On the molecular basis of adsorbed solution behaviour

Chemical Engineering Science, Vol. 42, No. Printed in Great Britain.

ON

THE

3. pp. 505-513.

1987. 0

MOLECULAR

BASIS OF ADSORBED BEHAVIOUR

elm-2509/87 $3.00 + 0.00 1987 Pergarnon Journals Ltd.

SOLUTION

P. A. MONSON Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A. (Received

13 February 1986; accepted

24 June 1986)

-A theoretical study, based on statistical thermodynamics, of the adsorbed solution behaviour of binary gas monolayers on a homogeneous solid surface is presented. The adsorbate-solid interactions are modelled via the summed 104 potential and the adsorbate-adsorbate interactions as those of a twodimensional fluid mixture in which the molecules interact via Lennard-Jones 124 potentials. The thermodynamic properties of the two-dimensional mixture are obtained from the van der Waals one-fluid model. We present results from Monte Carlo computer simulations of two-dimensional fluid mixtures which support the accuracy of this procedure. The model can be used to study the relative importance of adsorbate-solid and adsorbate-adsorbate interactions in determining the adsorbed solution behaviour. Comparisons with experimental adsorption equilibria data for ethane-propane mixtures adsorbed on graphitized carbon black show that the theory gives excellent predictions of the adsorption equilibria, without adjustable parameters. For this system at 298 K and 700 Torr the adsorption selectivity is dominated by the difference in the Henry’s law constants. However, it is shown that the adsorbate-adsorbate interactions and nonideal adsorbed solution behaviour become more or less important depending on the conditions in relation to the two-dimensional phase diagram.

Abstract

1. INTRODUCTION

A problem separation

of

some

processes

the development

importance based

in

on selective

of accurate

the

design

of

adsorption

is

thermodynamic

models

for adsorption equilibria in multicomponent systems (Ruthven, 1984). In particular, it is desirable to be able to predict multicomponent adsorption equilibria on the basis of pure component data. A useful starting point for treating the thermodynamics of mixed gas adsorption is the concept of the ideal adsorbed solution introduced by Myers and Prausnitz (1965). This adapts the ideas of bulk solution thermodynamics to the adsorption problem. A crucial feature of ideal adsorbed solution theory is the fact that ideal mixing is assumed to occur at constant spreading pressure and temperature. Since the spreading pressure as a function of coverage, temperature and composition is the natural counterpart of the thermal equation of state for a bulk system, mixing at constant spreading pressure is analogous to mixing at constant pressure for bulk fluids. Generally speaking, the predictions of ideal adsorbed solution theory are quite accurate. Exceptions to this include systems with a high degree of heterogeneity, such as activated carbons (Myers, 1968; 1983) and a number of mixtures adsorbed on zeolites (see, for example, Holborow and Loughlin, 1977). Whether or not an adsorbed solution behaves as an ideal solution is determined by a number of factors. To begin with, the adsorbate-adsorbate interactions may or may not be substantially different in the solution from those in the pure component adsorbates at the same spreading pressure. This is the counterpart of the effects which control ideality in bulk solutions. Of course,

the

interactions

magnitude

of

may be quite

the small

adsorbate-adsorbate in relation

to that of

the adsorbate-surface interactions. Under these circumstances, the adsorption equilibrium will be dominated by the absorbate-surface interactions to the arising from the extent that nonidealities adsorbate-adsorbate interactions may not be perceptible. Surface heterogeneity may also play a significant role. If, for example, the length scale of the heterogeneity is of the same order as the adsorbate molecular size, then this may have a screening effect on the adsorbate-adsorbate interactions. Myers (1983) has recently shown that apparent departures from ideal solution behaviour can occur on heterogeneous significant the surfaces even in absence of adsorbate-adsorbate interactions if there is a large relative variation in the strengths of the adsorbate-surface interactions for the different components with heterogeneity. This gives rise to selectivities which may vary with adsorbate with

coverage.

sitewise

A

adsorption

similar may

composition

mechanism

also

in

terms

and of

to the nonon zeolites (Ruthven,

contribute

idealities observed in adsorption 1984). Although the arguments presented above are quite plausible, none of them has yet been given a rigorous foundation in molecular theory. The present paper represents a first step in this direction. We present an analysis of adsorbed solution behaviour from the perspective of statistical thermodynamics. Using wellmodels for the adsorbate-solid and defined adsorbate-adsorbate interactions, we make calculations of adsorption equilibrium from gases on homogeneous surfaces. The present approach treats the adsorbed solution as a two-dimensional fluid mixture. The adsorbatesurface interactions are modelled using the summed l&4 potential. We show how the two-dimensional fluid mixture properties may be calculated via con505

