$
Xenon/Krypton$ Selectivity$ in$ a$ MetalKOrganic$ Framework$ with$ Small$ Pores$ and$
Strong$Adsorption$Sites.$Microporous%Mesoporous%Mater.%2013,$169,$176K179.$
11.$World$ Statistics:$ Nuclear$ Energy$ Around$ the$ World.$ Nuclear% Energy% Institute.$
(accessed$2016).$
12.$Stop$Dithering$on$Nuclear$WasteKThree$Decades$After$Chernobyl,$the$US$Needs$
to$Tackle$its$Own$Ominous$Nuclear$Safety$Problem.$Sci%Am.%2016,$314,$10.$
13.$Stuckless,$ J.$ S.;$ Levich,$ R.$ A.$ The$ Road$ to$ Yucca$ MountainKEvolution$ of$ Nuclear$
Waste$Disposal$in$the$United$States.$Environ.%Eng.%Geosci.%2016,$22,$1K25.$
14.$Processing$ of$ Used$ Nuclear$ Fuel.$ http://www.worldKnuclear.org/informationK
library/nuclearKfuelKcycle/fuelKrecycling/processingKofKusedKnuclearKfuel.aspx$
(accessed$2016).$
15.$Banerjee,$ D.;$ Cairns,$ A.$ J.;$ Liu,$ J.;$ Motkuri,$ R.$ K.;$ Nune,$ S.$ K.;$ Fernandez,$ C.$ A.;$
Krishna,$R.;$Strachan,$D.$M.;$Thallapally,$P.$K.$Potential$of$MetalKOrganic$Frameworks$
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16.$Independent$ Statistics$ and$ Analysis.$ U.S.% Energy% Information% Administration.$

13
17.$Tabuchi,$H.$An$Energy$Coup$for$Japan:$'Flammable$Ice'.$New%York%Times,%%Mar.$
12,$2013,$B1.$
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(accessed$2016).$
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2001,$93,$135K224.$

14
Chapter 2

Fundamentals of Physisorption
1. Background
Early work into determining a universal equation of state for all gases led to the
development of the ideal gas law:
!" = !"#

(1)

This law, derived by Emile Clapeyron in 1834, brings together 3 linear gas relationships:
Boyle’s Law, Charles’s Law and Avogadro’s Law. The ideal gas law is simple, functional and
fairly accurate at dilute conditions. However, at high pressures or low temperatures, the
assumptions of the ideal gas law, namely that gases are composed of non-interacting point
particles, break down. This shortcoming was addressed later in the 19th century by introducing
nonlinear gas equations of state such as the van der Waals equation:
!! !

! + !!

! − !" = !"#

(2)

Introduced by Johannes Diderick van der Waals in 1873, this equation of state
incorporates two gas-dependent parameters, a, and b, which account for the attractive
interactions between gas molecules, and the finite volume of real gas molecules, respectively.
Johannes van der Waals adamantly believed that gases collided as hard spheres and did not
possess any other repulsive interactions. This is now known to be false.
Real gases exhibit both attractive and repulsive interactions that are strongly correlated
with intermolecular spacing. For electrically neutral molecules, these forces (e.g. van der Waals)
typically fall into one of 4 categories: Keesom Forces, Debye Forces, London Dispersion
forces and Pauli repulsive forces. The first three result from some combination of permanent

15
and/or induced multipole interactions while the Pauli repulsive force is a purely quantum
effect.
The sum of the attractive and repulsive intermolecular interactions forms an
“interaction potential.” While early on it was generally agreed upon that van der Waals
interactions fall off quickly with increasing intermolecular spacing, the precise function was
unknown. A number of pair potentials were proposed including the commonly used LennardJones Potential, VLJ:
!!" = 4!

! !"

! !

(3)

where r is the intermolecular separation and σ and ε are the Lennard-Jones parameters, specific
to each gas. The Lennard-Jones potential balances the longer-range attractive interactions (that
-6

fall as r ) with the very short-range Pauli repulsion forces (arbitrarily modeled as falling with r
12

-6

-12

). While the r dependence derives from the London dispersion force, the r

repulsive

term has no physical basis. Thus while a useful heuristic, the Lennard-Jones potential is not
rigorously accurate in describing potentials between two molecules. In physisorption, gas
molecules interact with an adsorbent surface, which is generally considered to be much wider
than the molecule itself. Patchwise, these interactions may be modeled as between a flat
crystalline material and a small molecule, by generalizing the ideas of the Lennard-Jones
potential into a new form called the Steele potential1:

! ! = 2!!!" !! ! ! Δ

! ! !"

! !

!!
!! !!!.!"! !

(4)

16
where εsf is the solid-fluid well-depth given by Berthelot mixing rules2, ρs is the density of the
solid, Δ is interplanar spacing of the crystalline material, and σ is a Lennard-Jones type distance
parameter determined by Lorentz mixing rules2.
In a dynamic view, gases may collide with a solid interface, either elastically or
inelastically. Occasionally a gas molecule that collides inelastically will undergo an interaction
with the surface, wherein it is briefly localized by the surface potential. This is the essence of
physisorption at the microscale. The surface potential may vary over the surface due to
impurities, defects, or overall structural features. In microporous materials, the surface
potential heterogeneity is largely dictated by the pore-size distribution. Physisorptive systems
typically have only shallow (weak) surface potentials that allow adsorbed molecules to explore
multiple sites on the two-dimensional potential surface before reentering the gas phase. This is
known as a mobile adsorption, as opposed to localized adsorption, which is typically associated
with deeper potential wells and chemisorption.
From a fundamental thermodynamics perspective, adsorption compresses a 3dimensional gas phase into a 2-dimensional adsorbed phase. This presents a significant drop in
the molar entropy of the adsorptive species. The difference in molar entropy between the
adsorbed and gas phases (at constant coverage) is called the isosteric entropy of adsorption
(ΔSads). The gas-phase entropy (Sg) may be read from data tables. The adsorbed-phase entropy
(Sa) depends on a number of factors including coverage.
In order to establish equilibrium, the isosteric entropy of adsorption must be offset by
a comparable decrease in molar enthalpy upon adsorption. The difference between the
adsorbed-phase enthalpy (Ha) and the gas-phase enthalpy (Hg) (at constant coverage) is called

17
the isosteric enthalpy of adsorption (ΔHads). Physisorption is always exothermic, yielding a
negative isosteric enthalpy of adsorption. By convention, the isosteric heat of adsorption (qst) is
defined as a positive quantity as follows
!!" ≡ − !! − !!

(5)

The isosteric heat of adsorption may be thought of as a proxy metric of the binding energy
between the adsorbent and adsorbate that results from the interaction potential.
Over time an adsorbed molecule may explore many adsorption sites, but due to
energetic constraints, the most favorable sites (with the largest isosteric heats) will have the
highest average occupation. For conventional adsorbents, the highest isosteric heat values are
observed at the lowest coverage. As coverage is increased, the most favorable adsorption sites
become saturated. This leads to a decreasing isosteric heat (an average quantity) with increasing
coverage.

2. History of Adsorption
Simple adsorptive applications have been employed since at least 1550 BC, when
records indicate that the Egyptians made use of charcoals to adsorb putrid gases expelled
during human dissection.3 Scientific adsorptive experiments are more recent. Scheele in 1773,
followed by Fontana in 1777, were the first scientists to measure the uptake of gases by porous
solids.3 Saussure built upon this work, and in 1814 determined that adsorption was exothermic
in nature.3 A theoretical understanding of adsorption followed far behind experiments. It
wasn’t until 1888 that Bemmelen made the first known attempt at fitting adsorption data,
introducing a fitting equation now known as the “Freundlich Equation.”3 More precise

18
terminology soon followed when Bois-Reymond and Kayser introduced the term
“adsorption” into standard scientific lexiconography.3
By the 20th century, the field of adsorption was full steam ahead. 1903 saw the
discovery of selective adsorption (Twsett)4. In 1909 McBain introduced the term “absorption”
to differentiate bulk uptake from the surface phenomenon of “adsorption.”5 Within the next
few decades physisorption had been cast into a number of rigorous theoretical frameworks,
namely Eucken-Polanyi Theory (1914)6, the Langmuir Isotherm (1918)7, BET Theory (19351939)8,9,10 and Dubinin Theory (1946)11.

3. Adsorption Theory
Apart from limited calorimetric work, the “adsorption isotherm” forms the
fundamental basis of adsorption measurement and theory. Constant-temperature isotherms
may be measured in a number a ways but ultimately yield the same information: Gibbs excess
adsorption as a function of pressure. A simple model for adsorption may be drawn up as
follows (Figure 1).

Figure 1. Cartoon depiction of adsorption. The gray rectangle represents the adsorbent

19
surface. All blue molecules represent absolute adsorption molecules. The dark blue circles
represent excess adsorption molecules. The red line indicates a dividing surface. The green
circles represent the gas-phase molecules.
Here section 1 represents the solid adsorbent surface, section 2 represents the
adsorbed phase (densified molecules near the interface), and section 3 represents the molecules
that remain unaffected by the adsorbent and remain free in the gas phase. The quantity of
absolute adsorption (na) comprises all of the molecules in the adsorbed phase. The volume of
the adsorbed phase (Vads), however, is not rigorously established. Thus absolute adsorption
cannot be directly measured via experiment. Instead Gibbs worked around this problem by
defining excess adsorption (ne) as follows:
!! = !! − !!"# !(!, !)

(6)

Gibbs excess adsorption differs from absolute adsorption by an amount equal to the volume
of the adsorption layer multiplied by the density of the free gas phase (ρ(P,T)). Here the
volume of the adsorbed phase is defined as the volume between the adsorbent surface and a
dividing surface. The quantity of Gibbs excess adsorption measures the amount of adsorbate
in the vicinity of the adsorbent surface that is in excess of the free gas phase density. For this
reason, the Gibbs excess adsorption is directly measurable by volumetric or gravimetric
methods without need for assumptions about the volume of the adsorbed layer. Absolute
adsorption, however, cannot be easily and directly measured. Instead it is often crudely
assumed that absolute adsorption equals excess adsorption. This assumption is only valid
when the gas phase is dilute, and quickly breaks down at high pressures and low temperatures.
Thus in this thesis, we instead use a robust fitting method (presented in Chapter 3) to back out

20
reasonable values of the absolute adsorption.
The first attempts to fit physisorption isotherms came as early as 1888 when
Bemmelen introduced what is now known as the “Freundlich Equation”12,3:

= !!!

(7)

where n is the uptake, m is the mass of the adsorbent, P is equilibrium pressure, and K and η
are adsorbent-specific parameters. This equation is only empirical and does not purport to
capture or contain the physics of adsorption. Nonetheless, it displays key behaviors that are
common to fit functions for type 1 isotherms. At low pressures, uptake increases linearly with
pressure per Henry’s Law:
! = !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(8)!
where KH is the Henry’s Law constant. Henry’s Law was first formulated by William Henry in
1803 and can be derived from ideal gas assumptions. If we assume that the adsorbed phase
takes the form of a two-dimensional ideal gas following
!! !! = !!"

(9)

where Pa and Aa are the spreading pressure and surface area, respectively, and also assume that
the equilibrium spreading pressure is proportional to the equilibrium bulk gas pressure, then
! !

!"!

! = !!"! = !"! = !! !

(10)

Accordingly, the low-pressure regime of an isotherm where uptake is proportional to applied
gas pressure is commonly referred to as the Henry’s Law regime. At higher pressures, the
adsorption isotherm saturates as all of the available adsorption sites are filled. This correlation,
however, is not exact as unlike absolute adsorption, experimentally measured excess
adsorption data behaves in a non-monotonic fashion at high pressures. Nonetheless, most type

21
1 adsorption fit functions subsume both the Henry’s Law regime behavior and the saturation
regime behavior.
The Langmuir isotherm, derived by Irving Langmuir in 19187, provides a fundamental
model for gas adsorption that incorporates both the Henry’s Law regime the saturation regime
behavior. Here the adsorbed phase is assumed to exist as a monolayer of adsorbate directly
above the adsorbent surface. Langmuir’s model makes a number of simplifying assumptions as
follows:
1. The adsorbent surface is perfectly flat.
2. The gas adsorbs into an immobile state.
3. There is a finite number of adsorption sites that can each by filled by no more than
one adsorbate molecule.
4. All adsorption sites are energetically identical.
5. Adsorbate molecules do not interact.

With these simplifying assumptions in hand, the Langmuir adsorption isotherm may be
derived from kinetic theory, statistical mechanics, or from a phenomological perspective, and
takes the form:
!"

! = !!!"

(11)

where θ is fractional occupancy and K is an equilibrium constant given by an Arrhenius-type
equation (Equation 12):
!! =

!!

!!!

! !"

where Ai is a prefactor and Ei is an energy of the ith isotherm.

(12)

22
Unfortunately, the five simplifying assumptions above are almost never entirely
satisfied and the Langmuir isotherm cannot be applied over broad ranges of conditions. Many
of the drawbacks of the Langmuir isotherm may be overcome by fitting excess adsorption data
with a weighted superposition of Langmuir isotherms (see Chapter 3).
In particular, the Langmuir model breaks down when multilayer adsorption is possible,
as found in larger micropores, mesopores, and macropores. In 1938 Stephen Brunauer, Paul
Emmett, and Edward Teller extended the Langmuir model to consider multilayer adsorption.8
They realized that in multilayer adsorption, molecules do not successively fill one complete
monolayer after another. Rather, fragments of multilayer stacks of varying sizes dot the
adsorbent surface. Each layer is in dynamic equilibrium with the layers above and below it,
much in the same way that the Langmuir model assumes a dynamic equilibrium between the
adsorbed monolayer and the gas phase above it. The Brunauer, Emmett, Teller, or BET
method has been elaborated on in detail in literature13,14 and will not be rederived here. Rather,
the results and key insights are elucidated from the BET equation:

!!
!!

=!

!!!

!"# ! !!

+!

!"# !

(13)

where n is uptake, P is equilibrium pressure, Po is saturation pressure, nmax is maximum possible
uptake, and C is the BET constant. BET Theory assumes that infinite layers may be adsorbed
successively on a surface. Moreover, these layers do not interact with one another and each
follow the Langmuir model. Two additional assumptions are made:
1. The E1 parameter is the isosteric heat between the adsorbent and first adsorbed layer.
2. All higher layers have an EL parameter equal to the heat of liquefaction of the
adsorbate.

23
BET theory has proven particularly useful at measuring the specific surface areas of
porous carbons. For high quality surface area determinations, nitrogen, argon, carbon dioxide,
and krypton have been used. In particular, a plot of
line in the relative pressure range of 0.05<

!!

!!
!!

vs

!!

should yield a straight

<0.3. Using linear regression, the slope and y-

intercept of this line are determined. The parameters nmax and C are determined by:

!!"# = !"#$%!!"#$%&$'#
!=

!"#$%
!"#$%&$'#

+1

(14)
(15)

The surface area may then be determined from nmax, using the established cross-sectional area
of the probe molecule.
While the Langmuir isotherm rapidly gained popularity and contributed to Irving
Langmuir’s 1932 Nobel Prize, it competed with Polanyi’s theory of adsorption, which has now
earned its place in annals of science history. Whereas Langmuir conceptualized adsorption as a
monolayer effect localized at the adsorbent surface, Polanyi’s approach was more amenable
holistic pore filling with longer-range effects. Polanyi reasoned that the density of adsorptive
molecules near a surface diminishes with distance from the attractive surface, much the way
the atmosphere of a planet thins out at high altitudes. For adsorption this requires a longerrange interaction potential, now called the Polanyi Adsorption Potential6.

Polanyi recognized that at equilibrium, the chemical potential (µ) of the adsorbed
phase at an arbitrary distance, x, from the interface and a corresponding pressure Px, must

24
equal the chemical potential of the gas phase at an infinite distance and corresponding bulk
pressure, P.
! !, !! = ! ∞, !

(16)

! !,!!
! !,!

(17)

!" =! ! !, !! − ! ∞, ! = 0

Moreover,
!" = −!"# + !"# + !"

(18)

where S is entropy, V is volume and U is the potential energy. Constant temperature
(isothermal) conditions yield
!" = !"# + !"

(19)

! !,!!
! !,!

(20)

!" =! ! ! !"#! + ! ! − ! ∞ = 0

where U(x) is the potential energy at a distance x from the surface and U(∞) is the potential
energy at an infinite distance, which Polanyi took to be zero:

−! ! = ! ! ! !"#

(21)

By substituting in the ideal gas law
! !"

−! ! = ! ! ! ! !"

(22)

or
! ! = !!"#$

!!

≡!

(23)

where A is the Polanyi potential. While the Polanyi potential went unappreciated for many
years, it was given new life in 1946 when Dubinin and Radushkevich introduced the “theory of
the volume filling of micropores (TVFM)”.11,15,16 In this theory the Polanyi potential is the
negative of the work done by the sorption system:

25
! = −∆!

(24)

This insight extended the Polanyi potential to broad thermodynamic analysis, codified in the
Dubinin-Radushkevich equation:
! = !!"!! !

! !
!!!

(25)

where n is the uptake, nmax is the maximum possible uptake, β is the affinity coefficient, and Eo
is the standard characteristic energy. Other modified and more generalized forms were later
introduced, such as the Dubinin-Astakhov equation17

! = !!"#! ! !!!

(26)

where χ is an adsorbent-specific heterogeneity parameter.
If uptake is plotted as a function of the Polanyi potential, the Dubinin-Radushkevich
equation15 yields a single characteristic curve for each gas-adsorbent system. In theory the
characteristic curve may be used to predict uptake over a wide range of temperatures and
pressures, and its accuracy has been generally confirmed by experiment.18,19 Moreover, plotting

ln(n) as a function of − !!

yields a linear trend, wherein the y-intercept gives the maximal

uptake (nmax) and the slope gives the characteristic energy (Eo). The parameter nmax may be
used to determine the total micropore volume of the adsorbent by multiplying by the
established molecular volume of the adsorbate. The parameter Eo may be used to estimate an
average micropore width.

26

References:
1.$Siderius,$D.$W.;$Gelb,$L.$D.$Extension$of$the$Steele$10K4K3$Potential$for$Adsorption$
Calculations$ in$ Cylindrical,$ Spherical,$ and$ Other$ Pore$ Geometries.$ J.% Chem.% Phys.%
2011,$135,$084703K1K7.$
2.$Boda,$D.;$Henderson,$D.$The$Effects$of$Deviations$from$LorentzKBerthelot$Rules$on$
the$Properties$of$a$Simple$Mixture.$Mol.%Phys.%2008,$106,$2367K2370.$
3.$Dabrowski,$ A.$ AdsorptionKFrom$ Theory$ to$ Practice.$ Adv.% Colloid% Interface% Sci.%
2001,$93,$135K224.$
4.$Berezkin,$V.G.$(Compiler).$Chromatographic$Adsorption$Analysis:$Selected$Works$
of$M.S.$Tswett.$Elis%Horwood:%New$York,$1990.$
5.$McBain,$ J.$ W.$ The$ Mechanism$ of$ the$ Adsorption$ ("Sorption")$ of$ Hydrogen$ by$
Carbon.$Philos.%Mag.%1909,$18,$916K935.$
6.$Polanyi,$M.,$Potential$Theory$of$Adsorption.$Science.%1963,$141,$1010K1013.$
7.$Langmuir,$ I.$ The$ Adsorption$ of$ Gases$ on$ Plane$ Surfaces$ of$ Glass,$ Mica$ and$
Platinum.$J.%Am.%Chem.%Soc.%1918,$40,$1361K1403.$
8.$Brunauer,$ S.;$ Emmett,$ P.$ H.;$ Teller,$ E.$ Adsorption$ of$ Gases$ in$ Multimolecular$
Layers.$J.%Am.%Chem.%Soc.%1938,$60,$309K319.$
9.$Brunauer,$ S.;$ Emmett,$ P.$ H.$ The$ Use$ of$ Low$ Temperature$ van$ der$ Waals$
Adsorption$ Isotherms$ in$ Determining$ the$ Surface$ Areas$ of$ Various$ Adsorbents.$ J.%
Am.%Chem.%Soc.%1937,$59,$2682K2689.$
10.$Brunauer,$ S.;$ Emmett,$ P.$ H.$ The$ Use$ of$ van$ der$ Waals$ Adsorption$ Isotherms$ in$
Determining$the$Surface$Area$of$Iron$Synthetic$Ammonia$Catalysts.$J.%Am.%Chem.%Soc.%
1935,$57,$1754K1755.$
11.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR%Phys.%Chem.$1947,$55,$331K337.$
12.$Skopp,$ J.$ Derivation$ of$ the$ Freundlich$ Adsorption$ Isotherm$ from$ Kinetics.$ J.%
Chem.%Educ.%%2009,$86,$1341K1343.$
13.$Gregg,$ S.$ J.;$ Jacobs,$ J.$ An$ Examination$ of$ the$ Adsorption$ Theory$ of$ Brunauer,$
Emmett,$ and$ Teller,$ and$ Brunauer,$ Deming,$ Deming$ and$ Teller.$ T.% Faraday% Soc.%
1948,$44,$574K588.$
14.$Legras,$ A.;$ Kondor,$ A.;$ Heitzmann,$ M.$ T.;$ Truss,$ R.$ W.$ Inverse$ Gas$
Chromatography$ for$ Natural$ Fibre$ Characterisation:$ Identification$ of$ the$ Critical$
Parameters$ to$ Determine$ the$ BrunauerKEmmettKTeller$ Specific$ Surface$ Area.$ J.%
Chromatogr.%A.%2015,$1425,$273K279.$
15.$Nguyen,$ C.;$ Do,$ D.$ D.$ The$ DubininKRadushkevich$ Equation$ and$ the$ Underlying$
Microscopic$Adsorption$Description.$Carbon.%2001,$39,$1327K1336.$
16.$Dubinin,$ M.$ M.$ Generalization$ of$ the$ Theory$ of$ Volume$ Filling$ of$ Micropores$ to$
Nonhomogeneous$Microporous$Structures.$Carbon.%1985,$23,$373K380.$
17.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
18.$Saeidi,$N.;$Parvini,$M.$Accuracy$of$DubininKAstakhov$and$DubininKRadushkevich$
Adsorption$ Isotherm$ Models$ in$ Evaluating$ Micropore$ Volume$ of$ Bentonite.$ Period.%
Polytech.%Chem.%2016,$60,$123K129.$

27
19.$Do,$ D.$ D.;$ Nicholson,$ D.;$ Do,$ H.$ D.$ Adsorption$ in$ Micropores$ (Nanopores):$ A$
Computer$Appraisal$of$the$Dubinin$Equations.$Mol.%Simul.%2009,$35,$122K137.$

28
Chapter 3

Experimental Methods and Analysis
1. Equipment
The fundamental basis of quantitative adsorption analysis is the measurement of
excess adsorption isotherms. Each isotherm comprises a series of excess adsorption uptake
values, measured stepwise at increasing pressures and a constant temperature. At high
pressure, the excess adsorption becomes non-monotonic as a function of pressure, after
reaching an excess adsorption maximum. A desorption isotherm may be measured in reverse,
by starting with a preloaded adsorbent at high pressures and reducing the pressure stepwise.
Observed hysteresis between pairs of adsorption and desorption isotherms yields valuable
information about the adsorptive system, often indicating the presence of capillary
condensation in mesopores.
In this thesis, excess adsorption isotherms were measured by the volumetric method
also known as the Sieverts’ method using a custom Sieverts apparatus designed and tested for
accuracy up to 10 MPa (Figure 1).1 The Sieverts apparatus comprises a number of rigid,
stainless steel, and leak-proof compartments, each interconnected with controllable on-off
valves (either hand-turned or pneumatic). The volume of each compartment is known with
high precision (+ 0.01 mL). For standard adsorption measurements, only two of the
compartments are of interest: the manifold and the reactor. The manifold is equipped with a
midrange (3000 PSI) MKS Baratron (Model 833) pressure transducer for high-pressure
measurements and an MKS Baratron (Model 120AA) for low-pressure measurements of
higher resolution. The temperature of the gas in the manifold was measured with platinum

29
resistance thermometers. The temperature of the reactor was monitored with K-type
thermocouples. In a preparatory step the adsorbent sample of interest was weighed and sealed
within the reactor, which seals by the tightening of a conflat flange with a copper gasket.
Additional nickel filter gaskets with a 0.5-micron mesh size prevented the adsorbent from
escaping from the reactor. After sealing the reactor, each sample was degassed at ~520 K
-3

under a vacuum of less than 10 Pa prior to testing. The Sieverts apparatus is equipped with a
-4

molecular drag pump capable of achieving a vacuum of 10 Pa and vacuum pressures were
verified using a digital cold cathode pressure sensor (I-MAG, Series 423). To obtain low
temperature isotherms, the reactor was submerged in a circulated chiller bath or cryogenic bath
with temperature fluctuations no larger than + 0.1 K. High temperature isotherms were
obtained by encasing the reactor in a copper heat conductor wrapped with insulating fiberglass
heating tape. Using a proportional integral derivative (PID) controller, the reactor temperature
was maintained with fluctuations no larger than + 0.4 K. Prior to measurements, the entire
Sieverts was purged multiple times with the gas of interested to eliminate any impurity
residues. On each sample, multiple adsorption/desorption isotherms were taken to ensure
complete reversibility and identical measurements were found to be reproducible to within 1%
error.

