Predicting the Heats of Adsorption for Gas

Physisorption from Isotherm Measurements

Peter B. Whittaker BSE, MS

This thesis is presented for the degree of Doctor of Philosophy

of The University of Western Australia

School of Mechanical and Chemical Engineering

2015

1

Abstract

This thesis consists of an introduction which includes a brief survey of literature

relevant to the topic of predicting the heats of gas physical adsorption, three chapters

that have been written as a series of papers on the topic and a conclusion which

discusses the significance of each of the chapters to the field of adsorption. In Chapter I

a new method of predicting the heats of adsorption is presented based on forming the

Tóth potential function from isotherm models. This new method requires the

measurement of only a single isotherm, in contrast to methods based on Clapeyron or

Clausius-Clapeyron which require multiple isotherms. The method is compared to

calorimetrically obtained isosteric heats for a variety of adsorbate/adsorbent systems

showing type I and type II isotherm behavior; the method of predicting the heats of

adsorption is shown to agree with the measured values to within ±10-15%. In Chapter

II the previously introduced method of predicting the heats of adsorption is compared to

predictive methods predicated on the Clausius-Clapeyron equation, assuming ideal gas

behavior and negligible adsorbed phase volume and a modified Clapeyron approach that

incorporates real gas behavior but still neglects the adsorbate specific volume. Three

adsorbate/adsorbent systems are considered for which multiple isotherms and

corresponding integral heats of adsorption have been measured. Based on comparison

with the data, the new method is found to be superior to both Clausius-Clapeyron and

the modified Clapeyron approach. In Chapter III the new method for predicting the

heats of adsorption is adapted to model the adsorbate specific heat capacity and is

compared to a model derived from Clausius-Clapeyron. Models for residual,

integrated-isosteric heat are fit to data for an argon on rutile phase titanium dioxide

system to obtain parameters for specific heat capacity modeling. The specific heat

capacity model derived from the Tóth potential function is better at predicting the trend

found in the data of decreasing heat capacity with increasing uptake than is the

Clausius-Clapeyron inspired model and predicts values for the heat capacity closer to

the data than would be found from assumptions commonly found in the literature such

as the adsorbed phase heat capacity being equal to that of a saturated liquid or an ideal

gas.

2

Table of Contents

Abstract 1

Acknowledgements 4

Publications arising from this thesis 5

Statement of Candidate Contribution 6

Introduction 8

What this thesis is about 8

A few words about terminology 10

A very brief review of literature 11

Issues unaddressed by the literature 15

Structure 17

References 18

Chapter I: Predicting Isosteric Heats for Gas Adsorption 21

Introduction 23

Theory 24

Results and Discussion 27

Conclusions 43

Acknowledgement 44

Funding 44

References 44

Supporting Information 46

Chapter II: Predicting the Integral Heat of Adsorption for Gas Physisorption

on Microporous and Mesoporous Adsorbents

49

Preface 49

Introduction 52

Theory 54

Experimental 57

Results 61

Discussion 68

Conclusions 69

Acknowledgement 69

References 70

Supporting Information 73

3

Chapter III: Predicting Adsorbed Phase Heat Capacity: A Study of Argon

on Rutile

92

Preface 92

Introduction 94

Theory 95

Results 100

Discussion 105

Conclusion 106

Acknowledgements 106

References 107

Supporting Information 110

Conclusion 119

How does this thesis contribute to the field of adsorption? 119

Possible future work 120

Appendix A: Predicting Isosteric Heats for Gas Adsorption 122

Appendix B: Predicting the Integral Heat of Adsorption for Gas

Physisorption on Microporous and Mesoporous Adsorbents

133

Appendix C: Geothermal air conditioning: typical applications using deep-warm and shallow-cool reservoirs for cooling in Perth, Western Australia

143

Appendix D: Application of geothermal absorption air-conditioning system: A case study

150

4

Acknowledgments

My family, my Mother, my Father and my Oma. Without your continual support over

the years I wouldn’t have gotten here.

My supervisors, Hui Tong Chua, Klaus Regenauer-Lieb and Dongke Zhang.

Joe Pieri, one time Dean at my alma mater Case Western Reserve University. When I

was applying to be a Ph.D. at the University of Western Australia, the admissions office

couldn’t figure out my undergraduate transcript. I contacted Joe and he wrote to UWA

to explain how it was possible for me to have screwed up so badly as an undergraduate

and to express his support for my candidature.

The friends I’ve made over the last four years. I’ve never shied away from doing things

on my own but I’m so glad I haven’t been doing this Ph.D. alone. In particular I am

thankful to Alexander Christ with whom I’ve shared so much more than just supervisors

and to Ishita Chatterjee - Li’l Fella without you I simply would have been lost; who’d

have guessed that living with a filthy-fish eating Bengali would have been so agreeable.

Head pat!

All of the girls I’ve met down at the Railway Hotel and in the Swan Basement, who

have been a frequent source of inspiration in trying times.

Kriszti – Barátom, bárcsak itt lennél!

5

Publications arising from this thesis

1 Whittaker, P. B.; Wang, X; Regenauer-Lieb, K.; Chua, H. T. Predicting Isosteric

Heats for Gas Adsorption. Phys. Chem. Chem. Phys. 2013, 15, 473-482.

Appears in this thesis as Chapter I and in as published format as appendix A.

2 Whittaker, P. B.; Wang, X.; Zimmermann, W.; Regenauer-Lieb, K.; Chua, H. T.

Predicting the Integral Heat of Adsorption for Gas Physisorption on

Microporous and Mesoporous Adsorbents. J. Phys. Chem. C 2014, 118, 8350-

8358.

Appears in this thesis as Chapter II and in as published format as appendix B.

3 Whittaker, P. B.; Wang, X.; Regenauer-Lieb, K.; Blair, D.; Chua, H. T.

Geothermal Air Conditioning: Typical Applications Using Deep-Warm and

Shallow-Cool Reservoirs for Cooling in Perth, Western Australia. Int. J. Simul.

Multisci. Des. Optim. 2014, 5, A10.

Appears in this thesis in as published format as Appendix C.

4 Wang, X.; Bierwirth, A.; Christ, A; Whittaker, P; Regenauer-Lieb, K.; Chua, H.

T. Application of Geothermal Absorption Air-Conditioning System: A Case

Study. App. Therm. Eng. 2013, 50, 71-80.

Appears in this thesis in as published format as Appendix D.

Chapter 3 has been written as manuscript for a research paper and has been submitted to

The Journal of Physical Chemistry C.

6

Statement of Candidate Contribution

Chapter I: The candidate wrote this chapter as a published research paper co-authored

by Xiaolin Wang, Klaus Regenauer-Lieb and Hui Tong Chua. The candidate himself is

responsible for writing the the manuscript, producing all figures and tables and doing all

of the data analysis however before journal submission Hui Tong Chua (in the presence

of the candidate) made several minor alterations to the grammar and word choice.

Xiaolin, Klaus and Hui Tong all saw the manuscript numerous times throughout the

writing process and made numerous suggestions over the course of roughly a dozen

revisions. The theory is based off of an observation initially made by the candidate and

then developed jointly by the candidate and Hui Tong Chua.

Chapter II: The candidate wrote this chapter as a published research paper co-authored

by Xiaolin Wang, Wolfgang Zimmermann, Klaus Regenauer-Lieb and Hui Tong Chua.

The data for this paper was collected by Xiaolin Wang and Wolfgang Zimmermann.

Xiaolin wrote a first draft of the experimental section, Wolfgang made minor

corrections and the candidate wrote the final draft based on their earlier work.

Wolfgang wrote the first draft of the experimental error analysis which appears in the

supporting information and the candidate wrote the final draft based on Wolfgang’s

draft. The remainder of the paper was written by the candidate with only minor

alterations made at the time of submission by Hui Tong Chua in the presence of the

candidate. The other co-authors saw the manuscript numerous times during its

development and made suggestions during the course of nine revisions. All data

analysis was carried out by the candidate and with the exception of figure 1, all figures

and tables were produced by the candidate. The theory follows from what is presented

in chapter I and was developed jointly between the candidate and Hui Tong Chua.

Chapter III: The candidate wrote this chapter as a manuscript to be published with Hui

Tong Chua as a co-author. All analysis, figures and the table are the work of the

candidate. The text is primarily the work of the candidate with some revisions made by

Hui Tong Chua. The theory was developed jointly by the candidate and Hui Tong

Chua.

Appendix C: The candidate wrote this appendix as a research paper co-authored by

Xiaolin Wang, Klaus Regenauer-Lieb, David Blair and Hui Tong Chua. The paper was

written entirely by the candidate with some suggestions being made by the co-authors.

7

Tables 1 and 2 were adapted from Appendix D, by the candidate. Figures 1 and 3 were

generated by the candidate while figure 2 was supplied by David Blair and modified by

the candidate.

Appendix D: This appendix was written as a paper by Xiaolin Wang, adapting a

Bachelors thesis written by Alex Bierwirth, the candidate is listed as a co-author along

with Alexander Christ, Klaus Regenauer-Lieb and Hui Tong Chua. The candidate

supervised Alex Bierwirth in the collection, interpretation and analysis of the data.

8

Introduction

What this thesis is about

The research described in this thesis deals with the physical adsorption of single

component gases on to solid adsorbents and posits a new method by which the heats of

adsorption may be predicted from isotherm measurements.

Gas physical adsorption (physisorption) is a phenomenon in which molecules in a gas

bulk phase (the adsorptive) concentrate onto a surface (the adsorbent) to form a

condensed phase (the adsorbate). In physisorption it is assumed that the adsorbent

surface undergoes no change to its state properties during adsorption. Adsorption

occurs at temperatures and pressures inconsistent with vapor-liquid equilibrium, due to

van der Waals interactions between the adsorbent and adsorbate which help to stabilize

the condensed phase. Because the adsorbate layer on the adsorbent is rarely more than a

few molecules thick (multilayer adsorption) and often does not even entirely cover the

adsorbent in a layer one molecule thick (monolayer adsorption) it is difficult if not

impossible to measure the properties of the adsorbate directly.

The specific volume of the adsorbed phase is often treated as being similar to a liquid of

the same species as the adsorbate1, 2 however liquid properties refer to a bulk phase and

so caution must be shown particularly when trying to describe adsorption at less than a

complete monolayer. On the other hand for thermodynamic calculations the adsorbate

specific volume is often treated as negligible in comparison to the adsorptive. Yet gas

adsorption principles are often used to describe high pressure systems in which the

adsorptive is not a classical gas but instead a supercritical fluid and in which case the

specific volume of the adsorptive may approach the adsorbate specific volume.

As in vapor-liquid equilibrium, gas-adsorbed phase equilibrium involves a loss of

degrees of freedom for molecules transitioning from the gas to the condensed phase

which results in a change in energy state. Because of the additional stabilizing

influence of the adsorbent the amount of heat released during monolayer adsorption is

generally greater than the heat released during vapor-liquid condensation. Both

9

experiment3 and theory4 show that for multilayer adsorption the heat released quickly

approaches the heat of vapor-liquid condensation.

In this thesis the heats of adsorption refer to the integral heat, the differential heat and

the isosteric heat (constant amount adsorbed). Paraphrasing the explanation given by

Hill5, assume a well-insulated, constant temperature bath inside of which is a box with a

partition. On one side of the partition is an ideal gas and on the other side of the

partition there is adsorbent in a vacuum. When the partition is removed and equilibrium

is established the molecules of the gas have redistributed to become adsorptive and

adsorbate, taking them to be the system and the adsorbent, walls of the box and bath to

be the surroundings, the integral heat of adsorption is equal to the internal energy

change of the surroundings which is equal to the internal energy change of the adsorbed

gas multiplied by the amount of gas adsorbed

���� = ∆�� = ��� � − �� 1.

Differentiating with respect to the amount adsorbed ns gives the differential heat

�� = �������� 2.

The isosteric heat is defined by analogy to the Clapeyron equation for vapor-liquid

equilibrium,4

��� = �(�� − ��) �������

3.

As has been previously remarked on, the adsorbed phase specific volume vs is usually

considered negligible. As the differential heat comes from a change in internal energy

and the isosteric heat from a change in enthalpy, then neglecting the adsorbed phase

volume

��� = �� + ��� = �� + ��� 4

where R is the ideal gas constant and Z is the gas phase compressibility factor.

10

A few words about terminology

To describe the specific amount adsorbed ns the terms coverage and uptake are used

interchangeably throughout the thesis, generally with the associated units: moles of

adsorbate per kilogram of adsorbent (mol kg-1). In this introduction the words

adsorptive, adsorbate and adsorbent have been used in accordance with IUPAC

recommendations6 however chapters one and two were written and published before the

author had come across the IUPAC nomenclature recommendations. Instead of

adsorptive, phrases like “unadsorbed adsorbate” and “gas phase adsorbate” are used.

Since the chapters in question passed peer review without comment, it is assumed that

the meaning is clear from context.

11

A very brief review of literature

The work of predicting the heats of adsorption begins with Michael Polanyi7, 8 the father

of potential theory. According to this theory a stabilizing potential field exists near the

adsorbent surface such that the adsorbate can form as a condensed phase at temperatures

and pressures which are not vapor-liquid saturation conditions for the bulk phase

adsorptive. Although the theory does not explain why this potential field exists, it is

conceptually instructive. Key among Polanyi’s observations was that the adsorption

potential could be related to the vapor pressure of the adsorbate

� = �� ln ��∗" 5

in which ε is the adsorption potential, R is the universal gas constant, T and p are the

adsorption equilibrium temperature and pressure respectively and p* is the vapor

pressure of the adsorbate at T. Later authors8 approximated the adsorbate as a liquid

and replaced p* with psat, the saturated vapor pressure. In this modified form the

relationship would go on to inspire both Tóth as well as Dubinin and Astakhov in the

formulation of their eponymous adsorption isotherm models.

Working at the same time as Polanyi, Irving Langmuir9 derived the most fundamental

adsorption isotherm model. Langmuir considered a crystalline solid with fixed

adsorption sites on to which gas molecules would collide. In the conceptual model gas

molecules that collide with the crystal at points where there are available adsorption

sites collide inelestically and condense, on the other hand collisions with filled sites

result in the gas molecules rebounding (or evaporating) from the surface. In such a

model the overall rate of condensation is proportional not only to the rate of collision

(µ) and capture (α) but also to the fraction of available adsorption sites (θ) and the rate

of evaporation is proportional to both the rate of evaporation if all sites are filled (ν) and

the fraction of filled sites (θ1). Equating these two quantities for equilibrium conditions

and noting that θ + θ1 =1 results in the Langmuir isotherm model

12

#$ = %&' + %& 6.

The rate of molecules colliding per unit area of the crystal (µ) can be related to the

pressure of the gas. Though Langmuir derived his model from kinetic considerations,

the same result can be arrived at through thermodynamic considerations.10, 11 The

Langmuir isotherm remains widely used as it can describe data even in cases where the

assumption of fixed adsorption sites is inconsistent with real world adsorbate

distribution and mobility.12 Because of Langmuir’s assumptions about the nature of

adsorption (fixed adsorbent sites on a uniform surface with no interaction between

adsorbate molecules) the heat of adsorption is constant in Lagmuir’s model and this is

generally consistent with what has been observed experimentally for systems that can be

described by the Langmuir model (although chapter I contains an exception to this

observation, sulfur hexafluoride on NaX zeolite).

In their statistical mechanics treatment of adsorption Fowler and Guggenheim10

introduced two concepts that are rarely discussed explicitly in the literature but which

are of great importance to measuring adsorption and discussing the thermodynamics of

adsorption. The first concept is one of excess adsorption, that is, the amount of

adsorption should be regarded as the number of adsorbed molecules at the interfacial

boundary in excess of the number of molecules that would be in the same volume if the

adsorbent were not present. The second concept popularized by Fowler and

Guggenheim is the assumption that in physisorption the adsorbent remains completely

inert allowing all changes in thermodynamic state of the adsorptive-adsorbate-adsorbent

system to be ascribed to the adsorptive-adsorbate phase transformation.5, 10

From its formulation the Langmuir isotherm is limited to the description of monolayer

adsorption. Brunauer, Emmett and Teller (BET) used similar reasoning to Langmuir

but considered that gas molecules upon hitting a patch of adsorbent surface that already

contained adsorbate might form additional layers rather than immediately evaporating.

From which they developed the relationship

��(��(� − �) = 1

�* + + − 1�*+

���(�

7

13

in which v is the volume of the adsorbed phase, vm is the volume of an adsorbed

monolayer and c was postulated to be related to the difference in enthalpies between

adsorptive-adsorbate monolayer adsorption and vapor-liquid condensation.4 Later

authors have suggested that c is better regarded as merely a fitting constant.13 Though it

is often the case that other isotherm models will fit measured data better than the BET

equation, the equation remains in common use as a means of determining adsorbent

surface area (sometimes called BET surface area) based on finding the BET monolayer

capacity for nitrogen adsorbed at 77K and assuming that close packing of the adsorbed

molecules occurs in the monolayer. This method of determining the surface area is

believed to be accurate to within about 20%.13,14 BET also lent their name to an

isotherm classification system15 which is still commonly referenced although it has

since been subsumed into the IUPAC classification system.13

From the mid nineteen forties to the early fifties, Terrell L. Hill published a series of

papers in the Journal of Chemical Physics which followed Fowler and Guggenheim in

applying statistical mechanics to adsorption but addressed more complex systems than

had previously been dealt with in this manner such as rederiving the BET equation and

generalizing it for multicomponent adsorption,16 calculating that the fixed adsorption

sites, assumed by Langmuir and BET, in fact exist only for adsorption at very low

temperature17 and most pertinent to this thesis, laying out the thermodynamics of

adsorption as pertains to the heats of adsorption.5 This last mentioned paper explains

why experimentalists trying to measure the heat of adsorption obtain different

calorimetric values for similar working pairs due to differences in the design of the

calorimeters and how these different heats are related and can be reconciled.