P. A.

506

MONS~N

formal solution theory. Monte Carlo computer simulations are used to assess the accuracy of this procedure. Predictions of the model as a whole are compared with experimental data for ethane-propane mixtures adsorbed on graphitized carbon black. The approach is then used to assess the relative contributions of the adsorbate-solid and adsorbate- adsorbate interactions in determining adsorption equilibria. The remainder of the paper is organized as follows. In Section 2 we review some relevant features of the statistical thermodynamics of adsorption and describe the principal approximations used in our model. In Section 3 we describe the modelling of the molecular interactions, and in Section 4 we discuss the calculation of the two-dimensional fluid mixture properties. In Section 5 we compare the theoretical results with experimental data, and, finally, in Section 6 we present our conclusions and discuss the general significance of our results in relation to the modelling of adsorption equilibria. 2. THEORY

We begin with a review of the fundamental equations describing the statistical mechanics of multicomponent gas adsorption on a solid surface. Our equations are a generalization of the singlecomponent formalism given by Steele (1974, 1976). We treat the adsorbed layer as a separate phase in equilibrium with the bulk gas and we assume that the bulk pressure is sufficiently low that the bulk phase behaves as an ideal gas mixture. Using statistical mechanics in the canonical ensemble and neglecting internal molecular degrees of freedom, we may then write that for the gas phase pig)= kTln

[

g

1

+kTln

and is assumed to be pairwise additive, i.e. uN

=

c

a<8

(4)

u(ra8)

where rap is the distance between the centres of molecules OL and j?. aN denotes the interactions with the solid surface and is treated as a sum of the interaction of the individual absorbate molecules with the surface via (5)

Here, n is the number of components. Combining eqs (1) and (2) and rearranging yields

Gl =-(~)T,vll,N,+i +I$ t6) where xi and yi are the mole fractions of component i in the adsorbed layer and the bulk gas, respectively. This equation shows that the adsorption equilibrium is controlled by the configurational chemical potential of the adsorbed layer and is determined by the intermolecular interactions through the configurational partition function. Contributions come from both the adsorbate-adsorbate and the adsorbate-solid interactions. To make further progress we have to approximate the configurational partition function. The essential approximation in our approach is to treat the adsorbate-adsorbate interactions as those of a twodimensional fluid and to neglect the influence of the surface structure. Using this simplification we find that 2,

=

ZE”’ Ij ZTi

where

Pi

where k is Boltzmann’s constant, T is the absolute temperature, Ai is the thermal de Broglie wavelength for component i and Pi is the partial pressure of component i in the bulk. For the adsorbed layer we obtain &‘=kTlnA~+kTlnNi-kT

(2) T, W,Nj

z i

where Ni is the number of molecules of component i in the adsorbed phase and ZN is the configurational partition function and is given by

7(2D)= ‘N

s

. . . . A@,

exp [ - U$D’/kT]

dr, .

. drN

s

is the configurational partition function of a twodimensional mixture and Zi,i = &

s

exp C- +i

WkTl dr.

(9)

In eq. (8) the integrations are performed over the twodimensional position vectors T with the limits determined by the area, A(‘), of the adsorbed layer. Combining eqs (6) and (7) gives

ev C- U-J, + Q,)/kT]

dr, . . . dr,.

(3)

.The integration is performed over the position vectors ri of the adsorbate molecules with limits determined by the volume, Y(‘), of the adsorbed phase. U,,, is the sum of interactions between the molecules in the monolayer

-lnZ,,i+lnX’.

Yi

(IO)

A more sophisticated treatment which includes the effect of the surface structure is possible (Monson er al., 1981). However, at this stage the simple version of the theory is sufficient for our purposes.

