A New Classification of Isotherms for Gibbs Adsorption of Gases On

Ž .Fluid Phase Equilibria 158–160 1999 557–563

A new classification of isotherms for Gibbs adsorption of gases onsolids

M.D. Donohue ), G.L. Aranovich

Chemical Engineering Department, The Johns Hopkins UniÕersity, 3400 North Charles Street, Baltimore, MD 21218, USA

Received 12 April 1998; accepted 25 January 1999

Abstract

A systematic analysis of adsorption behavior has been performed. The results are presented here as a newclassification of isotherms for fluidrsolid equilibria. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Gas–solid equilibria; Lattice theory; Gibbs adsorption; Classification of isotherms

1. Introduction

The first systematic attempt to interpret adsorption isotherms for gasrsolid equilibria wasŽ . w xintroduced by S. Brunauer, L.S. Deming, W.E. Deming and E. Teller BDDT in 1940 1 . These

authors classified isotherms into five types. The BDDT classification became the core of the modernw xIUPAC classification of adsorption isotherms 2,3 . Type I isotherms characterize microporous

adsorbents. Types II and III describe adsorption on macroporous adsorbents with strong and weakadsorbate–adsorbent interactions, respectively. Types IV and V represent adsorption isotherms withhysteresis. Type VI has steps. The BDDT and IUPAC classifications have two deficiencies: they areincomplete and they give the incorrect impression that adsorption isotherms are always monotonicfunctions of pressure.

Here, we present a brief summary of a comprehensive analysis of adsorption behavior. Thisanalysis is based on experimental results, molecular simulations, and lattice theory concepts proposed

w xby Ono and Kondo 4 in 1960. The Ono–Kondo approach was derived to describe density gradientsat a vapor–liquid interface and it has become a classical part of molecular theory for liquid surfacesw x w x5 . This theory has been developed for gas–solid and liquid–solid equilibria 6–11 .

) Corresponding author. Tel.: q1-410-516-7761; fax: q1-410-516-5510; e-mail: [email protected]

0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0378-3812 99 00074-6

( )M.D. Donohue, G.L. AranoÕichrFluid Phase Equilibria 158–160 1999 557–563558

2. Ono–Kondo equations

Here we consider lattice theory for a one-component adsorbate with ´ being the energy ofinteraction between nearest neighbors, and ´ being the energy of interaction between surface ands

adsorbate molecule in the first layer. Consider taking an adsorbate molecule at a layer i and moving itto an empty site in an infinitely distant layer. This is equivalent to the exchange of a molecule with a

w xvacancy 7 ,

M qV ™ V qM 1Ž .i ` i `

Ž .where M is the adsorbate molecule, and V is the vacancy that it fills and vice versa . If thisexchange occurs at equilibrium, then:

D HyTDSs0 2Ž .where D H and DS are the enthalpy and entropy changes, and T is the absolute temperature. Thevalue of DS can be represented in the form:

DSsk ln W yk ln W 3Ž .B 1 B 2

where W is the number of configurations where site in the layer i is occupied by an adsorbate1

molecule and the site in the infinitely distant layer is empty, and W is the number of configurations2

where the site in the infinitely distant layer is occupied by an adsorbate molecule and site in the layeri is empty, and k is Boltzmann’s constant.B

If the overall number of configurations for the system is W , then:0

W rW sx lyx 4Ž . Ž .1 0 i `

and

W rW sx lyx 5Ž . Ž .2 0 ` i

where x is the fraction of sites occupied by molecules in layer i, and x is the fraction of sitesi `

occupied with fluid molecules in the bulk.Ž . Ž . Ž .Substituting Eqs. 4 and 5 into Eq. 3 we obtain

DSsk T ln x lyx r lyx x 6Ž . Ž . Ž .B i ` i `

The change in enthalpy can be calculated by considering the number of neighboring sites that areoccupied near the surface compared to the bulk:

D Hsy´ z x qz x qz x yz x 7Ž . Ž .1 iq1 2 i 1 iy1 `

where z is the volume coordination number, z is the monolayer coordination number, and2Ž . Ž . Ž . Ž .z s zyz r2. From Eqs. 2 , 6 and 7 it follows that for iG2:1 2

ln x lyx r lyx x q ´rk T z x yx qz x yx qz x yx s0� 4Ž . Ž . Ž . Ž . Ž . Ž .i ` 1 ` B 1 iq1 ` 2 i ` 1 iy1 `

8Ž .Ž .For is1, Eq. 7 must be modified to include the effect of the interface. Therefore,