Starting from observations about how Polanyi’s adsorption potential field could be

related to the differential heat of adsorption for Langmuir systems,2 Jozsef Tóth applied

graphical differentiation to measured isotherm slopes to develop a correction factor to

the Polanyi potential function which gave an improved prediction for the differential

heat of adsorption on heterogeneous adsorbents.18 From this observation Tóth later

drew inspiration for forming his own isotherm model as a modification of the Langmuir

model but with an additional term m which would vary the slope of the isotherm to

account for heterogeneity in the adsorbent19

�� = �,� �(- + �*)$ *⁄

8

14

where b is a fitting parameter which can be linked to the kinetics of adsorption and �,�

is the theoretical maximum uptake capacity of the adsorbate/adsorbent pair. Tóth’s

isotherm remains one of the most commonly used models for monolayer gas adsorption,

see Valenzuela and Myers20 for instance. However, to the best of the author’s

knowledge, Tóth did not take his observations about the relationship between adsorption

heats and isotherm shapes further to develop a model for predicting the heats of

adsorption. Because it is a modification of the Langmuir isotherm, if a Clapeyron type

equation is applied to the Tóth isotherm model only a single value (regarded as the

average value) is obtained as a prediction of the isosteric heat, which is contrary to what

is expected for adsorption on a heterogeneous adsorbent.

Inspired by the characteristic curves found in Polanyi’s potential theory, Mikhail M.

Dubinin and co-workers developed the theory of volume filling of micropores.1,21

Several isotherms were developed from this theory,1 the most notable of which is the

Dubinin-Astakhov (D-A) equation22

�� = /0�� exp 4− 5��∆6 ln 7��(�

� 89*

: 9

here W0 is the specific volume of the adsorbent micropores, ∆E is the characteristic

energy of adsorption and m is a parameter found through fitting. This model describes

adsorption on heterogeneous, microporous adsorbents – which are very common in

engineering and industrial applications. Unlike the Tóth model, when a Clapeyron type

equation is applied to the D-A model the result is a prediction of the isosteric heat which

varies with coverage; this has made the model popular with experimentalists.23-27

15

Issues unaddressed by the literature

The standard approach to predicting the heats of adsorption is to use the Clapeyron or

Clausius-Clapeyron equation to determine the isosteric heat from which the other heats

are easily calculated. This method can be applied directly to isotherm data28 or to an

isotherm model if it fits the data in question.23-27 However this procedure is not as

straight forward as it sounds and has drawn frequent criticism for being inaccurate or

hard to apply.29-33 Moreover, the use of Clapeyron type equations requires adsorption

isotherm data at multiple temperatures in order to make a statement about the pressure-

temperature gradient however it is sometimes the case, when working with data from

literature, that data from only a single isotherm is available.

Chapter I of this thesis addresses the problem of how to predict the heats of adsorption

when only a single isotherm is available by making use of Tóth’s correction factor to

the Polanyi potential function. By picking a suitable isotherm model and then

performing the same differentiation on the isotherm model that Tóth previously had

performed graphically on isotherm data, it is possible to form new models for the heats

of adsorption. By fitting an isotherm model to data and then using the values obtained

for the fitting parameters in the corresponding heat model, a prediction is made about

the heat of adsorption. The predictions made in chapter 1 are compared against

calorimetric data from literature with the result that the predictions are generally within

± 15-20% of the calorimetric values.

Chapter II compares a heat model derived from Tóth’s correction to the Polanyi

potential function to models derived from Clapeyron type equations – namely Clausius-

Clapeyron which considers the adsorbate volume to be negligible and the adsorptive to

act as an ideal gas and a modified Clapeyron equation which uses a real gas equation of

state for the adsorptive but still disregards the adsorbate volume. The predictions of the

various models are judged against calorimetric data with the verdict that the predictions

made by the Tóth’s corrected potential function model and the modified Clapeyron

equation are a substantial improvement over the Clausius-Clapeyron model. Over the

three working pairs considered, the predictions made by the Tóth’s corrected potential

model are more consistent than the predictions made with the modified Clapeyron

equation model and on this basis the former is considered to be superior.

16

Chapter III examines heat capacity of the adsorbed phase by adapting a Tóth corrected

potential function derived model for the isosteric heat of adsorption to form a new

model for the heat capacity of the adsorbate. No attempt is made to predict the heat

capacity because no data set could be found that included both multiple isotherms (for

establishing the uptake-temperature gradient) and adsorbate heat capacity data. Instead,

the hope is that fitting the model directly to the heat capacity data will give some insight

into the temperature dependent behavior of the various terms in the underlying heat

model as well as further validating the potential function method approach to modeling

the heats of adsorption.

17

Structure

This thesis is structured as a series of papers, which is an option within the regulations

of the University of Western Australia for candidates for the Degree of Doctor of

Philosophy. Chapters one and two have been previously published in peer reviewed

journals (Physical Chemistry Chemical Physics and The Journal of Physical Chemistry

C respectively). Chapter three has been written as a “to be published” manuscript and

has been submitted for publication to The Journal of Physical Chemistry C. To increase

continuity a few paragraphs of preface have been added to chapters two and three.

Because the thesis is structured as a series of papers, each chapter is referenced

individually. The series of papers are summed up in the conclusion to the thesis with a

discussion of their significance and possible future work.

In the appendix, chapters one and two are reproduced as they appear in the journals in

which they were published as appendices A and B respectively. Appendices C and D

are also published papers, one of which the candidate is first author on, but they did not

fit the overall theme of this thesis.

18

References

1 Dubinin, M. M. Physical Adsorption of Gases and Vapors in Micropores;

chapter in Progress in Surface and Membrane Science vol. 9, Academic Press:

New York, 1975.

2 Halász, I.; Schay, G.; Szőnyi, S. Inferences from the Analogy Between

Adsorption and Condensation of Vapours. Acta Chim. Acad. Sci. Hung. 1955, 8,

143-156.

3 Drain, L. E.; Morrison, J. A. Thermodynamic Properties of Argon Adsorbed on

Rutile. Trans. Faraday Soc. 1952, 48, 840-847.

4 Brunauer, S.; Emmett, P. H.; Teller, E. Adsorption of Gases in Multimolecular

Layers. J. Am. Chem. Soc. 1938, 60, 309-319.

5 Hill, T. L. Statistical Mechanics of Adsorption. V. Thermodynamics and Heat of

Adsorption. J. Chem. Phys. 1949, 17, 520-535.

6 Everett, D. H. Manual of Symbols and Terminology for Physicochemical

Quantities and Units, Appendix II Definitions, Terminology and Symbols in

Colloid and Surface Chemistry Part I.; Butterworths: London, 1972.

7 Polanyi, M. Über die Adsorption vom Standpunkt des dritten Wärmesatzes.

Verh. Dtsch. Phys. Ges. 1914, 16, 1012-1016

8 Polanyi, M. Adsorption von Gasen (Dämpfen) durch ein festes nichtflüchtiges

Adsorbens. Verh. Dtsch. Phys. Ges. 1916, 18, 55-80.

9 Langmuir, I. The Constitution and Fundamental Properties of Solids and

Liquids. J. Am. Chem. Soc. 1916, 38, 2221-2295.

10 Fowler, R.; Guggenheim, E. A. Statistical Thermodynamics, 2nd Impression,

University Press: Cambridge, 1960.

11 Hill, T.L. Statistical Mechanics, McGraw-Hill: New York, 1956.

12 Savara, A.; Schmidt, C. M.; Geiger, F. M.; Weitz, E. Adsorption Entropies and

Enthalpies and Their Implications for Adsorbate Dynamics. J. Phys. Chem. C

2009, 113, 2806-2815.

13 Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.;

Rouquérol, J.; Siemieniewska, T. Reporting Pyhsisorption Data for Gas/Solid

Systems with Special Reference to the Determination of Surface Area and

Porosity. Pure Appl. Chem. 1985, 57, 603-619.

19

14 Rouquérol, J.; Avnir, D.; Fairbridge, C. W.; Everett, D. H.; Haynes, J. H.;

Pernicone, N.; Ramsay, J. D. F.; Sing, K. S. W.; Unger, K. K. Recommendations

for the Charicterization of Porous Solids. Pure Appl. Chem. 1994, 66, 1739-

1758.

15 Brunauer, S.; Deming, L.; Deming, E.; Teller, E. On a Theory of the van der

Waals Adsorption of Gases. J. Am. Chem. Soc. 1940, 62, 1723-1732.

16 Hill, T. L. Statistical Mechanics of Multilayer Adsorption. I. J. Chem. Phys.

1946, 14, 263-267.

17 Hill, T. L. Statistical Mechanics of Multilayer Adsorption II. Localized and

Mobile Adsorption and Absorption. J. Chem. Phys. 1946, 14, 441-453.

18 Tóth, J. Gas-(Dampf-) Adsorption An Festen Oberflächen Inhomogener

Aktivität, I. Acta Chim. Acad. Sci. Hung. 1962, 30, 415-430.

19 Tóth, J. State Equations of the Solid-Gas Interface Layers. Acta Chim. Acad. Sci.

Hung. 1971, 69, 311-328.

20 Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook;

Prentice Hall: Englewood Cliffs, 1989.

21 Dubinin, M. M. The Potential Theory of Adsorption of Gases and Vapors for

Adsorbents with Energetically Nonuniform Surfaces. Chem. Rev. 1960, 60, 235-

241.

22 Dubinin, M. M.; Astakhov, V. A. Development of the Concepts of Volume

Filling of Micropores in the Adsorption of Gases and Vapors by Microporous

Adsorbents. Bull Acad. Sci. U.S.S.R. 1971, 20, 3-7.

23 Ozawa, S.; Kusumi, S.; Ogino, Y. Physical Adsorption of Gases at High

Pressure IV. An Improvement of the Dubinin-Astakhov Adsorption Equation. J.

Colloid Interface Sci. 1976, 56, 83-91.

24 Saha, B. B.; Koyama, S.; El-Sharkawy, I. I.; Habib, K.; Srinivasan, K.; Dutta, P.

Evaluation of Adsorption Parameters and Heats of Adsorption Through

Desorption Measurements. J. Chem. Eng. Data 2007, 52, 2419-2424.

25 Wang, X.; French, J.; Srinivasan, K.; Chua, H. T. Adsorption Measurements of

Methane on Activated Carbon in the Temperature Range (281 to 343) K and

Pressures to 1.2 MPa. J. Chem. Eng. Data 2010, 55, 2700-2706.

26 El-Sharkawy, I. I.; Saha, B. B.; Koyama, S.; Srinivasan, K. Isosteric Heats of

Adsorption Extracted from Experiments of Ethanol and HFC 134a on Carbon

Based Adsorbents. Int. J. Heat Mass Tran. 2007, 50, 902-907.

20

27 Akkimaradi, B. S.; Prasar, M.; Dutta, P.; Saha, B. B.; Srinivasan, K. Adsorption

of Nitrogen on Activated Carbon-Refit of Experimental Data and Derivation of

Properties Required for Design of Equipment. J. Chem. Eng. Data 2009, 54,

2291-2295.

28 Dutta, B. K. Principals of Mass Transfer and Separation Processes; Prentice-

Hall of India Private Limited: New Dheli, 2007.

29 Herdes, C.; Ferreiro-Rangel, C. A.; Düren, T. Predicting Neopentane Isosteric

Enthalpy of Adsorption at Zero Coverage in MCM-41. Langmuir 2011, 27,

6738-6743.

30 Zimmermann, W.; Keller, J. U. A New Calorimeter for Simultaneous

Measurement of Isotherms and Heats of Adsorption. Thermochim. Acta. 2003,

31-41.

31 Siperstein, F.; Gorte, R. J.; Myers, A. L. A New Calorimeter for Simultaneous

Measurements of Loading and Heats of Adsorption from Gaseous Mixtures.

Langmuir 1999, 15, 1570-1576.

32 Birkett, G. R.; Do, D. D. Correct Procedures for the Calculation of Heats of

Adsorption for Heterogeneous Adsorbents from Molecular Simulation.

Langmuir 2006, 22, 9976-9981.

33 Pan, H.; Ritter, J. A.; Balbuena, P. B. Examination of the Approximatations

Used in Determining the Isosteric Heats of Adsorption from the Clausius-

Clapeyron Equation. Langmuir 1998, 14, 6323-6327.

21

Chapter I

This chapter was published as

Whittaker, P. B.; Wang, X; Regenauer-Lieb, K.; Chua, H. T. Predicting Isosteric

Heats for Gas Adsorption. Phys. Chem. Chem. Phys. 2013, 15, 473-482.

and is included in as published format in Appendix A.

22

Predicting isosteric heats for gas adsorption

Peter B. Whittakera , Xiaolin Wang

b, Klaus Regenauer-Lieb

c and Hui Tong Chua

a

a School of Mechanical and Chemical Engineering, The University of Western

Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

b School of Engineering, The University of Tasmania, Hobart TAS 7001, Australia

c School of Earth and Environment, The University of Western Australia, 35 Stirling

Highway, Crawley, WA 6009, Australia

Corresponding Author

*Email: huitong.chua@uwa.edu.au

*Fax: 61-8-6488-1024

*Phone: 61-8-6488-1828

Other Authors

peter.whittaker@uwa.edu.au, xlwang@mech.uwa.edu.au, klaus.regenauer-

lieb@uwa.edu.au

23

A method of predicting the isosteric heat of gas adsorption on solid materials is

developed which requires the measurement of a single isotherm – where previous

methods, such as the Clausius-Clapeyron approach, required measuring either multiple

isotherms or complex calorimetric measurement. The Tóth potential function,

stemming from the Polanyi potential function, is evaluated using the Langmuir and Tóth

isotherm equations to generate new equations for the isosteric heat. These new isosteric

heat equations share common parameters with the isotherm equations and are

determined from isotherm fitting. This method is demonstrated on literature data for

gas adsorption onto solid adsorbates including zeolites of various surface charge

character and non-porous rutile phase titanium dioxide. Predictions are made using the

new isosteric heat equations and then compared to calorimetric data.

Keywords: adsorption, isotherm, isosteric heat, prediction

1. INTRODUCTION

There are two established methods for studying heats of adsorption. The first is an

indirect method by taking adsorption isotherm data at different temperatures,

extrapolating points of constant coverage ns from the isotherms and either

approximating the differential of Clapeyron’s equation by a finite difference

ratio or by graphing against and using the slope in the Clausius-

Clapeyron equation. The second method of studying heats of adsorption is by direct

measurement using adsorption calorimetry.

Various authors have mentioned the difficulty in getting reliable results from the

indirect method [1-3] and this method has been described as being “prone to large

calculation errors” [4]. Furthermore the common use of the Clausius-Clapeyron

equation is invalid at higher pressures because it both approximates the gas phase as

ideal and ignores the volume of the adsorbed phase; Pan et al. have shown that the error

caused by these assumptions can be on the order of 10% at 1 MPa and 300 K [5]. On

the other hand calorimetric determinations require a purpose built measurement system

[3, 6, 7], adding complexity to the research.

Recent works have shown molecular simulation to be a tool for predicting the isosteric

heat of adsorption [1, 4]. However molecular simulation still requires experimental data

for model calibration and verification – leading back to the two methods already

mentioned.

sn

dTdp

sn

Tp ∆∆ ( )pln T1

24

We have studied adsorption on zeolite adsorbents for a variety of gases exhibiting type-I

isotherm behavior as well as for neon and argon on rutile phase titanium dioxide which

exhibit type-II behavior. In this paper we will demonstrate a simple method for

predicting the isosteric heat of adsorption for based only on isotherm data, which can

easily be obtained through gravimetric or volumetric measurement methods [8, 9].

The method developed herein comes from potential theory. We have used the

Langmuir and Tóth isotherm equations in the development of this method; these

equations are simple and consistent with potential theory and can describe the type-I

isotherms we have analyzed. The resultant isosteric heat equations are also simple and

appropriate for engineering calculations. We have not used the Dubinin-Astakhov (D-

A) isotherm equation here because, while it is consistent with potential theory and often

used for microporous adsorbents, the data used consisted of a single isotherm for each

adsorbate-adsorbent pairing. This precluded the use of the four parameter form of the

D-A equation which has a parameter found through fitting multiple isotherms at

different temperatures. A three parameter version of the D-A equation was considered

but found not to fit the isotherm data as well as the Langmuir and Tóth equations.

For those isotherms that could not be described by either the Langmuir or Tóth models,

we tried several other methods (described in the supporting material) but found that

fitting the isotherms with a polynomial was most successful. Although this method of

isotherm fitting is not related to potential theory we were able to adapt our method to

allow reasonable predictions of the isosteric heat of adsorption even when an isotherm

model was unavailable.

2. THEORY

Polanyi [10, 11] observed an adsorption potential for monolayer adsorption on

homogeneous adsorbents expressed as

(1),

where R is the universal gas constant, T is the equilibrium temperature, p the

equilibrium pressure and p* is the vapor pressure of the adsorbed adsorbate.

Halász et al. [12] assumed that on the surface of the adsorbent, the adsorbate acted as

a liquid and that p* could be replaced by psat, the saturation pressure of the adsorbate at

the equilibrium temperature. With this assumption they showed that the Polanyi

potential function could be approximately related to the differential heat of adsorption

on a homogenous adsorbent by

=

p

pRT

*

lnε

25

λε +≈dq (2),

where qd is the differential heat of adsorption per mole adsorbate and λ the latent heat of

vaporization.

Using graphical methods to analyze isotherm curves Tóth [13] developed a correction

factor to the Polanyi potential to account for adsorbent heterogeneity, to wit

(3),

where ns denotes the amount of adsorbate, adsorbed on to the adsorbent, per unit

adsorbent. For monolayer coverage equation 3 can be related to the Gibb’s spreading

pressure [14].

Tóth’s corrected adsorption potential is expressed as

(4),

and the differential heat of adsorption on a heterogeneous adsorbent is accordingly

approximated as:

λλ +∆≈dq (5).