30

Figure 1. Sieverts Apparatus

2. Methodology
For each isotherm data point, a predetermined pressure of research-grade gas was
introduced into and cached in the manifold. Upon reaching equilibrium, the temperature and
pressure of this gas were measured with high precision and the gas density was determined
from REFPROP data tables2. Given that the manifold volume is known, the moles of gas in
the manifold were thus determined. Next, a valve was opened to allow the gas to occupy both
the manifold and reactor volumes. The reactor housed the porous sample of known mass.
Upon opening the valve, the gas was allowed to occupy a volume given by the sum of the

31
manifold and reactor volumes minus the volume of the sample. The volume of the sample
was given by the product of the sample mass and the sample skeletal density (obtained by
helium pycnometry). Once equilibrium was reestablished, pressure and temperature
measurements were taken and a final gas-phase density determined for the step. The moles of
gas left in the gas phase were thus calculated. Any quantity of gas introduced into the manifold,
but no longer contributing to the gas-phase pressure was considered to be in the adsorbed
phase. The sample within the reactor was held at a constant temperature over the course of a
successive set of pressure measurements, resulting in an excess adsorption isotherm. Upon
completion of an isotherm, the temperature was adjusted for the next isotherm, in order to
measure a multitude of isotherms over a wide temperature range.
At equilibrium the chemical potential of the gas phase (μg) and the adsorbed phase (μa)
are equal
!! = !!

(1)

By taking the total differential of both sides of Equation 1:
−!! !" + !! !" = −!! !" + !! !"
!"
!"

! !!

= !! !!!

(2)
(3)

Thus the derivative of pressure with respect to temperature is related to the change in entropy
(upon adsorption) divided by the change in volume (upon adsorption). At constant coverage,
the difference between the entropy of the adsorbed phase (Sa) and the entropy of the gas
phase (Sg) is the isosteric entropy of adsorption (ΔSads) and is given by Equation 4.
∆!!"# = !! − !!

(4)

32
!"

∆!!"# = !! − !! !"

(5)

The isosteric entropy of adsorption is in turn related to isosteric enthalpy of adsorption (ΔHads)
by Equation 6
∆!!"#

= ∆!!"#

(6)

At constant coverage, the difference between the adsorbed and gas-phase enthalpies is thus
given by Equation 7
!"

∆!!"# = ! !! − !! !"

(7)

This is the Clapeyron equation, and is fundamental to the calculation of the isosteric enthalpy
of adsorption. A number of simplifying assumptions may be made in dealing with the
Clapeyron equation. The two most common are as follows:
1. That the volume of the gas phase is significantly larger than that of the adsorbed phase
such that the overall change in volume is well approximated as the gas-phase volume.
2. That the gas follows the ideal gas law.
Together these assumptions transform the Clapeyron equation into Equation 8
!! ! !"

∆!!"# !! = − !

!" !!

(8)

This may in turn be rearranged into the Van’t Hoff form:
∆!!"# !! = !

!"#!!

(9)
!!

By plotting ln(P) vs (1/T), a Van’t Hoff plot is formed. The slope of the Van’t Hoff plot
multiplied by R gives the isosteric heat. In common practice, adsorption measurements directly
determine excess adsorption (ne), not absolute adsorption (na). It is thus common to hold
excess adsorption, not absolute adsorption, constant in Equation 9. This results in an isoexcess

33
enthalpy of adsorption that approximates the isosteric enthalpy of adsorption, but only at
low gas densities. The previously mentioned assumptions fail when applied broadly to gas data.
Thus we must consider alternatives to these oversimplifying assumptions.
First, the ideal gas assumption may be avoided by inserting values from gas data tables
directly into the Clapeyron equation. This is the course of action followed in this thesis. Using
gas data tables enables thermodynamic calculations outside of the ideal gas regime. Second, the
simplified Clausius-Clapeyron equation assumes that the adsorbed phase has virtually zero
volume and that the net difference between the molar volume of the gas phase and the
adsorbed phase equals the molar volume of the gas phase. In reality, the adsorbed phase has a
finite molar volume that approaches that of the liquid phase of the adsorbed species. Thus the
assumption of a zero molar volume adsorbed phase may be replaced with one of two options.
Either the adsorbed-phase molar volume is assumed to be equal to that of the liquid phase
molar volume, or, fit functions are used to approximate the adsorbed-phase molar volume.
In applying the Clapeyron equation, it is necessary to take the derivative of pressure
with respect to temperature at constant coverage conditions. Where absolute adsorption is
held constant, this determines the isosteric enthalpy of adsorption and where the excess
adsorption is held constant, this determines the isoexcess enthalpy of adsorption. In practice,
however, neither absolute nor excess adsorption is an experimentally tunable variable. The
uptake quantity is never directly selected, rather the pressure is roughly selected. Thus
obtaining data points at constant coverage conditions is not simple. In some cases, with
sufficient data, constant coverage conditions across a number of temperatures may be
achieved for select data points by pure coincidence. This can be thought of as “analysis
without fitting”. Fortuitously positioned data points, however, are sparse and unreliable.

34
In general it is necessary to establish an interpolation function to interpolate between
the measured data points. Interpolation functions of a variety of forms have been employed in
literature. The simplest entails linear interpolation between data points.
While simple, this approach fails to accurately capture the adsorption behavior, leading
to scatter and errors. In this thesis research I employ a superposition of Langmuir isotherms as
the fitting function of choice. Specifically, starting from the definition of excess adsorption
(ne):
!! = !! − !! !(!, !)

(10)

I fit the absolute adsorption (na) and the volume of the adsorption layer (Va) with
superpositions of Langmuir isotherms with appropriate prefactors:
!! (!, !) = !!"#

! !!

!! (!, !) = !!"#

! !!

!! !

(11)

!!!! !
!! !

(12)

!!!! !

th

where ai is the respective weight of the i isotherm

! ∝! = 1 , P is the pressure, and Ki is an

equilibrium constant given by an Arrhenius-type equation such that:
!! =

!!

!!!

! !"

(13)

where Ai is a prefactor and Ei is an energy of the ith isotherm. Altogether the fit function
becomes:
!! (!, !) = !! − !!"# !(!, !)

! !!

!! !
!!!! !

(14)

One particular advantage of this fitting procedure is that the absolute adsorption is
easily accessible. It is one of the quantities that is directly fitted and given by Equation 11.

35
Absolute adsorption is a more fundamental quantity than excess adsorption, and its
determination is critical for in-depth analysis of the adsorption physics.
In addition to producing high quality fits, the dual-Langmuir fitting method used in
this thesis gives physically realistic fitting parameters. In particular the nmax and vmax parameters
have been found to be in reasonable agreement with independently determined physical data
(See Chapter 9). The nmax parameter gives the maximum possible absolute adsorption as
determined by fits. This is directly comparable to an estimate of the maximum possible
adsorption as obtained by multiplying the measured micropore volume by the liquid density of
the adsorptive species2. Here we assume that the entirety of the micropores is filled with
adsorbate at a density equal to the liquid density (upon maximal adsorption). Furthermore, the
vmax parameter indicates the maximum possible volume of the adsorbed phase, which may be
directly compared to the measured micropore volume of the adsorbent.
While the isosteric enthalpy of adsorption (ΔHads) is a popularly cited proxy metric of
binding site energy, a more fundamental metric exists in the adsorbed-phase molar enthalpy
(Ha) given by Equation 15. The isosteric enthalpy of adsorption is the difference between the
adsorbed-phase and gas-phase (Hg) molar enthalpies and thus retains a dependence on gasphase properties.
!! = !! + Δ!!"#

(15)

For nonideal gas conditions, this dependence can obscure interesting phenomenon occurring
strictly within the adsorbed phase. The gas-phase enthalpy is obtained from reference tables2.
The adsorbed-phase enthalpy gives critical insight into the nature of the adsorbent-adsorbate

36
binding-site energy. Moreover, the constant pressure molar heat capacity of the adsorbed
phase (CP) follows directly from the adsorbed-phase enthalpy (Equation 16):
!!

!! = !"!

(16)

The adsorbed-phase molar heat capacities provide critical qualitative insight into the adsorbedphase layer. For a monatomic gas like krypton, comparison to theoretical estimates of the heat
capacity provides further means to peer into the underpinnings of the adsorbed-phase
thermodynamics. Furthermore, the isosteric entropy of adsorption (ΔSads) is directly accessible
from the isosteric enthalpy of adsorption:
Δ!!"# =

!!!"#

(17)

By adding the isosteric entropy of adsorption to the gas-phase molar entropy (from
REFPROP2), we obtain the adsorbed-phase molar entropy (Sa).
!! = !! + ∆!!"#

(18)

37

References:
1.$Bowman,$R.$C.;$Luo,$C.$H.;$Ahn,$C.$C.;$Witham,$C.$K.;$Fultz,$B.$The$Effect$of$Tin$on$the$
Degradation$ of$ LaNi5KYsNy$ MetalKHydrides$ During$ Thermal$ Cycling.$ J.% Alloys% and%
Compd.%1995,$217,$185K192.$
2.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$version$8.0;$
National$Institute$of$Standards$and$Technology:$Gaithersburg,$MD,$2007;$CDKROM.$

38
Chapter 4

Materials

Three carbonaceous materials, MSC-30, CNS-201, and zeolite-templated carbon (ZTC) were
studied repeatedly with different gases and under differing conditions in this work. These
materials were obtained and characterized in varied ways described herein.

1. MSC-30
MSC-30 (Maxsorb) is a microporous superactivated carbon obtained from Kansai
Coke & Chemicals Company Ltd. (Japan) and is an “AX21-type” superactivated carbon. MSC30 is synthesized by activating petroleum coke with molten KOH in a process patented by
Standard Oil Company (later Amoco Corporation).
Nitrogen adsorption isotherms were carried out at 77K in a Micromeritics ASAP 2420.
The specific surface area was determined by applying BET theory to the data as implemented
by Micromeritics ASAP 2420 version 2.02 software. The BET surface area of MSC-30 was
determined to be 3244+ 28 m2 g-1. Using nonlocal density functional theory (NLDFT)1 and a
slit-pore model, the MSC-30 pore-size distribution was determined as shown in Figure 1.
MSC-30 has a broad range of pore sizes (from 6 to 35 Å). Over 40% of the micropore volume
is contained in pores of greater than 21 Å (width). The micropore volume was found to be
1.54 cm3 g-1 by the Dubinin-Radushkevich method2,3. The skeletal density was measured by
helium pycnometry and the skeletal density determined to be 2.1 g cm-3.

39
Cu Kα X-ray diffraction of MSC-30 on a PANalytic Pro powder diffractometer gave
a broad peak at 2θ= 34 degrees, in accordance with that reported for AX21. The elemental
composition (CHN) was determined by the Dumas method4 in triplicate combustion
experiments, indicating that 1.16 wt% of MSC-30 is hydrogen. X-Ray Photospectroscopy
(XPS) measurements were made on a Kratos AXIS Ultra DLD spectrometer and the results
are summarized in Table 1. Electron Energy Loss Spectroscopy (EELS) measurements were
made on a FEI Technai F20 with a Gatan Imaging Filter system. MSC-30 has an sp3
hybridized carbon content of 16%. Transmission electron microscope (TEM) images were
taken with a Tecnai TF30 with a LaB6 filament and 80 keV electrons (Figure 2).

Figure 1. The pore-size distribution (left) of MSC-30 (red), CNS-201 (black), and ZTC (purple)
as calculated by the NLDFT method.

40

Figure 2. TEM image of MSC-30 and an accompanying fast Fourier transform of the image.
Table 1. Summary of XPS data on MSC-30 and ZTC
peak
position
(eV)
component
ZTC
MSC-30

285.0
285.7 286.4
287.3 288.1
289.4 290.2 291.5
C-C sp C-C sp C-OR C-O-C C=O COOR
53.4
18.0
8.6
6.0
1.1
4.2
1.0
7.7
48.0
18.8
6.8
4.8
6.1
4.2
3.6
7.7

2. CNS-201
CNS-201 is a microporous activated carbon obtained from A. C. Carbone Inc.
(Canada). It is synthesized by pyrolysis of coconut shells. Nitrogen adsorption isotherms were
carried out at 77K in a Micromeritics ASAP 2420. The specific surface area was determined by
applying BET theory to the data as implemented by Micromeritics ASAP 2420 version 2.02
software. As determined by nitrogen adsorption and BET analysis, CNS-201 has a surface area
of 1095 + 8 m2 g-1. Using the nitrogen adsorption data and a slit pore model, NLDFT1 pore-

41
size analysis was conducted to determine the pore size distribution (Figure 1). CNS-201 has
a three dominant pore widths of 5.4, 8.0, and 11.8 Å, containing roughly 50%, 20%, and 15%
of the total micropore volume respectively. The micropore volume was found to be 0.45 cm3
g-1 by the Dubinin-Radushkevich method2,3. The skeletal density was measured by helium
pycnometry and the skeletal density determined to be 2.1 g cm-3. Transmission electron
microscope (TEM) images were taken with a Tecnai TF30 with a LaB6 filament and 80 keV
electrons (Figure 3).

Figure 3. TEM image of CNS-201 and an accompanying fast Fourier transform of the image.

42

3. Zeolite-Templated Carbon
Zeolite-Templated Carbon (ZTC) is a microporous templated carbon synthesized in
multi-gram quantities with the following procedure at HRL Laboratories.
3.1 ZTC Synthesis
Faujasite-type Zeolum® zeolite Nay, HSZ-320NAA (faujasite structure, Na cation,
SiO2/Al2O3 = 5.5 mol/mol) (NaY) was obtained from Tosoh Corporation. 6.0 grams of
zeolite NaY were dried under vacuum at 450 °C for 8 hours. This powder was cooled and
mixed with 12 mL of furfuryl alcohol (98% Sigma Aldrich) and stirred under Argon for 24
hours. The resulting zeolite-furfuryl alcohol mixture was separated by vacuum filtration and
rinsed four times in 100 mL aliquots of xylenes.
Next the powder was loaded in a quartz boat and placed in quartz tube-furnace/CVD
reactor. The tube furnace was purged with argon and held under argon at 80 °C for 24 hours.
The reactor was heated to 150 °C (under argon) for 8 hours to induce polymerization. Next
the temperature was ramped up at a rate of 5 degrees °C per minute to a final temperature of
700 °C whereupon the gas flow was switched to a 7% propylene/ 93% nitrogen mixture for 4
hours. After 4 hours, the reactor was purged with argon at 700 °C for 10 minutes. Next the
temperature was increased to 900 °C and held for 3 hours under argon.
This product was then cooled and transferred to a PTFE beaker. 200 milliliters of
aqueous hydrofluoric acid (48% Sigma Aldrich) were added. The solution was left for 16 hours
before collecting the ZTC by vacuum filtration and rinsing 10 times with 50 mL aliquots of
water. Finally the ZTC product was dried at 150 °C under vacuum. Careful control of the

43
inert atmosphere and thorough drying were found to be critical to obtaining high surface
area product.
3.2 ZTC Characterization
Nitrogen adsorption isotherms at 77K were measured on a BELSORP-max
volumetric instrument (BEL-Japan Inc.). This data was analyzed with BET theory to
determine a specific surface area of 3591 + 60 m2 g-1 (among the highest to date for
carbonaceous materials)5. Using nitrogen adsorption data and a slit-pore model, NLDFT1
pore-size analysis was conducted to determine the pore-size distribution (Figure 1). Other
geometrical models including a cylindrical pore model were also tried, but none fit the data
better than the slit-pore model. ZTC has a narrow pore-size distribution centered at 12 Å.
Over 90% of the micropore volume is contained in pores of widths between 8.5 and 20$ Å.
The micropore volume was found to be 1.66 cm3 g-1 by the Dubinin-Radushkevich method2,3.
The skeletal density was measured by helium pycnometry and determined to be 1.8 g cm-3.
This is lower than most carbonaceous adsorbents (2.1 g cm-3), likely owing to a higher percent
of hydrogen terminations. The elemental composition (CHN) was determined via the Dumas
method4 in triplicate combustion experiments, indicating that 2.44 wt% of ZTC is hydrogen.
This higher percentage of hydrogen terminations may result from the hydrofluoric acid
treatment during synthesis.
Cu Kα X-ray diffraction of ZTC was measured with a PANalytic X’Pert Pro powder
diffractometer and produced a single sharp peak at 2θ= 6 degrees, indicative of the template
periodicity (14 Å). No signal from the original zeolite material was registered in the final
product. The absence of other peaks suggests that the product is (as expected) amorphous

44

carbon without any remnant zeolite. Applying the Scherrer equation to the peak (with a
Scherrer constant K=0.83 for spherical particles) suggests an ordering length scale of 24 nm.
X-Ray Photospectroscopy (XPS) measurements were made on a Kratos AXIS Ultra
DLD spectrometer and the results are summarized in Table 1. No significant differences were
noted as compared to MSC-30. Electron Energy Loss Spectroscopy (EELS) measurements
were made on a FEI Technai F20 with a Gatan Imaging Filter system. Similar to MSC-30,
ZTC has an sp3-hybridized carbon content of 18%. Transmission electron microscope (TEM)
images were taken with a Tecnai TF30 with a LaB6 filament and 80 keV electrons (Figure 4).
The samples were prepared by placing the powder on an aluminum foil square and putting a
lacey carbon copper TEM grid on top to pick up some particles. Unlike MSC-30 and CNS201, ZTC shows nongraphitic crystalline order in the Fourier transform of the TEM images.
In the fast Fourier transform of the images (Figure 4), ZTC shows spots indicative of a
spacing of 1.2 to 1.3 nanometers, in agreement with other measurements of the periodicity of
the pores. The spots also suggest hexagonal symmetry in the pore arrangement.

Figure 4. TEM image of ZTC and an accompanying fast Fourier transform of the image.

45

References:
1.$Tarazona,$P.;$Marconi,$U.$M.$B.;$Evans,$R.$PhaseKEquilibria$of$Fluid$Interfaces$and$
Confined$ Fluids$ K$ Nonlocal$ Versus$ Local$ Density$ Functionals.$ Mol.% Phys.% 1987,$ 60,$
573K595.$
2.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR%Phys.%Chem.$1947,$55,$331K337.$
3.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
4.$Dumas,$J.$B.$A$Method$of$Estimating$Nitrogen$in$Organic$Material.$Ann.%Chim.%Phys.$
1833,$58,$171K173.$
5.$Yang,$ Z.$ X.;$ Xia,$ Y.$ D.;$ Mokaya,$ R.$ Enhanced$ Hydrogen$ Storage$ Capacity$ of$ High$
Surface$Area$ZeoliteKLike$Carbon$Materials.$J.%Am.%Chem.%Soc.%2007,$129,$1673K1679.$
6.$Langford,$ J.$ I.;$ Wilson,$ A.$ J.$ C.$ Scherrer$ After$ 60$ Years$ K$ Survey$ and$ Some$ New$
Results$in$Determination$of$Crystallite$Size.$J.%Appl.%Crystallogr.%1978,$11,$102K113.$

46
Chapter 5

Methane on Microporous Carbons
N.P. Stadie, M. Murialdo, C.C. Ahn, and B. Fultz, “Unusual Entropy of Adsorbed Methane on
Zeolite-Templated Carbon,” J. Phys. Chem. C, 119, 26409 (2015).
DOI: 10.1021/acs.jpcc.5b05021
N.P. Stadie, M. Murialdo, C.C. Ahn, and B. Fultz, “Anomalous Isosteric Enthalpy
of Adsorption of Methane on Zeolite-Templated Carbon,” J. Am. Chem. Soc.
135, 990 (2013).
DOI: 10.1021/ja311415m

1. Adsorptive Uptake
The ability to store the necessary quantities of natural gas in inexpensive and
moderately sized onboard tanks remains a significant challenge to the wide-spread adoption of
natural gas as a fuel in the transportation sector. By filling the onboard tanks with specially
designed physisorptive materials, the storage capacity can be significantly improved and the
volumetric energy density significantly increased. Here we study high-pressure methane
adsorption on three microporous carbons (ZTC, MSC-30, and CNS-201). The zeolitetemplated carbon (ZTC) is of particular interest due to its unique synthesis and morphology (a
narrow pore-size distribution centered at 1.2 nm).
Methane isotherms were measured at 13 temperatures from 238 to 526K and up to
pressures of ~10 MPa in a volumetric Sieverts apparatus, commission and verified for accurate
measurements up to 10 MPa1. Multiple adsorption runs were completed at each temperature

47
with research-grade methane (99.999%) and errors between cycles were less than 1%.
Adsorption/desorption cycles demonstrated full reversibility of the isotherms. The excess
adsorption isotherms are plotted in Figure 1. Unlike absolute adsorption, excess adsorption
reaches a maximum at high pressures. This maximum is a readily accessible figure of merit for
the gravimetric performance of a material at a fixed temperature. The excess maximum is
similar for ZTC-3 and MSC-30 at room temperature, but slightly higher for MSC-30 at 14.5
mmol g-1 at 8 MPa. While excess adsorption increases faster for MSC-30 at pressures between
0 and 0.8 MPa, uptake in ZTC-3 increases fastest between 0.8 and 5.7 MPa. CNS-201 has
much lower maxima due to its significantly smaller surface area. The highest measured excess
uptake of this study is for ZTC-3 at 238 K: 22.1 mmol g-1 (26.2 wt%) at 4.7 MPa, despite a
gentler initial increase at low pressure. Interestingly, the excess uptake in ZTC-3 is also greater
than for MSC-30 at high temperatures, although neither reaches a maximum between 0 and 9
MPa. At all temperatures, methane uptake in ZTC-3 is characterized by a gradual initial rise
and delayed increase at pressures between 0.2 and 2 MPa, leading to higher eventual methane
capacity than MSC-30, a material of comparable specific surface area. The measured excess
adsorption maxima (at a sample temperature of 298K) scale linearly with specific surface area
and are consistent with the reported linear trend for methane uptake at 3.5 MPa and 298 K.2

48

Figure 1. Measured methane excess adsorption as a function of temperature and pressure on
CNS-201 (top), MSC-30 (middle), and ZTC (bottom). The curves indicate the best fit obtained
with a dual-Langmuir fitting function.

49

2. Analysis
The excess adsorption data were fitted with a superposition of two Langmuir
isotherms, as detailed in Chapter 3. In general, the best-fit parameters obtained correlate with
physical properties of the materials studied. For example, the parameter indicative of the
maximum volume of the adsorbed layer, Vmax, can be independently verified through
comparison to the micropore volume of the adsorbent as measured with the DubininRadushkevich method3,4. In ZTC, if taken to be proportional to surface area, the Vmax
parameter corresponds to half of the mean pore width of the material: a thickness of 0.6 nm.
Likewise, the maximum possible adsorption quantity, given by parameter nmax, correlates well
with an estimate determined by the product of the micropore volume and the molar liquid
density of the adsorbate (methane)5.