D Hsy´ z x qz x yz x q´ 9Ž . Ž .2 1 1 2 ` s

Ž . Ž . Ž .Combining Eqs. 2 , 6 and 9 , we obtain:

ln x lyx r lyx x q ´rk T z x qz x yz x q´ rk Ts0 10� 4Ž . Ž . Ž . Ž . Ž .1 ` 1 ` B 2 1 1 2 ` s B

( )M.D. Donohue, G.L. AranoÕichrFluid Phase Equilibria 158–160 1999 557–563 559

3. Analysis of adsorption isotherms

3.1. Monolayer adsorption

Ž . Ž .Eqs. 8 and 10 represent a set of coupled equations. For monolayer adsorption we have x sxi `

Ž . Ž .for iG2. In the low concentration limit, Eq. 10 gives x sx exp y´ rk T which is Henry’s1 ` s BŽ .law. When there are no adsorbate–adsorbate interactions, ´s0, it follows from Eq. 10 that

w Ž . Ž .xx sx r x q 1yx exp ´ rk T which is analogous the Langmuir isotherm. If x <x then Eq.1 ` ` ` s B ` 1Ž . w Ž . Ž .x10 gives x sx r x q 1yx exp ´ rk Tqz x ´rk T . This is similar to the Frumkin adsorp-1 ` ` ` s B 2 1 B

tion isotherm.

3.2. Multilayer adsorption

Ž . ` Ž .The Gibbs adsorption for multilayer adsorption is G x sÝ x yx . Calculation of isotherm` is1 i `

Ž . Ž .from Eqs. 8 and 10 shows that increasing ´ rk T leads to an increase in the Henry’s constant ands B

a convex shape of the isotherm at small densities. This is a characteristic feature of Type II and IVw xisotherms in the IUPAC classification 2,3 . For small ´ rk T , the isotherm becomes concave ats B

small densities. This behavior is known for Type III and Type V isotherms in the IUPACŽ . Ž .classification. For low temperature calculations, Eqs. 8 and 10 exhibit steps in the isotherms

resulting from two-dimensional phase transitions. This kind of isotherm is considered Type VI in thew xIUPAC classification 2,3 .

3.3. Adsorption at supercritical conditions

Fig. 1 illustrates the behavior of the Gibbs adsorption near and above the critical temperature. Fig.Ž . Ž .1a gives theoretical predictions obtained from Eqs. 8 and 10 . Fig. 1b shows experimental

Ž .Fig. 1. Isotherm of the Gibbs adsorption in a wide range of conditions. a Theoretical isotherms for different sets ofparameters: 1—for ´ rkT sy0.7 and ´ rkT sy5.2; 2—for ´ rkT sy2r3 and ´ rkT sy5.2; 3—for ´ rkT sy2r3s s

Ž .and ´ rkT sy2r3; 4—for ´ rkT sy0.6 and ´ rkT sy2r3; b Experimental isotherms for ethylene on carbon blacks s

at different temperatures: 1—263 K, 2—273 K, 3—283 K, 4—285 K, 5—288 K, 6—293 K, 7—303 K, 8—313 K, and 9w x—323 K. Data from Ref. 12 .


Žadsorption isotherms of ethylene on carbon black for temperatures from 263 K to 323 K data fromw x.12 . As can be seen from Fig. 1b, adsorption isotherms for subcritical ethylene are Type II in theIUPAC classification. However, for supercritical ethylene, the adsorption isotherms show maxima thatrange from being very pronounced to rather weak. Subcritical isotherms are monotonically increasingfunctions of the density. Supercritical isotherms have maxima, and at a higher density the Gibbsadsorption goes down as the density goes up. Subcritical isotherms show a singularity at the saturation

Žconditions. The remnants of this singularity lead to the peak at near-critical temperatures curves 2.and 3 in Fig. 1a . Supercritical isotherms do not have steps.

Fig. 1 demonstrates the theoretical roots and experimental evidence for those types of adsorptionisotherms which are beyond the current IUPAC classification. In particular, curve 1 in Fig. 1a is for

Ž .subcritical conditions mean-field critical point is at ´rk Ts2r3 . Curve 2 in Fig. 1a is at criticalB

conditions, and at high affinity to the surface. Curve 3 in Fig. 1a is at the same bulk conditions but forlow affinity to the surface. Curve 4 in Fig. 1a illustrates the adsorption behavior at the wellsupercritical temperature and low affinity to the surface. Fig. 1b shows experimental data forsubcritical, near-critical, and supercritical conditions. As seen from Fig. 1a and b, at subcriticalconditions we have standard BET-like adsorption isotherms. However, near the critical temperaturethese isotherms change dramatically to non-monotonic behavior showing sharp maxima, and furtherincrease in temperature leads to isotherms with smooth maxima. These types of behavior are predictedtheoretically and observed in experiment, but not included in the IUPAC classification.