Molecular basis of adsorbed solution hehaviour The remaining tasks are to find an expression for the configurational chemical potential of the components in a two-dimensional fluid mixture and to choose a model for the interaction potential of a molecule with the solid with which to calculate Z, ,i. We address these problems in Sections 3 and 4. To conclude this section, we note that in the absence of absorbate-adsorbate interactions or, more generally, in the limit of low coverage eq. (10) reduces to Pi=

Zl,iPi kT

= KiPi

(11)

where pi is the surface density of component i. This equation will be recognized as Henry’s law for an adsorbed solution. In summary, we are in a position to model multicomponent adsorption equilibrium on a homogeneous surface at three levels of approximation. We may treat the adsorbed phase, firstly, as a nonideal solution; secondly, as an ideal solution; and finally, as an ideal gas mixture. Such modelling reveals the relative importance of the adsorbate-solid and adsorbateadsorbate interactions in determining the adsorption equilibria. 3. INTERACTION

POTENTIAL

MODELS

Interaction potentials which do full justice to the complexities of the gas-solid problem have yet to be developed. The approach used here is aimed at capturing much of the important molecular physics of the problem with minimum complexity. To describe the interaction of a gas molecule with the surface we use the summed l&4 potential (Steele, 1974, 1978). The model has been used to make quite accurate calculations of the Henry’s law constants for simple molecules adsorbed on graphite. The potential is derived by assuming a Lennard-Jones 12-6 potential acting between the gas molecule and each molecule of the solid and averaging the interactions over each layer of solid molecules. The result is a sum of the form

507

To model the adsorbate-adsorbate interactions we have also used a 12-6 potential. The parameters for this potential were chosen to give the best fit to bulk fluid critical temperature and density, using values of the critical density and temperature of the 12-6 fluid from computer simulations (for a review, see Barker and Henderson, 1976). Parameters for argon, ethane and propane obtained in this manner are shown in Table 1. Traditionally, parameters for the 12-6 potential are determined on the basis of a fit to the second virial coefficient of the fluid (Hirschfelder et al., 1964). However, if this is done one finds that the well depths for ethane and propane are quite similar, although one would expect the propane well depth to be significantly larger. Evidently this reflects the inadequacy of the spherical 12-6 potential in modelling the interactions between substantially nonspherical molecules. It is noteworthy that the parameters for argon obtained via the present method are quite close to those which yield the best fit to the second virial coefficient. In our mixture calculations we have estimated the unlike interaction parameters using the Lorentz-Berthelot combining rules (i.e. an arithmetic mean rule for the collision diameters and a geometric mean rule for the well depths). Having obtained the absorbate-adsorbate interaction parameters, we approximate the adsorbate-solid interactions using the Lorentz-Berthelot combining rules and with values of ccc = 3.37 A and E,, = 28 K as suggested by Steele (1978). Values of the adsorbate-solid interaction parameters and the calculated Henry’s constants are given in Table 2. We emphasize that no attempt was made to optimize these parameters to improve agreement with the experimental data. We have neglected the nonsphericity of the molecules in these calculations and this undoubtedly will be a significant source of error. However, at the level of approximation used in the rest of the theory we do not regard this as particularly important. 4. TWO-DIMENSIONAL

FLUID

MIXTURE

PROPERTIES

There exist a number of approaches to the statistical mechanics of bulk fluids which have been adapted to Table 1. Adsorbate-adsorbate interaction parameters In this expression cr‘gs,iand E~,~ are the collision diameter and well depth in the Lennard-Jones 126 potential between a gas molecule of species i and a molecule of the solid. z is the distance of the gas molecule from the surface plane of the solid, d is the interplane spacing and a, is the surface area per carbon atom in the basal plane of graphite. Each term in the summation represents the interaction of a gas molecule with an entire basal plane and the summation is over all planes. We have found that 15 terms gives an accurate representation of the sum in eq. (12) for all values of z of interest. Using this potential we have made calculations of the Henry’s law constants via eq. (9).

Ethane Propane Argon

&sg(K)

U.&A)

235 284 116

4.33 4.81 3.45

Table 2. Adsorbate-solid interactionparametersand calcuHenry’slaw constantsfor ethaneand propaneadsorbed on graphitizedcarbon black

lated

Ethane Propane

ess(K)

Q&.,(A)

2, A@) (cm’/g)

81 89

3.85 4.09

0.849 4.841

P. A. MONSON

508

two dimensions, including thermodynamic perturbation theories (Steele, 1976; Henderson, 1977) and integral equation theories (Glandt and Fitts, 1978; Glandt et al., 1978). These may be extended to mixtures in a straightforward manner at the expense of some increase in complexity. Much simpler, if less physically appealing, approaches can be developed by adapting the conformal sohttion theory of bulk fluid mixtures and, of these, the most satisfactory is the van der Waals one fluid (vdW1) theory (Leland et al., 1968; McDonald, 1973). This is the approach taken in this work. In the vdW1 theory the nonideal part of the configurational Helmholtz free energy of a binary fluid mixture is related to that of a single-component reference fluid in which the molecules interact via a pair potential with a collision diameter crX and well depth E, determined from

uf = cc

i j

xixjuj21

(13)

XiXjEijUjll.