50

Figure 2. The isosteric heats of methane adsorption on CNS-201 (top), MSC-30 (middle), and
ZTC (bottom) as a function of temperature and fractional site occupancy (θ).
The isosteric enthalpy of adsorption is the molar change in enthalpy of the adsorptive
species upon adsorption. While adsorption is an exothermic process, the isosteric heat of
adsorption is reported as a positive quantity by convention, as shown in Figure 2. These curves
were obtained by applying the Clapeyron relationship to the dual-Langmuir fits. It is necessary
to use the general form of the Clapeyron relationship for methane adsorption at high pressure

51
because of the significant nonideality of methane gas-state properties. Its derivation and
explanation with respect to the ideal-gas form of the equation are given in Chapter 3.
The Henry’s Law value of adsorption heat, -ΔH0, is calculated by extrapolating the
heat of adsorption to zero pressure. The Henry’s law values for CNS-201, MSC-30, and ZTC3 are 18.1-19.3, 14.4-15.5, and 13.5-14.2 kJ mol-1, respectively. The isosteric heats of methane
adsorption as a function of fractional occupancy, θ, in the activated carbons (CNS-201 and
MSC-30) are typical of other carbon materials, with the isosteric heats decreasing with θ. In the
range 0 < θ < 0.6, the more graphitic CNS-201 shows a more gradual decrease of isosteric
heat than MSC-30, indicative of more heterogeneous site energies in MSC-30. Surprisingly, the
isosteric heat of adsorption in ZTC increases to a maximum at θ = 0.5-0.6 at temperatures
from 238 to 273 K. This increase is anomalous compared to previous experimental reports of
methane adsorption on carbon.
This anomalous effect results from adsorbate-adsorbate intermolecular interactions, as
suggested by theoretical work.6,7,8 We have reported similar effects for ethane (Chapter 6) and
krypton (Chapter 7). Accurately assessing the contribution of intermolecular interactions to the
isosteric heat requires knowledge of the adsorption binding-site energies. A heterogeneous
distribution of site energies, as in MSC-30, is reflected in the relatively rapid decrease of the
isosteric heat with θ. This behavior is common as the most favorable sites are filled first (on
average). The material properties of ZTC, including a narrow distribution of pore width,

periodic pore spacing, and high content of sp -hybridized carbon, suggest a high degree of
homogeneity of the binding-site energies. We expect that the measured increase of 0.5 kJ mol-1
in the isosteric heat at 238K reflects most of the contribution from favorable intermolecular

52
interactions, and this increase is in good agreement with calculations of lateral interactions of
methane molecules on a surface.6,7An increasing isosteric heat, as seen with methane on ZTC,
is highly desirable as it enhances deliverable storage capacity. This effect enables a larger
fraction of the adsorption capacity at pressures above the lower bound of useful storage.
Indeed, the deliverable gravimetric methane capacities of ZTC at temperatures near ambient
are the highest of any reported carbonaceous material.
A clearer picture of adsorption thermodynamics can be achieved by evaluating the
specific enthalpy of the adsorbed phase, as shown in Figure 3. This removes the gas-phase
dependency of the isosteric heat and focuses solely on adsorbed-phase thermodynamics. This
is particularly useful because at a constant temperature, it is a reasonable approximation that
the specific properties of the adsorbed phase as a function of increasing site occupancy do not
depend on contributions from internal (intramolecular) phenomena. Here, a decreasing
enthalpy of the adsorbed phase as a function of uptake, as seen on ZTC, corresponds to an
“increasing” (or, decreasing negative) isosteric heat of adsorption.

53

Figure 3. Adsorbed-phase enthalpy of CNS-201 (top), MSC-30 (middle), and ZTC (bottom) as
a function of temperature and fractional occupancy (θ).
The adsorbed-phase entropy was determined in a similar manner that is described in
more detail in Chapter 3. Adsorbed-phase entropy isotherms are shown in Figure 4. While all

54
three adsorbed phases show qualitatively similar entropies, a notable difference is seen
between the smaller pore materials (CNS-201 and ZTC) and MSC-30, which has a significant
fraction of pores of widths >2 nm. The molar entropy of methane adsorbed on CNS- 201 and
ZTC at 238 K approaches the value of the liquid reference state rather closely (within 12 J K-1
mol-1), indicating a liquid-like character of the adsorbed layer, unlike on MSC-30 (reaching a
minimum of 22 J K-1 mol-1 at 238 K). We must note, however, the relatively arbitrary nature
of the reference state; the entropy of saturated liquid CH4 varies by ~45 J K-1 mol-1 along its
liquidus phase boundary.

55

Figure 4. Adsorbed-Phase Entropy of methane on CNS-201 (top), MSC-30 (middle), and ZTC
(bottom) as a function of temperature and fractional occupancy. Curves indicate
corresponding statistical mechanics estimates.
For comparison, statistical mechanics calculations were carried out to independently
estimate the adsorbed-phase entropy. The only experimental parameter used in the theoretical
calculations was the material’s specific surface area. A remarkable consistency between theory

56
and experiment is observed across all three adsorbents, especially in the limit of high
temperature and low pressure where the approximations in the theoretical model are most
justified9. Of particular note, the theoretical calculations very closely reproduce the measured
molar entropies of the adsorbed phase on MSC-30, where the largest errors between θ = 0-0.2
are <3.5% at all temperatures measured (see Figure 4). The agreement between experimental
data and the statistical mechanical calculations for the adsorbed phase on CNS-201 is similarly
close in the dilute limit, but strays significantly beyond θ = 0.35. On ZTC, however, the
estimated adsorbed-phase entropy exceeds measured values, especially at low temperatures and
moderate coverage. This suggests the mechanism by which enhanced adsorbate-adsorbate
interactions are promoted in the adsorbed phase on ZTC, namely clustered configurations.
For ZTC, confinement of an adsorbed phase in narrow pores is likely to lead to
clustering as a result of enhanced lateral interactions. The formation of such clusters, or
adsorbate “islands”, on an adsorbent surface due to attractive intermolecular interactions, is a
well-known feature of physisorption of strongly interacting molecular species (e.g., methanol
on indium-tin-oxide glass10) and moderately interacting molecular species (e.g., subcritical CO2
on several MOFs and zeolites11,12) and also of chemisorption of fairly weakly interacting atomic
species (e.g., oxygen on Pt(111)13). This clustering behavior results in a reduction in entropy
due to the reduced number of accessible configurations of the cluster(s) in the same total
number of sites. This is consistent with the reduced entropy measured for methane on ZTC as
compared to the statistical mechanics estimate. The topic of clustering accounting for
enhanced adsorbate-adsorbate interactions is investigated in more detail in Chapter 8.
As a general note, the entropies of methane adsorbed between 238 and 526 K on the

57
various carbon materials measured in this work are very high as compared to some historical
theoretical estimates, approaching values of bulk gaseous methane in the dilute limit. In fact,
this observation has recently been made across many materials and adsorbed molecular
species.14 The ratio of adsorbed-phase entropy to the gas-phase entropy for CH4 on MSC-30,
for example, spans from 0.3 (at 238 K and 2 MPa) to 0.8 (at 0.1 MPa and 521 K), similar to the
value reported for methane on graphite(0001) in the dilute limit: 0.76 at 55 K.14,15

58

References:
1.$Bowman,$R.$C.;$Luo,$C.$H.;$Ahn,$C.$C.;$Witham,$C.$K.;$Fultz,$B.$The$Effect$of$Tin$on$the$
Degradation$of$LaNi5KySny$MetalKHydrides$During$Thermal$Cycling.$J.%Alloys%Compd.%
1995,$217,$185K192.$
2.$Sun,$ Y.;$ Liu,$ C.$ M.;$ Su,$ W.;$ Zhou,$ Y.$ P.;$ Zhou,$ L.$ Principles$ of$ Methane$ Adsorption$
and$Natural$Gas$Storage.$Adsorption.%2009,$15,$133K137.$
3.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR%Phys.%Chem.%Sect.$1947,$55,$331K337.$
4.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
5.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:%Reference%Fluid%Thermodynamic%and%Transport%PropertiesDREFPROP,$Version$8.0$
[CDKROM],$2007.$
6.$Salem,$ M.$ M.$ K.;$ Braeuer,$ P.;$ von$ Szombathely,$ M.;$ Heuchel,$ M.;$ Harting,$ P.;$
Quitzsch,$ K.;$ Jaroniec,$ M.$ Thermodynamics$ of$ HighKPressure$ Adsorption$ of$ Argon,$
Nitrogen,$ and$ Methane$ on$ Microporous$ Adsorbents.$ Langmuir.% 1998,$ 14,$ 3376K
3389.$
7.$Sillar,$ K.;$ Sauer,$ J.$ Ab$ Initio$ Prediction$ of$ Adsorption$ Isotherms$ for$ Small$
Molecules$ in$ MetalKOrganic$ Frameworks:$ The$ Effect$ of$ Lateral$ Interactions$ for$
Methane/CPOK27KMg.$J.%Am.%Chem.%Soc.%2012,$134,$18354K18365.$
8.$AlKMuhtaseb,$ S.$ A.;$ Ritter,$ J.$ A.$ Roles$ of$ Surface$ Heterogeneity$ and$ Lateral$
Interactions$on$the$Isosteric$Heat$of$Adsorption$and$Adsorbed$Phase$Heat$Capacity.$
J.%Phys.%Chem.%B.%1999,$103,$2467K2479.$
9.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Unusual$ Entropy$ of$ Adsorbed$
Methane$on$ZeoliteKTemplated$Carbon.$J.%Phys.%Chem.%C%2015,$119,$26409K26421.$
10.$Wang,$L.;$Song,$Y.$H.;$Wu,$A.$G.;$Li,$Z.;$Zhang,$B.$L.;$Wang,$E.$K.$Study$of$Methanol$
Adsorption$ on$ Mica,$ Graphite$ and$ ITO$ Glass$ by$ Using$ Tapping$ Mode$ Atomic$ Force$
Microscopy.$Appl.%Surf.%Sci.%2002,$199,$67K73.$
11.$Krishna,$R.;$van$Baten,$J.$A.$Investigating$Cluster$Formation$in$Adsorption$of$CO2,$
CH4,$and$Ar$in$Zeolites$and$Metal$Organic$Frameworks$at$Suberitical$Temperatures.$
Langmuir.%2010,$26,$3981K3992.$
12.$Krishna,$R.;$van$Baten,$J.$M.,$Highlighting$a$Variety$of$Unusual$Characteristics$of$
Adsorption$and$Diffusion$in$Microporous$Materials$Induced$by$Clustering$of$Guest$
Molecules.$Langmuir.%2010,$26,$8450K8463.$
13.$Parker,$ D.$ H.;$ Bartram,$ M.$ E.;$ Koel,$ B.$ E.$ Study$ of$ High$ Coverages$ of$ Atomic$
Oxygen$on$the$Pt(111)$Surface.$Surf.%Sci.%1989,$217,$489K510.$
14.$Campbell,$C.$T.;$Sellers,$J.$R.$V.$The$Entropies$of$Adsorbed$Molecules.$J.%Am.%Chem.%
Soc.%2012,$134,$18109K18115.$
15.$Tait,$S.$L.;$Dohnalek,$Z.;$Campbell,$C.$T.;$Kay,$B.$D.$NKAlkanes$on$Pt(111)$and$on$
C(0001)/Pt(111):$ Chain$ Length$ Dependence$ of$ Kinetic$ Desorption$ Parameters.$ J.%
Chem.%Phys.%2006,$125,$234308K1K15.$

59
Chapter 6

Observation and Investigation of Increasing Isosteric Heat of
Adsorption of Ethane on Zeolite-Templated Carbon
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, "Observation and Investigation of
Increasing Isosteric Heat of Adsorption of Ethane on Zeolite-Templated Carbon," J. Phys.
Chem. C, 119, 994 (2015).
DOI: 10.1021/jp510991y
Abstract
Ethane adsorption was measured on zeolite-templated carbon (ZTC) and compared to
superactivated carbon MSC-30. Isotherms measured at temperatures between 252 and 423 K
were fitted using a superposition of two Langmuir isotherms and thermodynamic properties
were assessed. Unlike typical carbon adsorbents, the isosteric heat of adsorption on ZTC
increases by up to 4.6 kJ mol-1 with surface coverage. This increase is attributed to strong
adsorbate-adsorbate intermolecular interactions, a hypothesis that is shown to be consistent
with fundamental estimates of intermolecular interactions. Furthermore, the molar entropy of
the adsorbed phase was measured and compared to an estimate derived from statistical
mechanics. While the measured and estimated entropies of the adsorbed phase of ethane on
MSC-30 are in agreement, they differ significantly on ZTC at high coverage, indicative of the
atypical properties of ethane adsorption on ZTC.

60

1. Introduction
Ethane is the second most abundant component in natural gas and an important
petrochemical feedstock. It is a common reactant for the synthesis of ethylene, and its
separation from natural gas has been an important process for many years. Currently, the
separation of ethane from natural gas is predominantly carried out via cryogenic distillation, an
energy intensive process.1 Physisorption materials have been proposed as a more efficient
separation solution.2,3,4 Physisorption materials also hold promise in improving the volumetric
energy density of stored ethane.5,6 An understanding of ethane adsorption is thus essential for
natural gas storage and separation processes.
Physisorption occurs when weak physical interactions between a solid adsorbent and a
gas induce the formation of a locally densified adsorbate layer at the solid surface. This
interaction depends sensitively on the surface chemistry and surface structure of the
adsorbent.7,8,9,10 The isosteric heat of adsorption, qst, is often reported as a critical figure of
merit for physisorption. This proxy measure of binding energy determines the equilibrium
uptake quantity at finite temperatures and pressures.
Microporous carbons have gained significant attention as potential adsorbents due
their light weight, low cost, and wide variability.11,12,13 However, these materials often exhibit
binding energies that are below desired optimal values.14,15 Furthermore, the isosteric heat of
adsorption typically decreases with surface coverage due to binding site heterogeneity, further
reducing the deliverable gas storage capacity in the range of practicality for applications. We
recently reported the observation of increasing isosteric heat of adsorption of methane on
zeolite-templated carbon,16 a unique and anomalous behavior with respect to methane which
typically has very weak intermolecular interactions. Recently, Yuan et al. reported the synthesis

61
of a mesoporous carbon wherein the isosteric heat of ethane adsorption increases as a
function of coverage.17 This was attributed to favorable ethane-ethane intermolecular
interactions and a relatively homogeneous adsorbent surface. While intriguing, the latter results
have limited applications for gas storage and separation due to limitations of the mesoporous
carbon in question. With an average pore width of 48 Å and a specific surface area of 599 m2
g-1, this mesoporous carbon has only a small total excess uptake capacity of ~2.5 mmol g-1 at
278 K. This effect has not been investigated or observed for ethane adsorption on a microporous
carbon with a large specific surface area, a system that would have superior potential for
advanced applications. In this work, we report that a zeolite-templated carbon (ZTC) with a
narrow distribution of pore widths centered at 12 Å and a large specific surface area of 3591
m2 g-1 exhibits an increasing isosteric heat of ethane adsorption as a function of coverage. This
material has an exceptional uptake capacity of 22.8 mmol g-1 (at 252K), owing to its very large
surface area and optimized structural properties. Furthermore, several novel analysis methods
including comparisons to methane adsorption, Lennard-Jones parameters and statistical
mechanics calculations are implemented to corroborate and assist in the understanding of the
phenomenon of increasing isosteric heat of adsorption.

2. Experimental Methods
2.1 Materials Synthesis
Two materials were chosen for comparison in this study: MSC-30 and ZTC. The
superactivated carbon “Maxsorb” MSC-30 was obtained from Kansai Coke & Chemicals
Company Ltd. (Japan). The zeolite-templated carbon (ZTC) was synthesized in a multistep
process that was optimized to achieve high template fidelity of the product.18 The faujasite-

62
type zeolite NaY (obtained from Tosoh Corp., Japan) was impregnated with furfuryl alcohol
which was subsequently polymerized at 150 °C, augmented by a propylene CVD step at 700
°C, and carbonized at 900 °C. The ZTC product was freed by dissolution of the zeolite
template in HF. ZTC was confirmed to exhibit very high fidelity with the zeolite template and
outstanding microporous periodicity by X-ray diffraction and transmission electron
microscopy, described in detail elsewhere.16
2.2 Materials Characterization
Equilibrium nitrogen adsorption isotherms were measured at 77 K using a BELSORPmax volumetric instrument from BEL-Japan Inc. The Dubinin-Radushkevich (DR) method19,20
was employed to calculate micropore volumes and the Brunauer-Emmett-Teller (BET)
method21 was used to calculate specific surface areas. Pore-size distributions were determined
by non-local density functional theory (NLDFT) analysis implemented by software provided
by Micromeritics Instrument Corp., and a carbon slit-pore model was utilized.22 Skeletal
densities of the materials were determined by helium pycnometry. Finally, the Dumas
method23 was employed to determine the elemental composition (CHN) of MSC-30 and
ZTC.24
2.3 Measurements
Equilibrium ethane adsorption isotherms were measured on ZTC and MSC-30 at 9
temperatures between 252 and 423 K. Research grade ethane (99.999%) obtained from
Matheson Tri-Gas Inc. was used in a custom Sieverts apparatus that was tested for accuracy up
to 10 MPa.25 The reactor containing the sample was held at a constant set temperature while
the remaining gas manifold always remained at room temperature. For low temperature
isotherms, the reactor was submerged in a circulated chiller bath leading to temperature

63
deviations no larger than +0.1 K. For high temperature isotherms, the reactor was encased
in a copper heat exchanger and wrapped with insulating fiberglass-heating tape. A PID
controller was used to maintain a constant temperature with fluctuations of less than +0.4 K.
In both setups, the temperature was measured by K-type thermocouples placed in direct
contact with the reactor. The manifold temperature was measured with a platinum resistance
thermometer in contact with the outer wall of the manifold.
High-pressure measurements were made with an MKS Baratron (Model 833) pressure
transducer. For degassing, vacuum pressures were ensured with a digital cold cathode pressure
sensor (I-MAG, Series 423). The Sieverts apparatus is equipped with a molecular drag pump
capable of achieving vacuum of 10-4 Pa. Each sample was degassed at 520 K under vacuum of
less than 10-3 Pa prior to measurements. Multiple adsorption/desorption isotherms were
measured to ensure reversibility, and errors between identical runs were less than 1%. Gas
densities were determined using the REFPROP Standard Reference Database.26

3. Results
3.1 Adsorbent Characterization
BET specific surface areas of ZTC and MSC-30 were found to be 3591 + 60 and 3244
+ 28 m2 g-1, respectively. Likewise, both materials have similar DR micropore volumes: 1.66
mL g-1 (ZTC) and 1.54 mL g-1 (MSC-30). The distribution of the pore sizes, however, differs
significantly between the two materials. ZTC was determined to have a uniform pore-size
distribution centered at 12 Å (see Figure 1). Over 90% of the micropore volume of ZTC is
contained in pores with widths between 8.5 and 20 Angstroms. MSC-30 exhibits a wide range
of pore sizes from 6 to 35 Å. 40% of the pore volume of MSC-30 is contained in pore widths

64
greater than 21 Angstroms. Furthermore, while MSC-30, like most activated carbons, has a
skeletal density of 2.1 g mL-1, ZTC has an unusually low skeletal density of 1.8 g mL-1 (in
agreement with other ZTCs).24 This discrepancy can be explained by the significantly higher
hydrogen content found in ZTC by elemental analysis experiments. Hydrogen was found to
account for 2.4% (by weight) of ZTC but only 1.2% of MSC-30.24

Figure 1. Pore-size distribution and relative pore volume of ZTC (orange) and MSC-30
(purple).

65
3.2 Adsorption Measurements:
Equilibrium ethane adsorption isotherms were measured at 9 temperatures between
252 and 423K and at pressures of up to 32 bar (see Figure 2). The maximum excess adsorption
quantities measured on ZTC and MSC-30 (at ~253K) were 22.8 and 26.8 mmol g-1,
respectively. At room temperature (297K), the maximum uptake quantities measured on ZTC
and MSC-30 were 19.2 and 22.1 mmol g-1, respectively. Thus at both temperatures, MSC-30
exhibits greater excess adsorption capacities, with differences between the maximums being
less than 15%. This is in contrast to methane adsorption on the same two materials, where
ZTC exhibited higher excess adsorption capacities than MSC-30 at low temperatures.16
Moreover, the differences in excess adsorption capacities between the two materials are
smaller (less than 5%) for methane adsorption.

66

Figure 2. Equilibrium excess adsorption isotherms of ethane on ZTC and MSC-30. The lines
indicate the best-fit analysis using a superposition of two Langmuir isotherms.

4. Discussion
4.1 Fitting Methodology
Thermodynamic analysis of adsorption requires interpolation of the adsorption data
points, generally with a fitting function. It is common in literature to assume that the excess

67
adsorption well approximates the absolute adsorption. This assumption, while valid at low
pressures and high temperatures, is invalid at temperatures near the critical point, particularly
in high-pressure studies. To determine the absolute adsorption quantities and avoid the welldocumented errors associated with equating excess adsorption and absolute adsorption, we
follow a method initially described by Mertens.27 We extend this method by modifications for
the nonideal gas regime.16
The Gibbs excess adsorption28 (ne) is a function of the bulk gas density (ρ):$

!! = ! !! − ! !! !

$$$$$$$$$$$$$$$$$$$$$

(1)

Determining the absolute adsorption quantity (na) is simplest when the volume of the
adsorption layer (Va) is known. We left Va as an independent fitting parameter, and assessed it
later. The measured excess adsorption quantities were fitted with a generalized (multi-site)
Langmuir isotherm:
!! (!, !) = !!"# − !!"# !(!, !)

!! !
! !! !!! !

(2)

Excess adsorption and density are functions of pressure (P) and temperature (T). The
independent fitting parameters in this fitting model are nmax, a scaling factor indicative of the
maximum absolute adsorption, αi, a weighting factor for the ith Langmuir isotherm(

! !! =

1), Va, which scales with coverage up to Vmax (the maximum volume of the adsorption layer),
and Ki , an equilibrium constant for the ith Langmuir isotherm. The parameter Ki is defined by
an Arrhenius-type equation:
!! =

!!

! !!! !"

(3)

68
Here, Ai is a prefactor and Ei is a binding energy associated with the ith Langmuir
isotherm. Using two superimposed isotherms (i=2), we obtained satisfactory results while
limiting the number of independent fitting parameters to 7: nmax, Vmax, α1, A1, A2, E1, and E2.
For ZTC and MSC-30, accurate fits were obtained with residual mean square values of 0.21
and 0.13 (mmol g-1)2. These fits are shown in Figure 2 and the best-fit values of the fitting
parameters are given in Table 1. For comparison, fitting parameters for methane adsorption on
ZTC and MSC-30 were obtained using the same fitting procedure16, and are also shown in
Table 1.
Table 1. Least-squares minimized fitting parameters of the excess adsorption isotherms of
ethane on MSC-30 and ZTC described by a two-site Langmuir isotherm.
Ethane on
ZTC
Ethane on
MSC-30
Methane on
ZTC
Methane on
MSC-30

nmax
25 mmol/g
36 mmol/g
36 mmol/g
41 mmol/g

Vmax
1.6
mL/g
2.6
mL/g
2.0
mL/g
2.3
mL/g

α1
0.82
0.71
0.46
0.70

A1
2.1E-7
K1/2/MPa
0.086
K1/2/MPa
0.059
K1/2/MPa
0.068
K1/2/MPa

A2
0.044
K1/2/MPa
0.0065
K1/2/MPa
0.00018
K1/2/MPa
0.0046
K1/2/MPa

E1
41
kJ/mol
20
kJ/mol
12
kJ/mol
13
kJ/mol

E2
18
kJ/mol
18
kJ/mol
20
kJ/mol
13
kJ/mol

Many of the independent fitting parameters in this method have physical significance.
For example, nmax represents the maximum specific absolute adsorption of the system. If the
entire micropore volume of the material is assumed to be completely filled at this condition,
then it follows that it should be approximately comparable to the value obtained by
multiplying the density of the liquid phase of the adsorbate by the total micropore volume.