4. Adsorption in pores and adsorption hysteresis

When there is adsorption in porous adsorbents, there is a limit to the number of adsorbed layers. Inslit-like pores with n layers there are two surfaces. For nG3 equations of equilibrium can berepresented in general form:

n

ln x 1yx r 1yx x q ´rk T A x yB x qC ´ rk Ts0 11Ž . Ž . Ž . Ž .Ýi ` i ` B i k i k ` k s Bž /is1

where 1FkFn.Ž .Eq. 11 indicates the possibility of having stepped isotherms for microporous adsorbents. Such

steps have been observed experimentally in high resolution measurements for argon and nitrogen onw x w xmicroporous adsorbents 13 . The current IUPAC classification of adsorption isotherms 2,3 does not

include this type of behavior.Ž .Eq. 11 also has multiple solutions at low temperatures. When this occurs, a particular bulk

density can be in equilibrium with two densities in adsorbed phase. However, this does not lead tohysteresis in the adsorption isotherm. Rather, the requirement that phases in equilibrium have equal

w xspreading pressures results in steps in the adsorption isotherm 14 .

5. New classification of adsorption isotherms

w xThe current IUPAC classification of adsorption behavior for gasrsolid equilibria 2,3 reflects thew xideas of Brunauer 15 based on knowledge in the 1930s–1940s. Though not stated explicitly in the


Fig. 2. New classification of adsorption isotherms.

w xIUPAC publications, the IUPAC classification is limited to condensable vapors 16 . However, manyŽimportant gasrsolid systems fall outside this classification for example, nitrogen, oxygen, fluorine,

hydrogen, carbon dioxide, carbon monoxide, nitric oxide, methane, ethylene, some freons at a room.temperature . Moreover, this restriction is ambiguous because gases can be non-condensable in the

w xbulk, but condensable in pores of an adsorbent. In our opinion, the current IUPAC classification 2,3is obsolete.

Fig. 2 shows a new classification of adsorption isotherms. While general, the classification shownin Fig. 2 is meant to be qualitative and does not show all possible details. In this classification, Type Ishows adsorption isotherms on microporous adsorbents for subcritical, near critical and supercriticalconditions. At supercritical conditions, the isotherm is not monotonic. Types II and III give adsorptionisotherms on macroporous adsorbents with strong and weak affinities, respectively. For low tempera-tures these Types have steps, but increasing temperature transforms them into the smooth monotoniccurves which are like those in Types II and III of the IUPAC classification. However, near the criticaltemperature these isotherms change dramatically to non-monotonic behavior showing sharp maxima,and further increase in temperature leads to isotherms with smooth maxima. Types IV and Vcharacterize mesoporous adsorbents with strong and weak affinities, respectively. For lower tempera-tures they show adsorption hysteresis. We also have included supercritical isotherms for mesoporousadsorbents which are predicted by lattice theory and the logic of this classification scheme. However,we are not aware of any experimental data showing the disappearance of hysteresis for mesoporoussystems at supercritical temperatures.

6. Discussion

Many advances have been made over the last sixty years in understanding adsorption behavior. Ofw xparticular note are advances that have been made using density functional theory 17,18 , self-con-

w x w x w x.sistent field theory 19 , molecular dynamics 20 , and Monte Carlo simulations 21,22 . Here, wehave presented an analysis made by a much simpler theoretical approach. We have shown that the


Ono–Kondo model is simple, flexible with respect to various boundary conditions, and able to predictdifferent adsorption behavior in a systematic way.

Ono–Kondo lattice theory is able to predict all known types of adsorption behavior including stepsw x w x w xin the isotherms 9 , scaling behavior near saturation conditions 7 , supercritical behavior 8 , and

w x Žadsorption hysteresis 6 . By changing two energetic parameters energies for adsorbate–adsorbate.and adsorbate–adsorbent interactions , one can obtain smooth or stepped multilayer adsorption

isotherms for macroporous adsorbents. By adopting appropriate boundary conditions, it is possible tow xdescribe adsorption on microporous adsorbents 11 . The monolayer version of the theory gives all

w xknown types of isotherms for monolayer adsorption of gases on solids 23 . Three-dimensionalw xequations allow one to predict adsorption on heterogeneous surfaces 10 and to analyze adsorption

w xhysteresis 6 .However, it should be noted that the new classification of adsorption isotherms shown in Fig. 2 is

not based entirely on the model. Rather, we have used this simple model to organize a wide spectrumof experimental results in a logical and systematic way. This could have been done using any ofseveral different approaches. Our conclusion from this analysis is that the IUPAC classification isobsolete. In particular, experiments and calculations shown in Fig. 1 demonstrate isotherms with

Žmaxima at supercritical conditions which is ‘forbidden’ in the framework of the current IUPACw x.classification 2,3 . Monte Carlo simulation at these conditions confirms this type of adsorption

w xbehavior 22 . This kind of behavior is beyond the current IUPAC classification, and it is included inthe new classification shown in Fig. 2.