(14)

and E,O; = cc i

j

The configurational Helmholtz ecule is then given by

free energy per mol-

xi In xi

A = A, + kTx

I

(15)

where A, is the configurational Helmholtz free energy per molecule of the reference fluid. A systematic derivation of these results via thermodynamic perturbation theory for a bulk mixture has been given by Smith (1972). By differentiation, the component chemical potentials may be obtained from eq. (15) (see, for example, Shing and Gubbins, 1983). For the residual chemical potential, we obtain, after some algebra,

(16) In this equation U, is the configurational internal energy per molecule in the reference fluid and P, is the pressure of the reference fluid. The approach requires that we have results for the thermodynamic properties of the reference fluid which in this case is the twodimensional 12-6 fluid. For this purpose we make use of a parameterized fit to the equation of state from Monte Carlo computer simulation results given by Henderson (1977). Henderson (1977) applied the Barker-Henderson thermodynamic perturbation theory, through second order, to the two-dimensional 12-6 fluid. Only by applying an empirical correction term was he able to achieve good agreement with Monte Carlo computer simulation data. However, the resulting expressions provide a very convenient and accurate method of calculating the thermodynamic properties of the two-dimensional 12-6 fluid. This

Table

3. Parameters used in the two-

dimensional 124

Mixture 1 Mixture 2

mixture simulations

61 I /u*2

El I /En

2.0 1.0

2.0 2.0

The Lorentz-Berthelot combining rules were used for unlike interaction parameters. All simulations were carried out with 100 particles at a reduced temperature kT/c,, = 0.5 and a reduced spreading pressure no:, /8, 1 = 0.25. Properties were averaged over 2 x lo6 configurations. provides

us with an essentially

exact description

of the

the only approximations used in our results for two-dimensional mixture properties come from the use of eqs (13) and reference

fluid

thermodynamics.

Thus

(141. We have made some tests of this approach by performing Monte Carlo computer simulations of two-dimensional 12-6 mixtures. These simulations were carried out in the isothermal-isobaric ensemble, by a technique similar to that described by Wood (1968) and McDonald (1972) for three-dimensional systems. The pressure being held constant here is the two-dimensional or spreading pressure. Two systems were studied at several compositions. The potential parameters used and some other technical details about the simulations are given in Table 3. The quantities calculated were the density, the configurational energy, the enthalpy and the residual component chemical potentials. These latter quantities were obtained using the potential distribution or test particle method (Widom, 1963; Jackson and Klein, 1964; Knight and Monson, 1985, 1986). Similar calculations for bulk fluid mixtures have been presented by Shing and Gubbins (1982). Figures 14 show comparisons of the vdW-1 theory with the simulation results for the two mixtures considered. Figure 1 shows a comparison of the predicted vs. simulated densities at constant pressure as a function of the composi tion. The agreement is very good. The maximum in the curve for the mixture with unequal sizes is somewhat striking. However, this is associated with a large negative area of mixing which arises from the effect of a large size difference between the components coupled with the strong attractive interactions. Figure 2 shows results for the configurational energy and once again the agreement is excellent. Finally, Figs 3 and 4 give a comparison of the predicted and simulated residual chemical potentials. Evidently, the vdW-1 theory provides a good representation of the residual chemical potentials for each component. Other comparisons with simulations at a higher temperature and spreading pressure have been given elsewhere (Knight and Monson, 1985). The simulation results for the density and configurational energy are subject to uncertainties of no more than a few per cent. However, the chemical potential calculations via the test particle method are subject to somewhat greater uncertainties, which may be as much

Molecular basis of adsorbed solution behaviour

-05

J..&’

’

509

I\ \

-10

\

a\

/ 01

I

0

02

I

I

04

06

08

1

\

10

X,

0

\

Fig. 1. Density as a function of the composition at constant spreading pressure and temperature for a two-dimensional 12-6 fluid mixture. The curves are the results from the vdW-1 theory and the points from the simulations. The full curve and the filled circles arc for mixture 1 of Table 2. The dashed curve and unfilled circles are for mixture 2.