69
26

Using the density of liquid ethane and liquid methane near the triple point, the maximum
possible adsorption quantities estimated in this simplified way for ethane and methane on
MSC-30 are 33 and 43 mmol g-1, respectively. These values are within 10% of the nmax values
determined through fitting. For ethane and methane adsorption on ZTC, the estimated and
fitted values of nmax both deviate more significantly, with estimated values greater by 44% and
32% respectively.
The Vmax parameter approximates the maximum volume of the adsorbed layer.
Dividing by surface area, this gives an average width for the adsorbed layer. For ethane and
methane adsorption on ZTC, this gives average adsorbed layer widths of 4.5 and 5.5 Å, both
of which are in agreement with the measured ZTC micropore half-width of 6 Å. This suggests
fairly effective filling of the ZTC micropores. Likewise, the ethane and methane Vmax
parameters on MSC-30 give adsorbed layer widths of 8 and 7 Å, which are in agreement with
the average measured micropore half-width, 7 Å. Furthermore, for ethane on ZTC, Vmax
equals 1.6 mL g-1, which is in good agreement with the micropore volume measured using the
DR method and nitrogen isotherms, 1.66 mL g-1. The Vmax values for ethane on MSC-30 and
methane on both materials deviate more significantly from measured DR micropore volumes
with discrepancies of up to 41%. This, however, is not unexpected as different adsorbates with
differing size and shape are confined and adsorbed in micropores distinctly.
4.2 Isosteric Enthalpy of Adsorption
The isosteric enthalpy of adsorption (ΔHads) is a widely used figure of merit that is
indicative of the strength of binding interactions at a fixed temperature, pressure, and
coverage. Typically it is determined using the isosteric method and reported as a positive value,

70
qst, the so-called isosteric heat (a convention that is followed herein), defined according to the
Clapeyron equation:

!!" = −∆!!"# = −!

!"
!" !!

Δ!!"#

(4)

The molar change in volume of the adsorbate upon adsorption (Δvads) is given by the
difference between the gas-phase molar volume and

!!"#
!!"#

. The fitting method used in this

work is especially convenient for thermodynamic calculations with the Clapeyron equation
because the generalized Langmuir equation with i=2 can be analytically differentiated.27 The
isosteric heats of adsorption of ethane on ZTC and MSC-30 calculated in this way are shown
in Figure 3.

71

Figure 3. Isosteric heats of ethane adsorption on ZTC and MSC-30.
It can be observed that the isosteric heat of adsorption of ethane on ZTC increases as
a function of coverage while MSC-30 (and a majority of other known systems) exhibits the
typical decreasing isosteric heat with coverage (in this case represented as the absolute uptake
quantity). This effect is most pronounced at low temperatures where the isosteric heat rises by
4.6 kJ mol-1 above its Henry’s Law value of 20.6 kJ mol-1 (at 252 K). This effect (an increasing
isosteric heat of adsorption) has also been observed for methane adsorption on the same
material (ZTC)16 and is dependent on gas properties as well as structural and surface properties

72
of the adsorbent. In particular, the effect is expected to increase with the strength of
intermolecular interactions of the adsorbate when adsorbate molecules are on nanostructured
surfaces that promote intermolecular interactions. ZTC is an ideal candidate adsorbent for
observing such a phenomenon due to its uniform pore-size distribution with pore widths
centered at 12 Å and homogeneous chemical nature24.
In the absence of intermolecular interactions and binding site heterogeneity, the
isosteric heat should be constant at all coverages and temperatures. The increase in isosteric
heat (as a function of coverage) reported in this work is hypothesized to result from attractive
intermolecular interactions between ethane molecules. Assuming random site occupancy in the
low coverage regime (e.g. less than 50% of the available sites filled), the probability of any site
being occupied is equal to the fractional site occupancy, θ. If z is the number of nearest
neighbor adsorption sites, on average an adsorbate molecule will have zθ occupied nearest
neighbor adsorption sites. By assuming that nearest neighbor interactions have a binding
energy of ε and higher order neighbors have a negligible binding energy, the average
adsorbate-adsorbate

binding

energy

per

molecule,

!!!

!= !

ξ,

is

given:

(5)

Taking the derivative of ξ with respect to θ gives the estimated slope of the isosteric heat as a
function of coverage (resulting from adsorbate-adsorbate interactions):
!(!)

!"

= !
!!

(6)

The Lennard-Jones parameter ε (which describes the well depth of the Lennard-Jones 12-6
interaction potential) is 1.7 kJ mol-1 for ethane-ethane interactions29. For adsorbed molecules
on a two-dimensional surface the number of nearest neighbor adsorption sites, z, is posited to

73
-1

be 4. This results in an estimated slope for ethane on ZTC of 3.4 kJ mol . The average slope
of the measured isosteric heat of adsorption of ethane on ZTC (Figure 3) has a similar value of
3.3 kJ mol-1 (at the lowest measured temperature 252K, up to 50% coverage). Interestingly,
these measured and estimated slopes are of a similar magnitude to the slope of the isosteric
heat of ethane adsorption on the mesoporous carbon of Yuan et al. (~4-5 kJ mol-1 at
~298K).17
This very simple model (Equation 6) also gives a reasonable estimate for the increase in the
isosteric heat of methane adsorption on ZTC. Here, ε for methane-methane interactions is 1.2
kJ mol-1,29 giving a predicted slope of 2.4 kJ mol-1. This is in agreement with the measured
slope of 2.2 kJ mol-1 (up to 50% coverage at 255K).16 It is important to note that z=4, while
intuitively reasonable, is a posited value. The true value of z is difficult, if not impossible, to
obtain, and likely varies for different gases.
A number of metrics suggest that ethane has stronger attractive intermolecular
interactions than methane by a factor of ~1.4 to 1.6. These metrics include Lennard-Jones
potential (adsorbate-adsorbate) well depth, boiling point, and critical temperature, and are
shown in Table 2.
Table 2. Gas properties of ethane and methane and their ratios.

ε (Lennard-Jones)
Boiling Point (1 atm)
Critical Temperature

ethane
1.7
kJ/mol29
184.57 K26
305.32 K26

methane
1.2
kJ/mol29
111.67 K26
190.56 K26

ratio
1.4
1.6528
1.6022

In agreement with the ratios in Table 2, the ratio of the slopes of the ethane and
methane heats of adsorption (as a function of coverage) on ZTC is 1.5. Stronger

74
intermolecular interactions within ethane correspond to an isosteric heat that increases more
steeply than methane on the same material. Furthermore, the average Henry’s Law values (zero
coverage limit) of the isosteric heat were determined to be 20 and 21 kJ mol-1 for ethane on
ZTC and MSC-30, respectively, and 14 and 15 kJ mol-1 for methane. For both ZTC and MSC30, these ratios of Henry’s law values are 1.4 for ethane and methane.
4.3 Entropy
At equilibrium, the Gibbs free energy of the adsorbed phase (Ga) equals the Gibbs free
energy of the gas phase (Gg). The isosteric entropy of adsorption (ΔSads) is
Δ!!"# = !" − !" = Δ!!"# − !Δ!!"# = 0
Δ!!"# =

!!!"#

(7)
(8)

The isosteric entropy of adsorption is the change in entropy of the adsorbate upon adsorption
and the measured values for ethane adsorption on MSC-30 and ZTC are shown in Figure 4
(reported as positive values).

75

Figure 4. The isosteric entropy of ethane adsorption on ZTC and MSC-30 between 252 and
423 K.
By adding the isosteric entropy of adsorption to the gas-phase entropy
(from Refprop26) we obtain the molar entropy of the adsorbed phase (Figure 5).

76

Figure 5. The molar adsorbed-phase entropies of ethane on ZTC and MSC-30. The curves
indicate measured data and the points are calculated values (from statistical mechanics).

4.4 Statistical Mechanics
For comparison, the adsorbed-phase molar entropies were also calculated using
statistical mechanics. Adsorbed ethane has numerous entropic contributions including
vibrational, rotational, and configurational components. Each contribution was accounted for

77
as follows using standard partition functions and established values for characteristic
frequencies. The effects of intermolecular interactions between the adsorbed molecules were
not included, and the resulting discrepancies are discussed following the analysis.
Ethane in the gas phase has 12 internal vibrational modes with well-known characteristic
vibrational frequencies and degeneracies,30 and minimal changes in these frequencies are
expected upon adsorption.

Near ambient temperature, these vibrational modes are of little

significance to the adsorbed-phase entropy, but their influence increases with temperature (see
Figure 6). The partition function for vibrational modes (qvib i) is:
!!"#!! ! =

! !!!"#!! !!
!!! !!!"#!! !

(9)

Here, Θvib i is the characteristic vibrational temperature of the ith vibrational mode.
Adsorbed gases also vibrate with respect to the adsorbent surface. While the partition
function for these vibrations is also given by Equation 9, the characteristic frequency is not
readily accessible. Instead we have estimated these characteristic harmonic frequencies by

!=

!!

(10)

Here K is the force constant and m is the molecular mass (of ethane). For simplicity (in the
absence of detailed knowledge of the adsorbent surface geometry) the adsorbate-adsorbent
potential was modeled as a Lennard-Jones potential with ε given by the average Henry’s law
value of the isosteric heat per molecule. For a Lennard-Jones potential, the force constant and
frequency are:
!"!

!=! !

(11)

78
!=

!"!

!!

!(!! )!

(12)

Here rm is the distance wherein the potential reaches its minimum. Rotationally, ethane is a
symmetric top with characteristic frequencies Θrota=Θrotb=0.953K, Θrotc=3.85K.31 For ethane,
the high temperature approximation of the partition function for the rotational modes (qrot) is:

!!

!!"# = !

!!"#$

!!"#$

(13)

The configurational entropy of the adsorbed phase was determined using the partition
function:
!!"# =

!!

!!

(14)

Here λ is specific surface area divided by specific absolute uptake, and Λ is the thermal de
Broglie wavelength. Using the partition functions above, individual entropy contributions were
calculated by taking the negative temperature derivative of the Helmholtz free energy. These
individual contributions were summed to obtain the total molar entropy of the adsorbed phase
(shown in Figure 5). The relative contributions from each component to the total entropy at a
representative fixed adsorption quantity are shown in Figure 6.

79

Figure 6. Relative contributions of each component to the total adsorbed-phase entropy of
ethane on MSC-30 at the absolute uptake value of 2 mmol g-1.

As shown in Figure 5, there is good agreement between the measured and calculated
values of the adsorbed-phase entropy of ethane on MSC-30. Errors are less than 10% (without
applying any fitting or offset) throughout the measured regime, and are especially low at high
temperatures. At high coverages the measured entropy of ethane on MSC-30 gradually levels
out, but maintains positive concavity. In contrast, the measured entropy of ethane adsorbed on
ZTC deviates significantly from calculated values. At high temperatures, the measured entropy
is in agreement with the calculation from statistical mechanics, as in MSC-30. However, at low
temperatures and high coverages, the measured entropy is well below the calculated value
(with discrepancies of up to 42%). This is associated with the increase in isosteric heat in this
regime; ethane adsorption on ZTC has an anomalously increasing isosteric heat, and likewise
an anomalously decreasing entropy in the adsorbed phase. This is expected to be caused by
enhanced adsorbate-absorbate interactions on the surface of ZTC. Stronger intermolecular
interactions correspond to stiffer vibrational modes, hindered rotational motion, and inhibited

80
molecular motion/rearrangement, all of which can lead to a decrease in the entropy of the
adsorbed phase.
The increase in isosteric heat above the Henry’s Law value is largest at low
temperatures. We expect that temperature will disrupt the adsorbate-adsorbate lateral
interactions, suppressing the increase in isosteric heat. The thermal behavior may be a
cooperative one, where the loss of lateral interactions makes other lateral interactions less
favorable. There appears to be a critical temperature around 300 K where the effect of lateral
interactions between ethane molecules is lost.

5. Conclusions
Ethane adsorption was measured on a zeolite-templated carbon material (ZTC) that
has an exceptionally high surface area and narrow and uniform microporosity. An increasing
isosteric heat of adsorption as a function of coverage was observed. The isosteric heat rises by
4.6 kJ mol-1 from a Henry’s law value of 20.6 kJ mol-1 at low coverage to a peak of 25.2 kJ
mol-1 at a coverage of 21.4 mmol g-1. By comparing ethane adsorption on ZTC to methane
adsorption on the same material, it was found that the slope of the isosteric heat of adsorption
with coverage approximately scales with the strength of the adsorbate-adsorbate
intermolecular interactions. A control material, superactivated carbon MSC-30, behaved as a
normal microporous carbon adsorbent, exhibiting a monotonically decreasing isosteric heat
with coverage. The measured adsorbed-phase entropy of ethane on MSC-30 was also
successfully estimated with a statistical mechanics based approach (without intermolecular
interactions), exhibiting discrepancies of less than 10%. The measured entropy of ethane
adsorbed on ZTC deviated significantly from this standard model prediction at high coverage

81
and low temperature, indicating atypical adsorption properties in this system. The behavior
of both the adsorbed-phase entropy and the isosteric heat of ethane on ZTC can be explained
by attractive adsorbate-adsorbate interactions promoted by the nanostructured surface of
ZTC.

82

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Tee,$L.$S.;$Gotoh,$S.;$Stewart,$W.$E.$Molecular$Parameters$for$Normal$Fluids$K$$
LennardKJones$12K6$Potential.$Ind.%Eng.%Chem.%Fundam.%1966,$5,$356K363.$
30.$Shimanouchi,$ T.$ Tables$ of$ Molecular$ Vibrational$ Frequencies$ Consolidated$
Volume$2.$J.%Phys.%Chem.%Ref.%Data%1977,$6,$993K1102.$
31.$Chao,$ J.;$ Wilhoit,$ R.$ C.;$ Zwolinski,$ B.$ J.$ Ideal$ Gas$ Thermodynamic$ Properties$ of$
Ethane$and$Propane.$J.%Phys.%Chem.%Ref.%Data$1973,$2,$427K437.$

84
Chapter 7

Krypton Adsorption on Zeolite-Templated Carbon and Anomalous
Surface Thermodynamics
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “Krypton Adsorption on Zeolite-Templated
Carbon and Anomalous Surface Thermodynamics,” Langmuir, 31, 7991 (2015).
DOI: 10.1021/acs.langmuir.5b01497
Abstract
Krypton adsorption was measured at 8 temperatures between 253 and 433 K on a zeolitetemplated carbon and two commercial carbons. The data were fitted using a generalized
Langmuir isotherm model and thermodynamic properties were extracted. Differing from that
on commercial carbons, krypton adsorption on the zeolite-templated carbon is accompanied
by an increasing isosteric heat of adsorption, rising by up to 1.4 kJ mol-1 as a function of
coverage. This increase is a result of enhanced adsorbate-adsorbate interactions promoted by
the ordered, nanostructured surface of the adsorbent. An assessment of the strength and
nature of these adsorbate-adsorbate interactions is made by comparing the measured isosteric
heats of adsorption (and other thermodynamic quantities) to fundamental metrics of
intermolecular interactions of krypton and other common gases.

85

1. Introduction
High surface area carbon materials have garnered interest for a variety of adsorptive
applications1,2,3,4 ranging from hydrogen storage5,6 to carbon capture7,8 and many others. The
recently emerged class of templated carbon materials9,10,11 exhibiting controlled pore-size
distributions that depend on the template, have shown exceptional performance in many
adsorptive applications owing to their uniquely ordered structure. Zeolite-templated carbon
(ZTC) is one of the highest surface area carbonaceous materials known12, and therefore
exhibits a high specific adsorptive capacity toward small molecular species including
hydrogen13,14 nitrogen15, carbon dioxide16, methane17, and water18. Further, our recent work has
shown that ZTC exhibits not only a high specific adsorptive capacity, but also hosts an
adsorbed phase with highly unusual properties; both ethane19 and methane20 exhibit isosteric
heats of adsorption on ZTC that increase with increasing surface coverage, a particularly rare
and advantageous property for deliverable storage capacity. Due to its chemical homogeneity20
and narrow pore-size distribution centered at a width of 12 Å, the surface of ZTC is optimized
for promoting lateral interactions between adsorbed molecules, even when these interactions
are exceedingly weak (e.g., as for methane).
Krypton, the fourth noble gas, is an unreactive monatomic gas that otherwise bears
many similarities to methane. The two gases share a similar size (Kr: 3.9 Å, CH4: 4.0 Å)21 and
approximately spherical symmetry, as well as similar boiling points (120 K and 112 K,
respectively)22 and critical temperatures (209 K and 190 K, respectively)22. Conveniently,
monatomic krypton allows for very simple calculations of thermodynamic properties such as
entropy, since rotational and internal vibrational modes do not exist. Krypton has applications
in the photography, lighting23, and medical industries,24,25 and is commonly used as an

86
26,27

adsorbate for characterizing low-surface-area materials

. There is also active interest in

finding adsorbent materials that can effectively separate krypton from xenon, especially the
radioactive isotope 85Kr. 28,29 These gases are off-gassed from spent nuclear fuel and their
separation is crucial to the development of “closed” nuclear fuel cycles30. Nevertheless,
krypton adsorption across a wide range of temperatures and pressures is a relatively
unexplored topic, and the results can have relevant implications for many other more complex
adsorptive systems.

2. Experimental
2.1 Materials Synthesis
Three microporous carbons were chosen for this study: MSC-30, CNS-201 and ZTC.
MSC-30 was obtained from Kansai Coke & Chemicals Company Ltd. (Japan) and CNS-201
was obtained from A. C. Carbone Inc. (Canada). ZTC is a zeolite-templated carbon that was
synthesized in a multistep process31 based on a previously reported approach designed to
achieve high template fidelity of the product13. The template used was a NaY zeolite (faujasite)
obtained from Tosoh Corp. (Japan). Briefly, the zeolite was first impregnated with furfuryl
alcohol, which was polymerized at 423 K, before undergoing a 973 K propylene chemical
vapor deposition step, and finally carbonization at 1173 K. The zeolite template was removed
by dissolution in 48% hydrofluoric acid. The synthesis is described in detail elsewhere31.

2.2 Materials Characterization
Nitrogen adsorption isotherms were measured at 77 K using a BELSORP-max
instrument (BEL-Japan Inc.). From these measurements, micropore volumes (Dubinin-

87
32,33

Radushkevich (DR) method

) and specific surface areas (Brunauer-Emmett-Teller (BET)

method34) were determined. Pore-size distributions were obtained using non-local density
functional theory (NLDFT) analysis35,36 with a carbon slit-pore model and software from
Micromeritics Instrument Corp. The skeletal density of each material was determined by
helium pycnometry.
2.3 Measurements
Equilibrium adsorption isotherms of krypton on the three carbon adsorbents were
measured at 8 temperatures between 253 and 433 K. Research-grade krypton (99.998%) was
obtained from Air Liquide America Corp. and used in a custom Sieverts apparatus designed
and tested for accuracy up to 10 MPa.37 Measurements were made up to high pressures using
an MKS Baratron (Model 833) pressure transducer. Each of the samples was degassed at 520
K under a vacuum of less than 10-9 MPa prior to testing. The Sieverts was equipped with a
molecular drag pump capable of achieving a vacuum of 10-10 MPa and vacuum pressures were
verified using a digital cold cathode pressure sensor (I-MAG, Series 423). The adsorbent was
loaded into a stainless steel reactor, sealed with a copper gasket, and held at a constant
temperature. To obtain low temperature isotherms, the reactor was submerged in a circulated
chiller bath with temperature fluctuations no larger than + 0.1 K. High temperature isotherms
were obtained by encasing the reactor in a copper heat exchanger wrapped with insulating
fiberglass heating tape. Using a PID controller, the reactor temperature was maintained with
fluctuations no larger than + 0.4 K. The temperature of the reactor was monitored with Ktype thermocouples while the temperature of the gas manifold was measured with platinum
resistance thermometers. For calculations of excess uptake, bulk phase gas densities were
obtained from the REFPROP Standard Reference Database22. Multiple adsorption/desorption

88
isotherms were taken to ensure complete reversibility and identical measurements were
found to be reproducible to within 1% error.

3. Results
3.1 Adsorbent Characterization
The BET surface areas of ZTC, MSC-30, and CNS-201 were determined to be 3591 +
60, 3244 + 28, and 1095 + 8 m2 g-1, respectively. ZTC and MSC-30 also have similar
micropore volumes of 1.66 and 1.54 cm3 g-1 while CNS-201 has a much smaller micropore
volume of 0.45 cm3 g-1. Despite their similarities, ZTC and MSC-30 have very different poresize distributions (Figure 1). Due to its templated nature, ZTC exhibits a single, sharp peak in
its pore-size distribution, corresponding to a pore width of 12 Å. This has been determined by
NLDFT pore-size analysis and further evidence is given by X-ray diffraction and transmission
electron microscopy (TEM) investigations20. Based on NLDFT pore-size analysis, over 90% of
the micropore volume of ZTC is contained in pores with widths between 8.5 and 20 Å. MSC30 on the other hand has a broad pore-size distribution with micropore widths ranging from 6
to 35 Å and over 40% of its micropore volume is in pores of widths greater than 21 Å. CNS201, has three prominent pore widths at approximately 5.4, 8.0, and 11.8 Å, which contain
roughly 50%, 20%, and 15% of the micropore volume respectively. The skeletal density of
both activated carbons (MSC-30 and CNS-201) was found to be 2.1 g cm-3, which is close to
the ideal density of graphite (~2.2 g cm-3). The templated carbon ZTC, however, was found to
have a skeletal density of 1.8 g cm-3, which is in agreement with other zeolite-templated
carbons.20,31

89

Figure 1. Pore-size distributions of the three carbon materials (CNS-201, MSC-30, and ZTC)
derived from NLDFT analysis of nitrogen adsorption measurements at 77 K.
3.2 Adsorption Measurements
Equilibrium excess adsorption isotherms of krypton on ZTC, MSC-30 and CNS-201
are presented in Figure 2. At high pressures and low temperatures, excess adsorption reaches a
maximum, a well-known phenomenon for Gibbs excess adsorption38. At 253 K, ZTC, MSC30, and CNS-201 have excess adsorption maxima of 22.6, 23.3, and 7.9 mmol g-1, respectively.
At 298 K, the excess adsorption maxima are 16.3, 17.7, and 6.6 mmol g-1, respectively. MSC-30
exhibits a greater excess adsorption maximum than ZTC at all temperatures measured. This is
in contrast to methane adsorption on the same materials where excess adsorption quantities on
ZTC exceeded those on MSC-30 at low temperatures (238-265 K). CNS-201 exhibits the
smallest excess adsorption uptake of the three materials due to its lower surface area.

90

Figure 2. Equilibrium excess adsorption isotherms of krypton on ZTC, MSC-30, and CNS201. The lines indicate the best fit as determined using a generalized (two-site) Langmuir
isotherm model.,

91

4. Data Analysis
4.1 Fitting Methodology
Thermodynamic analysis requires fitting the adsorption data points to a continuous
function. While it is common to assume that excess adsorption is equivalent to absolute
adsorption at low pressures, this assumption becomes invalid at high pressures and low
temperatures. Our method for both fitting and determining the absolute quantity of
adsorption from experimental data is based on a generalized-Langmuir model. Briefly, a
previously described method39 has been further modified to account for phenomena that are
relevant to the nonideal gas regime; the complete details of this methodology are described
elsewhere.40
Gibbs excess adsorption (ne) is a function of both the absolute adsorption (na) and the
gas density in the bulk phase (ρ):
!! = ! !! − ! !! !

(1)

If the volume of the adsorption layer (Va) is known, determining absolute adsorption is trivial
(given excess adsorption). However, as there is no generally accurate method for determining
Va, we have left it as an independent fitting parameter. Excess adsorption quantities were
fitted with the following generalized (multisite) Langmuir isotherm:
!! (!, !) = !!"# − !!"# !(!, !)

!! !
! !! !!! !

(2)

Here the independent fitting parameters are nmax, the maximum absolute adsorption which
serves as a scaling factor, Va, which scales with coverage up to the maximum volume of the
adsorption layer (Vmax), αi which weights the ith Langmuir isotherm (Σi αi=1), and Ki the

92
equilibrium constant of the ith Langmuir isotherm. Ki is given by an Arrhenius-type equation
where Ai is a prefactor and Ei is the binding energy of the ith Langmuir isotherm:
!! =

!!

! !!! !"

(3)

Pressure and temperature are denoted by P and T respectively. By setting the number of
Langmuir isotherms equal to two (i=2) we limit the number of independent fitting parameters
to seven while still obtaining highly accurate fits. The residual mean square values of the fits on
ZTC, MSC-30, and CNS-201 are 0.067, 0.070, and 0.0078 (mmol g-1)2 respectively. Individual
fitting parameters for adsorption on the three materials are given in Table 1.