Various sources [15-17] give the derivation for both the differential heat and the

isosteric heat of adsorption. When the gas phase adsorbate can be approximated as an

ideal gas, the isosteric heat of adsorption differs from the differential heat by the amount

RT. For high pressure adsorption, ZRT is used instead where Z is the compressibility

factor. The general form of the isosteric heat equation for low to moderate pressures is

then typically approximated as

RTqst ++∆≈ λλ (6),

The λ in equations 2, 5 and 6 has long been regarded as the latent heat of evaporation

at the isotherm temperature. By observing a large body of data, we posit that the latent

heat should instead be a function of pressure and therefore,

RTq pst ++∆≈ λλ (7).

We will demonstrate the validity of this in the results section by comparison with direct

calorimetric measurements performed by Dunne and co-workers [18, 19], Sircar et al.

[20], Siperstein et al [21], Tykodi et al. [22] and Drain and Morrison [23].

For some adsorbates, common adsorption equilibrium pressures are below the triple

point pressure of the species; in making predictions of the isosteric heat below the triple

point pressure we have used the latent heat of vapor-liquid equilibrium at the triple point

as a constant value rather than the variable latent heat of vapor-solid equilibrium, as we

1d

d−=

T

s

s

n

p

p

nψ

=∆

p

pRT

satψλ ln

26

neither expect a sudden change in the surface force field nor is such a change observed

in the data [18-23].

The saturation pressure in equation 4 is at the isotherm temperature. For isotherms

above the adsorbate critical point temperature a pseudo saturation pressure as suggested

by Ozawa [26] has been used: . Here psat is the pseudo-saturation

pressure, pc is the critical point pressure, Tc is the critical point temperature and T is the

isotherm temperature.

Equation 7 is the general form of the isosteric heat equation; by applying equations 3,

4 and 7 to the Langmuir and Tóth isotherm equations we have developed corresponding

isosteric heat equations specific to those isotherms – this is shown in table 1.

Table 1 Isotherm equations and corresponding new equations for isosteric heat.

Isotherm equations relate adsorbent coverage sn to pressure p using modeling

parameters s

mn or sn∞ , b and m. The isosteric heat equations relate isosteric

heat of adsorption stq to dimensionless coverage ssnn ∞=θ , using saturation

pressure satp at the isotherm temperature T, the latent heat of liquid-vapor

phase change at equilibrium pressure pλ , the universal gas constant R and

parameters b and m from the corresponding isotherm model.

Isotherm Equation Isosteric Heat Equation

Langmuir (8)

RTbpRTq p

sat

st ++≈ λ)/ln( (9)

Tóth

(10) RTbpRTq p

mmmmmsat

st ++−≈ − λθθ })]1/()[/ln{( /)1(/1

(11)

Referring to table 1, for the Langmuir isotherm and isosteric heat equations (8 & 9), s

mn

and b are model parameters determined by fitting the isotherm equation to data; s

mn

denotes the amount adsorbed at monolayer coverage while b is an equilibrium constant.

In the Tóth expressions (equations 10 & 11), sn∞ , b and m are model parameters; s

n∞ is

the predicted coverage at infinite pressure, m is related to the heterogeneity of the

adsorbent surface, usually having a value less than 1, θ is ss nn ∞/ and b is again an

( )2

cc

sat TTpp =

)/( pbpnns

m

s +=

mmsspbpnn

/1)/( += ∞

27

equilibrium constant. For the special case of m = 1, the Tóth isosteric heat equation

reduces to the Langmuir isosteric heat equation, just as the Tóth isotherm reduces to the

Langmuir isotherm under the same condition.

For both the Langmuir and Tóth isotherms, it has been customary to assume that the

parameter b can be related to the heat of adsorption, qst, through an exponential

function, with a pre-exponential constant [13, 28, 29], to wit

])/exp(//[ pRTqAnpnn st

s

m

s

m

s += (12),

for Langmuir and

mmm

st

ssspRTqAnpnn

/1})]/exp(//{[ += ∞∞ (13),

for Tóth. Applying the Clausius-Clapeyron equation to equations 12 and 13 will

demonstrate that this common assumption is indeed thermodynamically consistent at

low to moderate pressures, where the gases behave ideally and when the adsorbed phase

volume is negligible. To find this assumed qst however, it is still necessary to have

multiple isotherms at different temperatures. In the simplest evaluation, a single value

for qst is extracted. The equations for isosteric heat given in table 1 capture the

possibility of qst varying with coverage and follow calorimetric data. On a separate

front, Savara et al. [24] investigated the behaviour of the entropy at the standard state of

adsorption systems subjected to certain bounds of the enthalpy of adsorption. Our

present findings on the behaviour of the isosteric heat will serve to better inform that

area of pursuit.

3. RESULTS AND DISCUSSION

Dunne and colleagues [18, 19] gave adsorption isotherm and isosteric heat data for a

variety of gases on the zeolite adsorbents silicalite, H-ZSM-5, Na-ZSM-5 and NaX.

Data for some light gases (all argon and oxygen isotherms and some nitrogen and

methane isotherms) could not be used because the isotherms were in the Henry’s Law

region where the effect of saturation was not observed and therefore neither the

Langmuir nor Tóth isotherms could be fitted to give a unique set of equation

parameters. For the isotherm data that extended beyond the Henry’s Law region we

have fitted equation 8 or 10, using least squares fitting to determine the value of the

equation parameters. To avoid statistical ambiguity, the simpler equation 8 with only

two parameters was favored whenever fitting resulted in a similar goodness of fit for

both equations. The values for the model parameters obtained from the isotherm fitting

28

were then used in the corresponding isosteric heat equation (either 9 or 11) to predict

the isosteric heat for each isotherm.

Figures 1 – 4 present first the isotherm data and fitting lines and second the isosteric

heats and predictions. Figure 5 presents the percentage over or under prediction of the

isosteric heat for gases on each of the four adsorbents considered by Dunne and

colleagues [18, 19]. The model parameters derived from fitting equations 8 and 10 to

the isotherm data are given in table 2.

29

Fig. 1 Adsorption isotherms and isosteric heats for gases on silicalite [18]. All

symbols are taken from experimental data, lines in 1A and 1B are fits to that

data, while lines in 1C and 1D are calculated from the fits and shown alongside

measured values. 1A adsorption isotherms and 1C isosteric heats: ∆ C2H6 at

296.18 K, ◊ C2H6 at 296.46 K, □ C2H6 at 296.10 K, + CH4 at 296.22 K. 1B

adsorption isotherms and 1D isosteric heats: ▲ SF6 at 304.94 K, ♦ CO2 at

305.45 K. Dashed lines are from equations 8 and 9, dotted lines from equations

10 and 11. The reported error in the isotherm data (0.6%) is smaller than the

symbols used; hence no error bars are shown. Error bars on the isosteric heat

primary plots show 2% error; error bars on isosteric heat inset plot shows 5%

error.

30

Fig. 2 Adsorption isotherms and isosteric heats for gases on H-ZSM-5 [19]. All

symbols are taken from experimental data, lines in 2A and 2B are fits to that

data, while lines in 2C and 2D are calculated from the fits and shown alongside

measured values. 2A adsorption isotherms and 2C isosteric heats: ▲ SF6 at

294.5 K, ♦ CO2 at 297.1 K, ж CO2 at 295.7 K. 2B adsorption isotherm and 2D

isosteric heat: ∆ C2H6 at 294.5 K. Dotted lines are from equations 10 and 11.

The reported error in the isotherm data (0.6%) is smaller than the symbols

used; hence no error bars are shown. Error bars on the isosteric heat plots

show 2% error.

31

Fig. 3 Adsorption isotherms and isosteric heats for gases on Na-ZSM-5 [19]. All

symbols are taken from experimental data, lines in 3A and 3B are fits to that

data, while lines in 3C and 3D are calculated from the fits and shown alongside

measured values. 3A adsorption isotherms and 3C isosteric heats: ▲ SF6 at

295.4 K, ♦ CO2 at 297.3 K, + CH4 at 296.5 K. 3B adsorption isotherms and 3D

isosteric heats: ∆ C2H6 at 295.8 K, ◊ C2H6 at 297.3 K, ä N2 at 295.1 K. Dotted

lines are from equations 10 and 11. The reported error in the isotherm data

(0.6%) is smaller than the symbols used; hence no error bars are shown. Error

bars on the isosteric heat primary plots show 2% error; error bars on isosteric

heat inset plot shows 5% error.

32

Fig. 4 Adsorption isotherms and isosteric heats for gases on NaX [19]. All

symbols are taken from experimental data, lines in 4A are fits to that data, while

lines in 4B are calculated from the fits and shown alongside measured values.

4A adsorption isotherms and 4B isosteric heats: ♦ CO2 at 304.6 K, ж CO2 at

306.0 K, ▲ SF6 at 304.8 K, ∆ C2H6 at 305.6 K. Dashed lines are from

equations 8 and 9, dotted lines from equations 10 and 11. The reported error in

the isotherm data (0.6%) is smaller than the symbols used; hence no error bars

are shown. Error bars on the isosteric heat plot shows 2% error.

33

Table 2 Range of experimental data from Dunne and colleagues [18, 19] and

parameters derived from data fitting with equations 8 and 10, for use with

equations 9 and 11. Derived fitting parameters include and b for the

Langmuir equations and , b and m for the Tóth equations.

Adsorbate - Adsorbent

Adsorbent mass /g

Isotherm temperature /K

Pressure range /kPa

Fitting equation

Parameter

or /

mol kg-1

/kPa m

C2H6 - Silicalite

0.8752 296.18 0.145 – 53.726

Tóth 2.05 11.4 1.12

C2H6 - Silicalite

0.6332 296.46 0.355 – 14.883

Tóth 1.92 13.3 1.19

C2H6 - Silicalite

0.6332 296.10 0.337 – 14.379

Tóth 1.92 13.3 1.20

CO2 - Silicalite

2.3556 303.75 4.456 – 75.579

Tóth 4.81 17.8 0.626

SF6 - Silicalite

2.3556 304.94 0.129 – 14.043

Tóth 2.67 3.68 0.672

CH4 - Silicalite

0.5542 296.22 0.696 – 93.464

Langmuir 2.28 238 N/A

CO2 -

H-ZSM-5

1.079 297.1 0.55 – 102.34

Tóth 5.27 5.02 0.459

CO2 -

H-ZSM-5

1.079 295.7 0.316 – 18.792

Tóth 10.9 3.39 0.328

C2H6 -

H-ZSM-5

1.079 294.5 0.251 – 72.203

Tóth 2.06 5.22 0.891

SF6 -

H-ZSM-5

1.06 294.5 11.23 – 107.52

Tóth 2.03 2.36 0.724

C2H6 -

Na-ZSM-5

1.551 295.8 0.099 – 68.020

Tóth 2.22 3.03 0.758

s

mn

sn∞

sn∞

s

mn b

34

C2H6 -

Na-ZSM-5

1.551 297.3 0.135 – 3.620

Tóth 3.71 2.47 0.521

CO2 -

Na-ZSM-5

1.551 297.3 0.013 – 71.518

Tóth 372 0.771 0.0823

SF6 -

Na-ZSM-5

1.551 295.4 0.080 – 33.444

Tóth 2.05 2.94 0.786

CH4 -

Na-ZSM-5

1.551 296.5 2.614 – 82.488

Tóth 5.43 15.4 0.500

N2 -

Na-ZSM-5

1.551 295.1 4.818 - 1122.8

Tóth 26.8 12.5 0.317

CO2 -

NaX

0.558 304.6 0.012 – 68.733

Tóth 9.93 0.671 0.301

CO2 -

NaX

0.558 306.0 0.011 – 28.514

Tóth 9.90 0.703 0.291

C2H6 - NaX 0.558 305.6 1.431 – 75.030

Tóth 3.44 1010 1.97

SF6 -

NaX

0.558 304.8 0.371 – 66.548

Langmuir 2.59 9.18 N/A

The predicted isosteric heats from equations 8 and 10 generally follow the data well,

reproducing the trends if not the exact values. Only in the case of sulfur hexafluoride

do the predictions not capture the trend and even here, the values given by equations 11

and 9 respectively are generally representative of the measured isosteric heat values.

The adsorption pressures reported for both sulfur hexafluoride and carbon dioxide were

below their triple points and as noted in the theory section the latent heat at the triple

point was used for λp in making the isosteric heat predictions.

As previously mentioned the Tóth model parameter m is usually less than 1. With m <

1, as coverage goes to zero, equation 11 goes to infinity. In reality isosteric heat must

always have a finite value, however as seen in figures 1 – 4 equation 11 can quickly

converge from infinity to representative values on relatively homogenous adsorbents

like zeolites. On the other hand figure 4C shows that on NaX, the isosteric heat of

adsorption for ethane increases with increasing coverage and this is described by

equation 11 with m > 1.

35

Based on repeat measurements of some isotherms, Dunne et al. [19] estimated the

average experimental error in the isotherm data at about 0.6%. The repeat

measurements however do not cover the entire range of pressure and coverage as the

primary measurements, which lead to different fitting parameters being calculated (table

2). For carbon dioxide on H-ZSM-5 and ethane on Na-ZSM-5 the repeat measurements

reach less than half the coverage and only about 20% and 10% respectively, of the

pressure range of the primary measurement, resulting in a significant difference in both

the fitting parameters (table 2) and the isosteric heat predictions (figures 2 and 3). The

model parameters derived from fitting over a greater range of pressure and coverage

give a better prediction of the isosteric heat at low coverage than did model parameters

derived only from lower coverage data (figures 2, 3 and 9). This suggests that the

ability to describe an isotherm over a greater range gives a better ability to predict the

isosteric heat.

Sircar et al. [20] had measured adsorption isotherms and isosteric heats of adsorption on

silicalite for nitrogen and carbon dioxide. Figure 5A shows the adsorption isotherm

data along with isotherm model best fits; figure 5B contains calorimetric isosteric heat

data corresponding to the isotherms in figure 6A as well as lines showing the predicted

isosteric heats using the fitting parameters from the isotherm model best fits and

equations 9 and 11. The isosteric heat values for carbon dioxide adsorption on silicalite

in figure 5B are clearly a function of coverage, unlike the values in figure 1d which are

nearly constant. Sircar et al. speculated that the reason for the difference between their

data and data reported by Dunne et al. [18] was that while Dunne’s group was using

pure silicalite, Sircar’s group used a pelletized silicalite that included binders and that

these binders provided a second site for adsorption to take place.

36

Fig. 5 Adsorption isotherms and isosteric heats for gases on pelletized

silicalite [20]. All symbols are taken from experimental data, lines in 5A are fits

to the data, while lines in 5B are calculated from the fits and shown alongside

the measured data. 5A adsorption isotherms and 5B isosteric heats for ♦ CO2

at 305.2 K and ä N2 at 305.2 K. Dashed lines are from equations 8 and 9;

dotted lines from equations 10 and 11.

Table 3 Range of experimental data from Sircar et al. [20] and parameters

derived from data fitting with equations 8 and 10, for use with equations 9 and

11. Derived fitting parameters include and b for the Langmuir equations and

, b and m for the Tóth equations.

Adsorbate - Adsorbent

Adsorbent mass /g

Isotherm temperature /K

Pressure range /kPa

Fitting equation

Parameter

or /

mol kg-1

/kPa m

N2 - Silicalite

Unreported 305.2 14.965 – 460.17

Langmuir 1.92 1200 N/A

CO2 - Silicalite

Unreported 305.2 1.108 – 62.95

Tóth 6.37 4.00 0.365

Adsorption isotherms and corresponding isosteric heats for methane and sulfur

hexafluoride on an MFI zeolite adsorbent have been reported by Siperstein et al. [21].

Markers in figure 6A show the isotherm data and the dashed lines are best fits with the

Langmuir isotherm (equation 8). Figure 6B shows the reported isosteric heat values as

well as predictions made with equation 9 using the fitting parameters from the

isotherms.

s

mn

sn∞

sn∞

s

mn b

37

Fig. 6 Adsorption isotherms and isosteric heats for gases on an MFI zeolite

[21]. All symbols are taken from experimental data, lines in 6A are fits to the

data, while lines in 6B are calculated from the fits and shown alongside the

measured data. 6A adsorption isotherms and 6B isosteric heats for ▲ SF6 at

297 K and + CH4 at 296 K. Lines were calculated from equations 8 and 9.

38

Table 4 Range of experimental data from Siperstein et al. [21] and parameters

derived from data fitting with equation 8, for use with equation 9. Derived fitting

parameters are and b for the Langmuir equation.

Adsorbate - Adsorbent

Adsorbent mass /g

Isotherm temperature /K

Pressure range /kPa

Fitting equation

Parameter

or /

mol kg-1

/kPa m

CH4 - MFI Unreported 296 5.025 – 109.78

Langmuir 2.44 258 N/A

SF6 - MFI Unreported 297 0.295– 107.2

Langmuir 1.928 4.69 N/A

Figures 7 and 8 show the adsorption of neon [22] and argon [23] respectively on rutile

phase titanium dioxide. The isotherm data seems to follow type-II behavior according

to the BET classification system [25], indicative of multi-layer adsorption. This type of

adsorption is not described by either the Langmuir or Tóth isotherm models (see

supplementary material for further discussion on fitting multilayer adsorption). Figure

7A indicates that over a limited range of coverage equation 10 may describe the data

reasonably well. However as the method of predicting the isosteric heat of adsorption

outlined in this paper is predicated on accurate estimation of the isotherm gradient

(equation 3) we have tried using a polynomial function (given in table 5) to describe the

isotherms (figures 7A and 8A) as well.

s

mn

sn∞

s

mn b

39

Fig. 7 Adsorption isotherm and isosteric heats for neon on rutile TiO2 [22]. All

symbols are taken from experimental data, lines in 7A are fits to the data, while

lines in 7B are calculated from the fits and shown alongside the measured data.

7A adsorption isotherm and 7B isosteric heats for ○ Ne at 30 K, dotted lines are

from equations 10 and 11, dashed and dotted lines come from the polynomial

correlation described in table 5.