oL

0;2

0;

x’o;6

0;8

.-o

o\

NY

\_O_.-

Fig. 3. Residual chemical potentials of the components in units of .sI1 as a function of the composition at constant spreading pressure and temperature for mixture 1. The curves are the results from the vdW-1 theory and the points from the simulations. The full line and filled circles are for component 2 and the dashed curve and unfilled circles for component 1.

ip

-05-

X, 02

04

06

08

10

-1.0-

UC

-I.*-

-2.o-

Fig. 2. Configurational energy per molecule in units of E, , as a function of the composition at constant spreading pressure and temperature for a two-dimensional fluid mixture. The legend is the same as that for Fig. 1.

as 10 ‘A in some cases. Naturally, not represent

an exhaustive

these comparisons do test of the theory; never-

theless, they clearly indicate that the vdW-1 theory is at least an excellent first approximation to the twodimensional fluid mixture properties. Having assembled our techniques, we proceed now to a comparison with experimental results and an

analysis

of the nonidealities

in adsorbed

solutions.

Fig. 4. Residual chemical potentials of the components as a function of the composition at constant spreading pressure and temperature for mixture 2. The curves are the results from the vdW-1 theory and the points from the simulations. The full line and filled circles are for component 2 and the dashed curve and unfilled circles for component 1.

510

P. A. MONSON

5. ADSORBED

SOLUTIONS

ON

GRAPHITIZED

CARBON

BLACKS

Using the formalism have made calculations simple

gases

Some

technical

in

the

on

details

Appendix.

graphitized

carbon

of the calculations

The

present

we for

black.

dimensional

fluids.

Ideally,

for pure component

made

we

should

for the present

first

spherical studies

test

the

molecules. have been

systems of this type (Thorny

et al., 1981), no suitable

data are available

for mixtures.

However,

ago,

and

(1972)

some

time

reported

data

for

Friederich

simple

binary

Mullins

hydrocarbon

carbon blacks. We have made data for ethane-propane mix-

mixtures on graphitized comparisons with their

Sterling FTG DS at 298 K. 5 shows a y-x diagram for

the

propane

mixture

and

approximation and of

with the

are in good

ideal

solution equation

component dicted

12-6

data

are

to

and

at all three

agreement

theory state

at the

theory.

used

solution

Friederich

data. of

fluid

by the vdW1

nonideal

by

Predictions

the experimental

Henderson’s

tion

studied

at 700 Torr.

obtain ideal

based

(1977)

for

the

on

using

the

pure

pressure

pre-

experimental

solution

of

implementation

spreading No

Muilins levels

with each other

Our is

ethane-

adsorp-

parameters.

The

results

vir-

are

What

system,

equilibrium

contributions. that excellent with

ideal

of

is important,

principally

is dominated

Friederich predictions

adsorbed

in view

however,

because by

is

clearly that the be very accurate the

theory

the adsorpHenry’s

and Mullins for this system

solution

of the

the intermolecular

law

(1972) report may be made

using

the

pure

these are the first predictions of multicomponent adsorption equilibrium purely from molecular parameters obtained without reference to adsorption data. In Fig. 6, we show results for the spreading pressure vs. gas-phase composition at two values of the bulk pressure. The agreement between theory and exper-

component

iment values of

To our knowledge,

isotherms.

in this calculation

the surface

tures on Figure (1972)

in the system.

tion

chosen

theories,

modelling

that the theoretical results demonstrate ideal adsorbed solution theory should

were

systems

with data for simple extensive experimental

forces

refined

in our

are described

because the surface has only a small degree of heterogeneity and the adsorbed phases tend to behave as twoapproach Although

in the more

uncertainties

described in Sections 24 of adsorption equilibria

adsorbed

of errors

of the surface

these

automatic

is very sensitive

area used. We have given was

area:

determined

surface

to the value of

the results

for two

13.1 and 11 .O m2/g. The first by

area analyser

Friederich

using

an

and the second was the

value quoted by the manufacturer of the carbon black (Friederich, 1970). The lower surface area gives a much closer agreement between theory and experiment. In view of the uncertainties in the surface area measurements it is difficult to reach firm conclusions concerning the agreement between theory and experiment as far as the spreading pressure is concerned. It should be noted, however, that knowledge of the surface area is

from the ideal gas mixture (Henry’s law) results. The fact that the Henry’s law results give the best agreement with experiment should not be regarded as indicative tually

indistinguishable

02

.

but only slightly different

i

J Fig. 5. Adsorption equilibrium for a mixture of ethane and propane on graphitized carbon black at 298 K and 700 Torr. The ordinate gives the gas-phase composition and the abscissa the adsorbed-phase composition. The points are the experimental data of Friederich and Mullins (1972). The other curves are . -, nonideal adsorbed solution theory (the ideal adsorbed solution results are identical on the scale of the plot); -. -. -, Henry’s law.