Table 1. Least Squares Minimized Fitting Parameters of Krypton Excess Adsorption

ZTC
MSC-30

A1
nmax
Vmax
(K1/2
-1
3 -1
(mmol g ) (cm g ) α1
MPa-1)
39
2.0
0.31 0.092
58
3.0
0.73 0.11

A2
(K1/2
MPa-1)
1.8E-6
0.0031

CNS-201

11

0.0059

0.069

0.49

0.46

E1
(kJ mol-1)
10
12

E2
(kJ mol-1)
30
13

15

16

The optimized fit parameters were found to be in reasonable agreement with
independent estimates of physical quantities. For example, Vmax corresponds to the maximum
micropore filling volume. Dividing Vmax by the BET surface area of the adsorbent gives an
average maximum adsorption layer width. For ZTC, CNS-201, and MSC-30, the maximum
adsorption layer widths determined from Vmax are 5.6, 4.5, and 9.2 Å. These are in reasonable
agreement with the average micropore half-widths for ZTC, CNS-201, and MSC-30 as
determined by NLDFT analysis of the nitrogen adsorption uptake at 77K, which are 6, 4, and

93
7 Å, respectively. Additionally, estimates of the maximum possible absolute adsorption can
be made by multiplying the measured micropore volume by the density of liquid krypton (28.9
mmol cm-3)22. For each material, nmax was within 30% of the estimated maximum possible
absolute adsorption.

4.2 Determination of Isosteric Enthalpy of Adsorption
The isosteric enthalpy of adsorption (ΔHads) is a commonly used metric for assessing
the strength of adsorbent-adsorbate interactions at constant coverage conditions. Here it is
evaluated via the isosteric method and reported as a positive value, qst, the isosteric heat
defined by the Clapeyron equation:
!"

!!" = −∆!!"# = −! !"

!!

Δ!!"#

(4)

The molar change in volume of the adsorbate upon adsorption (Δvads) is determined by taking
the difference between the gas-phase molar volume and the average adsorbed-phase molar
volume (the average is approximated as

!!"#
!!"#

). The isosteric heats of krypton adsorption on

ZTC, MSC-30, and CNS-201 calculated in this way are shown in Figure 3.

94

Figure 3. Isosteric heats of krypton adsorption on ZTC, MSC-30, and CNS-201.

95

5. Discussion
5.1 Isosteric Heat of Adsorption
The isosteric heats of krypton adsorption on MSC-30 and CNS-201 decrease as a
function of absolute adsorption, or equivalently surface coverage, as shown in Figure 3. This is
the typical behavior of gas adsorption on a heterogeneous surface, where binding sites are
filled according to energetic favorability. CNS-201 has significantly higher isosteric heat of
adsorption Henry’s Law (zero coverage) values due to its small average pore width (8 Å).
Krypton adsorption on ZTC, however, is accompanied by an initially increasing isosteric heat
with coverage. At 253 K the isosteric heat rises to 14.6 kJ mol-1, 1.4 kJ mol-1 above its Henry’s
Law value of 13.2 kJ mol-1 (an 11% increase). This effect has also been observed in both
ethane and methane adsorption investigations on ZTC.19,20 The increasing isosteric heat is a
result of adsorbate-adsorbate interactions promoted by the nanostructured surface of ZTC, an
effect that becomes larger at low temperatures. As temperature is increased, the effect is
severly diminished. At temperatures above 300 K no increase in the isosteric heat is observed.
This suggests that the adsorbate-adsorbate interactions responsible for the increasing isosteric
heat of adsorption have cooperative behavior that can be thermally disrupted.

5.2 Slope of Increasing Isosteric Heat of Adsorption
The slope of the increasing isosteric heat as a function of fractional coverage roughly
scales with the strength of the intermolecular interactions, as determined by fundamental
metrics such as the critical temperature (CT), boiling point (BP), and the Lennard-Jones well
depth (ε). For krypton, methane, and ethane on ZTC, the average slopes of the isosteric heat

96
up to 50% surface coverage are reported alongside the CT, BP, and ε parameters for each
gas (see Table 2).$
Table 2. Slopes of Isosteric Heats of Adsorption as a Function of Fractional Coverage on ZTC
at the Lowest Measured Temperature and Gas Properties of Krypton, Methane, and Ethane.
Slope (kJ mol-1)
Krypton
Methane
Ethane

ε (kJ mol-1)
CT (K)
BP (K)
2.7
20922
12022
1.321
2.2
19022
11222
1.221
3.3
30522
18522
1.741

The ratios of the krypton/methane and krypton/ethane slopes are 1.2 and 0.82 respectively.
These ratios are similar to the krypton/methane and krypton/ethane ratios of CT, BP, and ε.
Furthermore, the slopes of the isosteric heat are in good agreement with a simplistic model
that we have previously proposed19:
!(!)

!"

= !
!!

(5)
!(!)

The left hand side of Equation 5 (

!!

) is the slope of the isosteric heat as a function of

fractional coverage while z represents the number of nearest neighbors (posited to be 4) and ε
is the Lennard-Jones potential well depth of the gas. Using Equation 5 the slopes of the
krypton, methane and ethane isosteric heats on ZTC are estimated to be 2.6, 2.4, and 3.4 kJ
mol-1 (all within 10 percent of the average measured slope for each gas).

5.3 Isosteric Heat of Adsorption Maxima
At high coverage the isosteric heat of krypton adsorption on ZTC reaches a maximum
and decreases with further coverage. In this regime the adsorbed-phase interatomic
interactions are dominated by short-range repulsion due to the high density of adsorbates. The

97
optimal density for promoting adsorbate-adsorbate interactions is the adsorbate density at
the maximum of the isosteric heat (for a given temperature). Here we label this optimal
adsorbate density “ρΔHmax” and make comparisons to the bulk gas phase via the
compressibility factor (Z). The compressibility factor provides a good metric of the nonideality
of a gas under specific conditions.
!"

! = !"#

(6)

While an ideal gas has a compressibility factor of 1, attractive intermolecular interactions
decrease Z and repulsive interactions increase Z. For a van der Waals gas, the compressibility
factor can be recast in terms of the coefficients of the van der Waals equation of state (a and
b):

!"

! = !!!" − !"#

(7)

In this representation the minimum of the compressibility factor (where attractive interactions
are most dominant) occurs at:

!=

!! !"# !
!!

(8)

As temperature is increased at a fixed volume (V), the minimum point of the compressibility
factor shifts to a lower number of particles (n) and hence to a lower density. The actual
compressibility factor of krypton22 shows similar behavior (see Figure 4).

98

Figure 4. Compressibility factor of krypton between 253-433 K, as calculated by REFPROP22.
The minima are indicated by orange circles.
The importance of the minimum in the compressibility factor is that it represents a
critical point after which repulsive interactions begin to dominate over attractive interactions in
the gas. The density at the compressibility factor minimum (ρZmin), shown in Figure 4, can
therefore be expected to correlate with the density of the adsorbed phase at the maximum in
isosteric heat of adsorption (ρΔHmax). Low temperature values (253-273 K) of ρΔHmax (for
krypton on ZTC) were determined by dividing the absolute adsorption quantity at the isosteric
heat maximum by the ZTC micropore volume (1.66 cm3 g-1). There is reasonable agreement
between ρZmin and ρΔHmax at low temperatures (less than 12% discrepancy) (see Figure 5).

99

Figure 5. Comparison of ρZmin (squares) and ρΔHmax (triangles).
5.4 Adsorbed-Phase Enthalpy
The adsorbed-phase enthalpy (Ha) of krypton on ZTC, MSC-30, and CNS-201 was
determined as a function of coverage by adding the isosteric enthalpy of adsorption to the gasphase enthalpy (Hg) (determined by REFPROP22) (see Figure 6).

!! = !! + Δ!!"#

(9)

100

Figure 6. Adsorbed-phase enthalpy of krypton on ZTC, MSC-30, and CNS-201.

101
Due to favorable adsorbate-adsorbate interactions, the adsorbed-phase enthalpy of
krypton on ZTC decreases towards a minimum (most favorable) enthalpy with coverage.
Conversely, the adsorbed-phase enthalpy of krypton on MSC-30 and CNS-201 increases with
coverage. The adsorbed-phase enthalpy may also be determined as a function of average
interatomic distance. For example, the average interatomic distance of adsorbed krypton (xavg)
at a given state of surface coverage can be determined by dividing the micropore volume, Vmic,
(1.66 cm3 g-1 for ZTC) by the quantity of absolute adsorption (na), and taking the cube root:

!!"# =

!!"# !
!!

(10)

The adsorbed-phase enthalpy at 253 K on ZTC as a function of average interatomic
distance is comparable to the 12-6 Lennard-Jones potential between two krypton atoms21 (see
Figure 7).

Figure 7. Comparison of the adsorbed-phase enthalpy of krypton on ZTC at 253 K (red) and
the 12-6 Lennard-Jones potential of krypton (dashed blue). The inset shows both curves
translated and superimposed for easier comparison.

102
At the lowest measured temperature in this work (253 K), the magnitude and form
of the adsorbed-phase enthalpy of krypton on ZTC is remarkably similar to the 12-6 LennardJones potential. Conversely, the adsorbed-phase enthalpies of krypton on MSC-30 and CNS201 display no such behavior and are monotonically increasing functions. This provides
further evidence that the anomalous isosteric heat of adsorption of krypton on ZTC results
from enhanced interatomic interactions which can be rather accurately accounted for by the
classic 12-6 interaction potential. The apparent offset in energy seen in Figure 7 is a result of
the adsorbent-adsorbate interactions and the arbitrary nature of the enthalpy reference state (in
this case the reference state is saturated liquid krypton at its normal boiling point, 120 K). The
offset in interatomic distance (~0.07 nm) between the adsorbed-phase enthalpy and the 12-6
Lennard-Jones potential may result from clustering of the krypton atoms. While some of the
krypton atoms may temporarily cluster into more optimally spaced groupings which reproduce
the 12-6 Lennard-Jones potential shape and distance, the presence of non-clustered krypton
atoms with larger interatomic distances could shift the average interatomic spacing to higher
values resulting in the offset in interatomic distance.
The Henry’s Law values of the adsorbed-phase enthalpies are also indicative of the
atypical properties of ZTC as an adsorbent for krypton. The enthalpy of a two-dimensional
ideal gas is 2RT and therefore depends linearly on temperature with a slope of 2R (16.6 J mol-1
K-1). Correspondingly, the Henry’s Law values of the adsorbed-phase enthalpy of krypton on
MSC-30 and CNS-201 also depend linearly on temperature, with slopes of 15.6 and 16.7 J mol1

K-1, respectively (see Figure 8).

103

Figure 8. Henry’s law enthalpies of adsorbed krypton on ZTC (circles), MSC-30 (triangles) and
CNS-201 (squares) as a function of temperature. Lines are to guide the eye.
The Henry’s Law values of the adsorbed-phase enthalpy of krypton on ZTC, however,
do not vary linearly with temperature until beyond 350 K. At high temperatures (>350 K) the
slope converges to approximately 2R. At low temperatures, however, deviations between the
measured Henry’s Law values and those predicted using the ideal gas slope of 2R, are
observed. These deviations likely result from the unique structure of ZTC and may in part be
due to a loss of favorable interactions with increasing temperature.

5.5 Entropy
The isosteric entropy of adsorption (ΔSads) is the change in entropy upon adsorption: the
difference between the entropy of the adsorbed phase and the entropy of the gas phase at
isosteric conditions. At equilibrium, the isosteric entropy of adsorption and the isosteric
enthalpy of adsorption (ΔHads) are related by:
Δ!!"# =

!!!"#

(13)19

104
As for the enthalpy, the entropy of the adsorbed phase can be determined by adding the
isosteric entropy of adsorption to the entropy of krypton gas (calculated using REFPROP22).
The molar entropy of adsorbed-phase krypton as a function of coverage on the three materials
in this study is shown in Figure 9. The reference state in this case is solid krypton at absolute
zero.

105

Figure 9. The entropy of adsorbed-phase krypton on ZTC, MSC-30 and CNS-201 derived by
experiment (lines) and calculated using statistical mechanics (asterisks).

106
For comparison to the experimental data, the adsorbed-phase entropy of krypton
was also calculated using statistical mechanics (shown in Figure 9). A basic statistical mechanics
model based on a two-dimensional lattice gas was used, as described elsewhere19. Since krypton
is a monatomic gas with spherical symmetry and no internal vibrational modes, only partition
functions for the surface vibrational modes and configurational modes were considered.$The
entropies corresponding to these individual contributions were determined and summed to
obtain the total entropy of the adsorbed phase (see Figure 9).
For krypton adsorbed on MSC-30 and CNS-201, agreement between the measured
and calculated adsorbed-phase entropies is good, with discrepancies of 5% and 15%
respectively. The small pores and high isosteric heat of adsorption of krypton on CNS-201
result in less accurate statistical mechanics approximations of surface vibrational modes and
hence somewhat larger deviations than for MSC-30. Moreover, the general temperature
dependence is preserved in both cases, especially at low quantities of uptake. For krypton on
ZTC, however, discrepancies are in excess of 23% despite moderate pore sizes; the
experimental adsorbed-phase entropy is much lower than estimated values due to enhanced
interatomic interactions. It is reasonable to attribute a large fraction of this discrepancy to
clustering effects (reduced configurations of the adsorbed phase due to interatomic
interactions)40, and further investigation of such phenoma is warranted.

6. Conclusions
Equilibrium excess adsorption uptake of krypton was measured on three microporous
carbon materials: ZTC, MSC-30 and CNS-201. By fitting the data using a robust generalized
Langmuir isotherm model, absolute adsorption quantities were determined along with

107
physically realistic fitting parameters and thermodynamic quantities of adsorption. While the
isosteric heat of adsorption decreases with coverage on MSC-30 and CNS-201 (the typical
case), it increases by over 10% on ZTC, reaching its maximum at a surface coverage of 19.6
mmol g-1 at 253 K. This previously unreported effect for supercritical krypton adsorption on a
high surface area carbon results from the enhancement of favorable krypton-krypton
interactions on the ZTC surface due to its uniquely ordered porous nanostructure. Moreover,
the magnitude of the increase is dependent on the strength of the interatomic interactions of
krypton, a result that is corroborated by comparisons to ethane and methane. Additional
analysis of the isosteric heat of adsorption maxima, adsorbed-phase enthalpy, and adsorbedphase entropy provide further evidence and insight into the nature of the interactions
responsible for the anomalous surface thermodynamics reported in this paper.

Acknowledgement:
This work was sponsored as a part of EFree (Energy Frontier Research in Extreme
Environments), an Energy Frontier Research Center funded by the US Department of Energy,
Office of Science, Basic Energy Sciences under Award Number DE-SC0001057.

108

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111
Chapter 8

A Generalized Law of Corresponding States for the Physisorption of
Classical Gases with Cooperative Adsorbate-Adsorbate Interactions
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “A Generalized Law of Corresponding
States for the Physisorption of Classical Gases with Cooperative Adsorbate-Adsorbate
Interactions,” J. Phys. Chem. C, 120, 11847 (2016).
DOI: 10.1021/acs.jpcc.6b00289
Abstract
The Law of Corresponding States for classical gases is well established. Recent attempts at
developing an analogous Law of Corresponding States for gas physisorption, however, have
had limited success, in part due to the omission of relevant adsorption considerations such as
the adsorbate volume and cooperative adsorbate-adsorbate interactions. In this work, we
modify a prior Law of Corresponding States for gas physisorption to account for adsorbate
volume, and test it with experimental data and a generalized theoretical approach.
Furthermore, we account for the recently-reported cooperative adsorbate-adsorbate
interactions on the surface of zeolite-templated carbon (ZTC) with an Ising-type model, and in
doing so, show that the Law of Corresponding States for gas physisorption remains valid even
in the presence of atypically enhanced adsorbate-adsorbate interactions.

112

1. Introduction
Gas physisorption on microporous carbons has been extensively explored for a variety
of applications ranging from improved gas storage to efficient gas separation. At high
pressures, nonideal effects can significantly influence both the gaseous and adsorbed phases.
In bulk gases, the Law of Corresponding States generalizes the gas nonideality and provides a
simplified equation of state. According to this law, nonideal gases behave similarly and have
similar compressibility factors at corresponding conditions (i.e., when the conditions of the
gases relative to the critical point are equal). The Law of Corresponding States has been shown
to be highly accurate for bulk gases through a number of studies1,2.
In 2002 Quinn hypothesized an extension to the Law of Corresponding States for
gas physisorption, based on empirical evidence.3 Quinn posited that gases have approximately
equal excess adsorption uptake quantities (ne) at corresponding conditions on the same
adsorbent, which we call “Quinn’s hypothesis of corresponding states.” Quinn provided
evidence to support this hypothesis by comparing argon, carbon monoxide, methane,
nitrogen, and oxygen physisorption on four microporous carbons. Quinn found general
agreement among the corresponding excess adsorption uptake quantities, although
discrepancies often exceeded 50%. Hydrogen physisorption did not follow the trend, a fact
that was initially attributed to hydrogen being able to better penetrate the adsorbent
micropores, but later explained more satisfactorily as a quantum effect.4 Recently, others have
expanded Quinn’s hypothesis to include gate-opening MOFs and have noted deviations for
larger molecules,5 but to date this issue has not been resolved. In this work we present a novel
Law of Corresponding States for gas physisorption that includes adsorbate volume

113
considerations. It builds upon previous work but is more successful, especially for larger
molecules such as ethane.

2. Quinn’s Hypothesis of Corresponding States
Excess adsorption uptake of methane, krypton and ethane on three microporous
carbons are compared at corresponding conditions in Figure 1 using values from Ref. 6,7,8 along
with new measurements. These three carbons (ZTC, MSC-30 and CNS-201) have significant
differences in synthesis, specific surface area, and pore-size distribution, but all contain
predominantly micropores (see Supporting Information, S1). For each adsorbent,
experimentally measured isotherms of the three gases are compared at reduced temperatures
(TR) of 1.25 + 0.02 and 1.38 + 0.03 and at corresponding reduced pressures (PR), which are
defined as

!! = !

(1)

!! = !

(2)

!! = !

(3)

Here T, P and V are the system temperature, bulk phase pressure and bulk phase molar
volume, and Tc, Pc, and Vc are the critical temperature, pressure, and molar volume of the bulk
fluid.

114

Figure 1. Comparison of methane (squares), krypton (circles), and ethane (triangles) excess
adsorption at corresponding conditions as per Quinn’s hypothesis (at reduced temperatures of
1.25+0.02 (blue) and 1.38+0.03 (red)).

115
As seen in Figure 1, there is good agreement between the measured methane and
krypton isotherms using Quinn’s hypothesis of corresponding states, with only moderate
discrepancies (less than 25%). The ethane isotherms, however, deviate significantly from those
of the other gases, with discrepancies of ~50%.
It is notable that all of the adsorptive species used for empirical support of Quinn’s
hypothesis of corresponding states (argon, carbon monoxide, methane, nitrogen, and oxygen)
have similar molecular size. This is seen by comparing their 12-6 Lennard-Jones parameters, σ,
which serve as proxies for excluded-volume diameters (see Table 1). Argon, carbon monoxide,
methane, nitrogen, oxygen, and krypton all have σ values that are within 5% of their group
average, 0.3560 nm. Ethane, however, is a significantly larger molecule with a σ that is 26%
larger than the group average. Accordingly, ethane does not adhere to Quinn’s hypothesis of
corresponding states.
Table 1. Lennard-Jones σ Parameters of Relevant Gases

gas
argon
carbon monoxide
methane
nitrogen
oxygen
krypton
ethane

Lennard-Jones
σ (nm)
0.34059
0.36619
0.37379
0.36139
0.33829
0.36369
0.448010
$$

116

3. Law of Corresponding States Comparing Fractional Occupancy
In this work, we define fractional occupancy (θ) as the ratio of the absolute adsorbed
amount (na) to the maximum possible quantity of adsorption for the adsorbent at any
temperature or pressure (nmax):

!=! !

!"#

(4)

By comparing fractional occupancies instead of excess adsorption, we develop a Law of
Corresponding States for the physisorption of classical gases interacting through London
dispersion forces that better fits empirical data:
“At corresponding conditions on the same adsorbent, classical adsorbed gases have
the same fractional occupancy.”
Unfortunately, absolute adsorption and hence fractional occupancy are not easily accessible
through experimentation. Consequently we also define a proxy metric, the excess occupancy
(θe):

!! = ! !

!"#

(5)

At low gas densities, the excess occupancy well approximates the fractional occupancy, but
excess occupancies are more easily obtained by experiment. For this reason, here, we compare
excess occupancies instead of the more fundamental fractional occupancies, which must be
determined indirectly. For completeness, we also extrapolate fractional occupancies using a
fitting procedure and compare these values at corresponding conditions in the Supporting
Information (S3). The maximum adsorption quantity was estimated by multiplying the liquid

117
molar density of the adsorptive species at its triple point by the measured micropore volume
of the adsorbent (see Supporting Information S2).
The experimentally-derived excess occupancies of methane, krypton and ethane on
ZTC, MSC-30 and CNS-201 are compared at reduced temperatures of 1.25+0.02 and
1.38+0.03 (Figure 2, A-C). Using a robust fitting technique described in detail elsewhere11,
ethane excess occupancies have been extrapolated to higher reduced temperatures of
1.43+0.01, 1.80+0.02 and 2.07+0.05 and compared to experimentally measured methane and
krypton data at corresponding conditions (Figure 2, D-E).

118

A.$

B.$

C.$

119

D.

E.
Figure 2.

Comparison of excess occupancies at corresponding conditions for methane

(squares), krypton (circles), and ethane (triangles) adsorption. The data in A-C are at reduced
temperatures of 1.25+0.02 (blue) and 1.38+0.03 (red). The data in D-E are at reduced
temperatures of 1.43+0.01 (black), 1.80+0.02 (magenta), and 2.07+0.05 (red). The lines
indicate extrapolated ethane results, calculated from the fit parameters obtained by fitting
experimental data with a superposition of Langmuir isotherms.

120
Figure 2, which presents the experimentally-derived excess occupancies of methane,
krypton, and ethane at corresponding conditions, shows good correspondence between the
curves, with discrepancies of less than 25%, except for ethane at high temperatures on CNS201. This may result from a rotational hindrance of the ethane molecules within the very small
pores of CNS-201 (as small as ~0.54 nm in width). This comparison of excess occupancies
instead of excess uptake gives a significant improvement over Quinn’s hypothesis of
corresponding states where discrepancies of ~50% can be found with the same experimental
data. Likewise, the extrapolated excess occupancies of ethane on MSC-30 are in reasonable
agreement (discrepancies of less than 25%) with the experimentally derived excess occupancies
of methane and krypton at reduced temperatures of 1.43+0.01, 1.80+0.02, and 2.07+0.05. On
ZTC, deviations between the experimental and the extrapolated isotherms are larger and likely
derive from small fitting inaccuracies magnified over the huge extrapolation range (>220K).
The fitting and extrapolation procedure was not applied to ethane on CNS-201 due to an
insufficient number of available isotherms.

4. Law of Corresponding States for Physisorption
There are two fundamentally distinct approaches to understanding gas-solid
physisorption: the mono and multi-layer adsorption models developed by Langmuir, Brunauer,
Emmett, Teller and others12, and the pore-filling model developed by Euken, Polanyi,
Dubinin, and others12. Each model successfully treats relevant physisorption phenomenon
under differing conditions and both have widespread use. Here the Law of Corresponding
States for physisorption is justified in the context of each model. ,

121
4.1 Monolayer Adsorption Model,
Adsorbed molecules form a densified layer near the adsorbent surface, in a dynamic
equilibrium with the gas phase. The significant decrease in the molar entropy of the adsorptive
species upon adsorption is offset by a commensurate decrease in the molar enthalpy. At
equilibrium
∆!!"#

= ∆!!"#

(6)

Δ!!"# = !! − !!

(7)

Δ!!"# = !! − !!