Figure 7b shows the result of fitting equation 10 to neon isotherm data of Tykodi et

al.[22] and using the fitting parameters in equation 11 to predict the isosteric heat. Even

though the Tóth model may not be meant physically for this kind of adsorption

behavior, the resultant isosteric heat predictions are quite good, particularly at low

coverage. Figure 7C shows the result of fitting the isotherm with a polynomial (table 5)

and then applying equations 3, 4 and 7 to the polynomial. At coverage above 0.5 moles

per kilogram, the predictions based on the polynomial fit follows the trend of the

isosteric heat data better than the predictions made with equation 11 however at

coverage values below where there was isotherm data (below 0.12 moles per kilogram)

the polynomial fit was unable to provide predictions (the calculated values were

complex numbers).

40

Fig. 8 Adsorption isotherm and isosteric heats for Ar on rutile TiO2 [23]. All

symbols are taken from experimental data, lines in 8A are fits to the data, while

lines in 8B are calculated from the fits and shown alongside the measured data.

8A adsorption isotherm and 8B isosteric heats for ● Ar at 85 K; in 8A the line

with dashes and single dots is from the polynomial fit (table 5) to the lower

pressure data, the line with dashes and double dots is from the polynomial fit to

the higher pressure data, in 8B the dashed and dotted line is a prediction made

from values calculated from both the lower pressure and higher pressure

polynomial fits.

Figure 8b compares the isosteric heat predictions made by using a best fit polynomial to

describe an isotherm for argon adsorption on rutile phase titanium dioxide and the use

of. The polynomial fit given in table 5, was broken into two pieces with the split being

made at a point where the gradient of the function used for the lower pressure data is

nearly identical to gradient of the function used for the higher pressure data.

41

Table 5 For adsorption on rutile phase titanium dioxide a polynomial of the form

given in the table was used to correlate the data. For the neon data [22] a

single set of coefficients was used, on the other hand for the argon data [23] a

best fit was accomplished by using two different sets of coefficients. The

polynomial switches coefficients at the point 19.44 kilopascals and 1.11 moles

per kilogram, this point was used because both sets of coefficients predict very

similar values for the coverage at this pressure and result in nearly identical

slopes. The values used are given in the table.

Polynomial Correlation

pDpCBpAns *)*/()( +++≈

A B C D

Ne on Rutile -0.0284 1.08 1.032 0.0123

Ar on Rutile – lower pressure 0.0327 0.410 1.398 0.0213

Ar on Rutile – higher pressure -102 -145 1.92 0.0174

Dunne et al. [18] placed the uncertainty in the measurement of the isosteric heat

between 5% and 10% when the isosteric heat was measured to be below 20 kJ/mol. On

the other hand when the isosteric heat was measured to be above 30 kJ/mol they

reported the uncertainty as about 2%. The other authors did not offer estimates of the

error in their isosteric heat measurements. Deviation plots are shown in figure 9; plots

A-D are for the four adsorbents reported on by Dunne and colleagues [18, 19], plot E

shows the deviation for the data reported by the other authors mentioned [20-23].

42

Fig. 9 Deviation plots for the prediction of the isosteric heat of adsorption for

gases on solid adsorbents. On the ordinate ∆qst = predicted isosteric heat -

experimentally observed isosteric heat qst. On the abscissa coverage ns /

maximum observed coverage nsmax. Dashed lines show the band within which

95% of all predictions fall. 9A – Silicalite: ∆ C2H6 at 296.18 K, ◊ C2H6 at 296.46

K, □ C2H6 at 296.10 K, ▲ SF6 at 304.94 K, ♦ CO2 at 305.45 K, + CH4 at 296.22

K. 9B – H-ZSM-5: ▲ SF6 at 294.5 K, ♦ CO2 at 297.1 K, ж CO2 at 295.7 K, ∆

C2H6 at 294.5 K. 9C – Na-ZSM-5: ▲ SF6 at 295.4 K, ♦ CO2 at 297.3 K, + CH4

at 296.5 K, ∆ C2H6 at 295.8 K, ◊ C2H6 at 297.3 K, ä N2 at 295.1 K. 9D – NaX: ♦

43

CO2 at 304.6 K, ж CO2 at 306.0 K, ▲ SF6 at 304.8 K, ∆ C2H6 at 305.6 K. 9E: ♦

CO2 at 305.2 K and ä N2 at 305.2 K on pelletized silicalite, ▲ SF6 at 297 K and

+ CH4 at 296 K on an MFI zeolite, ○ Ne at 30 K on rutile TiO2 as predicted using

equations 10 and 11, ж Ne at 30 K on rutile TiO2 as predicted using a

polynomial best fit, ● Ar at 85 K on rutile TiO2.

4. CONCLUSIONS

Starting from the Tóth potential function, which is a modified Polanyi’s potential

function, we have posited a general relationship between the isosteric heats of

adsorption and adsorption isotherms for type-I and type-II adsorption onto microporous

and mesoporous adsorbents. We have derived specific equations for predicting the

isosteric heat based on fitting isotherm data to the Langmuir and Tóth isotherms. For

isotherm behavior that is not modeled by either the Langmuir or Tóth equations we have

shown that other approaches such as polynomial best fits can provide equally good

insight into isosteric heat trends. However without the guiding philosophy of an

isotherm model these other methods can be less reliable at low pressure and coverage

where relative error in the isotherm data is known to be large or where there is no

isotherm data.

Published data [18- 23] have been used to provide justification to our approach. Our

isosteric heat equations can reliably predict isosteric heats using only parameters

derived from fitting a single isotherm. Even in difficult scenarios such as sulfur

hexafluoride adsorption onto a range of zeolites, the isosteric heat equations predicted

representative values.

This predictive capability from a single isotherm is not exemplified either in previous

works using the Clausius-Clapeyron equation or elsewhere in the literature. The

usefulness and fundamental nature of Clausius-Clapeyron is beyond question however

equations 9 and 11 can give insight into the change in isosteric heats with coverage

where with a single isotherm Clausius-Clapeyron cannot be applied and where with

multiple isotherms Clausius-Clapeyron when used injudiciously can give rise to

ambiguous results as noted in the literature [1-4].

44

ACKNOWLEDGMENT

We thank Alexander Christ for his help with translations.

FUNDING

This project was supported by an ARC Linkage grant LP110100597. The grant in no

way influenced the theories, analysis or outcomes reported in this paper.

REFERENCES

[1] C. Herdes, C. A. Ferreiro-Rangel, T. Düren, Langmuir 27 (2011) 6738-6743.

[2] W. Zimmermann, J. U. Keller, Thermochim. Acta 405 (2003) 31-41.

[3] F. Siperstein, R. J. Gorte, A. L. Myers, Langmuir 15 (1998) 1570-1576.

[4] G. R. Birkett, D. D. Do, Langmuir 22 (2006) 9976-9981.

[5] H. Pan, J. A. Ritter, P. B. Balbuena, Langmuir 14 (1998) 6323-6327.

[6] B. E. Handy, S. B. Sharma, B. E. Spiewak, J. A. Dumesic, Meas. Sci. Technol. 4

(1993) 1350-1356.

[7] V. Solinas, I. Ferino, Catal. Today 41 (1998) 179-189.

[8] X. Wang, W. Zimmermann, K. C. Ng, A. Chakraboty, J. U. Keller, J. Therm.

Anal. Calorim. 76 (2004) 659-669.

[9] X. Wang, H. T. Chua, L. Z. Gao, J. Therm. Anal. Calorim. 90 (2007) 935-940.

[10] M. Polanyi, Verh. Dtsch. Phys. Ges. 16 (1914) 1012-1016.

[11] M. Polanyi, Verh. Dtsch. Phys. Ges. 18 (1916) 55-80.

[12] I. Halász, G. Schay, S. Szőnyi, Acta Chim. Acad. Sci. Hung. 8 (1955) 143-156.

[13] J. Tóth, Acta Chim. Acad. Sci. Hung. 30 (1962) 415-430.

[14] J. Tóth, J. Colloid Interface Sci. 163 (1994) 299-302.

[15] T. L. Hill, J. Chem. Phys. 17 (6) (1949) 520-535.

45

[16] J. Tóth, Adv. Colloid Interface Sci. 55 (1995) 1-239.

[17] C. Tien, Adsorption Calculations and Modeling, 1st ed. Butterworth-Heinemann,

Newton Massachusetts, 1994, Chapter 2.

[18] J. A. Dunne, R. Mariwala, M. Rao, S. Sircar, R. J. Gorte, A. L. Myers, Langmuir

12 (1996) 5888-5895.

[19] J. A. Dunne, M. Rao, S. Sircar, R. J. Gorta, A. L. Myers, Langmuir 12 (1996)

5896-5904.

[20] S. Sircar, R. Mohr, C. Ristic, M. B. Rao, J. Phys. Chem. B. 103 (1999) 6539-

6546.

[21] F. Siperstein, R. J. Gore, A. L. Myers, Langmuir 15 (1999) 1570-1576.

[22] R. J. Tykodi, J. G. Aston, and G. D. L. Schreiner, J. Am. Chem. Soc. 77 (1955)

2168-2171.

[23] L.E. Drain, J. A. Morrison, Trans. Faraday Soc. 48 (1952) 840-847.

[24] A. Savara, C. M. Schmidt, F. M. Geiger, E. Weitz, J. Phys. Chem. C 113 (2009)

2806-2815.

[25] S. Brunaur, L. S. Deming, N. S. Deming, E. Teller, J. Am. Chem. Soc. 62 (1940)

1723-1732.

[26] S. Ozawa, S. Kusumi, Y. Ogino, J. Colloid Interface Sci. 56 (1) (1976) 83-91.

[27] J. Tóth, Acta Chim. Acad. Sci. Hung. 69 (1971) 311-328.

[28] H. T. Chua, K. C. Ng, A. Chakraborty, N. M. Oo, M. A. Othman, J. Chem. Eng.

Data 47 (2002) 1177-1181.

[29] X. Wang, J. French, S. Kandadai, H. T. Chua, J. Chem. Eng. Data 55 (2010)

2700-2706.

46

Supporting Information

Derivation of Equations 9 and 11 for the Predictions of Isosteric Heat

from an Isotherm

Equations 9 and 11 were derived from equations 8 and 10, namely the Langmuir and

Tóth isotherm equations, by applying equations 3, 4 and 7.

For equation 9, the derivation is as follows.

Firstly, the Langmuir isotherm is expressed as

pb

pnn

s

ms

+=

(8).

Looking at the form of equation 3, first we take the derivative of pressure as a function

of coverage as

( )bn

pb

dn

dps

mT

s

2+

= (S.1)

and multiply it by the Langmuir isotherm for coverage divided by pressure to yield

( )1

1 2

+=+

=+

+=

b

p

b

pb

bn

pb

pb

pn

pdn

dp

p

ns

m

s

m

T

s

s

(S.2).

Subtracting 1 from equation S.2 and comparing it with equation 3, we arrive at the

Tóth’s correction (ψ ) to the Polanyi potential function that is consistent with the

Langmuir isotherm, namely

b

p

dn

dp

p

n

T

s

s

==− ψ1 (S.3).

Inserting this into equation 4 gives the Tóth potential function (∆λ) οr

=

=∆

b

pRT

p

pRT

satsat

lnlnψ

λ (S.4),

which can then be used in equation 7 to get

47

RTb

pRTq p

sat

st ++

≈ λln

(9).

Equation 9 encapsulates our method for predicting the isosteric heat of adsorption for

gases on solid adsorbents when the isotherm can be described by the Langmuir model

(equation 8).

Similarly for equation 11, the derivation is as follows.

Firstly the Tóth isotherm model is expressed as

( ) mm

s

s

pb

pnn

1+

= ∞ (10).

We begin by taking the derivative of pressure as a function of coverage

( )( )( )1

11

−

−−=

−

−

ms

mm

T

s n

b

dn

dp

θ

θ

(S.5)

and multiplying it by the Tóth isotherm divided by pressure to yield

( )( )( )( )

( ) 1

1

1

1

1

1

1 −−=

−

−

−−=

−−

−

− mms

mm

mm

s

T

s

s

n

b

b

n

dn

dp

p

n

θθ

θ

θ

(S.6).

In accordance with equation 3, the Tóth correction to the Polanyi potential then takes

the following form

11

11 −

−−==−

−m

T

s

s

dn

dp

p

n

θψ

(S.7).

From equation 4, the Tóth potential function (∆λ) is then expressed as

( )[ ]( )[ ]

( )

−=

−

−+−=

=∆

−

−

mm

m

m

m

sat

mm

msatsat

b

pRT

b

pRT

p

pRT

1

11 1ln

1/

1/11lnln

θ

θ

θ

θψλ

(S.8)

which, when inserted into equation 7 gives our function for predicting the isosteric heat

of adsorption when the isotherm conforms to the Tóth model (equation 10) or

48

( )

RTb

pRTq p

mm

m

m

m

sat

st ++

−≈

−

λθ

θ1

1 1ln

(11).

49

Chapter II

Preface

The preceding chapter addresses a significant problem in the study of gas adsorption,

predicting the heats of adsorption when only a single isotherm measurement is

available. In this circumstance the classical thermodynamics approach, namely using

the Clapeyron equation or more commonly the Clausius-Clapeyron equation, cannot be

applied because there is insufficient data to make a statement about the pressure-

temperature gradient.

The issue of having only a single isotherm available can arise when using literature data

but is of less concern to researchers taking their own measurements for whom obtaining

multiple isotherms at different temperature is only a matter of changing the temperature

of an isothermal bath and repeating the experiment, although this can be a time

consuming procedure.

This chapter examines the situation where multiple isotherms are available and answers

the question of whether there is still value in pursuing the new, Tóth potential function,

method of developing predictive heat of adsorption models when sufficient data exists

to fall back on Clapeyron type equations

The remainder of this chapter, was published as

Whittaker, P. B.; Wang, X.; Zimmermann, W.; Regenauer-Lieb, K.; Chua, H.T.

Predicting the Integral Heat of Adsorption for Gas Physisorption on

Microporous and Mesoporous Adsorbents. J. Phys. Chem. C. 2014, 118, 8350-

8358

and is included in as published format in appendix B.

50

Predicting the Integral Heat of Adsorption for Gas

Physisorption on Micro and Mesoporous Adsorbents

Peter B. Whittaker,a Xiaolin Wang,b Wolfgang Zimmermann,c Klaus Regenauer-Liebd

and Hui Tong Chua*a

a School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

b School of Engineering, The University of Tasmania, Hobart TAS 7001, Australia

c SPG-Steiner GmbH, Wittgensteiner Str.14, D-57072 Siegen

d School of Earth and Environment, The University of Western Australia, 35 Stirling

Highway, Crawley, WA 6009, Australia

*Corresponding author

Email: huitong.chua@uwa.edu.au

*Fax: 61-8-6488-1024

*Phone: 61-8-6488-1828

Other Authors

peter.whittaker@uwa.edu.au, xiaolin.wang@utas.edu.au, wolfgang.zimmermann@spg-

steiner.com, klaus.regenauer-lieb@uwa.edu.au

51

We have developed two predictive methods for the heats of adsorption that stem from

isotherm models and benchmarked them against the Clausius-Clapeyron equation.

These are the Tóth potential function model and the modified Clapeyron equation.

Three adsorbate/adsorbent working pairs are used as examples: n-butane/BAX 1500

activated carbon, i-butane/BAX 1500 activated carbon and ammonia/Fuji Davison type

RD silica gel, all of which are examples of gas physisorption on adsorbents with both

micro and mesopores. Isotherms and corresponding integral heats of adsorption were

measured in the range of 298 K – 348 K. For n-butane and i-butane, the pressures were

up to 235 kPa, and for ammonia, the pressures were up to 835 kPa. Our two predictive

methods consistently offer significant improvements over the Clausius-Clapeyron

equation. Between the two predictive methods, the Tóth model is more robust across all

three working pairs studied with predictions generally falling within 10-15 % of the

values of the measured heats.

Keywords: Isotherm, Dubinin-Astakhov, Clausius-Clapeyron, Clapeyron, Polanyi

potential, Tóth potential

52

Introduction

Most gas adsorption data in the literature are presented as experimentally obtained

isotherm measurements without measured heats of adsorption. This is presumably

because adsorption experiments that incorporate calorimetry are more expensive and

more complicated to both set up and operate than experiments based upon volumetric or

gravimetric measurements alone.1, 2

However knowledge of the heat of adsorption can lend insight into surface phenomena.3

Additionally thermal management is an integral part of the design and operation of

systems which make use of adsorption phenomena such as adsorption chillers,4, 5

adsorbed gas storage tanks 6 and adsorption separation units.7 All of which require

knowledge of the heat of adsorption.

Conventionally experimentalists circumvent the dearth of calorimetric data by invoking

the Clausius-Clapeyron equation, either through measuring many closely spaced

isotherms and applying the equation directly to the data (to wit. the isostere method)8, 9

or by fitting the isotherm data with an isotherm model and then applying the Clausius-

Clapeyron equation to the model.10, 11 The isostere method can lead to ambiguous

results 1, 2, 12 particularly if the number of isotherms obtained is low, if the isotherms are

widely spaced or where the uncertainty in the isotherm measurement is high. Moreover

even at moderate pressure (p < 1 MPa) the ideal gas assumption inherent in Clausius-

Clapeyron can lead to large errors in predicting the heat of adsorption.13

Theoretical models (e.g.14-17) have been developed for this problem by using numerous

isotherm models consistent with statistical mechanics and specific to various different

adsorbate/adsorbent modes of interaction. However it remains a challenge to know

which model to apply to get a correct prediction of the surface interactions and heat of

adsorption without significant prior knowledge of the behavior of the

adsorbate/adsorbent system. Savara et al.18 developed a rigorous method for selecting

between a number of 2D gas and Langmuir related models based on a van’t Hoff

analysis of the expected entropy and enthalpy changes for the various models. However

this method appears to be applicable only when the isotherm data can be described by a

Langmuir isotherm which is often not the case.

In this paper we will offer two improved models to the use of the Clausius-Clapeyron

equation so as to predict heat of adsorption based on isotherm measurements. The first

53

method applies the Tóth potential function to the isotherm model; using literature data

we had shown that this method was accurate to within ± 10-15% of calorimetric values

for predicting the isosteric heat of adsorption.19 The second method is a modified

Clapeyron equation, whereby we explicitly incorporate the full equation of state of

adsorbing gas.