7. List of symbols

k Boltzmann’s constantB

M Adsorbate moleculen Number of adsorbed layersT Absolute temperatureV VacancyW Overall number of configurations0

W Number of configurations where site in the layer i is occupied by an adsorbate molecule1

and the site in the infinitely distant layer is emptyW Number of configurations where the site in the infinitely distant layer is occupied by an2

adsorbate molecule and site in the layer i is emptyx Fraction of sites occupied by molecules in layer ii

x Fraction of sites occupied with fluid molecules in the bulk`

z Volume coordination numberz Monolayer coordination number2

Ž .z zyz r21 2

G Gibbs adsorptionD H Enthalpy changeDS Entropy´ Energy of interaction between nearest neighbors´ Energy of interaction between surface and adsorbate molecule in the first layers


Acknowledgements

MD would like to acknowledge support by the Division of Chemical Sciences of the U.S.Department of Energy, under Contract DE-FG02-87ER13777.

References

w x Ž .1 S. Brunauer, L. Deming, W. Deming, E. Teller, J. Am. Chem. Soc. 62 1940 1723.w x Ž .2 IUPAC Recommendations, Pure Appl. Chem., 57 1985 603.w x Ž .3 IUPAC Recommendations, Pure Appl. Chem., 66 1994 1739.w x4 S. Ono, S. Kondo, Molecular Theory of Surface Tension, Springer, Berlin, 1960.w x5 J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1982.w x Ž .6 G.L. Aranovich, M.D. Donohue, J. Colloid Interface Sci. 205 1998 121–130.w x Ž .7 G.L. Aranovich, M.D. Donohue, J. Colloid Interface Sci. 189 1997 101–108.w x Ž .8 G.L. Aranovich, M.D. Donohue, J. Colloid Interface Sci. 180 1996 537–541.w x Ž .9 G.L. Aranovich, M.D. Donohue, J. Colloid Interface Sci. 178 1996 204–208.

w x Ž .10 G.L. Aranovich, M.D. Donohue, J. Chem. Phys. 104 1996 3851–3859.w x Ž .11 G.L. Aranovich, M.D. Donohue, Carbon 33 1995 1369–1375.w x12 G.H. Findenegg, Fundamentals of Adsorption, Eng. Found., NY, 1984, pp. 207–218.w x Ž .13 E. Maglara, A. Pullen, D. Sullivan, W.C. Conner, Langmuir 10 1994 4167.w x14 R.H. Fowler, E.A. Guggenheim, Statistical Thermodynamics, Cambridge Univ. Press, Cambridge, England, 1952.w x15 S. Brunauer, The Adsorption of Gases and Vapors, Princeton Univ. Press, 1945.w x16 K.S.W. Sing, J. Rouquerol, Comments at the TRIrPrinceton Workshop on Characterization of Porous Materials, June

18–20, 1997, Princeton, NJ, USA.w x17 P.B. Balbuena, C. Lastoskie, K.E. Gubbins, N. Quirke, Proc. 6th International Conference on Fundamentals of

Adsorption, Kyoto, 1992.w x Ž .18 R. Evans, P. Tarazona, Phys. Rev. Lett. 52 1984 557.w x Ž .19 L.J.M. Schlangen, L.K. Koopal, Langmuir 12 1996 1863.w x Ž .20 M.W. Maddox, C.M. Lastoskie, N. Quirke, K.E. Gubbins, in: M.D. LeVan Ed. , Fundamentals of Adsorption, Kluwer

Academic Publishers, Boston, 1996, p. 571.w x Ž .21 V.A. Bakaev, W.A. Steele, Langmuir 8 1992 1372.w x Ž .22 J. Vermesse, D. Levesque, J. Chem. Phys. 101 1994 9063.w x Ž .23 T. Hocker, G.L. Aranovich, M.D. Donohue, J. Colloid Interface Sci. 211 1999 61.

Documents

A New Classification of Isotherms for Gibbs Adsorption of Gases On