(8)

where ΔHads is the isosteric enthalpy of adsorption and ΔSads is the isosteric entropy of
adsorption. It should be possible to predict the fractional occupancy of the adsorbed species at
a fixed temperature and gas pressure with knowledge of the gas-phase enthalpy (Hg) and
entropy (Sg), and knowledge of the constant-coverage, adsorbed-phase enthalpy (Ha) and
entropy (Sa) (and how they change with fractional occupancy at a fixed temperature).
The gas-phase molar entropy, Sg, (in reference to the boiling-point liquid molar
entropy, SL1), of monatomic gases with similar critical volumes is well approximated by a
function that depends only on reduced quantities, f(TR, VR) (Supporting Information, S4). We
assume that the molar entropy of the adsorbed phase (Sa) is given by the molar entropy of the
liquid phase (SL1) with the addition of a θ-dependent entropy of configurations of the
adsorbate molecules on the adsorbent, f(θ).13
!! = !!! + !(!)

(9)

122
The Sg (in reference to SL1) is approximated by f(TR, PR),

−Δ!!"# = ! ! ! , !! − !(!)!

(10)

For a monatomic gas, the right-hand side of Equation 6 (ΔSads) depends only on fractional
occupancy and reduced quantities. Although polyatomic gases have additional degrees of
freedom from internal vibrational and rotational modes, for many adsorbate-adsorbent
systems, these internal vibrational and rotational modes are only negligibly altered upon
physisorption13,14 and do not significantly contribute to ΔSads. The assumption that rotational
modes remain largely unchanged upon physisorption may break down in special
circumstances, particularly in pores small enough to inhibit rotational modes.
The left-hand side of Equation 6 depends on the isosteric enthalpy of adsorption, a proxy
metric of the physisorption binding-site energies. We first consider an idealized adsorbent with
completely homogeneous binding-site energies and no adsorbate-adsorbate interactions.
Under these assumptions, ΔHads is a constant for an ideal gas-adsorbent system, independent
of pressure, or fractional occupancy at a fixed reduced temperature. These assumptions are
later relaxed.
To begin, we assume that the isosteric enthalpy of adsorption is proportional to the
critical temperature of the adsorptive species,5 as detailed in the Supporting Information (S5,
S6).
!!!"# = !! !!

(11)

Here c1 is an undetermined (adsorbent specific) coefficient and the left-hand side of Equation
6 becomes

123
!!!"#

!! !!

(12)

By substituting Equation 1 into Equation 12
!!!"#

= !!!

(13)

Hence under the idealized assumptions above, the left-hand side of Equation 6 only depends
on reduced quantities. Since both sides of Equation 6 only depend on reduced quantities and
fractional occupancy, the fractional occupancy of distinct gases individually adsorbed on a
specific idealized adsorbent must be equal at corresponding conditions.

4.2 Heterogeneities and Adsorbate-Adsorbate Interactions
Real adsorbents typically exhibit a heterogeneous distribution of binding sites. Such a
distribution of binding-site energies leads to an isosteric heat (-ΔHads) that decreases as a
function of fractional occupancy as the most favorable sites are filled first. The distribution of
binding sites is unique to the adsorbent and depends on pore-size distribution, surface
structure, and chemical homogeneity. One may posit that each adsorbent has a characteristic
binding-site energy distribution that varies with fractional occupancy and is proportional to the
critical temperature of the adsorptive species, but is otherwise independent of the adsorptive
species at corresponding conditions (see Supporting Information, S5, S6). In this
approximation, c1 is no longer a constant in Equation 11. Instead c1 becomes a function of
fractional occupancy that is unique to each adsorbent, but independent of the adsorptive
species at corresponding conditions.
Furthermore, cooperative adsorbate-adsorbate interactions are important in some
physisorptive systems. In these systems, favorable interactions can lead to an isosteric heat that

124
6,7

increases as a function of fractional occupancy . The contribution of these interactions is
assumed to be proportional to the product of the critical temperature, Tc, and a function that
depends only on fractional occupancy, f(θ), at a fixed reduced temperature (see Supporting
Information, S7). The expression for ΔHads in Equation 11 is thus modified by adding a term
that is proportional to f(θ)Tc:
!!!"#

! (!)!! !!!!! ! ! !!

= !

(14)

Here c2 is an undetermined coefficient that is independent of the adsorptive species at
corresponding conditions. By substitution
!!!"#

! (!)!!! ! !

= !

!!

(15)

The left-hand side of Equation 6 remains approximately independent of the adsorptive
species, even upon taking into account binding-site heterogeneity and adsorbate-adsorbate
interactions (Equation 15). Consequently, Equation 6 only depends on reduced quantities and
the fractional occupancy, consistent with the proposed Law of Corresponding States for gas
physisorption.
4.3 BET Model
A simple extension of the Langmuir model to incorporate multiple layers of
adsorption was worked out by Brunauer, Emmett, and Teller and is known as BET Theory15.
This theory is widely used to determine the surface area of porous materials and gives the
fractional occupancy (θ) as a function of pressure (P), saturation pressure (Po), and a
parameter, cBET.

125

!=

!!"#
!!

!!

!!"# = !

!!

!!

(16)

!!!"#
!!
!!

!!!!"# !!!
!"

(17)

The parameter cBET depends on the heat of adsorption for the first layer, -ΔHads, the heat of
liquefaction of the adsorbate, Hl, and the temperature, T. The assumption that ΔHads is
proportional to Tc has been previously justified. We may similarly assume that Hl is
proportional to Tc (see Supporting Information, Section S10). Finally, we assume that for a
given gas, Po is proportional to Pc, allowing Equation 16 to be reduced as:
!!

!∝
!!! !

! !!!

!!

!!! ! !

!!
! !!!

(18)
!!

In this expression, c3 is the factor relating the heat of adsorption and the heat of liquefaction.
Equation 18 gives the fractional occupancy in terms of only reduced parameters, consistent
with the Law of Corresponding States for physisorption.
4.4 Pore-Filling Model
The similarity between the Law of Corresponding States and Dubinin-Polanyi theory
was recently noted by Sircar et al.5 Here we further develop this insight to show the
importance of fractional occupancy to the Law of Corresponding States for physisorption.

126
Specifically, we consider a pore-filling model of adsorption in the form of the DubininAstakhov equation:16

!=!

!"#$( ! )

(19)

In this equation, T is the temperature, P is the pressure, Po is the equilibrium vapor pressure, E
is the characteristic binding energy, and χ is an adsorbent-specific heterogeneity parameter.
! = !!!

!=

(20)

(21)

!!

The affinity coefficient (β) relates the characteristic binding energy of a sample adsorbate (E)
to that of the standard adsorbate (Eo) and depends on the ratio of their static polarizabilities, α
and αo, respectively. The adsorbate polarizability is assumed to be proportional to the critical
temperature of the adsorptive species (see Supporting Information, S5).
Equation 19 then reduces to:
(! !" ! )

!!

!"#( !
!)
!! !!

!!

! ! !"( !
!)
!!

(22)

where

!!! = !!

(23)

The undetermined constant, c4, is derived from the polarizability and characteristic binding
energy of the standard adsorbate. Both sides of Equation 22 are independent of the adsorptive

127
species at corresponding conditions, consistent with the Law of Corresponding States for
gas physisorption.

5. Anomalous Surface Thermodynamics
As we previously reported, both methane and krypton physisorption on ZTC yield
anomalous surface thermodynamics at supercritical temperatures.11,6,7 Methane and krypton
isotherms were fitted with a superposition of Langmuir isotherms to extract thermodynamic
quantities11,6,7. This yields analytically differentiable fits that are useful in determining the
absolute adsorption, and the isosteric enthalpy of adsorption.11,6,7 On ZTC, the isosteric heats
of methane and krypton adsorption increase with fractional occupancy, a property that is
attributed to enhanced adsorbate-adsorbate interactions within the uniquely nanostructured
pores.
From Equation 7, the molar enthalpy of the adsorbed phase, Ha, (as a function of
absolute adsorption) was determined by adding the isosteric enthalpy of adsorption (ΔHads) to
the gas-phase enthalpy values (Hg) at the same conditions (obtained from REFPROP17). On
ZTC, enhanced adsorbate-adsorbate interactions cause the low-temperature adsorbed-phase
enthalpies to decrease toward minimum, most favorable values, which are reached at moderate
fractional occupancies (Figure 3).

128

Figure 3. Adsorbed-phase enthalpies of krypton and methane on ZTC. Labels are
temperatures in K.
The adsorbed-phase enthalpies may also be plotted as a function of the average
intermolecular spacing in the adsorbed phase (xavg) (Figure 4). Here xavg was estimated by
dividing the micropore volume of ZTC (Vmic), as measured by the Dubinin-Radushkevich
method18,19), by the quantity of absolute adsorption (na) and taking the cube root.7

!!"# =

!!"# !
!!

(24)

129

Figure 4. Adsorbed-phase enthalpies of methane and krypton on ZTC, as a function of xavg.
Labels are temperatures in K.

As shown in Figure 4, the adsorbed-phase enthalpies (as a function of xavg) change
with temperature and resemble a 12-6 Lennard-Jones potential. The deepest “potential wells”
are observed at the lowest temperatures. The significant temperature dependence shows that
the enhanced adsorbate-adsorbate interactions are disrupted by thermal motion, and suggests

130
that they may be modeled with a cooperative interaction model such as the Ising model. The
heat capacity of the adsorbed phase at constant pressure (CP) is determined from the
adsorbed-phase enthalpy (Figure 5).

!! =

!!!
!" !

(25)

Figure 5. Constant pressure heat capacities of methane (squares) and krypton (circles)
adsorbed phases on ZTC (black) and MSC-30 (orange) at an example pressure of 2 MPa.

For both methane and krypton there is good agreement between adsorbed-phase heat
capacities at high temperatures for both ZTC and MSC-30. In the case of krypton, a
monatomic gas, the CP on ZTC at high temperatures and the CP on MSC-30 at all
temperatures are in good agreement with the CP for an ideal two-dimensional gas (16.6 J mol-1
K-1). At low temperatures, however, the CP of both gases are significantly larger on ZTC than
on MSC-30, with values that exceed even the liquid-phase heat capacities (methane~55.7 J

131
-1

-1

-1

-1 17

mol K and krypton~44.7 J mol K ) . These unexpectedly large adsorbed-phase heat
capacities may be attributed to a phase transition in the adsorbed phase on the surface of ZTC.

6. Model for Cooperative Interactions
As temperature is increased, the effects of adsorbate-adsorbate interactions decrease,
and disappear as the slopes in Figure 3 go to zero at a critical temperature (To) of ~300 K.
Figure 5 indicates the presence of a phase transition (in the adsorbed phase) around 270 K. We
assume that below To, two phases arise: a vacancy-rich phase with a low concentration of
occupied sites (α’), and an adsorbate-rich phase with a high concentration of occupied sites
(α’’) and more adsorbate-adsorbate interactions. We expect an unmixing phase diagram where
the concentration of occupied sites in the α’ and α’’ phases are temperature dependent. The
simplest phase diagram is symmetric, so these concentrations, cα’ and cα’’, are related by
!!! = 1 − !!!!

(26)

The average adsorbate-adsorbate interaction energy per adsorbed molecule (Uavg) should be
dominated by the interactions in the adsorbate-rich α’’ phase, with phase fraction Fα’’
!!!

!!

!!!! = !! !!!!!
!!!

(27)

For simplicity we consider the case when θ=0.5 and thus equal phase fractions of 0.5
for all temperatures. If we assume (as in the point approximation) that the arrangement of
molecules within the “clustered” α’’ phase is random and that each nearest-neighbor pair has
an interaction strength of ε (from the Lennard-Jones potential)9, then for θ=0.5

132
!"!!!! !

!!"# =

(28)

With the following definitions
! = 2!!!! − 1
!!!

=e
!!!

!!! !

(29)

(30)

the unmixing problem is transformed to the Bragg-Williams ordering problem with one order
parameter, L.20 By substitution

!!"# =

!"(

!!! !

(31)

In the context of this model, cooperative adsorbate-adsorbate interactions on ZTC at
temperatures below To are consistent with the Law of Corresponding States for physisorption
as explained in Section 4.2 and elaborated on in the Supporting Information, Section S7.
For experimental comparison, for each measured temperature, we determined the
difference between the minimum enthalpy (θ≈0.5) and the low coverage enthalpy (θ≈0) (in
Figure 3) as a proxy measure of Uavg. These “potential well” depths were then normalized by
!"

so they could be compared directly to (

!!! !

) in Figure 6, per Equation 31. In Figure 6,

we set To=300 K and assume z=4 (square-lattice).

133

Figure 6. Normalized “potential well” depths of methane and krypton on ZTC, compared to
calculated (

!!! !

) .

Figure 6 compares the measured adsorbate-adsorbate interaction enthalpies to thermal
trends from the Ising model. Qualitative similarities between the measured and modeled
temperature dependence support the hypothesis of cooperative adsorbate-adsorbate
interactions on ZTC. The observed deviations are not surprising for such a simple model and
uncertainty in the nmax, To and z parameters.

7. Conclusions
The principle that distinct gases have similar adsorptive fractional occupancies at
corresponding conditions on the same adsorbent is established as a novel Law of
Corresponding States for gas physisorption. This principle is tested empirically using
measurements of methane, krypton, and ethane physisorption on ZTC, MSC-30, and CNS201 at reduced temperatures of 1.25+0.02 and 1.38+0.03. Reasonable agreement is found

134
across different size gases with discrepancies of less than 25%. Accordingly this principle is
useful for estimating pure gas isotherms, though should not be considered a replacement for
experimental isotherm data. Further support is obtained from statistical mechanics wherein the
validity of this principle as a first approximation is established for a number of conditions. The
principle of corresponding states proves successful even in the presence of cooperative
adsorbate-adsorbate interactions, which were modeled using the Ising model in a low-level
approximation.

Acknowledgement:
This work was sponsored as a part of EFree (Energy Frontier Research in Extreme
Environments), an Energy Frontier Research Center funded by the US Department of Energy,
Office of Science, Basic Energy Sciences under Award Number DE-SC0001057.

135

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253K261.$
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Hill$Book$Company:$New$York,$1977.$
3.$Quinn,$ D.$ F.$ Supercritical$ Adsorption$ of$ 'Permanent'$ Gases$ under$ Corresponding$
States$on$Various$Carbons.$Carbon%2002,$40,$2767K2773.$
4.$Kim,$H.$Y.;$Lueking,$A.$D.;$Gatica,$S.$M.;$Johnson,$J.$K.;$Cole,$M.$W.$A$Corresponding$
States$ Principle$ for$ Physisorption$ and$ Deviations$ for$ Quantum$ Fluids.$ Mol.% Phys.%
2008,$106,$1579K1585.$
5.$Sircar,$ S.;$ Pramanik,$ S.;$ Li,$ J.;$ Cole,$ M.$ W.;$ Lueking,$ A.$ D.$ Corresponding$ States$
Interpretation$ of$ Adsorption$ in$ GateKOpening$ MetalKOrganic$ Framework$
Cu(dhbc)(2)(4,4$'Kbpy).$J.%Colloid%Interface%Sci.%2015,$446,$177K184.$
6.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Anomalous$ Isosteric$ Enthalpy$ of$
Adsorption$ of$ Methane$ on$ ZeoliteKTemplated$ Carbon.$ J.% Am.% Chem.% Soc.% 2013,$ 135,$
990K993.$
7.$Murialdo,$ M.;$ Stadie,$ N.$ P.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Krypton$ Adsorption$ on$ ZeoliteK
Templated$ Carbon$ and$ Anomalous$ Surface$ Thermodynamics.$ Langmuir% 2015,% 31,%
7991K7998.%
8.$Murialdo,$ M.;$ Stadie,$ N.$ P.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Observation$ and$ Investigation$ of$
Increasing$ Isosteric$ Heat$ of$ Adsorption$ of$ Ethane$ on$ ZeoliteKTemplated$ Carbon.$ J.$
Phys.%Chem.%C$2015,$119,$944K950.$
9.$Talu,$ O.;$ Myers,$ A.$ L.$ Reference$ Potentials$ for$ Adsorption$ of$ Helium,$ Argon,$
Methane,$and$Krypton$in$HighKSilica$Zeolites.$Colloid.%Surf.%A%Physicochem.%Eng.%Asp.%
2001,$187,$83K93.$
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LennardKJones$12K6$Potential.$Ind.%Eng.%Chem.%Fundam.%1966,$5,$356K363.$
11.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Unusual$ Entropy$ of$ Adsorbed$
Methane$on$ZeoliteKTemplated$Carbon.$J.%Phys.%Chem.%C%2015,$119,$26409K26421.$
12.$Dabrowski,$ A.$ Adsorption$ K$ From$ Theory$ to$ Practice.$ Adv.% Colloid% Interface% Sci.%
2001,$93,$135K224.$
13.$Campbell,$C.$T.;$Sellers,$J.$R.$V.$The$Entropies$of$Adsorbed$Molecules.$J.%Am.%Chem.%
Soc.%2012,$134,$18109K18115.$
14.$Lundqvist,$ B.$ I.$ Theoretical$ Aspects$ of$ Adsorption.$ In$ Interaction% of% Atoms% and%
Molecules% with% Solid% Surfaces;$ Bortolani,$ V.;$ March,$ N.$ H.;$ Tosi,$ M.$ P.,$ Eds.;$ Plenum$
Press:$New$York,$1990;$213K254.$
15.$Brunauer,$ S.;$ Emmett,$ P.$ H.;$ Teller,$ E.$ Adsorption$ of$ Gases$ in$ Multimolecular$
Layers.$J.%%Am.%Chem.%Soc.%1938,$60,$309K319.$
16.$Burevski,$ D.$ The$ Application$ of$ the$ DubininKAstakhov$ Equation$ to$ the$
Characterization$of$Microporous$Carbons.$Colloid%Polym.%Sci.%1982,$260,$623K627.$
17.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
23:% Reference% Fluid% Thermodynamic% and% Transport% Properties,$ version$ 8.0;$ National$
Institute$of$Standards$and$Technology:$Gaithersburg,$MD,$2007;$CDKROM.$

136
18.$Dubinin,$ M.$ M.;$ Radushkevich,$ L.$ V.$ Equation$ of$ the$ Characteristic$ Curve$ of$
Activated$Charcoal.$Proc.%Acad.%Sci.%USSR,$Phys.%Chem.%Sect.$1947,$55,$331K337.$
19.$Nguyen,$ C.;$ Do,$ D.$ D.$ The$ DubininKRadushkevich$ Equation$ and$ the$ Underlying$
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York,$2011.$

137
Chapter 9

Supporting Information for Chapter 8
M. Murialdo, N.P. Stadie, C.C. Ahn, and B. Fultz, “A Generalized Law of Corresponding
States for the Physisorption of Classical Gases with Cooperative Adsorbate-Adsorbate
Interactions,” J. Phys. Chem. C, 120, 11847 (2016).
(Supporting Information)
DOI: 10.1021/acs.jpcc.6b00289

S1. Adsorbent Materials
MSC-30
MSC-30 (Maxsorb) is a microporous superactivated carbon obtained from Kansai
Coke & Chemicals Company Ltd. (Japan). As determined by nitrogen adsorption and BET
analysis, MSC-30 has a surface area of 3244 + 28 m2 g-1. Using nitrogen adsorption data and a
slit-pore model, NLDFT pore-size analysis1 was conducted to determine the pore-size
distribution. MSC-30 has a broad range of pore sizes (from 6 to 35 Å). Over 40% of the
micropore volume is contained in pores of greater than 21 Å in width. The total micropore
volume was found to be 1.54 cm3 g-1 by the Dubinin-Radushkevich method. The skeletal
density was measured by helium pycnometry and determined to be 2.1 g cm-3. Cu Kα X-ray
diffraction of MSC-30 gave a broad peak at 2θ=34°, in accordance with that reported for AX21. The elemental composition (CHN) was determined via the Dumas method2 in combustion
experiments, indicating that 1.16 wt% of MSC-30 is hydrogen. Results from XPS experiments
on MSC-30 are summarized in Table S1.3

138
Table S1. Summary of XPS Data on MSC-30 and ZTC
peak
position
(eV)
285.0
285.7 286.4
287.3 288.1
289.4
C-C sp C-C sp C-OR C-O-C C=O COOR
component
ZTC
53.4
18.0
8.6
6.0
1.1
4.2
MSC-30
48.0
18.8
6.8
4.8
6.1
4.2

290.2
1.0
3.6

291.5
7.7
7.7

CNS-201
CNS-201 is a microporous activated carbon obtained from A. C. Carbone Inc.
(Canada). As determined by nitrogen adsorption and BET analysis, CNS-201 has a surface area
of 1095 + 8 m2 g-1. Using nitrogen adsorption data and a slit-pore model, NLDFT pore-size
analysis was conducted to determine the pore-size distribution. CNS-201 has a three dominant
pore widths of 5.4, 8.0, and 11.8 Å, containing roughly 50%, 20%, and 15% of the total
micropore volume respectively. The micropore volume was found to be 0.45 cm3 g-1 by the
Dubinin-Radushkevich method4,5. The skeletal density was measured by helium pycnometry
and determined to be 2.1 g cm-3.

Zeolite-Templated Carbon (ZTC)
ZTC is a templated carbon synthesized using zeolite NaY and a process described
elsewhere.6,7 As determined by nitrogen adsorption and BET analysis, ZTC has a surface area
of 3591 + 60 m2 g-1. Using nitrogen adsorption data and a slit-pore model, NLDFT pore-size
analysis was conducted to determine the pore-size distribution. ZTC has a narrow pore-size
distribution centered at 12 Å. Over 90% of the micropore volume is contained in pores of

139

-1

widths between 8.5 and 20$Å. The total micropore volume was found to be 1.66 cm g by
the Dubinin-Radushkevich method. The skeletal density was measured by helium pycnometry
and determined to be 1.8 g cm-3. Cu Kα X-ray diffraction of ZTC produced a sharp peak at
2θ=6°, indicative of the template periodicity of ~15 Å. The elemental composition (CHN) was
determined using the Dumas method in combustion experiments, indicating that 2.44 wt% of
MSC-30 is hydrogen. Results from XPS experiments on ZTC are summarized in Table S1.
TEM analysis has also provided evidence of the periodic structure of ZTC.3

S2. Approximation of nmax
All three materials tested are predominantly microporous. The maximum adsorption
quantity (nmax) was estimated as the product of the total micropore volume and the liquid
density of the adsorptive species at its triple point. This assumes complete micropore filling at
a maximal density given by the liquid-phase density. The micropore volumes of each adsorbent
were determined by applying the Dubinin-Radushkevich method to nitrogen adsorption
isotherms measured at 77 K.3 The liquid densities were obtained from REFPROP8. Both the
micropore volumes and the liquid densities are listed in Table S2 along with the as-determined
values for nmax.
Table S2. Liquid Molar Densities, Adsorbent Micropore Volumes,
and Estimated nmax Values
density (mol dm-3)
micropore volume (cm-3)
krypton
methane
ethane

29.2
28.1
21.7

ZTC MSC-30 CNS-201
1.66
1.54
0.45
48.5
45.0
13
46.7
43.3
13
36.0
33.4
9.8

140

S3. Comparison of Fractional Occupancies at Corresponding
Conditions
While absolute adsorption is of greater physical relevance, excess adsorption is the
experimentally-determined quantity in physisorption experiments. To obtain absolute
adsorption uptake quantities (na) from excess adsorption quantities (ne) we used Gibb’s
definition of excess adsorption
!! = !! − !!"# !!

(S1)

where ρg is the gas-phase density. We fit the unknown absolute adsorption uptake quantities
and adsorption volumes (Vads) with a superposition of Langmuir isotherms9 to obtain
!! = !!"# − !!"# !!

! !!

!! !

(S2)

!!!! !

where Ki is the equilibrium constant of the ith Langmuir isotherm, αi is the weighting factor
for the ith Langmuir isotherm (

! !! = 1), and Vmax is the maximum adsorption volume.

While the number of Langmuir isotherms (i) may be adjusted, we found that setting i=2 gives
high quality fits with a minimum number of parameters. Moreover many of the fitting
parameters hold physical significance that may be verified by comparison to independent
estimates. For i=2, the absolute adsorption is given by
!! = !!"#

1−!

!! !
!!!! !

! !

+ ! !!!

(S3)

This fitting procedure has been described in more detail elsewhere.9
Having determined absolute adsorption, we compare the fractional occupancies of
methane, krypton and ethane on ZTC and MSC-30 at corresponding reduced temperatures of

141
1.25+0.02 and 1.38+0.03 and corresponding reduced pressures (Figure S1). CNS-201 was
not considered due to an insufficient number of experimental isotherms.