To benchmark our two proposed methods of predicting the heat of adsorption against

the Clausius-Clapeyron equation we have measured isotherms for three

adsorbate/adsorbent working pairs: n-butane/BAX 1500, i-butane/BAX 1500 and

ammonia/silica gel. Additionally we have simultaneously and directly measured their

corresponding integral heat of adsorption so that the predictions can be compared to

calorimetric values.

In this paper, all three predictive methods (namely Tóth potential function model,

Clapeyron equation and Clausius-Clapeyron equation) involve deriving a model for the

heat of adsorption from an isotherm model. The coefficients for each heat of adsorption

model are then found from fitting the isotherm model to the isotherm data. Once the

value of the coefficients has been determined from the isotherm fitting, the coefficients

are then used in the corresponding heat of adsorption model to predict the heat. We

emphasize that no fitting has been done with respect to the heat data and that the heat

data are solely used as a benchmark against which the merits of all three methods can be

compared.

For n-butane/BAX 1500, i-butane/BAX 1500 and ammonia/silica gel, we have found

that the Dubinin-Astakhov (D-A) isotherm equation fit the isotherms very well. Dubinin

and Astakhov (D-A) developed the equation based on the theory of volume filling of

micropores.20 While activated carbon and silica gel may contain mesopores as well as

micropores, for energetic reasons the micropores are expected to fill first and in fact the

theory of volume filling of micropores was experimentally validated with data from

adsorption onto activated carbons. It should be noted that the D-A isotherm has the

following limitations: it cannot be used to describe low uptake (below 10-20% of the

adsorbent micropore capacity), it should only be used to describe a family of isotherms

for which the Gibbs free energy of adsorption can be considered temperature

independent and it is meant for adsorbate/adsorbent working pairs where the heat of

adsorption is expected to decrease with increasing uptake.20, 21

54

Theory

The Dubinin-Astakhov isotherm equation is 20

(1).

Where ns is the specific adsorbate uptake on the adsorbent, vs the specific volume of the

adsorbed phase, R the universal gas constant, T the isotherm temperature, p the

equilibrium pressure, psat the saturation pressure of the adsorbate at the isotherm

temperature. The fitting parameters are W0, ∆E and m. According to the theory of

volume filling of micropores W0 is the specific volume of the adsorbent micropores, ∆E

the characteristic energy of the adsorption process and m a coefficient related to the

distribution of adsorption sites. For the adsorbed phase specific volume, which cannot

be directly measured, the most commonly used model is the one first proposed by

Nikolayev and Dubinin 21:

)](exp[ 00 TTvvs −= α (2)

in which α is the thermal coefficient of limiting adsorption and v0 and T0 are the specific

volume and temperature of the adsorbate as a liquid at a known reference state,

respectively. Because of the availability of data, by far the most frequently used

reference state is the normal boiling point.

Various methods exist of dealing with the thermal coefficient of limiting adsorption α.

Common methods include:

a. using an arbitrary constant value (often 0.0025 K-1),22

b. making α equal to the inverse of the absolute temperature,23

c. making α a function of other physical properties such as the critical temperature

and Van der Waals volume11 and

d. if isotherm data at multiple temperatures are available α can be a fourth fitting

parameter.10, 24

Choosing a value for α prior to data fitting amounts to making an assumption about the

behavior of the adsorbed phase volume; that there are several methods in common

usage suggests that none of the methods are consistently useful or particularly

physically meaningful. Letting α be a fitting parameter may produce a better isotherm

∆−=

msat

s

s

p

p

E

RT

v

Wn lnexp0

55

fit than the methods that predetermine the value of α. However if the adsorbed phase

volume is only weakly a function of temperature across the isotherms being fitted, then

using a form of the D-A equation with α as a fourth fitting parameter risks the value of

the fitting parameters being statistically indeterminate.

As the three methods of predicting the heat of adsorption in this paper are dependent on

obtaining a good isotherm fit and getting the correct fitting constants out of that

isotherm fit, we follow Saha et al.25 in lumping the adsorbed phase volume vs together

with the fitting constant W0, to form a new fitting constant n* and writing the D-A

equation as

(3).

This avoids both making assumptions about the behavior of the adsorbed phase volume

and the statistical indeterminacy trap. As W0 was formulated to be the specific volume

of the available micropores of the adsorbent, n* can be regarded as a maximum uptake

capacity within that micropore volume.

Previously, we have shown that it is possible to predict the heats of gas adsorption to

within 10-15% by finding the Tóth potential function corresponding to an isotherm

equation.19 The method can be summarized as

ZRTq pst ++∆= λλ

(4)

Where qst is the isosteric heat of adsorption, is the Tóth potential

function, in which is Tóth’s correction factor to the Polanyi potential.

The latent heat of condensation at the isotherm equilibrium pressure is symbolized by λp

and the gas compressibility factor is Z. Inserting the pressure explicit form of equation

3,

∆−=

m

s

sat

n

n

RT

Epp

1*

lnexp

into equation 4, taking the derivative with respect to uptake and canceling terms leaves

∆−=

msat

s

p

p

E

RTnn lnexp*

=∆

p

pRT

satψλ ln

1−∂

∂=

T

s

s

n

p

p

nψ

56

ZRTmRT

n

nE

RTn

nEq p

m

sm

sst ++

−

∆

+

∆=

−

λ1

ln

lnln

11

*

1*

(5)

as the Tóth potential function model for the isosteric heat of adsorption. The

differential heat of adsorption can be found from the isosteric heat by subtracting out

ZRT 26or simply RT if the gas phase is considered to behave ideally. An analytical

expression for the integral heat corresponding to equation 5 could not be found;

therefore numerical integration is necessary in making integral heat predictions with this

model. For adsorption at pressures below the triple point we have found that for λp the

heat of vapor-liquid phase change at the triple point gives good results and that heats of

sublimation should not be used.19

The Clapeyron equation is a fundamental thermodynamic relationship between volume

change, the pressure-temperature gradient and heat released during phase change. For

gas adsorption it is written as

(6).

Where ∆v is the change in volume between the gas phase and the adsorbed phase. This

is commonly simplified to the Clausius-Clapeyron equation by assuming ideal gas

behavior for the gas phase adsorbate and negligible volume for the adsorbed phase, thus

sn

stT

p

p

RTq

∂

∂=

2

(7).

Taking the pressure explicit form of equation 3 and substituting it into equation 7 results

in

m

sTstn

nEq

1*

ln

∆+= λ

(8)

as the Clausius-Clapeyron expression for the isosteric heat. Here λT is the latent heat of

condensation at the isotherm temperature, which arises from the saturated pressure term

in equation 3.

sn

stT

pvTq

∂

∂∆=

57

The expression for the integral heat of adsorption corresponding to equation 8 is

( )

+Γ∆+−=

s

u

T

s

n

n

mEnRTnq

**

int ln,1

1λ (9).

Where Γu is the upper incomplete gamma function, or . In

developing this model for the integral heat, RT was subtracted from equation 8 rather

than ZRT before integration, in order to be consistent with the ideal gas assumptions of

Clausius-Clapeyron.

Because equation 3 has no adsorbed phase volume model associated with it, this form of

the D-A equation cannot be used with the classic form of Clapeyron’s equation (6). We

have accordingly modified the Clapeyron’s equation by explicitly treating the gas phase

adsorbate as a real gas, while neglecting the adsorbed phase volume. We will

demonstrate that this results in a substantial improvement over Clausius-Clapeyron.

Starting from the modified Clapeyron’s equation

sn

gstT

pTvq

∂

∂=

(10),

where vg is the gas phase specific volume and applying equation 10 to equation 3 yields

( )[ ]m

s

s

T

g

satTstn

nETTnpZ

v

v

p

pq

1*

ln,,

∆+

∆= λ

(11)

as the model for the isosteric heat. The compressibility factor Z is thermodynamically a

function of uptake and temperature. Depending on the functional form of Z used it may

be possible to integrate equation 11 analytically if not it can always be integrated

numerically.

In the supplementary material we show a complete derivation of equation 11.

Experimental

Using the Sensor Gas Calorimeter (SGC) constructed at the University of Siegen

Germany, we have measured adsorption isotherms and integral heats of adsorption for

pure (99.998%) n-butane and i-butane on the activated carbon BAX 1500, a wood based

( ) ∫∞

−−=Γx

atudttexa

1,

58

activated carbon manufactured by the Westvaco Corporation. Measurements were

made at temperatures of 298 K, 323 K and 348 K and over a range of pressures up to

235 kPa. Additionally we have measured adsorption isotherms and heats of adsorption

for pure (99.998%) ammonia on Fuji Davison type RD silica gel at temperatures of 298

K and 348 K and over a range of pressures up to 835 kPa.

The apparent density of BAX 1500 is between 0.27 – 0.35 g⋅⋅⋅⋅cm-3. The total pore

volume and micropore volume are 1.29 cm3⋅g-1 and 0.50 cm3⋅g-1 respectively. The

mean pore diameter is 1.32 nm and the BET surface area is 1350 m2⋅g-1. The total pore

volume and pore volume distribution are consistent with those reported by Wilhelm et

al.27 however they reported a significantly greater BET surface area of 2173 m2⋅g-1.

The physical properties of Fuji Davison type RD silica gel were studied extensively by

Chua et al.28. They found the BET surface area to be 838 ± 3.8 m2⋅g-1 and the total pore

volume and micropore volume to be 0.37 cm3⋅g-1 and 0.18 cm3⋅g-1 respectively.

A sketch of the Sensor Gas Calorimeter, for simultaneous measurements of gas uptake

and heats of gas adsorption is shown in Figure 1. It had previously been described by

Zimmermann and Keller 2 and its features are merely summarized here. The Sensor

Gas Calorimeter setup consists of a standard volumetric measurement system for

isothermal measurements and an enclosing sensor gas jacket for the measurement of

heat transferred during adsorption processes.

59

Figure 1. Schematic of a sensor-gas calorimeter adapted from Zimmermann and

Keller.2

60

For the isothermal measurement of gas uptake, its construction and working principles

are similar to the static volumetric measurement which has been described by Wang et

al.24. The volumetric part of the Sensor Gas Calorimeter consists of an adsorption

vessel and a connected gas reservoir vessel. The pressures and temperatures in these

two vessels are monitored by calibrated pressure transducers and temperature sensors

respectively. The temperature of the gas reservoir is maintained at 25oC by using a

thermoelectric cooler/heater while the temperature of the adsorption vessel is held

constant by means of a thermostatic bath. The adsorbent sample under investigation is

loaded into the adsorption vessel and is heated prior to uptake measurements for

possible outgassing and surface cleaning at a temperature of 423 K for around 3hrs at a

residual pressure of 10-3 Pa. The dead volume after loading the sample into the

calibrated adsorption vessel is determined by expansion of helium gas which is assumed

to adsorb to no more than a negligible degree on the BAX 1500.29 The adsorbate is first

charged into the gas reservoir from a gas cylinder and the amount of gas is quantified by

the measured pressure and temperature when the calibrated gas reservoir vessel reaches

thermal equilibrium. Thereafter the valve labelled 5 in figure 1 is opened such that a

certain amount of adsorbate is introduced into the adsorption vessel and closed again

after approximately 30s. After the two vessels return to thermal equilibrium, the

pressure and temperature of both vessels are recorded so that the adsorbent uptake on

the BAX 1500 sample can be calculated as the difference between the reduction of

adsorbate in the gas reservoir and the increase of adsorbate in the dead volume of the

sample vessel. The corresponding adsorption pressure is monitored by the pressure

sensor inside the adsorption vessel. During this measurement process, it is important

that the velocity of gas flow is small enough to allow thermal equilibration of the gas

temperature to thermostat temperature.

The heat released during the adsorption process is measured via the sensor gas jacket

which encloses the adsorption vessel. The sensor gas jacket is connected with a

reference gas pressure vessel of identical volume. All the heat generated during the

adsorption process must be rejected to the constant temperature bath via the sensor gas

jacket. In consequence the sensor gas in the thermal jacket is heated and its pressure

increases transiently. This temporary pressure rise is measured by a highly sensitive

differential pressure manometer against the constant pressure in the reference vessel

leading to an asymmetric peak signal. By calibrating the differential pressure response

against a known energy input from a resistance heater in the adsorption vessel prior to

61

adsorption measurements, the energy released from the adsorbate during the adsorption

process can be accurately calculated from the area beyond the differential pressure

signal peak. By stepwise increase of adsorbate pressure at constant temperature

isotherms can be obtained along with the simultaneous measurement of the differential

heat released during the adsorption process. In the same way, the heats of desorption

can also be evaluated by stepwise decrease of adsorbate pressure. See Zimmermann

and Keller 2 for a further explanation of how the SGC was calibrated.

We have performed a rigorous error propagation analysis, the methodology of which is

presented in the supporting information. For the adsorption isotherm measurements at

very low uptake the measurement uncertainty is about ±10% of the measured values and

as the pressure and number of measurement steps increase the uncertainty reduces to

±5%. Similarly for the heat of adsorption measurements the uncertainty is ±5-10% of

the measured value.

Results

The isotherm data were fitted with equation 1 with each of the adsorbed phase volume

models (equation 2 with the various methods of determining α) and with equation 3. Of

the various D-A models equation 3 gave a better fit to the data (as determined by least

squares fitting and visual inspection) than equations 1 and 2 with the various methods of

determining α. Letting α be a fourth fitting parameter does lead to a very slight

improvement in fit but as equation 3 gave a fit that was generally inside of our estimated

uncertainty of about ± 5% the improvement offered by the four parameter model was

deemed insignificant and therefore rejected. The results of the fitting are shown in

figure 2 and the fitting parameters are given in table 1. Original data recorded for the

pressure and uptake, as well as the integral heat data are furnished in the supplementary

information.

62

Figure 2. Adsorption isotherm data and D-A equation fits to that data. Fitting constants

are given in table 1.

Table 1. D-A isotherm (equation 3) parameter values found from fitting (figure 2) and

used in predictions shown in figures 3, 4 and 5.

63

Working pair Parameter Value Unit

n-Butane/BAX

1500

ns* 16.1 mol kg-1

∆E 6731 J mol-1

m 1.00 -

i-Butane/BAX

1500

ns* 12.5 mol kg-1

∆E 8420 J mol-1

m 1.25 -

Ammonia/Fuji

Davison type RD

silica gel

ns* 16.5 mol kg-1

∆E 6957 J mol-1

m 0.80 -

The corresponding integral heats of adsorption are presented in figures 3-5. Overlaying

the data are predictions made using the fitting parameter values from table 1 and the

three methods of predicting the heats of adsorption described in the theory section (Tóth

potential function heat model, modified Clapeyron’s equation heat model and Clausius-

Clapeyron heat model). Accompanying the integral heats and prediction in figures 3-5

are deviation plots showing the percentage error for each of the three methods of

prediction.

For n-butane/BAX 1500 (figure 3) at 298 K the Tóth heat model closely predicts the

data up to about 12 mol kg-1 uptake, beyond that value the measured heat begins to dip.

From looking at the corresponding isotherm (figure 2), we interpret that multilayer

adsorption begins taking place somewhere between 12 and 14 mol kg-1 uptake, which is

likely related to the dip in the integral heat. At 323 K and 348 K the modified

Clapeyron heat model best predicts the data.

64

Figure 3. Left column: measured heats of adsorption (symbols) and predicted heats

(lines) for n-butane on BAX 1500. Right column: error in predictions of the integral

heat for n-butane/BAX 1500. Error was calculated as Error = 100 × (fitting value –

measured value) / measured value.

For i-butane/BAX 1500 (figure 4) the Tóth heat model overall best predicts the data at

all temperatures. For the heats measured at 323 K the prediction of the Tóth heat model

agrees best with the data at low uptake (<6 mol kg-1) and beyond which the measured

65

heats are in between the predictions of Tóth model and that of the modified Clapeyron

heat model.

66

Figure 4. Left column: measured heats of adsorption (symbols) and predicted heats

(lines) for i-butane on BAX 1500. Right column: error in predictions of the integral

heat for i-butane/BAX 1500. Error was calculated as Error = 100 × (fitting value –

measured value) / measured value.

For ammonia/silica gel (figure 5) at 298 K the Tóth heat model best predicts the

measured values to within ~15%. Again at 348 K the Tóth heat model is the best,

providing predictive capability to within ~10% over most of the uptake range.

67

Figure 5. Left column: measured heats of adsorption (symbols) and predicted heats

(lines) for ammonia on Fuji Davison type RD silica gel. Right column: error in

predictions of the integral heat for ammonia/silica gel. Error was calculated as Error =

100 × (fitting value – measured value) / measured value.

68

Discussion

Examining all three adsorbate/adsorbent pairs, the predictions that stem from the Tóth

potential function heat model are the most consistent, at worst over predicting the

integral heat for n-butane/BAX 1500 at 348 K by 15-20% and under predicting the

integral heat for ammonia/silica gel at 298 K by 15-20% at low coverage and 10-15% at

higher coverage.

In all cases, for all the three working pairs studied, the Clausius-Clapeyron model

significantly under predicted the measured data for the integral heat of adsorption. The

difference between the results obtained with the Clausius-Clapeyron model and

modified Clapeyron model are entirely due to the assumption in the former model of

ideal gas behaviour rather than real gas behaviour. The use of real gas behaviour leads

to the latent heat term in equation 11 being modified by a temperature dependent

dimensionless group. For the adsorbate/adsorbent pairs presented here, this

dimensionless group leads to predictions of higher values for the heats of adsorption for

the modified Clapeyron model than in the Clausius-Clapeyron model from 5% to 20%.

This is explored in greater depth in the supporting material.