Fig. 6. Spreading pressure as a function of the gas-phase composition for a mixture of ethane and propane on graphitized carbon black at 298 K. The points are the experimental data of Friederich and Mullins (1972) and the curves are the results from the nonideal adsorbed solution theory using two values of the adsorbate surface area: ~ 13.1 m’/g; - - -, 11.0 m’/g. The data at the ton of the _era& . are for a bulk pressure of 700 Torr and those at the bottom for 100 Torr.

Molecular basis of adsorbed solution hehaviour

511

not required for the analysis of the experimental data or for the theoretical prediction of the equilibrium compositions presented above. Of course, the spreading pressure calculation is expected to be sensitive to inadequacy in the intermolecular force model used. Since we have ignored all contributions from the nonsphericity of the molecules, we might expect this to have a significant effect.

6. DISCUSSION

We turn now to a discussion of the conclusions which may be drawn from these results. Firstly, at the conditions studied by Friederich and Mullins these adsorbed solutions behave as ideal solutions. Secondly, the relative strengths of the interactions of the different components with the solid surface seem to be the dominant effect in determining selective adsorption. However, it is important to point out that such results can depend significantly upon the state conditions in relation to the phase diagram of the adsorbed layer, as we will now demonstrate. It is becoming well known that monolayers adsorbed on homogeneous carbon blacks exhibit twodimensional phase behaviour which is qualitatively similar to that seen in bulk fluids (Thorny et al., 1981). In particular, vapour-liquid coexistence may occur, with a critical temperature in two dimensions that is typically about 0.4 TA3D! Although much is known about the phase diagrams of pure adsorbates, almost no information is available for multicomponent systems. Nevertheless, we might assume reasonably that the mixed adsorbate critical temperatures are bounded by the pure adsorbate critical temperatures. To our knowledge, the critical temperatures for ethane and propane adsorbed on graphite have not been determined. Nevertheless, assuming that the value of 0.4 TL3D’is a reasonable one, we would find that for ethane TL2D’- 122 K and for propane TfD’ - 148 K. Thus on this basis the experiments of Friederich and Mullins were performed at roughly twice the twodimensional critical point of propane and 2.5 times that of ethane, i.e. well away from any two-phase behaviour in the adsorbed solution. To investigate the significance of this, we have repeated the calculations described in Section 5 at a temperature of 160 K and a pressure of 1 atm. The x-y diagram is given in Fig. 7. Note that there are now substantial differences between the three levels of approximation. In particular, there are significant nonidealities and the adsorption selectivity is no longer dominated by the Henry’s law contributions. This clearly demonstrates the importance of the adsorbate phase diagram in determining adsorption equilibrium. We believe that these results demonstrate that we have an accurate model, without adjustable parameters, for binary adsorption equilibrium on homogeneous carbon blacks and other graphite surfaces. We are currently using the theory to study the adsorption equilibrium over wider ranges of conditions, particularly in the regions of the adsorbate phase diagram

Fig. 7. Adsorption equilibrium for a mixture of ethane and propane on graphitizedcarbon black at 160 K and 760 Torr. The ordinate gives the gas-phase composition and the abscissa the adsorbed-phase composition. The curves are: - --, nonideal adsorbed solution theory; ---, ideal adsorbed solution theory; -.-.-, Henry’s law. where vapour-liquid coexistence might be expected to occur. Predictions and analysis of binary fluid phase equilibria in adsorbed layers using the model wilt be the subject of a future publication. work was supported by grants from the National Science Foundation (CPE-8307947) and Research Corporation. Generous allocations of computing resourcesfrom the College of Engineeringand University Computing Center at the University of Massachusetts are

Acknowledgements-This

also &km&edged NOTATION

a, *(a)