Figure S1. Comparison of krypton, methane and ethane fractional occupancies on ZTC and
MSC-30 at corresponding conditions.
As with the excess occupancies, the fractional occupancies compared at corresponding
conditions are in reasonable agreement with one another. Additional errors may have been
introduced in the fitting procedure.

142

S4. Assumption that Sg (in Reference to SL1) is Well Approximated
by f(TR, PR) for Monatomic Gases with Similar Critical Volumes
According to the Trouton-Hildebrand-Everett rule, “the entropy of vaporization for
normal liquids is the same when evaporated to the same concentration”.10 Despite notable
exceptions (e.g. due to hydrogen bonding), the Trouton-Hildebrand-Everett rule is generally
accurate.11,12 The entropy of vaporization (ΔSvap) is defined as the difference between the gasphase entropy (Sg1) and the liquid-phase entropy (SL1), where the “1” subscript indicates that
the entropies are measured at the normal boiling temperature.
Δ!!"# = !!! − !!!

(S4)

For a monatomic gas at dilute conditions, additional changes to the gas-phase enthalpy due to
increasing temperature are accounted for as

∆!!!!→! = ! !!!" !

!!

(S5)

where the subscript "!!1 → 2" indicates that the entropy change takes the gas from the
normal boiling temperature, 1, to a new temperature, 2. Noting the proportionality between
the normal boiling temperature and the critical temperature for gases with similar Vc,

∆! ∝ ! !!!" ! !

(S6)

For gases with similar Vc, the gas concentration, or equivalently the molar volume (V), is
approximately proportional to VR, and the difference between Sg2 and SL1 is well
approximated as a function that depends only on TR and VR, f(TR, VR). Figure S2 compares the
gas-phase entropies (measured in reference to SL1) of three monatomic gases with similar
critical volumes (argon, 0.096 dm3 mol-1; krypton 0.12 dm3 mol-1; xenon 0.15 dm3 mol-1) and

143

-1

one monatomic gas with a different critical volume (neon, 0.051 dm mol ) at corresponding
conditions.8 The gas-phase entropies of argon, krypton, and xenon are all in good agreement
with one another, while that of neon deviates significantly due to its smaller critical volume and
quantum effects.

Figure S2. Gas-Phase molar entropies of neon (+), argon (-), krypton (*), and xenon (x) at
reduced temperatures of 1.25 (blue) and 2.07 (red), at corresponding reduced pressures.

S5. Assumption that ΔHads is Proportional to Tc
For interactions from purely London dispersion forces, ΔHads is expected to be
proportional to the static polarizability (α) of the adsorbate following London’s theory13. This
assumption is valid for small and moderately-sized classical molecules (e.g., ethane) that have a
fairly uniform charge distribution (e.g., non-polar species without a strong quadrupole
moment).
For small and moderately-sized classical molecules that interact through

144
London dispersion forces, the critical temperature is found to be proportional to the square
root of the static polarizability14. In many cases, however, the curvature is minimal and the
trend is essentially linear. This is illustrated for noble gases in Figure S3.

Figure S3. Static polarizability of noble gases15 as a function of the gas critical temperature8. A
linear fit is shown.

Therefore, a simple approximation that the critical temperature is directly proportional to the
polarizability (α) is acceptable:
! ≈ !! !!

(S7)

Here, c5 is the proportionality constant. It then follows that:
!!!"# ≈ !! !! $
(!!!"# )
!!

≈ !!

(S8)
(S9)

The validity of this approximation is investigated in Table S3, wherein -ΔHads is derived
experimentally from adsorption measurements (at θ≈0 on ZTC, MSC-30, and CNS-201). We
find that c6 is reasonably constant across different gases on the same adsorbent. As is a

145
commonly employed assumption elsewhere, the effect of temperature on the isosteric
enthalpy of adsorption is assumed to be negligible compared to its absolute magnitude.

Table S3. Empirical Values of

(!!!!"# )
!!

for Krypton,

Methane and Ethane on MSC-30, ZTC and CNS-201

krypton
methane
ethane
mean and
spread

MSC-30
0.062
0.076
0.069

ZTC
0.058
0.072
0.066

CNS-201
0.085
0.097
0.080

0.069+10%

0.065+11%

0.087+11%

The validity of Equation S9 is also investigated by comparing

(!!!!"# )
!!

!as a function of

fractional occupancy on MSC-30 and ZTC (based on the availability of high pressure isosteric
enthalpy of adsorption data at corresponding temperatures). On MSC-30, the

(!!!!"# )
!!

!values

approximately track one another as they decrease monotonically with fractional occupancy due
to binding-site heterogeneity (Figure S4). On ZTC, methane and krypton exhibit anomalous
surface thermodynamics at temperatures below To due to cooperative adsorbate-adsorbate
interactions. This leads to

(!!!!"# )
!!

!values that initially increase with fractional occupancy, but

nonetheless approximately track one another (Figure S5) due to the considerations explained
in Sections 4.2 (Chapter 8) and S7 (Chapter 9). Additional discrepancies may result from
uncertainty in the nmax parameter.

146

Figure$ S4.$ Comparison$ of$

(!!!!"# )
!!

!$ of$ methane$ (orange),$ krypton$ (purple),$ and$

ethane$(black)$on$MSCK30$at$a$reduced$temperature$of$1.25.$

Figure S5. Comparison of

(!!!!"# )

reduced temperature of 1.25.

!!

! of methane (orange) and krypton (purple) on ZTC at a

147

S6. Generalizing the Proportionality
Between ΔHads and Tc to NonIdeal Gases
The molar enthalpy of an ideal gas depends solely on the temperature. This, along with
the assumptions in Section 4.1 of Chapter 8, allows for a constant

!!!!"#

for multiple gases

individually adsorbed on a single adsorbent at a fixed reduced temperature. The enthalpy of a
nonideal gas, however, depends on pressure and volume as well as temperature, as illustrated
by the case of a van der Waals gas (HvdW)
!! !

!!"# = ! !!! ! − ! + !"

(S10)

where N is the number of gas molecules, kB is Boltzmann’s constant, T is the temperature, P is
the pressure, V is the volume of the system, and a and b are the van der Waals parameters.
Expressed as a function of TR, PR, and VR:

!!"# =

! !!! !

!! ! ! !

!!

!"

(S11)

The van der Waals critical temperature, !! =

!!"# =

!"!! !!!

!!! !

− !! ! +

!!
!"!"!!

, can also be included giving:

!! ! ! !
!"

(S12)

The gas-phase molar enthalpy (Hg) of a van der Waals gas is therefore proportional to the
critical temperature and otherwise depends only on the reduced quantities as:
!! ∝ !! ! ! ! , !! , ! !

(S13)

148
Furthermore, we assume that the adsorbed-phase molar enthalpy (Ha) is
proportional to the critical temperature, and otherwise depends only on the fractional
occupancy and reduced quantities (as empirically supported by the data in Figure 3 of Chapter
8).
!! ∝ !! ! !, ! ! , !! , ! !

(S14)

Since the isosteric enthalpy of adsorption is the difference between the gas-phase and
adsorbed-phase enthalpies at constant coverage, Tc can be factored out of both:
Δ!!"# ∝ !! ! !! ! , !! , ! ! − ! !, ! ! , !! , ! !

(S15)

The isosteric enthalpy of adsorption of a van der Waals gas is therefore proportional to its
critical temperature, and otherwise depends only on the fractional occupancy and reduced
quantities.

S7. Assumption that the Contributions of Adsorbate-Adsorbate
Interactions to ΔHads are Proportional to f(θ)Tc
Methane and krypton adsorption on ZTC exhibit anomalous surface thermodynamics
where the isosteric heats of adsorption increase with coverage.16,17 This results from adsorbateadsorbate interactions that are enhanced by the surface nanostructure. We have previously
provided both enthalpic and entropic evidence of adsorbed-phase clustering on ZTC. 9,17 In
this work we draw a connection between the adsorbed-phase clustering and the Ising model by
analyzing the temperature dependence of the clustering as an unmixing phase transition. The
applicability of this model is supported by considerations of the adsorbed-phase heat capacity
(Figure 5 of Chapter 8) and the general temperature dependence of the effect (Figure 6 of
Chapter 8). To use the Ising model, we assume the critical temperature of the unmixing phase

149
transition (To) is proportional to the Lennard-Jones interaction potential (ε) of the adsorbate,
and hence proportional to the critical temperature of the adsorbate (Tc).18$We define a reduced
phase-transition critical temperature !!! such that

!!! =

!!

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(S16)$

By the transitive property
!!! ∝ ! !

(S17)

For !!! < 1, unmixing occurs, and the adsorbate-vacancy binary solid solution separates into a
vacancy-rich α’ phase and an adsorbate-rich α’’ phase. The adsorbate concentration in each
phase, cα’ and cα’’, respectively, depends on !!! . The relative phase fraction of the α’’ phase
(Fα’’) is given by the lever rule:
!!!

!!

!!!! = !! !!!!!

(S18)

!!!

The adsorbate-adsorbate interaction energy is predominantly from the adsorbate-rich α’’ phase.
Under the assumption of random ordering (the point approximation), the average adsorbateadsorbate energy per molecule in the α’’ phase, Uα’’, is
!!!! =

!"!!!!

(S19)

where z is the coordination number. The overall adsorbate-adsorbate energy per molecule,
Uavg, in both phases is given by a weighted average of the mean adsorbate energy in each phase
and the fraction of adsorbate molecules in each phase
! ! ! !!

!!"# = !! !! !! !!!! !!! !!!

(S20)

150
where the energy of the vacancy-rich phase, Uα’ is assumed to be zero. By substitution we
arrive at a general expression to model Uavg.
!!"# =

!"!!!!

!!!!!! !!

!!!!! !!

(S21)

For a fixed reduced temperature, cα’’ is a constant. The coordination number, z, is also
assumed to be a constant. Equation S21 is thus the product of a function proportional to ε and
a function f(θ) that only varies with θ. Notably this equation displays the expected qualitative
behaviors, increasing as a function of fractional occupancy at low coverage and leveling off at
high coverage. We may make the assumption that ε is proportional to Tc,18 leaving an equation
that only varies with f(θ)Tc.

S8. Justification for the Law of Corresponding States in Bulk Gases
The Law of Corresponding States in bulk fluids is well established and can be readily
understood by examining the van der Waals equation of state:
!"

! = !!! − ! !

(S22)

Here, P is pressure, T is temperature, V is volume, R is the gas constant, and a and b are the
van der Waals parameters (unique for each gas). By substituting in the reduced temperature,
reduced pressure and reduced volume:
!! ∗ !

!∗ !! = ! ∗ ! !!
− ! ∗!

The critical quantities in terms of the van der Waals parameters (a and b)19 are

(S23)

151
!!

!! = !"!"

(S24)

!! =
!"! !
!! = 3!

(S25)

(S26)

Substituting these critical quantities into Equation S23:
!! !

!! = !! ! !! −

!!

(S27)

Equation 11 depends only on reduced quantities, which are by definition identical for all gases
at corresponding conditions, thus providing a basis for the Law of Corresponding States in
bulk fluids.

S9. Additional Data Compared with Law of Corresponding States for
Physisorption
Using experimental physisorption data published elsewhere, the Law of Corresponding
States for physisorption is illustrated on three additional adsorbents, each representing a
unique class of materials. These examples showcase the importance of the adsorbate molecular
volume. Two plots are shown for each adsorbent, the first comparing excess adsorption at
corresponding conditions and the second comparing excess occupancies at corresponding
conditions. In each case, excess occupancies at corresponding conditions have better
agreement than excess adsorption at corresponding conditions. This is especially pronounced
for adsorbates that have very different liquid molar volumes (e.g. argon and cyclohexane).
Agot Grade Artificial Nuclear Graphite20
In Figure S6, argon and nitrogen adsorption isotherms on highly pure graphite are
compared at a reduced temperature of 0.6 + 0.01. Argon and nitrogen have similar van der
Waals molar volumes of 0.03201 and 0.0387 L mol-1 respectively. Thus switching from a

152
comparison of excess adsorption to excess occupancy only minimally improves the
agreement between the corresponding isotherms.

Figure S6. Comparison of excess adsorption (top) and excess occupancy (bottom) of argon
(circles) and nitrogen (+) adsorption on Agot grade artificial nuclear graphite at corresponding
conditions (TR= 0.6 + 0.01).20

Ni(bodc)(ted)0.5 Metal-Organic Framework21
In Figure S7, argon and cyclohexane adsorption isotherms on Ni(bodc)(ted)0.5 are
compared at a reduced temperature of 0.57+ 0.01. Argon and cyclohexane have dissimilar van
der Waals molar volumes of 0.03201 and 0.1424 L mol-1 respectively. This leads to large

153
discrepancies when comparing excess adsorption that are resolved by comparing excess
occupancy.

Figure S7. Comparison excess adsorption (top) and excess occupancy (bottom) of argon
(circles) and cyclohexane (*) adsorption on [Ni(bodc)(ted)0.5

at corresponding conditions (TR

= 0.57+ 0.01).21
Zeolite NaX22
In Figure S8, carbon dioxide, ethane, and sulfur hexafluoride adsorption isotherms on
zeolite NaX are compared at a reduced temperature of 0.94 + 0.02. The van der Waals molar
volumes of carbon dioxide, ethane, and sulfur hexafluoride are 0.04267, 0.0638, and 0.08786 L
mol-1, respectively. Comparing excess occupancies rather than excess adsorption improves the

154
agreement of the isotherms. It is notable that all three gases have similar critical temperatures
of 304, 305, and 319 K respectively, and yet the differences in the van der Waals molar
volumes yield significant discrepancies when comparing excess adsorption.

Figure S8. Comparison of excess adsorption (top) and excess occupancy (bottom) of carbon
dioxide (x), ethane (triangles), and sulfur hexafluoride (squares) adsorption on zeolite NaX at
corresponding conditions (TR = 0.94 + 0.02).22

155

S10. Heat of Liquefaction vs. Critical Temperature$

Figure S9. Plotting the heat of liquefaction as a function of critical temperature for moderatelysized classical gases shows an approximately linear correlation.8

156

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8.$Lemmon,$E.$W.;$Huber,$M.$L.;$McLinden,$M.$O.$NIST%Standard%Reference%Database%
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9.$Stadie,$ N.$ P.;$ Murialdo,$ M.;$ Ahn,$ C.$ C.;$ Fultz,$ B.$ Unusual$ Entropy$ of$ Adsorbed$
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990K993.$

157
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Templated$ Carbon$ and$ Anomalous$ Surface$ Thermodynamics.$ Langmuir% 2015,% 31,%
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Experimental$Considerations.$In$Dynamical%Processes%in%Condensed%Matter;$Evans,$M.$
W.,$Ed.;$John$Wiley$and$Sons:$New$York,$1985;$293K376.$
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1159.$

158
Chapter 10

A Thermodynamic Investigation of Adsorbate-Adsorbate
Interactions of Carbon Dioxide on Nanostructured Carbons
M. Murialdo, C.C. Ahn, and B. Fultz, “A Thermodynamic Investigation of AdsorbateAdsorbate Interactions of Carbon Dioxide on Nanostructured Carbons,” AIChE J.
(submitted, 2016).

Abstract
A thermodynamic study of carbon dioxide adsorption on a zeolite-templated carbon (ZTC), a
superactivated carbon (MSC-30), and an activated carbon (CNS-201) was carried out at
temperatures from 240 to 478 K and pressures up to 5.5 MPa. Excess adsorption isotherms
were fitted with generalized Langmuir-type equations, allowing the isosteric heats of
adsorption and adsorbed-phase heat capacities to be obtained as a function of absolute
adsorption. On MSC-30, a superactivated carbon, the isosteric heat of carbon dioxide
adsorption increases with occupancy from 19 to 21 kJ mol-1, before decreasing at high loading.
This increase is attributed to attractive adsorbate-adsorbate intermolecular interactions as
evidenced by the slope and magnitude of the increase in isosteric heat and the adsorbed-phase
heat capacities. An analysis of carbon dioxide adsorption on ZTC indicates a high degree of
binding-site homogeneity. A generalized Law of Corresponding States analysis indicates lower
carbon dioxide adsorption than expected.

159

1. Introduction
Anthropogenic contributions to carbon dioxide in the atmosphere have become an
ever-increasing concern, recently highlighted by the IPCC Fifth Assessment Report1. Given
the growing nature of world energy demands and our continued reliance on fossil fuels,
carbon-capture and storage (CCS) should be considered in any plan to mitigate climate
change.2,3
Physisorptive materials are promising for a variety of gas applications because they
require only low regeneration energies and are stable through thousands of cycles.4,5 In
physisorption, solid adsorbents induce a local densification of gas at the adsorbent surface by
weak physical interactions. Physisorptive materials have been widely used for gas storage
applications. 6,7 Because different gases adsorb to a given surface with differing selectivities,
physisorptive materials are also promising for gas separations8,9. A number of studies have
shown that physisorptive materials are viable candidates for separating carbon dioxide from
mixed-gas streams.10,11,12
An adsorbent’s selectivity of one gas over another is dependent on the thermodynamic
properties of the gas-adsorbent system. As gas molecules are adsorbed, their molar entropy
decreases. This is offset by a decrease in molar enthalpy from favorable interactions between
the adsorbate and the adsorbent. The differential change in molar enthalpy at constant surface
occupancy, defined as the isosteric enthalpy of adsorption (ΔHads), is an important and readily
accessible figure of merit for physisorptive processes.13 Optimizing the isosteric enthalpy of
adsorption has been the topic of active research.14,15,16 A variety of high surface area materials
including metal-organic frameworks, covalent-organic frameworks, zeolites, zeolite-templated

160
carbons, and activated carbons have been investigated as carbon dioxide adsorbents, and
-1

have exhibited isosteric heats of adsorption of up to 90 kJ!mol .17,18,19,20,21,22,23
In general the isosteric heat of adsorption (-ΔHads) decreases as a function of surface
loading due to heterogeneity of the binding sites.24 This has the effect of limiting the pressure
range over which gas separation and storage are optimal. However, recent investigations of
nonideal gases adsorbed on carbonaceous materials at high pressures have shown that, in some
systems, strong lateral intermolecular interactions between adsorbate molecules can lead to an
isosteric heat of adsorption that increases with increasing occupancy near ambient
temperatures.25,26,27,28 This effect is realized in adsorbent materials with optimal surface
structure and a narrow distribution of binding-site energies.29,30 At high pressures, carbon
dioxide gas has strong intermolecular interactions and nonideal behavior, and is therefore a
candidate for this effect. In this paper we report that the superactivated carbon MSC-30 has an
atypical, increasing isosteric heat of carbon dioxide adsorption, and we provide evidence that
this effect derives from adsorbate-adsorbate intermolecular interactions.

2. Experimental Methods
Carbon dioxide uptake was measured on three carbonaceous materials: CNS-201,
MSC-30, and ZTC. The activated carbon CNS-201 was obtained from A.C. Carbons Canada
Inc., and MSC-30, a superactivated carbon, was obtained from Kansai Coke & Chemicals
Company Ltd. ZTC was synthesized by infiltrating zeolite NaY with furfuryl alcohol followed
by polymerization, carbonization and the ultimate dissolution of the zeolite template. This
synthesis has been described in detail elsewhere31. Each material was degassed at 520 K under

161
-3

vacuum (less than 10 Pa) prior to measurements. The skeletal densities of the materials
were determined by helium pycnometry.
Equilibrium N2 adsorption isotherms were measured at 77 K with a BELSORP-max
volumetric instrument (BEL-Japan Inc.), and surface areas were calculated using the BrunauerEmmett-Teller (BET) method32. The Dubinin-Radushkevich (DR) method33,34 was used to
calculate micropore volumes. Pore-size distributions were calculated by the non-local density
functional theory (NLDFT) method35 from high-resolution data collected on a Micromeritics
ASAP 2020, using a carbon slit-pore model and software provided by Micromeritics.
Equilibrium carbon dioxide adsorption isotherms were measured at up to nine
temperatures between 240 and 478 K using a custom Sieverts apparatus that was designed and
tested for accuracy to 10 MPa36. The sample holder was submerged in an isothermal chiller
bath for low temperature isotherms, or placed inside a cylindrical copper heat exchanger and
wrapped with insulated fiberglass heating tape for high temperature isotherms. A proportionalintegral-derivative (PID) controller was used to maintain a constant temperature during
measurements; fluctuations were less than ±0.1 K at low temperatures and no higher than
±0.4 K at high temperatures.
The Sieverts apparatus is equipped with a digital cold cathode pressure sensor (I-4

MAG, Series 423) and a molecular drag pump capable of achieving a vacuum pressure of 10

Pa. High-pressure measurements were made using an MKS Baratron (Model 833) pressure
transducer. Temperature was measured on the wall of the manifold and on the outer wall of
the sample holder using K-type thermocouples and platinum resistance thermometers.
Supercritical fluid chromatography (SFC) grade carbon dioxide (99.995%) gas was obtained
from Air Liquide America Corporation, and multiple adsorption/desorption isotherms were

162
measured to ensure reversibility and reproducibility. Gas densities were determined from the
REFPROP Standard Reference Database37.

3. Results
3.1. Nitrogen Adsorption and Helium Pycnometry
The pore-size distributions calculated by nonlocal density functional methods of CNS201, MSC-30, and ZTC are shown in Figure 1. CNS-201 was found to have distinct, extremely
narrow pore sizes of < 1.5 nm in width. MSC-30, on the other hand, has a broad pore size
distribution with pore widths ranging from 0.6 to >3 nm. ZTC has a single dominant pore
width of 1.2 nm, consistent with the inverse of the zeolite structure. The BET surface areas of
MSC-30, CNS-201, and ZTC were found to be 3244+28, 1095+8, and 3591+60 m2!g-1,
respectively. ZTC is one of the highest specific surface area carbonaceous materials reported
to date. The micropore volumes of MSC-30, CNS-201, and ZTC were determined to be 1.54,
0.45, and 1.66 mL!g-1, respectively. The theoretical maximum possible carbon dioxide
adsorption (nm) was estimated as the product of the micropore volume and the density of
liquid carbon dioxide at its triple point,30 yielding estimated nm values of 41.3, 12, and 44.5
mmol!g-1 for MSC-30, CNS-201, and ZTC, respectively. MSC-30 and CNS-201 have skeletal
densities of 2.1 g!mL-1, while ZTC has a lower skeletal density of 1.8 g!mL-1 due to a higher
hydrogen content38.

163

Figure 1. The pore-size distribution of MSC-30, CNS-201, and ZTC as calculated by the
NLDFT method.

3.2. Carbon Dioxide Adsorption
Equilibrium excess adsorption isotherms of carbon dioxide on the nanostructured
carbons are shown in Figure 2. The low temperature isotherms display maxima at pressures
between 0.5-5 MPa, as expected for Gibbs excess quantities. At all measured temperatures,
MSC-30 has the highest maximum excess adsorption quantities. At room temperature (298 K),
the maximum excess uptakes on MSC-30, CNS-201, and ZTC were 22.7, 7.25, and 17.8
mmol!g-1 respectively, corresponding to 7.00, 6.62, and 4.96 mmol!(1000 m2) -1. This
correlation of BET surface area and Gibbs surface excess uptake maximum capacity is
consistent with other similar materials, which have an average “Chahine’s-type rule” for
carbon dioxide uptake at 298 K of ~7 mmol (1000 m2) -1 39.

164

Figure 2. Equilibrium excess adsorption isotherms of carbon dioxide on MSC-30 (top), ZTC
(middle), and CNS-201 (bottom). The curves show the best simultaneous fit of the measured
data for a given material, using Equation 2 below.

165
3.3. Adsorption Data Analysis
Adsorption measurements give excess adsorption (ne), not absolute adsorption (na).40
At low pressures, excess adsorption approximates absolute adsorption. At high pressures,
however, the absolute adsorption remains a monotonically increasing quantity, but excess
adsorption does not. By utilizing a fitting equation consistent with the definition of excess
adsorption, measured data can be fitted beyond the maximum of excess adsorption. Our
carbon dioxide excess adsorption isotherms were fitted using a generalized Langmuir equation,
recently described and applied to high-pressure methane adsorption.29 This fitting technique
was adapted from that implemented by Mertens41.
The fitting procedure incorporates the Gibbs definition of excess adsorption, which is
related to absolute adsorption as follows:
!! = ! !! − ! !! !(!, !)