As for why the Tóth potential function heat model outperformed both of the other two

models, we believe it is because of the difficulty of obtaining an accurate experimental

determination of the local pressure – temperature gradient along an isostere, which is

required by the other two models, as opposed to the simplicity of the Tóth-modified

Polanyi potential function which simply requires the local pressure-uptake gradient

along an isotherm.30

Considering the results of this paper and our previous work19 we have used Tóth

potential function heat models to make predictions about the heats of adsorption, based

only on isotherm data, for twenty-four different adsorbate/adsorbent pairs. The only

adsorbate/adsorbent pairings that we are aware of for which some caution may be

required are those featuring strong cooperative interactions between the binding sites,

for instance SF6 on a number of zeolites.19 Even for these systems, while the trend of the

differential or isosteric heat may not be well represented, the average heat values remain

representative.

69

Conclusions

We have developed two predictive heat of adsorption models based on the Dubinin-

Astakhov isotherm equation and benchmarked them against the Clausius-Clapeyron

heat of adsorption model. One stems from the Tóth potential function while the other is

a modification of Clapeyron’s equation, explicitly incorporating real gas behaviour.

To demonstrate the predictive capability of these models, we have experimentally

measured the adsorption isotherms and integral heats of adsorption for pure n-butane

and i-butane on activated carbon BAX 1500, and pure ammonia on Fuji Davison type

RD silica gel. The fitting parameters for the Dubinin-Astakhov isotherm equation were

calibrated with the measured isotherm data and were then used in the two predictive

heat of adsorption models we developed and in the Clausius-Clapeyron model.

The predictions made with the Clausius-Clapeyron model under predicted the measured

data for all three working pairs at every temperature measured. This is significant

because the use of Clausius-Clapeyron is very common in the literature and both the

Tóth heat model and modified Clapeyron heat model offer a significant improvement.

Based on the three working pairs studied the Tóth heat model is on the whole more

robust than the modified Clapeyron equation, with an overall accuracy of about 10-15%

which is consistent with our previous findings.19

Supporting Information Available: Derivation of the isosteric heat expression using

the modified Clapeyron’s equation, experimental data and deviation plots for the

isotherm fitting and an explanation of the error propagation analysis. This material is

available free of charge via the internet at http://pubs.acs.org

Acknowledgment: The authors gratefully acknowledge financial support from the

Western Australian Geothermal Centre of Excellence and ARC Linkage grant

LP110100597. The funding in no way influenced the theories, analysis or outcomes

reported in this paper.

70

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73

Supporting Information for the paper:

Predicting the Integral Heat of Adsorption for Gas

Physisorption on Micro and Mesoporous Adsorbents

Peter B. Whittaker,a Xiaolin Wang,b Wolfgang Zimmermann,c Klaus Regenauer-Liebd

and Hui Tong Chua*a

a School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

b School of Engineering, The University of Tasmania, Hobart TAS 7001, Australia

c SPG-Steiner GmbH, Wittgensteiner Str.14, D-57072 Siegen

d School of Earth and Environment, The University of Western Australia, 35 Stirling

Highway, Crawley, WA 6009, Australia

74

Derivation of the isosteric heat expression using the modified

Clapeyron’s equation

We start from the D-A isotherm equation in the form

(1S),

which can be algebraically rearranged to be explicit in pressure,

∆−=

m

s

sat

n

n

RT

Epp

1*

lnexp

(2S).

There are two terms in equation 2S which are temperature dependent: the saturated

pressure and the exponential term. The temperature derivative of the saturated pressure

is defined by the Clapeyron equation to be

T

T

sat

vTT

p

∆=

∂

∂ λ

(3S)

and the derivative of the exponential term is found to be

2

1*

1*

1*

lnexplnlnexp

RT

n

n

RT

E

n

nE

T

n

n

RT

E m

s

m

s

n

m

s

s

∆−

∆

=∂

∆−∂

(4S).

From the chain rule the derivative with respect to temperature of equation 2S is then

∆−

∆+

∆−

∆=

∂

∂ m

s

m

s

satm

s

T

T

n n

n

RT

E

n

nE

RT

p

n

n

RT

E

vTT

p

s

1*

1*

2

1*

lnexplnlnexpλ

(5S)

.

The modified Clapeyron equation for adsorption with negligible adsorbed phase volume

can be written as

∆−=

msat

s

p

p

E

RTnn lnexp*

75

sn

stT

p

p

ZRTq

∂

∂=

2

(6S).

Recognizing that p is given by equation 2S, combining equations 5S with 6S and

simplifying results in

m

s

T

sat

Tst

n

nEZ

vp

ZRTq

1*

ln

∆+

∆=

λ

(7S),

as the equation for the isosteric heat of adsorption. This can be rewritten as

m

s

T

g

satTstn

nEZ

v

v

p

pq

1*

ln

∆+

∆

= λ

(8S),

showing that the latent heat of vaporization at the isotherm temperature is modified by

the relative pressure and the ratio of the specific volume of the gas phase adsorbent, to

the volume change of pure adsorbent upon condensation at the isotherm temperature.

Deviation between the modified Clapeyron model and Clausius-

Clapeyron model

Equation 7S or 8S could alternatively have been written as

∆+

∆=

m

s

T

satTstn

nE

vp

RTZq

1*

lnλ

(9S),

a form which facilitates comparison with the isosteric heat model developed from the

Clausius-Clapeyron equation

m

sTstn

nEq

1*

ln

∆+= λ

(10S).

In the modified Clapeyron model (as opposed to the Clausius-Clapeyron derived model)

the latent heat λT is modified by the temperature dependent, dimensionless term

T

satvp

RT

∆the values for which are given in table S1 and the entire isosteric heat model

76

scales with the compressibility factor Z, for which select values are given in table S2.

Deviation plots are presented in figure S1 which show the percentage difference

(typically 5% to 20%) as a function of uptake for the isosteric heat predictions made by

the modified Clapeyron heat model (equation 9S) benchmarked against the Clausius-

Clapeyron heat model (equation 10S).

77

Table S1. Values of T

satvp

RT

∆.

n-Butane i-Butane Ammonia

298 K 1.093 1.128 1.147

323 K 1.167 1.229 1.259

348 K 1.288 1.402 1.453

Table S2. Selected values for the compressibility factor Z.

Compressibility factor (Z) for gases as a function of pressure and temperature 298 K 323 K 348 K Pressure n-Butane i-Butane Ammonia n-Butane i-Butane Ammonia n-Butane i-Butane Ammonia 1 kPa 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 50 kPa 0.985 0.987 0.995 0.989 0.990 0.996 0.991 0.992 0.997 100 kPa 0.970 0.974 0.990 0.977 0.980 0.992 0.983 0.984 0.994 150 kPa 0.955 0.960 0.984 0.966 0.970 0.989 0.974 0.976 0.991 200 kPa 0.939 0.946 0.979 0.954 0.959 0.985 0.964 0.968 0.989 250 kPa Condensed 0.932 0.973 0.942 0.948 0.981 0.955 0.960 0.986 300 kPa Condensed 0.917 0.968 0.929 0.937 0.977 0.946 0.951 0.983 350 kPa Condensed Condensed 0.962 0.916 0.926 0.973 0.936 0.943 0.980 400 kPa Condensed Condensed 0.957 0.903 0.914 0.969 0.926 0.934 0.977

79

Figure S1. Deviation plots for the prediction of isosteric heat via the modified

Clapeyron model (equation 9S) benchmarked against the Clausius-Clapeyron model

(equation 10S).

Experimental data

Table S3. Isotherm and integral heat data for n-butane/BAX 1500.

80

298.15 K 323.15 K 348.15 K

p /

kPa

ns /

mol kg-

1

qint/

kJ/kg

p /

kPa

ns /

mol kg-

1

qint/

kJ/kg

p /

kPa

ns /

mol

kg-1

qint/

kJ/kg

0.86 0.728 35.36 0.51 0.441 21.12 0.48 0.446 18.84

0.97 1.133 53.96 0.95 0.815 36.90 1.41 0.765 31.84

1.35 1.478 69.47 1.43 1.181 51.19 2.37 1.071 42.92

1.44 1.858 85.19 1.94 1.517 63.61 3.35 1.332 52.78

1.87 2.201 98.54 2.47 1.823 74.30 4.36 1.551 61.15

2.22 2.512 110.66 3.32 2.097 83.21 5.58 1.757 68.33

2.45 2.825 121.55 3.92 2.354 91.78 6.82 1.963 74.95

2.95 3.122 131.78 4.83 2.618 100.19 8.24 2.122 80.35

3.45 3.381 141.44 5.77 2.855 108.46 9.61 2.270 85.44

4.22 3.660 150.55 6.75 3.083 115.21 10.97 2.416 91.06

4.83 3.914 159.41 7.75 3.299 122.26 15.22 2.859 106.13

5.51 4.171 167.59 8.76 3.505 129.25 19.97 3.232 118.40

6.30 4.418 175.64 9.95 3.698 136.31 24.87 3.576 129.15

7.24 4.640 183.66 11.18 3.894 141.92 30.21 3.910 139.50

8.18 4.865 191.51 15.08 4.398 157.15 35.61 4.211 148.89

9.14 5.076 198.96 19.49 4.851 170.69 41.43 4.474 156.62

10.17 5.280 205.66 24.38 5.272 183.46 47.33 4.701 163.05

13.39 5.821 222.61 29.73 5.660 195.93 53.49 4.936 168.29

16.99 6.289 237.78 35.17 6.005 205.93 59.56 5.139 173.93

20.90 6.727 251.92 40.49 6.295 215.00 65.90 5.357 179.16

24.87 7.128 264.93 46.16 6.581 222.22 72.37 5.563 184.75

29.25 7.509 277.18 51.75 6.845 229.33 79.03 5.746 189.93

33.73 7.875 286.83 57.56 7.068 236.51 85.67 5.925 195.02

81

38.50 8.205 295.62 63.42 7.283 242.51 92.57 6.099 198.86

43.06 8.543 305.16 69.42 7.501 248.67 99.47 6.268 203.15

47.84 8.875 313.51 75.62 7.745 254.62 106.35 6.435 207.37

52.79 9.181 322.52 81.77 7.940 260.37 113.26 6.593 210.66

58.03 9.479 330.28 87.87 8.147 265.54 120.34 6.722 214.45

63.40 9.758 337.94 94.21 8.337 270.20 127.43 6.885 217.25

68.49 10.031 350.43 100.53 8.532 274.91 134.70 7.000 220.63

74.00 10.300 356.93 106.91 8.721 279.02 141.98 7.119 223.54

79.34 10.568 363.48 113.59 8.915 282.30 149.36 7.241 225.63

84.90 10.830 369.57 120.01 9.097 286.15 156.65 7.358 228.34

90.48 11.070 375.55 126.41 9.279 290.11 163.90 7.479 230.54

96.18 11.329 381.24 132.79 9.461 293.87 170.49 7.643 233.38

102.01 11.537 386.55 139.33 9.640 297.19 177.27 7.755 237.37

107.85 11.746 392.07 145.88 9.807 300.94 184.03 7.875 240.64

113.89 11.951 396.45 152.42 9.986 304.30 190.75 8.033 242.86

119.95 12.161 400.57 159.09 10.151 307.56 197.26 8.149 246.21

125.92 12.405 404.65 165.88 10.327 311.36 204.07 8.298 249.24

131.78 12.613 409.64 173.29 10.527 315.59 210.99 8.453 252.45

137.67 12.814 414.54 179.94 10.703 318.67 218.14 8.570 255.93

143.97 13.004 418.22 186.97 10.901 321.46 225.08 8.721 259.63

149.84 13.219 423.30 194.27 11.080 326.02 232.26 8.843 262.20

155.75 13.423 427.60 200.90 11.259 328.74

161.61 13.625 432.71 208.42 11.470 331.92

167.66 13.833 437.03 214.70 11.659 335.61

173.76 14.035 440.43 221.09 11.855 339.38

179.62 14.280 444.21

185.50 14.540 448.59

82

191.33 14.802 453.25

197.24 15.064 457.01

203.09 15.331 460.05

208.66 15.607 463.70

214.35 15.912 467.09

220.02 16.243 471.13

Table S4. Isotherm and integral heat data for i-butane/BAX 1500.

298.15 K 323.15 K 348.15 K

p /

kPa

ns /

mol kg-

1

qint/

kJ/kg

p /

kPa

ns /

mol

kg-1

qint/

kJ/kg

p /

kPa

ns /

mol

kg-1

qint/

kJ/kg

0.39 0.664 30.17 0.67 0.457 23.24 0.98 0.533 27.39

0.83 1.060 47.65 1.39 0.912 42.05 2.34 0.876 42.13

1.29 1.416 60.26 1.94 1.257 55.50 3.39 1.160 51.70

1.62 1.705 70.55 2.79 1.561 66.79 4.79 1.374 59.31

1.97 1.979 80.36 3.48 1.812 77.15 6.64 1.658 69.25

2.38 2.192 88.98 4.36 2.046 85.32 8.39 1.883 76.79

2.83 2.385 96.53 5.78 2.346 95.32 10.02 2.044 81.42

3.25 2.558 102.75 6.77 2.554 102.86 14.63 2.511 97.79

3.54 2.733 108.76 7.78 2.750 109.76 19.79 2.876 111.17

3.90 2.918 114.15 9.09 2.920 114.73 25.33 3.226 122.16

4.35 3.104 120.56 10.22 3.075 119.80 30.81 3.525 134.15

4.83 3.246 125.70 13.75 3.540 134.40 36.62 3.794 142.64

5.31 3.397 131.34 17.59 3.947 147.16 42.89 4.047 149.31

5.81 3.544 136.53 21.95 4.318 159.19 49.14 4.284 155.93

6.31 3.696 140.99 26.79 4.612 170.16 55.66 4.524 161.91

83

6.85 3.844 145.75 31.86 4.907 179.30 62.43 4.714 168.29

7.66 4.016 151.40 37.05 5.180 187.17 69.29 4.895 173.96

8.26 4.165 156.43 42.43 5.430 194.83 76.17 5.073 179.19

9.21 4.316 160.46 48.42 5.681 201.70 84.42 5.241 186.05

10.21 4.502 164.84 54.31 5.890 207.05 91.74 5.404 196.34

12.34 4.905 178.07 61.05 6.140 213.92 99.94 5.557 199.87

14.59 5.203 188.27 67.36 6.369 219.96 107.33 5.681 202.71

17.05 5.513 198.53 74.07 6.585 225.48 114.69 5.836 207.25

19.95 5.804 206.83 80.53 6.767 230.65 122.99 5.988 213.03

22.87 6.085 215.70 87.36 6.940 234.42 131.81 6.128 216.04

26.29 6.313 223.84 94.19 7.122 238.92 140.56 6.272 218.27

29.89 6.545 230.76 101.03 7.293 243.08 148.90 6.407 221.73

33.49 6.766 237.69 108.19 7.462 246.95 157.69 6.542 224.63

37.02 6.984 244.17 116.01 7.615 250.79 166.25 6.666 227.99

41.13 7.197 250.49 124.33 7.770 254.63 174.61 6.794 231.52

45.27 7.397 256.49 132.34 7.924 258.32 183.07 6.920 234.64

49.27 7.589 261.98 140.60 8.119 262.52 191.41 7.008 239.49

53.76 7.777 268.32 148.93 8.309 267.03 199.22 7.104 242.06

57.91 7.947 280.29 157.57 8.471 269.99 207.56 7.225 244.29

62.33 8.133 286.47 165.97 8.624 273.40 215.14 7.313 246.94

67.13 8.289 291.67 174.41 8.776 276.73 222.77 7.403 249.00

71.73 8.467 296.64 183.06 8.917 280.45 231.87 7.505 253.79

76.12 8.596 301.28 191.88 9.059 283.93

80.77 8.756 306.79 200.63 9.212 287.57

85.50 8.919 311.76 209.42 9.354 290.94

90.55 9.035 316.11 218.27 9.489 294.13

95.32 9.182 320.84 227.01 9.644 297.21

84

100.46 9.333 325.19 235.95 9.778 300.41

105.49 9.492 329.31

111.17 9.642 333.56

117.01 9.787 337.22

123.16 9.931 341.28

129.16 10.072 346.35

135.21 10.204 351.21

141.56 10.344 354.57

147.86 10.473 358.64

154.26 10.609 363.27

160.61 10.741 368.24

166.97 10.877 372.20

173.47 11.009 375.68

180.02 11.105 378.77

186.35 11.238 382.45

192.83 11.341 385.80

199.16 11.466 389.00

205.95 11.592 392.62

Table S5. Isotherm and integral heat data for ammonia/silica gel.

298.15 K 348.15 K

p /

kPa

ns /

mol kg-1

qint/

kJ/kg

p /

kPa

ns /

mol kg-

1

qint/

kJ/kg

0.43 0.359 29.52 0.48 0.387 28.37

0.44 0.754 57.81 0.85 0.739 51.62

0.45 1.110 82.72 1.03 1.028 68.75

85

0.48 1.461 106.32 1.93 1.473 95.50

0.57 1.778 126.35 3.36 1.842 117.49

0.81 2.094 143.95 5.34 2.153 135.13

0.94 2.381 160.06 8.21 2.415 149.49

1.37 2.612 174.13 11.56 2.650 160.52

2.38 3.055 197.91 30.08 3.282 189.96

4.24 3.394 214.61 53.60 3.737 207.45

6.77 3.701 228.58 79.16 4.081 219.08

9.73 3.952 239.98 105.81 4.385 229.05

13.24 4.178 248.07 133.10 4.632 237.05

29.76 4.940 276.66 160.55 4.856 243.22

49.09 5.541 298.05 188.23 5.063 249.52

69.78 6.066 316.03 217.24 5.273 255.09

91.08 6.524 331.01 244.42 5.486 259.55

113.35 6.972 343.22 272.96 5.676 264.03

135.87 7.376 354.39 300.62 5.880 268.24

158.42 7.751 364.82 328.61 6.089 272.58

181.14 8.141 374.95 355.96 6.277 278.22

204.38 8.481 384.60 384.21 6.453 282.36

227.04 8.837 393.95

250.43 9.172 402.48

273.07 9.512 411.49

295.95 9.839 419.80

318.84 10.177 427.88

341.78 10.511 435.97

364.40 10.847 443.93

386.17 11.196 452.46

86

Deviation plots

Figure S2 shows deviation plots from the isotherm fitting of n-butane/BAX 1500 as an

example of how the different D-A models compare. The trends were the same for all

three sets of working pairs examined. Isotherms models were fit to the data by

minimizing the sum of the square of the residuals between the calculated uptake and the

measured values; this fitting method was chosen over chi squared minimization because

the former method gives less weighting to the low pressure data which has higher

relative uncertainty. The deviation plots cover only the pressure range above 5 kPa

because below this value (Figure S3) the isotherm data show Henry’s Law behavior –

something which the D-A isotherm does not conform to and hence there should be no

expectation of a good fit at very low pressures.