A AX d k Ki N Ni P pi

Pp

r T TPD’ T (2D) C

area per carbon atom in the basai plane of graphite area occupied by the adsorbed phase configurational Helmholtz free energy per molecule configurational Helmholtz free energy per molecule of reference fluid in vdW-1 theory interplane spacing in graphite Boltzmann’s constant Henry’s law constant for component i total number of molecules number of molecules of component i bulk pressure partial pressure of component i in the bulk phase pressure of bulk gas in equilibrium with pure component adsorbate i at the spreading pressure of the adsorbed solution three-dimensional position vector of a molecule in the adsorbed phase absolute temperature critical temperature of the component bulk fluid critical temperature of two-dimensional condensation for pure component adsorbate

P. A. MONSON

512 adsorbate-adsorbate

u(r)

tial energy

intermolecular

poten-

function

N particle

intermolecular

potential

energy in

u (2W N

the adsorbed phase N particle intermolecular

potential

energy in

a two-dimensional

fluid.

u

configurational

ux

configurational internal energy per molecule of reference fluid in vdW-1 theory.

internal

zN Z’2W N

of

Greek

i in the adsorbed

component

i in the

bulk

N particle configurational

partition

function

N particle

partition

function

configurational

for a two-dimensional Z 1.1

phase

of comDonent

mole fraction phase

Yi

energy per molecule

of the adsorbed

mole fraction phase

single particle function

letters well

depth

molecular

fluid

configurational

in

partition

adsorbate-adsorbate

inter-

potential

well depth in adsorbate molecule/solid atom intermolecular potential for component i chemical

potential

of component

i in the gas

phase chemical

potential

adsorbed

phase

configurational

of

component

chemical

potential

i in the of com-

ponent i in the adsorbed phase residual chemical potential of component the adsorbed

i in

phase

spreading

pressure

adsorbate

density

collision diameter in adsorbate-adsorbate intermolecular potential collision

diameter

ecule/solid

atom intermolecular

component

molecule total

in

position

potential

mol-

potential

for

vector

energy of an adsorbate

of component

interaction

adsorbate

adsorbate

i

two-dimensional interaction

i with the solid

potential

energy

of

the

with the solid

Subscripts a, B i, j

summation

index

for molecules

summation

index

for species

X

reference

0

pure component adsorbate at the same spreading pressure and temperature as the adsorbed

fluid for vdW-1

solution

Superscripts (c) (HL) (ID)

(r)