(1)

The gas density (ρ) is a function of pressure (P) and temperature (T) and was estimated from
the modified Benedict–Webb–Rubin equation of state37. The volume of the adsorption layer,
Va, is the only unknown preventing direct calculation of absolute adsorption. Here Va, is left
as an independent fitting parameter. The Gibbs excess adsorption was fitted using the
following generalized Langmuir equation:
!! (!, !) = !!"# − !!"# !(!, !)
!! =

!!

! !!! !"

! !! = 1

!! !
! !! !!! !

(2)
(3)
(4)

The independent fitting parameters are αi (weighting factors) and Ki (equilibrium constants)
for the i Langmuir isotherms, the scaling factor nmax, and the maximum volume of the

166
adsorption layer Vmax. The equilibrium constants (Equation 3) depend on the prefactors
(Ai) and energies (Ei) of the Arrhenius-type exponentials and R, the universal gas constant. For
a good balance between number of fitting parameters and goodness of fit, the number of
superimposed Langmuir equations used was i=2. The residual mean square values of the
resulting fits of the adsorption data for MSC-30, CNS-201, and ZTC were 0.1, 0.07, and 0.7
(mmol g-1)2 respectively.
The best-fit parameters of carbon dioxide on MSC-30, CNS-201, and ZTC are
compared to the best-fit parameters of methane25, ethane26, and krypton27 on the same
materials (using the same fitting equation) in Table 1. In many cases, the fit parameters
correlate to physical properties, and may sometimes be validated by comparison to
independent estimates. For example, the adsorbent micropore volume can be determined by
the Dubinin-Radushkevich method33 and compared to Vmax. Likewise the product of the
adsorbent micropore volume and the liquid molar density of the adsorbate37 (nm) provides an
estimate of nmax. Rigorous comparisons of the fitted and independently estimated parameters
are shown in Table 2.
Table 1. Best-fit parameters from Generalized Langmuir Fits.
!!
CO2 /ZTC
CO2 /CNS-201
CO2/MSC-30
CH4/ZTC
CH4/CNS-201
CH4/MSC-30
C2H6/ZTC
C2H6/MSC-30
Kr/ZTC
Kr/CNS-201
Kr/MSC-30

nmax%
(mmol%gD1)%

45.4
12.9
81.1
35.6
9.77
41.0
25.0
36.1
39.3
10.9
57.8

α%

Vmax%
(mL%gD1)%

A1%
(k1/2%MPaD1)%

E1%
(kJ%molD1)%

A2%
(k1/2%MPaD1)%

E2%
(kJ%molD1)%

1.64E-12
0.800
0.207
0.460
0.580
0.700
0.827
0.713
0.686
0.462
0.726

12.4
3.41
10.0
2.04
0.490
2.30
1.58
2.60
2.02
0.490
2.98

0.00143
0.0456
0.000107
0.0590
0.0610
0.0680
2.14E-07
0.0865
1.81E-06
0.00590
0.112

21.6
23.0
23.6
11.6
17.2
13.4
41.0
19.8
30
15.1
11.6

0.121
0.00244
0.0635
0.000180
0.00440
0.00460
0.0444
0.00647
0.0924
0.0689
0.00306

3.94
22.0
14.3
20.4
16.4
12.9
18.5
17.8
10.0
16.3
12.8

167
Table 2. Comparison of V max and n max Parameters to Independent Estimates
!!

Vmax%(mL%gD1)%

CO2 /ZTC$
CO2 /CNS-201$
CO2/MSC-30$
CH4/ZTC$
CH4/CNS-201$
CH4/MSC-30$
C2H6/ZTC$
C2H6/MSC-30$
Kr/ZTC$
Kr/CNS-201$
Kr/MSC-30$

micropore% volume%
(mL%gD1)%

12.4$
3.41$
10.0$
2.04$
0.490$
2.30$
1.58$
2.60$
2.02$
0.490$
2.98$

1.66
0.45
1.54
1.66
0.45
1.54
1.66
1.54
1.66
0.45
1.54

nmax%
%(mmol%gD1)%

45.4$
12.9$
81.1$
35.6$
9.77$
41.0$
25.0$
36.1$
39.3$
10.9$
57.8$

nm%
%(mmol%gD1)%

44.5
12
41.3
46.6
13
43.3
36.0
33.4
48.5
13
45.0

Although more difficult to validate through independent comparison, the Ei
parameters give important insight into the adsorbent-adsorbate binding energies of the ith
isotherm, and the prefactor Ai gives insight about the relative number of adsorption sites with
energy Ei. Specifically, the product of αi, the ith isotherm weighting factor, and Ai gives an
overall “weight” of sites with energy Ei. For carbon dioxide, methane, ethane and krypton
adsorption on MSC-30 and CNS-201, the contributions of each isotherm in the fitted
superposition of isotherms are moderately well-balanced. No isotherm accounts for less than
1% of the overall “weight”, except for carbon dioxide on MSC-30. This may result from
carbon dioxide adsorbing on a more limited set of MSC-30 adsorption sites than other gases,
as noted on single-wall carbon nanotube bundles42.
For ZTC, each generalized Langmuir fit heavily favors just a single isotherm (see Table
3). This suggests that ZTC usually has a higher degree of binding-site homogeneity than MSC30 or CNS-201, consistent with the pore-size distributions of the three carbonaceous
adsorbents. Unlike MSC-30 and CNS-201, ZTC has a single sharply peaked pore width (1.2

168
nm). In microporous carbons, the predominant contribution to binding-site heterogeneity
often results from the spectrum of pore widths. ZTC eliminates much of this heterogeneity
with its micropores of approximately a constant width.

Table 3. Normalized Relative Weights of Isotherms 1 and 2 as Determined by
Multiplying the Isotherm Weighting Value (αi ) by the Isotherm Prefactor (A i )
Isotherm 1 Weight
Isotherm 2 Weight
1.94E-14
1.00E+00
9.87E-01
1.32E-02
4.40E-04
1.00E+00
9.96E-01
3.57E-03
9.50E-01
4.96E-02
9.72E-01
2.82E-02
2.30E-05
1.00E+00
9.71E-01
2.92E-02
4.28E-05
1.00E+00
6.85E-02
9.32E-01
9.90E-01
1.02E-02

CO2 /ZTC
CO2 /CNS-201
CO2/MSC-30
CH4/ZTC
CH4/CNS-201
CH4/MSC-30
C2H6/ZTC
C2H6/MSC-30
Kr/ZTC
Kr/CNS-201
Kr/MSC-30

The Clapeyron relation was used to determine the isosteric heat of adsorption:
!!" = −∆!!"# (!! ) = −!

!"
!" !!

Δ!!"#

(5)

Here the isosteric heat of adsorption (qst) is (by convention) a positive value when adsorption is
exothermic. The coverage-dependent change in enthalpy upon adsorption is ΔΗads(na). The
change in molar volume of the adsorbate upon adsorption is Δvads.

169

4. Discussion
4.1. Isosteric Heat of Carbon Dioxide Adsorption
The isosteric heats of carbon dioxide adsorption on MSC-30, CNS-201, and ZTC
derived according to Equation 5 are shown in Figure 3. The isosteric heat of carbon dioxide
adsorption on MSC-30 differs significantly from that on a conventional activated carbon like
CNS-201 in its dependence on absolute uptake. For carbon dioxide adsorption on CNS-201,
the isosteric heat displays typical behavior.$ It is a decreasing function of occupancy. On the
other hand, the isosteric heat of carbon dioxide adsorption on MSC-30 first increases as a
function of occupancy before reaching a maximum, and then decreases at high occupancy. On
ZTC, the isosteric heat of carbon dioxide adsorption decreases very gradually with loading,
-2

especially at low temperatures where the slope is ~ -25kJ!mol . This behavior is in agreement
with the high degree of binding-site homogeneity expected from the pore-size distribution and
fit parameters for ZTC. Carbon dioxide adsorption differs from methane, ethane, and krypton
adsorption where the isosteric heats on MSC-30 decrease with occupancy, while the isosteric
heats on ZTC increase with occupancy due to enhanced adsorbate-adsorbate interactions. This
suggests that carbon dioxide adsorbate-adsorbate interactions are better optimized for the
micropore distribution of MSC-30 than ZTC.

170

Figure 3. Isosteric heats of adsorption of carbon dioxide on CNS-201, MSC-30, and ZTC as a
function of absolute uptake.

171
For carbon dioxide adsorbed on MSC-30, the total increase in isosteric heat (peak
value minus low coverage value) is more pronounced at lower temperatures, reaching a
maximum measured increase of 2.1 kJ!mol-1 at 241 K. This energy is consistent with lateral
intermolecular interactions between adsorbed carbon dioxide molecules. For example, the well
depth of the Lennard-Jones potential between two carbon dioxide molecules is ε = 1.8
kJ!mol-1 43.
At low temperatures and low occupancy, the isosteric heat of adsorption of carbon
dioxide on MSC-30 increases approximately linearly with occupancy. This linear increase is
nearly identical for the lowest temperatures measured (e.g. 107 kJ!g!mol-2 at 241 K, 111
kJ!g!mol-2 at 247 K, 113 kJ!g!mol-2 at 262 K and 106 kJ!g!mol-2 at 283 K). Similar trends
hold true for methane, ethane, and krypton adsorption on ZTC.25,26,27 The gases with stronger
intermolecular interactions have larger slopes, consistent with the hypothesis that the increases
in the isosteric heats with loading result from adsorbate-adsorbate intermolecular interactions.
The slopes of the isosteric heats with respect to fractional occupancy,

!(!!!!"# )
!"

, may be

reasonably estimated with Equation 6 (see Table 4) in the low coverage regime

!(!∆!!"# )
!"

!"

= !

(6)

where the coordination number, z, is posited to be 5 and ε is the well depth of the LennardJones 12-6 interaction potential43. This simple first approximation (Equation 6) assumes that
adsorbed molecules are randomly situated and only interact with first nearest neighbors, each
interaction having strength ε.

172
Table 4. Measured and Estimated Slopes of Increasing Isosteric Heats of Methane,
Ethane, and Krypton on ZTC, and Carbon Dioxide on MSC-30 as a Function of
Fractional Occupancy (at Low Coverage and Low Temperature)
$$

Estimated$Slope$(kJ!molK1)$

Measured$Slope$(kJ!molK1)$

Methane$

3.0$

3.2$

Krypton$

3.3$

3.4$

Ethane$

4.3$

4.0$

Carbon$Dioxide$

4.5$

4.5$

A qualitatively similar increase in the isosteric heat of carbon dioxide adsorption on
Maxsorb® (from Kansai Netsu Kagaku Co.) was previously noted by Himeno et al.44, however
it is unclear to what degree those results are quantitatively accurate. Himeno et al. used the
Toth equation to fit three experimental isotherms (at 273, 298, and 323K). This fit was then
used to calculate an isosteric heat via a reduced form of the Clausius-Clapeyron Equation
(Equation 7).
∆!
!! !

!"#$
!"

(7)

The calculated “isosteric heat” (ΔH) was determined from the partial derivative of the
logarithm of the pressure with respect to temperature at constant coverage (N). In this case,
however, excess adsorption, not absolute adsorption was held constant and what was
calculated may more aptly be called an “isoexcess heat of adsorption”. At low coverage, excess
adsorption accurately approximates absolute adsorption, but deviations arise and become
significant as the equilibrium gas-phase density increases. In our work we employ a generalized
Langmuir-type fitting function that determines absolute adsorption and gives true “isosteric

173
25,26,27,29

heats of adsorption”. We have elaborated on this methodology in prior publications

Additionally, Equation 7, which is employed by Himeno et al. but not in our work, makes two
idealized assumptions that break down for nonideal gas conditions. Equation 7 assumes that
the volume of the adsorbed phase is zero, and further assumes the validity of the ideal gas law.
These assumptions lead to significant errors under nonideal gas conditions. Futhermore, our
analysis determines the temperature dependence of the isosteric heat, which is ignored by
Himeno et al. Himeno et al. report an increasing isosteric heat for carbon dioxide adsorption
-2

on Maxsorb® with a slope of ~300 (kJ!g!mol ). We report a more moderate slope (~110
kJ!g!mol-2) on the comparable superactivated carbon MSC-30 after considering isotherms
taken at eight temperatures. This more moderate increase is more consistent with the strength
of carbon dioxide intermolecular interactions as measured by the Lennard-Jones parameter,
ε=1.8 kJ/mol43, Equation 6 and the trend in Table 3.$

4.2 Adsorbed-Phase Enthalpies and Heat Capacities
The adsorbed-phase enthalpies for carbon dioxide adsorption on MSC-30, CNS-201, and ZTC
were calculated as a function of absolute adsorption by subtracting the isosteric heats in Figure
3 from gas-phase enthalpy values determined from data tables at equivalent conditions37.
Figure 4 shows that MSC-30 has different behavior than CNS-201 or ZTC in that the
adsorbed-phase enthalpy is not a monotonically increasing function of coverage. Rather the
adsorbed-phase enthalpy on MSC-30 decreases with coverage at low temperatures and low
coverages, consistent with enhanced favorable adsorbate-adsorbate interactions.

174

Figure 4. Adsorbed-phase enthalpies of carbon dioxide on MSC-30, ZTC, and CNS-201.

175
The adsorbed-phase constant pressure heat capacities of carbon dioxide on MSC-30, CNS201, and ZTC were calculated by taking the partial derivative of the adsorbed-phase enthalpies
with respect to temperature at constant pressure. Figure 5 shows the adsorbed-phase heat
capacities at a constant sample pressure of 2 MPa.

Figure 5. Adsorbed-phase molar heat capacities of carbon dioxide on MSC-30 ("), CNS-201
(+), and ZTC (•). Values are given at a constant sample pressure of 2 MPa.

Figure 5 shows that the constant pressure adsorbed-phase heat capacities of carbon
dioxide on CNS-201 and ZTC gradually increase with temperature as expected for a
polyatomic gas. On MSC-30 a different behavior is observed where the adsorbed-phase heat
capacity rises significantly around a temperature of 250K. This is suggestive of a phase
transition and indicates that the origin of the enhanced adsorbate-adsorbate interactions may
be an adsorbed-phase clustering transition as previously noted in other systems.29,30

176
4.3 Law of Corresponding States and Selectivities
It is well established that different nonideal gases behave similarly at corresponding
conditions, at equal reduced temperatures and reduced pressures.45 The reduced temperature
and reduced pressure are defined as the ratios of the system temperature to the gas critical
temperature, and the system pressure to the gas critical pressure, respectively. We have recently
reported an extension to the Law of Corresponding States that applies to physisorbed
molecules.30 Specifically:
Classical gases adsorb to the same fractional occupancy on the same adsorbent at corresponding conditions.
Fractional occupancy (θ) is defined as the ratio of the absolute adsorption (na) to
maximum possible adsorption (nm). To facilitate comparisons in this work and others,30 we
instead compare the more accessible quantity, excess occupancy (θe), defined as the ratio of
excess adsorption (ne) to maximum possible adsorption (nm). For each adsorbent, nm was
estimated by multiplying the adsorbent micropore volume by the liquid molar density of the
adsorbate at its triple point.30,37 Carbon dioxide, ethane, methane, and krypton excess
occupancy isotherms on ZTC and MSC-30 are compared at corresponding conditions in
Figure 6.

177

Figure 6. Comparison of carbon dioxide (+), ethane (∆), methane ("), and krypton (!) excess
occupancies at corresponding conditions (with reduced temperatures given in the key).
At low temperatures, there is reasonable agreement between carbon dioxide and
ethane isotherms at corresponding conditions. At higher temperatures, however, the carbon
dioxide isotherms are significantly smaller than corresponding isotherms of the other gases.
This may be due in part to the large quadrupole moment of carbon dioxide46, which is not
present in the other gases considered. The large quadrupole moment accounts for ~50% of
the cohesive energy of solid carbon dioxide.47,48 Notably the bulk critical temperature and
isosteric heats of carbon dioxide and ethane are similar in spite of carbon dioxide’s significantly

178
lower polarizability. This is also consistent with a strong quadrupole interaction of carbon
dioxide.49,50,51 It is possible that the short-range quadrupole-induced dipole interactions
between carbon dioxide and the adsorbent are disrupted with temperature, but further
investigation is needed.
The selectivities of carbon dioxide with respect to methane, ethane and krypton were
calculated as the ratio of the Henry’s Law constants. Henry’s Law constants were calculated
directly from the excess adsorption data by extrapolating to zero coverage the logarithm of
pressure divided by excess adsorption (ln(P/ne)).52 The measured room temperature Henry’s
Law constants and selectivities are given in Table 5. Methane and krypton have similar Henry’s
Law constants, as one might expect based on the similarity of the gases, particularly their
similar critical temperatures. While carbon dioxide and ethane have nearly identical critical
temperatures, ethane has significantly higher room temperature Henry’s Law constants for
each adsorbent. This is consistent with the trends noted in Figure 6.

Table 5. Henry’s Law Constants (mmol g-1 MPa-1) and Equilibrium Adsorption
Selectivities at Room Temperature
Henry’s Law Constants
C2H6
CO2
CH4
Kr
CO2/CH4
C2H6/CO2
CO2/Kr
C2H6/CH4
C2H6/Kr

ZTC
45.7
19.7
7.03
6.80
2.80
2.32
2.90
6.50
6.72

MSC-30
51.9
19.9
8.99
9.03
2.21
2.61
2.20
5.77
5.75

CNS-201
35.1
21.4
6.99
6.57
3.06
1.64
3.26
5.02
5.34

179

5. Conclusions
The excess uptakes of carbon dioxide on MSC-30, CNS-201, and ZTC were measured
volumetrically and fitted with a generalized Langmuir-type equation. The fitted data were used
in thermodynamic analyses that show how MSC-30 exhibits an atypical, increasing isosteric
heat of carbon dioxide adsorption, while CNS-201 has the thermodynamic properties of a
conventional carbon. The isosteric heat on ZTC suggests a high degree of binding-site
homogeneity. At near-ambient temperatures the isosteric heat on MSC-30 rises with uptake
from 19 to 21 kJ!mol-1. The measured adsorbed-phase enthalpies, adsorbed-phase heat
capacities, and comparisons to studies with other nonideal gases indicate that this increasing
isosteric heat results from enhanced adsorbate-adsorbate interactions within the pores of
MSC-30.

Acknowledgements
This work was supported as part of EFree, an Energy Frontier Research Center under Award
No. DE-SC0001057. A special thanks to Christopher Gardner for his contributions in the lab.

180

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183

Conclusions

This work has focused on fundamental aspects of gas adsorption, in particular how
adsorbent nanostructured architecture affects adsorption thermodynamics. By studying a wide
variety of adsorptive species (methane, ethane, krypton, carbon dioxide, etc.) we have noted
similarities and differences in adsorptive features arising from differences in adsorbate size,
shape, polarizability, etc. Moreover the storage and/or separation of these various gases from
mixed gas streams are of critical interest to a variety of energy sectors (including transportation
and nuclear). An improved understanding of these adsorptive species under various adsorption
conditions, and any anomalies that arise in the nonideal regime have proven and could further
prove valuable to addressing energy concerns.
Of note, we discovered anomalous thermodynamics in the adsorbed phase on zeolitetemplated carbon (ZTC). At sub ambient temperatures, methane, krypton, and ethane undergo
cooperative intermolecular interactions on the surface of zeolite-templated carbon. These
lateral interactions between adsorbed molecules affect the adsorbed-phase enthalpy, entropy,
and heat capacities, in largely beneficial ways. This is unprecedented for supercritical gases
adsorbed on carbonaceous materials. The resulting increase in the isosteric heat of adsorption
improves gas storage by increasing the pressure range over which gas storage and delivery is
effective. These effects owe to the gas nonideality, which allows for favorable adsorbateadsorbate interactions that are enhanced on the surface of ZTC, evidently due to its unique
surface homogeneity and appropriate pore-size confinement. These cooperative interactions of
the adsorbed species are highly temperature dependent and may be modeled with an Ising-type
model as a first approximation. Moreover, the prospect of tuning the nanostructure of future

184
adsorbents to enhance the adsorption of other gases holds promise as an area of future
inquiry.
Neither hydrogen nor carbon dioxide registered anomalous cooperative adsorbateadsorbate interactions on ZTC. Hydrogen has a much smaller polarizability than methane,
ethane and krypton, and likewise much weaker van der Waals interactions. This likely
precludes hydrogen from exhibiting strong effects of adsorbate-adsorbate interactions under
most circumstances. Carbon dioxide is distinct from the other gases studied due to its large
quadrupole moment. This short-range interaction may disrupt the optimized adsorption that
gases like methane and krypton experience on ZTC, though further investigation is needed.
Carbon dioxide did exhibit similar (subcritical) cooperative adsorbate-adsorbate interactions on
MSC-30, suggesting that carbon dioxide adsorption is better optimized in larger pores.
Furthermore, we have developed a generalized Law of Corresponding States for gas
physisorption to highlight the similarities of distinct nonideal gases under specific adsorption
conditions and as a means of unifying various aspects of adsorption theory. We assert and
demonstrate that “distinct classical gases adsorb to the same fractional occupancy on the same
adsorbent at corresponding conditions.” This generalization accounts for the distinct excluded
volume of a particular molecule as well as its nonideal van der Waals interactions. First we
explore the validity of this statement with our own data and with literature data. Next we show
how it may be derived from multiple theories of adsorption under appropriate approximations.
We develop our theoretical basis from three distinct theories, including two that are largely at
odds with one another, the theory of layer-by-layer adsorption and the theory of micropore
filling. Moreover, by developing a model for the anomalous thermodynamics discovered on
ZTC we show that even these cooperative adsorbate-adsorbate interactions may be reasonably
subsumed in the generalized Law of Corresponding States for physisorption.

185

Future Work
Mixed Gases
Currently we have only investigated single gas isotherms. From these we can attempt
to estimate mixed-gas quantities of interest such as adsorbent selectivities. There are, however,
a number of valid reasons to investigate gas mixtures as a future project. Gases of differing
size, shape, intermolecular interactions, etc. may behave unexpectedly when adsorbed together
in micropores that are of dimensions similar to that of the adsorptive species. Larger molecules
may be sieved by the adsorbent nanostructure or lead to pore blocking. Confinement of mixed
gases may lead to unexpected diffusivities or thermodynamics within the pores. In particular, it
is unclear how having more than one adsorptive species would affect the anomalous
cooperative adsorbate-adsorbate interactions noted in the adsorbed phases on ZTC. It is
possible that adding a few highly polarizable “dopant” adsorbate molecules to the adsorbed
phase could enhance (or possibly sterically hinder) cooperative interactions. This could
perhaps be even further investigated by adsorbing azeotropic mixtures on microporous
adsorbents. However, mixed gas adsorption is also of practical interest. Natural gas is not
composed of pure methane, but contains ethane and other impurities that vary by source.
Thorough testing of methane/ethane, methane/carbon dioxide, or even krypton/xenon (for
nuclear applications) mixed gas adsorption is vital before moving forward with engineering
applications of the adsorbents.

186
Electron-Rich Materials

We are presently engaged in investigating hydrogen sorption on materials with
nonstandard electron densities such as porous covalent electron-rich organonitridic
frameworks (PECONF). These amorphous organic materials have previously been reported to
have exceptional methane and carbon dioxide thermodynamics owing to their electron-rich
phenyl groups. While still under investigation, we have presently noted a higher than expected
hydrogen uptake. Carbonaceous adsorbents are prone to only 1 weight percent of hydrogen
excess adsorption at 77K for every 500 m2 of surface area. This is a well-established and rarely
broken rule of thumb known as “Chahine’s Rule”. The specific surface area of PECONF was
determined by applying BET theory to nitrogen adsorption isotherms measured at 77K and
argon adsorption isotherms measured at 87K. The specific surface area was established as 732
m2 g-1. We have measured reproducible hydrogen uptake isotherms on PECONF at multiple
temperatures, the highest uptake occurring at a value of 1.9 weight percent at 77K and 2.3
MPa. This yields a ratio of 1.3 weight percent of hydrogen per 500 m2 g-1, in excess of
Chahine’s rule. The low-pressure isosteric heat of adsorption was found to be 8 kJ mol-1, in
agreement with the small pore size distribution for PECONF (average pore width of 5 Å).
Further investigation of this material as a hydrogen adsorbent is warranted.