87

Figure S2. Deviation plots for isotherms fit to n-butane/BAX 1500 adsorption data.

Error was calculated as Error = 100 x (fitting value – measured value) / measured value.

88

The major trends are:

• Using equation 3 or letting the thermal coefficient of limiting adsorption α be a

fourth fitting parameter resulted in nearly identical fits.

• Assuming α to have a value of 0.0025 K-1 or that α is an inverse of the isotherm

temperature resulted in very similar fits and fits using these two methods were

more likely to fall outside the +/- 5% estimated uncertainty band than the other

methods considered.

• Assuming that α is a function of other physical constants like the critical

temperature and pressure and normal boiling point properties tended to fall

somewhere between the first two methods mentioned and the second two

methods.

• The fits from equation 3, as well as letting α be a fourth fitting parameter and

assuming that α is a function of other physical constants were all generally

within the uncertainty band.

Figure S3. Low pressure adsorption data for i-butane on BAX 1500 showing Henry’s

Law behavior.

89

Error Propagation Analysis for the Sensor Gas Calorimeter (SGC)

A. Errors in the calculation of amount of gas adsorbed

For the determination of the amount of adsorbed gas by means of the SGC errors in the

following measurements contribute to experimental uncertainty: absolute pressures PGR

(gas reservoir) and PSV (sample vessel), temperatures TGR (gas reservoir) and TSS

(sorbent sample) volumes VGR (gas reservoir), VP (piping volume between V5 and the

manually operated valve), VSV (sample vessel/adsorption vessel volume) and VSS

(sorbent sample) and m, the mass of the sample itself (please refer to the figure S4).

Figure S4. Sketch of Sensor Gas Calorimeter (SGC).

The absolute pressures are taken with an uncertainty of δp = ±100Pa. The uncertainty

of the temperature measurement is δT = ±0.1K. For the mass of the sample the

uncertainty of the analytical balance is δm = ±0.1mg. The errors in determination of the

different volumes have been estimated as root mean square deviations of the mean from

multiple trials of volume determination gas expansion experiments.

All together, the error in the determination/calculation of the amount of gas adsorbed

(∆nAds =∆nGR -∆nP -∆nSV) can be calculated using Gaussian error propagation as follows:

90

( )2

22

2

01

2

2 2

×

∆++×

−+

−=∆ SS

SS

SVSVSS

SS

SVSV

SS

SSSVSV T

T

nVV

RT

ppp

RT

VVn δδδδδ

(11S)

22

201

2

2 2

×

∆+

−+

=∆ GR

GR

GRGR

GR

GRGR

GR

GRGR T

T

nV

RT

ppp

RT

Vn δδδδ

(12S)

22

201

2

2 2

×

∆+

−+

=∆ GR

GR

GRGR

GR

SVSV

GR

PP T

T

nV

RT

ppp

RT

Vn δδδδ

(13S).

Applying these equations for the situation as it was during our measurements the error

in the determination of the change in the amount of gas in the sample vessel (equation

11S) can be considered to be 7%, the error in the change in the amount of gas in the

piping between the manual valve and valve 5 and the error in the change in the amount

of gas in the gas reservoir are both around 5% (equations 12S and 13S, respectively).

From those three different errors the total error for the amount of gas adsorbed in each

single step (i) can be calculated as follows:

222PGRSVi nnnn ∆+∆+∆= δδδδ (14S).

Using the results of equations 1-3, the uncertainty in the amount adsorbed for each step

is about ±10%.

Related to the mass of the sample the error is

( )

( )

2

2

2

+

∆

∆=

∑∑

m

m

n

nnn

i i

i i

adsads

δδδ

(15S).

For the adsorption measurements given in the main paper, the uncertainty falls in the

range of ±5-10% of the measured values.

B. Errors in the calculation of heats of ad- and desorption

The uncertainty of the heat measurement is determined by errors in the measurement of

the amount of gas adsorbed (or desorbed), the measurement of the area under the peak

of the differential pressure signal and by the calibration function. The error for the

amount adsorbed is calculated from equation 14S.

91

The SGC was calibrated by using an electrical heating element in the sample vessel to

dissipate known amounts of heat and measuring the responses of the differential

pressure sensor (the peak area for each heat input). The result was a linear relationship

between the heat dissipated and the peak area called the calibration function: A =

a1Q+a0 where A is the peak area in kPa-s, Q is the heat input in Joules and a1 and a0 are

calibration parameters. The error in the calibration function of the calorimeter is made

up of the error in the slope a1 and intercept a0 of the linear relationship. For a more

detailed discussion of how the SGC was calibrated please refer to Zimmermann and

Keller 2003 [S1]. The error functions related to the calibration line are

( ) xx

xxyy

SN

SaSa

2

21

1−

−=δ

(16S)

and

( ) ( )22

221

02 ∑∑

∑−−

−=

ii

ixxyy

xxN

x

N

SaSaδ

(17S)

with ( )2

∑ −= xxS ixx , ( )∑ −=2

yyS iyy and where N is the number of points to which

the calibration function is fitted.

The error for the peak area is assumed to be the average error from various repeated

calibrations (3%).

The total uncertainty in the measured heats of adsorption/desorption from the SGC is

thus given by

22

1

1

2

0

0

2

0

03.0

+

+

−+

−=

ads

ads

adsads n

n

a

a

aA

a

aA

A

n

Q

n

Q δδδδ

(18S).

The uncertainty of the measured heats of adsorption/desorption given in the main paper

are on the order of ±5-10% of the measured values.

References

(S1) Zimmermann, W.; Keller, J. U. A New Calorimeter for Simultaneous

Measurement

of Isotherms and Heats of Adsorption. Thermochim. Acta 2003, 405, 31-41.

92

Chapter III

Preface

The previous chapter demonstrates that the method developed in chapter 1, namely

deriving models for the heats of adsorption from isotherms via the Tóth potential

function, is not only useful in the limited scope of predicting the heats of adsorption in

cases where only a single isotherm is available and therefore standard methods

(Clapeyron or Clausius-Clapeyron) cannot be applied but more profoundly that in

situations where multiple isotherms are available, predictions made with Tóth potential

function derived models are better than the standard methods.

Though it is unremarked upon in the immediately preceding chapter, it is perhaps worth

noting that the modified Clapeyron equation (equation 10 in chapter II) predicts a heat

greater than the Clapeyron equation (equation 6 ibid.), because the volume change must

always be less than the volume of the gas (∆v<vg). Yet when compared to the

calorimetric data the heat predictions made with the modified Clapeyron model tend to

be low, in particular for isobutane they are generally about 10% low (at the very limit of

the uncertainty estimate) and for ammonia they are 20-30% low. So the conclusion that

the Tóth potential function derived models are preferable to Clapeyron derived models

(based on the available data) cannot be explained as being due to the necessary

simplifying assumption of negligible adsorbed phase volume. Rather the short comings

of the modified Clapeyron derived model in the preceding chapter is probably best

explained as being due to (and illustrative of) the difficult in making accurate

measurements of the local pressure temperature gradient, even in a well-controlled

experiment.

Buoyed by the results of chapters I and II the following chapter, a manuscript of yet

unpublished research work, is an attempt at modeling the adsorbed phase heat capacity

with an eye on describing the adsorbed phase entropy and giving the complete

thermodynamic property field of the adsorbate. The back bone of the heat capacity

calculations is a model for the integrated heat of adsorption derived from the Tóth

isotherm equation and Tóth potential function which is validated by the many working

pairs examined in chapters I and II.

93

Predicting Adsorbed Phase Heat Capacity: A Study of Argon on Rutile

Peter B. Whittaker1, Hui Tong Chua1*

1School of Mechanical and Chemical Engineering, The University of Western Australia,

35 Stirling Highway, Perth, WA 6009, Australia

*Corresponding Author

Abstract: Two models for the adsorbed phase heat capacity are developed from the

Dubinin-Astakhov isotherm model; one following the route of Clausius-Clapeyron and

the other via Tóth potential function with ideal gas assumptions. The associated heat of

adsorption models are fitted to literature data for the heat of adsorption of argon on

rutile phase titanium dioxide and the parameters obtained are used in the heat capacity

models to make predictions about the behavior of the heat capacity. These predictions

are compared with measured heat capacity data and show that the model developed

from the Tóth potential function is able to give both a reasonable estimate of the heat

capacity and correctly predict the trends of the heat capacity with changes in

temperature and adsorbate uptake. Moreover the fitting parameters found for the heat of

adsorption model based on the Tóth potential function are also able to correctly predict

isotherm behavior when compared to isotherm data published independent from heat of

adsorption and heat capacity data.

Keywords: Adsorption, Modeling, Prediction, Isotherm

94

Introduction

A better understanding of the adsorbed phase heat capacity would lend itself to a

complete description of the thermodynamic property field of the adsorbed phase1 and

would enhance the understanding, analysis and design of adsorption cycles.2 The heat

capacity of the adsorbed phase is a topic which has spawned numerous publications in

the literature1-7 yet no model has yet gained wide acceptance and the simplifying

assumptions used in the treatment of the adsorbed phase heat capacity can lead to the

conclusion that the adsorbate heat capacity can be treated as being the same as an ideal

gas6 or a saturated liquid8 even though this is incongruent to the available data. 9-20

The relationship between the isosteric heat of adsorption and the adsorbed phase heat

capacity is well known.21 Several models for the isosteric heat, derived from the

Dubinin-Astakhov (D-A) isotherm model22 and variations on the Clapeyron equation,

have appeared in the literature.23-26 However these isosteric heat models have rarely

been adapted to describe the adsorbed phase heat capacity; one exception by Rahman et

al.5 presents a heat capacity model based on the D-A isotherm and Clausius-Clapeyron

equation but lacks data for comparison. Another approach to heat capacity modeling is

found in the work of Al-Muhtaseb and Ritter,7 in which isotherms for localized

(Langmuir) and non-localized (2-D fluid) adsorption, consistent with statistical

mechanics, are transformed into isosteric heat models via Clausius-Clapeyron and then

further developed into models for heat capacity –the results presented are purely

conceptual and are not compared to measured data.

A thorough review of the literature9-20 uncovered only a single paper, by Morrison et

al.,20 in which heat capacity data for the adsorbed phase had been reported at multiple

temperatures and a range of coverage along with heats of adsorption data (for an

argon/rutile phase titanium dioxide system). Morrison et al.20 did not provide

adsorption isotherms to go along with their heat capacities however a separate paper by

Drain and Morrison27 does provide data for a single isotherm for argon adsorption on

rutile phase titanium dioxide at 85 K.

95

Theory

The complete range of uptake data of the single available isotherm27 as measured by

Drain and Morrison shows type II behavior in the IUPAC classification.28 However in

the limited range of adsorbate loading corresponding to the heat capacity data20 the D-A

isotherm model,22 normally reserved for microporous adsorbents, can be used in a

superficial manner to fit the isotherm data and by way of either Clausius-Clapeyron or

the Tóth potential function method the isotherm model may be transformed into a model

for the isosteric heat of adsorption.29, 30

The D-A equation is written as

�� = /0�� exp 4− 5��∆6 ln 7��(����� 89*: (1)

where the specific adsorbate uptake is symbolized by ns, R is the universal gas constant,

T is the isotherm temperature, p is the equilibrium pressure and psat(T) is the saturation

pressure of pure adsorbate, a function of the isotherm temperature. The fitting constants

are W0, ∆E and m: according to the theory of volume filling of micropores31 ∆E is the

characteristic energy of the adsorption process, m is related to the distribution of

adsorption sites and W0 is the specific volume of the micropores of the adsorbent.

According to the same theory, vs is the adsorbed phase specific volume which cannot be

directly measured but can be modeled as31

�� = �0 exp=%�� − �0�> (2)

in which v0 is the specific volume of pure liquid adsorbate at known reference

temperature T0 (often the normal boiling point) and α is the thermal coefficient of

limiting adsorption, the value of which is found through fitting.

Previously we have shown how to derive models for the heats of adsorption from

isotherm models using the Clapeyron equation,29 Clausius-Clapeyron equation29 and the

Tóth potential function.29, 30 The data and the way in which the data have been

reported20 are consistent with assuming that the adsorptive behaves as an ideal gas and

so we consider here only models derived from Clausius-Clapeyron and the Tóth

potential function devoid of the gas compressibility factor.

The Clausius-Clapeyron equation as it relates to adsorption is

96

��� = ��?����, �������, ���� A�� (3)

in which qst symbolizes the isosteric heat of adsorption; p(ns, T) can be found by

rearranging equation 1.

Applying equation 3 to the D-A isotherm leads to

��� = BC exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G +

∆6 HIln JKL MNOPQ�CLRC�S��TL UV WX + QC* Iln JKL MNOPQ�CLRC�S��TL UV WXR$Y

(4)

as the model for the isosteric heat; here λT symbolizes the latent heat of vaporization for

pure adsorbate at the isotherm temperature.

The general expression of the isosteric heat model following from the Tóth potential

function and ideal gas assumption is

��� = ΔB + BZ + ��

with

ΔB = �� ln 5[��(��������, �� 9 and in which

[ = ������, �� �����, ����� AC − 1

(5)

where the final expression is Tóth’s correction to the Polanyi potential function. In

equation 5, λp is the latent heat of vaporization at the equilibrium pressure p(ns, T).

Applying equation 1 to equation 5 results in

��� =�� ln \exp H]^_C Iln JKL MNOPQ�CLRC�S��TL UV WXY H ]^*_C Iln JKL MNOPQ�CLRC�S��TL UV WXR$ − 1Y`

+BZ + ��

(6)

as the model for the isosteric heat stemming from the Tóth potential function.

The heat capacity of the adsorbed phase at constant uptake is related to the heat capacity

of the absorptive gas and the isosteric heat of adsorption by

97

+��� = +��� − ������ ��� (7)

where +��� is the adsorbed phase specific heat capacity at constant uptake and +��� the gas

phase specific heat capacity under the same conditions. For an ideal gas +��� = +Z�, the

constant pressure heat capacity.

The derivative of the Clausius-Clapeyron based isosteric heat model (equation 4) with

respect to temperature is given as equation 8.

������ ��� = �BC�� ��� exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G +exp D−Δ6�� aln bF0 exp=%��0 − ��>���0 cd$*Ge?��? aln bF0 exp=%��0 − ��>���0 cd? ∆6 7ln 4F0 exp=%��0 − ��>���0 :8 $*

×ghhiexp jk

kkl−∆6 aln bF0 exp=%��0 − ��>���0 cd$*�� mn

nno �e − 1���?%? + BCe ln 4F0 exp=%��0 − ��>���0 :7�% + e ln 4F0 exp=%��0 − ��>���0 :8

pqqr

(8)

The term BC refers to the difference between the enthalpies of the pure adsorbate species in saturated vapour and saturated liquid states. As these are

saturated properties of a pure fluid the state is specified by only a single variable; for BC that variable is the saturation temperature (� = ��(�) and

therefore

�BC���(���� A�� = sBC���(��s� = +Z��(� − +Zt�(�

(9).

Equation 10 is the derivative with respect to temperature for the isosteric heat model based on the Tóth potential function (equation 6). The BZ term is

similar to BC except that the variable which specifies BZ is the saturation pressure p(ns, T) = psat which leads to equation 11. In the event that p(ns, T) is

below the triple point, saturated vapor and liquid properties at the triple point should be used.29, 30

������ ��� = −u6 aln bF0 exp=%��0 − ��>���0 cd $*R$ a�% +e ln bF0 exp=%��0 − ��>���0 cde�+ �?� a−�e − 1��% + e ln bF0 exp=%��0 − ��>���0 cde�� ln bF0 exp=%��0 − ��>���0 c − u6 aln bF0 exp=%��0 − ��>���0 cd $*

+ �ghi �e − 1��%e ln bF0 exp=%��0 − ��>���0 c + lnvwx

wyexpjkkklu6 aln bF0 exp=%��0 − ��>���0 cd $*�� mn

nnoghiu6 aln b

F0 exp=%��0 − ��>���0 cd $*R$e�� − 1pqrzw{w|pqr

+ �BZ�� ���

(10)

�BZ�� ��� = �+Z��(� − +Zt�(�� ��(�����(� − �t�(���ℎT�(� − ℎt�(�� �����, ���� A�� (11)

In equation 11, ��(� is the saturation temperature, ℎ�~�� and ℎ�~�� are the enthalpies of pure adsorbate in the saturated vapor and saturated liquid states,

respectively; all saturation properties are calculated based upon psat = p(ns, T).

100

A more detailed derivation of equations is given in the Supporting Information.

Results

Morrison et al.20 provided observed heats (integrated-isosteric heats multiplied by

adsorbent mass) from which a polynomial function had been subtracted to leave a

residual heat value, at multiple temperatures. The models for isosteric heat (equations 4

and 6) do not have analytic solutions for their integrals but can be integrated

numerically. Working in Mathematica we multiplied the numerical integration of the

isosteric heat models by the mass of adsorbent reported24 for the experiment (i.e. 0.039

kg) to get observed heat models and then subtracted the polynomial � = 513.49 × �� −98.64 × ��? to obtain a function to fit the residual heat data. The results of the fitting

and original residual heat data are shown in figure 1.

101

Figure 1. Residual heat data20 and best fits of numerically integrated forms of equation 4

(Clausius-Clapeyron model) and equation 6 (Tóth potential function model) to those

data.