configurational Henry’s

law limit

ideal adsorbed restdual

pure

component

spreading

UN

volume

0

solution

theory

adsorbed

pressure

adsorbate and

at

the

temperature

same as the

solution REFERENCES

Barker, J. A., and Henderson, D., 1976, What is liquid? Rev. mod. Phys. 48, 587-671. Costa, E. J. L., Sotelo, J. L., Calleja, G. and Marron, C., 1981, Adsorption of binary and ternary hydrocarbon gas mixtures on activated carbon: experimental determination and theoretical prediction of the ternary equilibrium data. A.i.Ch.E. J. 27, 512. Friederich, R. O., 1970, Ph.D. Thesis, Clemson University. Friederich, R. 0. and Mullins, J. C., 1972, Adsorption equilibria of binary hydrocarbon mixtures on homogeneous carbon black at 25°C. Ind. Engng Chem. Fundam. 11, 439445. Glandt, E. D. and Fitts, D. D., 1978, Percus-Yevick equation of state for the two-dimensional Lennard-Jones fluid. J. Chem. Phys. 68, 45034508. Gland& E. D., Myers, A. L. and Fitts, D. D., 1978, Physical adsorption of gases on graphitized carbon black. Chem. Engng Sci. 33, 1659-1665. Henderson, D., 1977, Monte Carlo and perturbation theory studies of the equation of state of the two-dimensional Lennard-Jones Auid. Mol. Phys. 34, 301-315. Hirschfelder, J. O., Curtiss, C. F. and Bird. R. B.. 1964. Molecular Theory of Gases and Liquids. Wiley, New York: Holborow, K. A. and Loughlin, K. F., 1977, Multicomponent adsorption equilibria of binary hydrocarbon gases in 5Azeolite. A.C.S. Symp. Ser. 40, 379-392. Jackson, J. L. and K&n, L. S., 1364, Potential distribution method in equilibrium statistical mechanics. Phys. Fluids 7, 228-23 1. Knight, J. F. and Monson, P. A., 1985, Potential distribution theory applied to adsorption, A.I.Ch.E. Symp. Ser. No. 242 81, 45-50. Knight, J. F. and Monson, P. A., 1986, Computer simulation of adsorption equilibrium for a gas on a solid surface using Phys. 84, the potential distribution theory. J. Chem. 1909-1915. Leland, T. W., Rowlinson, J. S. and Sather, G. A., 1968, Statistical thermodynamics of mixtures of molecules of different size. Trans. Faraday Sot. 64, 1447-1460. McDonald, 1. R., 1972, N, p. T ensemble Monte Carlo calculations for binary liquid mixtures. Mol. Phys. 23, 41-58. McDonald, I. R., 1973, Equilibrium theory of liquid mixtures. In Statistical Mechanics (Edited by Singer, K.), Vol. 1, Chemical Society Specialist Periodical Reports. Monson, P. A., Steele, W. A. and Henderson, D. 1981, Theory of monolayer physical adsorption. II. Fluid phases on a periodic surface. J. Chem. Phys. 74, 6431-6439. Myers, A. L., 1968, Adsorption of gas mixtures. A thermodynamic approach. Ind. Engng Chem. 60, 4-9. Myers, A. L., 1983, Activity coefficients of mixtures adsorbed on heterogeneous surfaces. A.I.Ch.E. J. 29, 691493. Myers, A. L. and Prausnitz, J. M., 1965. Thermodynamics of mixed gas adsorption. A.I.Ch.E. J. 11, 121-127. Ruthven, D. M., 1984, Principles ofAdsorDtion and Adsorotion Processes. Wiley-Interscience,-New York. Shing, K. S. and Gubbins, K. E.. 1982. The chemical potential in dense fluids and fluid mixtures via computer simulation. Mol. Phys. 46, 1109-l 128. Shing, K. S. and Gubbins, K. E., 1983, The chemical potential in non-ideal liquid mixtures. Computer simulation and theory. Mol. Phys. 49, 1121-113X. Smith, W. R., 1972, Perturbation theory and one-fluid corresponding states theories of fluid mixtures. Can. J. them. Engng 50, 271-274. Steele, W. A., 1974, The Interaction of Gases with Solid Surfaces. Pergamon Press, Oxford. Steele, W. A., 1956, Theory of monolayer physical adsorption.

513

Molecular basis of adsorbed solution behaviour I. Flat surface. J. Chem. Phys. 65, 52565266. Steele, W. A., 1978, The interaction of rare gas atoms with graphitized carbon black. J. phys. Chem. 82, 817-821. Thorny, A., Duval, X. and Regnier, J., 1981, Two-dimensional phase transitions as displayed by adsorption isotherms on graphite and other lamellar solids. Surf. Sci. Rep. 1, l-38. Widom, B., 1963, Some topics in the theory of fluids, J. Chem. Phys. 39, 2808-28 12. Wood, W. W., 1968, Monte Carlo studies of simple liquid models. In Physics of simple Liquids (Edited by Rowlinson, J. 8, Temperley, H. N. V. and Rushbrooke, G. S.). NorthHolland, Amsterdam. APPENDIX Here we briefly outline equilibrium calculations.

the steps in our

adsorption

(1) General (non-ideal) case The adsorption equilibrium calculations are carried out using an iterative method. (a) The temperature, bulk pressure and adsorbed-phase composition are chosen. (b) The Henry’s law constants are calculated using eq. (9). (c) An initial estimate of the surface density is made. (d) The gas-phase composition at this surface density is calculated using

where Ap = (j.$) --F))/kT-lnKi+lnKj.

(A2)

(e) The bulk pressure P’ is determined using eq. (10). (f) Steps (c) through (e) are repeated iteratively until a value of the adsorbate density at which P’ = P is found. (g) The spreading pressure, x, is calculated for the final density, temperature and surface composition. (2) Ideal adsorbed solution (a) The pure adsorhate chemical potentials, c(!$, at the spreading pressure calculated in 1(g) are determined. (b) The pressure, Pp. of the bulk phase in equilibrium with the pure component adsorbate at the spreading pressure of the adsorbed solution is calculated using lng

= &!/kT-In

Ki.

(A3)

(c) The gas-phase composition is determined using (Myers and Prausnitz, 1965) yi (ID) =-

X,PP P

(3) Henry’s law The gas-phase composition

is found from

AS Yi

=&

(AlI

WI