The model parameters found from fitting the integrated form of equation 3 to the

residual data are presented in table 1.

102

Table 1. Model parameters determined by fitting to the residual heat data.24

model parameter values from fitting

Clausius-

Clapeyron derived

model

values from fitting

Tóth potential

function derived

model

units

∆E 5.06×102 2.26×103 J mol-1

W0 3.61×10-3 5.63×10-5 m3 kg-1

m 8.03×10-1 1.25

α 7.50×10-2 3.60×10-3 K-1

The parameters from table 1 column 2 were used in equation 8 and the values from table

1 column 3 were used in equation 10, along with the various thermodynamic properties

for pure argon incorporated into those equations to calculate ���� ��⁄ |��. These

calculated values were then used in equation 7 along with the ideal gas constant

pressure heat capacity for a monatomic gas (5/2 R) to calculate the constant coverage

specific heat capacity of the adsorbed argon. The results are presented in figure 2

alongside data reported by Morrison et al.20 for three uptake values.

103

Figure 2. Heat capacity models alongside reported heat capacity data20 as well as the

heat capacity of a monatomic ideal gas and pure saturated liquid argon which are shown

as familiar points of reference for the reader. Tile A shows the heat capacity model

derived with the Clausius-Clapeyron equation (equation 8 inserted into equation7) while

tile B shows the heat capacity model derived with the Tóth potential function (equation

10 inserted into equation 7).

104

Using the same two sets of fitting parameters from table 1, together with equation 1,

allows us to make predictions about the isotherms that correspond to the residual heat

and heat capacity data. Figure 3 presents those predictions for each set of fitting

parameters from Table 1, for the 85 K isotherm (the only temperature for which there is

corresponding data) alongside data that was published separately27 and obtained

independently from the heat capacity and residual heat data.20

Figure 3. 85 K isotherm, predictions and data. For the isotherm predictions, pressures

were calculated as a function of uptake by rearranging equation 1.

105

Discussion

Figure 1 shows that both models for the residual heat (the numerically integrated forms

of equations 4 and 6) were able to fit the data with similar fidelity while table 1 shows

that the fitting parameters obtained from fitting the two models to the residual heat data

are very different.

Figure 2 on the other hand shows that in spite of the apparently similar fits to the

residual heat data, the two models predict quite different behavior for the adsorbed

phase heat capacity. The model derived following the route of Clausius-Clapeyron (tile

A of figure 2) is unable to resolve the trend of decreasing heat capacity with increasing

adsorbate uptake between 85 K and 90 K and shows that with increasing temperature

the heat capacity decreases rapidly. This is inconsistent with the argon on rutile data.20

In contrast to this, the model developed via the Tóth potential function (tile B of figure

2) maintains the trend of decreasing heat capacity with increased uptake over the full

temperature range which is consistent with the data; both the data and the predictions

show an increase in heat capacity with temperature however while the data show only a

modest increase, the model predicts a more marked one.

Figure 3 shows that only the model coming forth from the Tóth potential function is

able to give a reasonable prediction of the isotherm behavior. Although we have only a

single isotherm for comparison, this reinforces our previous findings 29 which have

shown that heat of adsorption models derived from isotherm models with the Tóth

potential function approach are superior at connecting isotherms to heats of adsorption.

We tried to compare the models developed in this work with those presented by Al-

Muhtaseb and Ritter,7 however we were unable to get a good fit to the heat data using

their models. We believe this is because the isotherm models they started from

(Langmuir and 2-D fluid) are unable to describe the isotherm behavior of the

argon/rutile TiO2 system in the range of interest.

We also closely examined the work of Rahman et al.5 They began with the same

isotherm as we have used (equation 1) and applied the Clausius-Clapeyron procedure

(equation 3) but their expression for the isosteric heat of adsorption was incorrect and

hence so was their heat capacity model, making any comparison pointless.

106

Conclusion

Heat of adsorption models derived from the Dubinin-Astakhov isotherm with either the

Clausius-Clapeyron equation or the Tóth potential function are able to fit the residual

heat data for this system (argon adsorbed onto rutile phase TiO2). However only the

latter model provides a consistent description of the adsorbed phase heat capacity when

compared to the data in terms of behavior with increasing uptake and temperature.

Furthermore the model parameters found from fitting the heat of adsorption model

derived from the Tóth potential function to the residual heat data are also able to predict

the isotherm behavior when inserted into the D-A isotherm model (equation 1). The

superior ability of the Tóth potential function derived model in comparison to the

Clausius-Clapeyron derived model to link isotherms and their associated heats of

adsorption is consistent with our previous work.30

Although the predictions made by the heat capacity model derived via the Tóth potential

function are imperfect, estimating the heat capacity of the adsorbed phase based on

model predictions is a step forward from blindly assuming the adsorbed phase heat

capacity will behave as an ideal gas or saturated liquid.

Supporting Information

More thorough derivations of the isosteric heat and heat capacity models are given in

the supporting information. This information is available free of charge via the Internet

at http://pubs.acs.org

Acknowledgments

The authors acknowledge financial support from the Western Australian Geothermal

Centre of Excellence, ARC Linkage Grant LP110100597 and the University of Western

Australia via the Completion Scholarship. The funding in no way influenced the

theories, analysis or outcomes reported in this paper. The authors thank Oleksandr R.

for help with model fitting in Mathematica.

107

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110

Supporting Information

All symbols are as defined in the text of the main paper unless otherwise stated.

Isosteric Heat Models

Clausius-Clapeyron

The Dubinin-Astakhov isotherm is

�� = /0�� exp 4− 5��∆6 ln 7��(����� 89*:

with

�� = �0 exp=%�� − �0�>

(S1a)

and can be rewritten in pressure explicit form as

���~, �� = �~����� exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G (S1b).

The Clausius-Clapeyron equation for adsorption is

��� = ��?����, �������, ���� A�� (S2).

Applying equation S1b to the second term on the right hand side of equation S2 and

following from the product rule for derivatives

�����, ���� A�� = ��~������� A�� exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G +

��(���� ���exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>�~�0 :81eG��~

(S3);

111

the first derivative is defined by the Clausius-Clapeyron equation applied to phase

change in a pure substance

��~������� A�� = BC����, ����? (S4)

and the second derivative in equation S3 comes out to

��� exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G���

= exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G ×

H ]^_C� Iln JKL MNOPQ�CLRC�S��TL UV WX + Q]^*_C Iln JKL MNOPQ�CLRC�S��TL UV WXR$Y

(S5).

Using the results of equations S4 and S5 in equation S3 and then substituting the result

into equation S2 and cancelling terms results in the isosteric heat model:

��� = BC exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G +

∆6 HIln JKL MNOPQ�CLRC�S��TL UV WX + QC* Iln JKL MNOPQ�CLRC�S��TL UV WXR$Y

(S6).

112

Tóth Potential Function

In its most general form the Tóth potential function isosteric heat model is

��� = ΔB + BZ + ���

with

ΔB = �� ln 5[��(��������, �� 9 in which

[ = ������, �� �����, ����� AC − 1

(S10)

Where Z is the compressibility factor of the adsorptive gas and is equal to 1 when the

ideal gas assumption is invoked. Substituting equation 1b into the pressure-uptake

derivative leads to

�����, ����� AC = Δ6��(������e��

× exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G 7ln 4F0 exp=%��0 − ��>���0 :8 $*R$

(S11)

which in turn leads to

[ = Δ6e�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*R$ − 1

(S12)

and ultimately to

��� =�� ln \exp H]^_C Iln JKL MNOPQ�CLRC�S��TL UV WXY H ]^*_C Iln JKL MNOPQ�CLRC�S��TL UV WXR$ − 1Y`

+BZ + ���

(S13).

113

Heat Capacity Models

We start by defining the heat capacity of the adsorbed phase as

+��� ≡ �ℎ��� ��� = �ℎ��� A�� −������ ��� with

�ℎ��� A�� =�ℎ��� AZ +

�ℎ������, ��AC �����, ���� A��

= +Z� + ���1 − %��������, ���� A��

(S14a)

where hg is the enthalpy of the gas phase adsorptive, hs is the enthalpy of the adsorbed

phase adsorbate and %� is the gas phase volume expansivity. The expansivity %�, where the subscript X can either be g (gas phase) or l (liquid phase), will be

distinguished from the fitting parameter % (the thermal coefficient of limiting

adsorption) by the use of subscripts and where needed superscripts. We note that for an

ideal gas the enthalpy is a function of temperature alone

�ℎ���������, ��AC = 0

∴ +��� = +Z�−������ ���

(S14b).

114

Clausius-Clapeyron

This derivative is quite straight forward. As shown in the main paper

������ ���= �BC�� ��� exp D−Δ6�� 7ln 4F0 exp=%��0 − ��>���0 :8 $*G

+exp D−Δ6�� aln bF0 exp=%��0 − ��>���0 cd $*Ge?��? aln bF0 exp=%��0 − ��>���0 cd? ∆6 7ln 4F0 exp=%��0 − ��>���0 :8 $*

×ghhiexp jk

kkl−∆6 aln bF0 exp=%��0 − ��>���0 cd $*�� mn

nno �e − 1���?%?

+ BCe ln 4F0 exp=%��0 − ��>���0 : 7�% + e ln 4F0 exp=%��0 − ��>���0 :8pqqr

with

�BC����� ��� = sBC���s� = sℎ��(�s� − sℎt�(�s� = +Z��(� − +Zt�(�

(S15)

The term BC refers to the difference between the enthalpies of the pure adsorbate species

in saturated vapour and saturated liquid states. As these are saturation properties of a

pure fluid the state is specified by only a single variable; for BC that variable is the

saturation temperature (� = ��(�) and all other saturation properties are determined by

��(�.

Tóth Potential Function

The heat capacity of the adsorbed phase is the sum of the temperature derivatives of the three components of the isosteric heat model (equation S10 or

S13):

������ ��� = �ΔB�� ��� + �BZ�� ��� + ������ ��� (S16)

The ΔB derivative with respect to temperature is dependent on the isotherm model, for equation 1 it has the form

�ΔB�� ��� = −� − u6 aln bF0 exp=%��0 − ��>���0 cd $*R$ a�% + e ln bF0 exp=%��0 − ��>���0 cde�+ �?� a−�e − 1��% + e ln bF0 exp=%��0 − ��>���0 cde�� ln bF0 exp=%��0 − ��>���0 c − u6 aln bF0 exp=%��0 − ��>���0 cd $*

+ �ghi �e − 1��%e ln bF0 exp=%��0 − ��>���0 c

+ lnvwxwyexp

jkkklu6 aln bF0 exp=%��0 − ��>���0 cd $*�� mn

nnoghiu6 aln b

F0 exp=%��0 − ��>���0 cd $*R$e�� − 1pqrzw{w|pqr

(S17)

The BZ term is similar to BC in that it refers to a property of a pure, saturated substance except that it is a function of saturation pressure. The saturation

pressure is equal to the equilibrium pressure of the isotherm, which is itself a function of uptake and temperature, that is ��(� = ����, ��. �BZ�����, ����� A�� =

sBZs� �����, ���� A�� = 7sℎT�(�s� − sℎt�(�s� 8�����, ���� A��= P���(��1 − %��(���(�� − �t�(��1 − %t�(���(��S �����, ���� A��

alternatively, following from the fact that fixing the saturation pressure determines the saturation temperature a change of

variables is possible

�BZ�����, ����� A�� =sBZ���(����(���s� s��(�s��(� �����, ���� A�� = 7sℎT�(�s� − sℎt�(�s� 8��(�∆��(�BZ �����, ���� A��

= �+Z��(� − +Zt�(�� ��(�����(� − �t�(���ℎT�(� − ℎt�(�� �����, ���� A��

(S18)

with all saturated properties calculated from ��(�. Here %�~�� and %�~�� are the saturated vapor and saturated liquid volume expansivities, respectively.

For the Z R T term, for an ideal gas where Z=1 the derivative with respect to the temperature is just R. Otherwise we substitute Z R T = p vg and take

the derivative as follows:

������ A�~ =������ = ���~, ���������~, ��, ���� A�~ + �� ����

~, ���� A�~ where the total derivative of �� leads to

�������~, ��, ���� A�~ =����� A� + 7 �������~, ��8C 7����

~, ���� 8�~ = ��%� − ���� 7����~, ���� 8

(S19)

in which �C is the isothermal compressibility of the adsorptive and %� is the gas expansivity.

119

Conclusion

How does this thesis contribute to the field of adsorption?

Chapter I introduced a method of calculating the heats of adsorption based on the isotherm

measurement. The author cannot claim this method to be completely new as it based on the

earlier, published observations of Tóth, however to the best of the authors knowledge

neither Tóth nor anyone else has used this observation to form predictive models for the

heats of adsorption from isotherm models. This method is substantially different from the

classical method used in literature, which is constructed around the Clapeyron equation.

The method presented in Chapter I requires knowledge of the uptake-pressure gradient,

something that can be obtained from a single isotherm measurement, unlike the Clapeyron

equation which requires knowledge of the pressure-temperature gradient and therefore the

careful measurement of multiple, closely spaced isotherms. This method thus gives new

information about literature data in the case where only a single isotherm was published

and could also be useful to busy experimentalists who want an estimate of the heat of

adsorption but do not want to invest the time to measure multiple isotherms or are

prevented from making such measurements by other constraints.

Chapter II reinforces the value of the method presented in Chapter I by providing a direct

comparison with models formed from the Clausius-Clapeyron equation which incorporates

the common assumptions of adsorptive ideal gas behavior and negligible adsorbate volume

and a more rigorous modified Clapeyron model that treats the adsorptive as a real gas.

Based on measured isotherms at several temperatures for three different

adsorbate/adsorbent systems and corresponding integral heats obtained through calorimetric

measurement, it was shown that the method of predicting the heats of adsorption posited in

this thesis was superior, for the examined systems, to the Clausius-Clapeyron method and

the modified Clapeyron method. The method presented in this thesis is thus not only of use

when only a single isotherm is available but should be considered alongside or even

120

preferred to Clapeyron based methods in the case where multiple isotherms have been

measured.

Chapter III adapts the method to modeling the specific heat capacity of the adsorbed phase.

Both heat of adsorption models derived from Clausius-Clapeyron and the Tóth potential

function were able to fit the available residual heat data reasonably well, however the

predictions about the adsorbed phase heat capacity coming from the Tóth potential function

method were better at reproducing the trends of the measured data. What’s more, only the

fitting parameters stemming from the Tóth potential function derived model were able to

predict the shape of a measured isotherm. In addition to the success of the modeling

method in its own right, this chapter is significant because although there are other

published models for the adsorbate specific heat capacity, the one presented in Chapter III

appears to be the only one that has ever been presented alongside measured values, a fact

which is notable in and of itself. No doubt this in part because of the scarcity of adsorbed

phase heat capacity data in the literature.

Possible future work

In Chapter II it is suggested that the reason for the superiority of the Tóth potential function

method to methods with Clapeyron type equations is the difficulty of accurately measuring

the localized pressure-temperature gradient. It may be worth pondering an alternative

explanation. The modified Clapeyron method does quite well predicting the integral heats

of adsorption for n-butane and isobutane on activated carbon but significantly under

predicts the heat for ammonia adsorption on silica gel. Silica gel is known to be a weak

acid and anhydrous ammonia is known to be a proton accepter, it may therefore be the case

that it is wrong to assume that the adsorbent is completely inert in the adsorption process as

is generally assumed for physical adsorption and that the enthalpy change upon adsorption

should not be completely ascribed to the volume change of the adsorptive-adsorbate phase

transformation. This should be investigated further.

121

At present the method put forth in this thesis for predicting the heats of adsorption are

applicable only to adsorption of pure gas, which is of interest in the field of adsorption

chillers but not applicable for gas separation by adsorption or adsorbed natural gas storage.

Further work could be carried out to adapt the method to mixed gas adsorption scenarios.

Recently the author has been involved in the design and construction of a Tian-Calvet type

calorimeter for gas adsorption studies. In addition to filling the hole in the literature for

data sets containing isotherms, measured heats and heat capacities, this calorimeter will be

used to measure the heats of adsorption at high pressure (up to 5 MPa). At UWA the

author has been part of a project which has measured adsorption isotherms for methane on

activated carbon up to 5 MPa, the isotherms can be modeled by the Langmuir equation

(equation 8 of Chapter I). Whether the method advanced in this thesis is applied (equation

9 of Chapter I) or the Clapeyron equation is used, the predicted behavior of the isosteric

heat at high pressure is not consistent with widely accepted Langmuir behavior (a constant

isosteric heat of adsorption). With regard to equation 9 of Chapter I, as the equilibrium

pressure increases λp the latent heat at the equilibrium pressure declines and goes to zero as

the vapor-liquid critical point is reached. On the other hand if a Clapeyron type model is

used with real gas behavior, the gas compressibility factor lowers the predicted isosteric

heat as pressure increases and at supercritical pressures treating the adsorbate specific

volume as negligible compared to the adsorptive specific volume is no longer valid –

though how to measure or model the adsorbate volume remains an open question.

122

Appendix A

Due to copyright this appendix cannot appear in the publically available version of

Predicting the Heats of Adsorption for Gas Physisorption from Isotherm

Measurements

The appendix is available at dx.doi.org/10.1039/C2CP41756A or via www.rsc.org/pccp

133

Appendix B

Due to copyright this appendix cannot appear in the publically available version of

Predicting the Heats of Adsorption for Gas Physisorption from Isotherm

Measurements

The appendix is available at dx.doi.org/10.1021/jp410873v or via pubs.acs.org/JPCC

143

Appendix C

The appendix is available at dx.doi.org/10.1051/smdo/2013010 or via www.ijsmdo.org

150

Appendix D

Due to copyright this appendix cannot appear in the publically available version of

Predicting the Heats of Adsorption for Gas Physisorption from Isotherm

Measurements

The appendix is available at dx.doi.org/10.1016/j.applthermaleng.2012.05.011 or via

www.journals.elsevier.com/applied-thermal-engineering/