Statistical thermodynamics of physisorption

STATISTICAL THERMODYNAMICS OF P H Y S I S O R P T I O N * J. G. DAsN

Department of Physics, Universityof Washington, Seattle, Washington98195, U.S.A.

CONTENTS 1. Introduction 2. Complete Adsorption System and Its Division into Separate Components A. Energies B. Entropies C. Helmholtz Free Energy D. Forces and Displacements 3. Film-Vapor Equilibrium 4. Homogeneity of Film Vapor 5. Vapor Pressure 6. Heat Capacity and Entropy 7. Surface Phases 8. Heats of Adsorption 9. Summary

References

119 121 122 122 123 123 127 131 136 138 143 150 154 155

1. INTRODUCTION Thermodynamics plays a particularly important part in the study of physical adsorption. Traditionally, adsorption studies have dealt with simple thermodynamic quantities, such as pressure, volume and heat capacity, and their variations with temperature. For many years these were the only variables accessible to experiment. More recently, a variety of modern techniques have begun to probe more detailed and microscopic properties of gas-surface interactions. The newer methods complement, rather than supplant, traditional techniques: thermal measurements continue to be useful and, in certain cases, remain the principal means of investigation.

*Research supported by The National Science Foundation. 119

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J.G. DASH

The formal development of adsorption thermodynamics can be said to have begun with Gibbs's great work "On the equilibrium of heterogeneous substances"3 t~ Subsequent contributions to the theory were made by many people, and a short list of some papers t2-9~ that continue to be cited with regularity is given at the end of this review. This abbreviated list only hints at what is a long and very extensive literature, ~1°) but, nevertheless, the thermodynamics of adsorption does not yet appear to be a completely closed subject. The lack of a precise boundary between film and vapor, the strong gradients in film properties near the solid surface, and the difficulties of putting theory to experimental tests, are persistent problems contributing to the continuing discussion and controversy. These questions are made more troublesome by the lack of a formalism and notation consistent within the field. Separate styles of notation is typical of thermodynamics, but such individuality seems exaggerated in the case of adsorption thermodynamics. Here one even finds disagreements as to the proper formulation of basic thermodynamic functions, as, for example, the Helmholtz and Gibbs free energies. In this situation, it seems advisable to inspect the basic connections between thermodynamic and fundamental statistical quantities, i.e. to develop a general thermodynamic theory of adsorption beginning with statistical mechanics. This approach seems to offer several advantages: it provides clear connections with other branches of statistical thermodynamics, it yields unequivocal definitions of proper free energies, it simplifies the derivation of many "standard" formulae, and it provides a natural formalism for any subsequent model of monolayers and multilayers. I do not claim to be the originator of this approach to surface thermodynamics: Guggenheimt4) has asserted its statistical foundations. Perhaps its most familiar exposition is the chapter on surface layers in the text by Fowler and Guggenheim.t~l) But whereas Fowler and Guggenheim focus their interest on a few special models of monomolecular films, we derive general thermodynamic relationships which serve as a framework for the subsequent assumption of specific microscopic models. For the most part, we use the general style and notation of Landau and Lifshitz" 2) with the usage of nomenclature specific for adsorption systems wherever they appear reasonably accepted and not in conflict with the wider field of thermodynamics.

STATISTICAL THERMODYNAMICS OF PHYSISOKPTION

121

2. COMPLETE ADSORPTION SYSTEM A N D ITS DIVISION INTO SEPARATE C O M P O N E N T S Consider a vessel containing definite quantities of adsorbent (which will be equivalently termed substrate) and adsorbate. It is not necessary, at this stage, to define these terms, but we may anticipate later specification and take their meanings qualitatively: Substrate refers to a relatively stable solid material having an appreciable surface area, and adsorbate to a mixture of a vapor and its film. Although the thermodynamic properties of adsorbent and adsorbate appear clearly separable in most cases of physical interest, their distinction cannot, in principle, be exact, and it will therefore be useful to develop the theory, at least at the outset, with greater generality. A division of the adsorbate between film and vapor is even more diffuse. We will therefore begin by considering substrate, film and vapor as distinctions in name only, the thermal properties of the system being due to a mixture of all three. Their later subdivision into components having separable properties will come about only after the adoption of specific physical assumptions. The mixed system will, in general, have a minimum energy E o and a set of higher levels E~with level density F(E). If the system is in thermal equilibrium with a reservoir at temperature T, then the probability that it is in a state with energy E~ is e - $El

coi = Xe_#n;

~.co i = 1

(2.1)

i

i

where fl = (kT) -1. The canonical partition function Z for a fixed number of particles in the total system is the sum over states Z = ~., e -pE' = ~ e-a~r(E)dE. i

(2.2)

Eo

Now Z can be formally written as the product of separate terms for film, vapor and substrate: Z = ZfZvZ r

(2.3)

This factoring does not imply any special properties or distinguishability of the parts. It is always permitted by the convolution property

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of the der, sity of states: for any composite system A + B, (2.4)

F a ÷B(E) = ~ F~(E') FB(E - E') dE'.

The statistical properties of film, vapor and substrate can be formally related through the factoring of the total Z and the familiar general results of statistical mechanics. The derivations are outlined below.

A. Energies

The ensemble average energy E of the total mixed system is the weighted sum over states: E = ~ o~iE~.

(2.5)

i

Substituting (2.1) for o9~:

E _

don Z)

Y e-PE'Ei

(2.6)

Writing Z in terms of the separate factors (2.3) we see that E is the sum of component energies, viz., (2.7)

E = Ef + Ev + E

where EI =

c3(ln Zs)

E

a(ln Z ) ,

E

c~(lnZ )

B. Entropies

The equilibrium entropy S of any system is related to the statistical distribution by S = - klno9 = - k ~ o 9 iln~o i, i

(2.8)

STATISTICAL THERMODYNAMICS OF PHY$ISORPTION

123

which, with (2.1), (2.2) and (2.6), yield S = ~,, o i ~ J E i -4- In Z) = E / T + k In Z.

(2.9)

i

Using the additivity of energy (2.7) and (2.3) for Z, S is the sum S = S f ~- S v -~- S s

(2.10)

S.f = E f / T + k In Z f

(2.11)

where

and similarly for vapor and substrate.

C. Helmholtz Free Energy The fundamental connection between statistical and thermodynamic properties of the canonical ensemble is through the Helmholtz free energy F = - k T In Z.

(2.12)

With the factoring of Z, we immediately obtain the free energies of the components: F = Fy + F v + F s

(2.13)

k T In Zy, etc.

(2.14)

where FI

=

-

Combining relations of the type (2.14) with (2.11), we obtain connections between the energy, entropy and free energy of the total system and of each component: F = E - T$, F f = Ey - TSy, etc.

(2.15) (2.16)

D. Forces and Displacements The first law of thermodynamics for reversible processes involving

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generalized forces X~ and displacements x~ is dE = dQ - ~ X~, axe, = T dS - ~ X~, a x e . ot

(2.17)

ot

With the expression for the total differential of energy obtained from (2.15), we have dF = - S d T

(2.18)

- Z X~, dx~, .

Equation (2.18) is perfectly general for any type of system, whether homogeneous or composed of different parts, as is the adsorption system. It is just at this point that one can effect a definite separation between the thermal properties of film and vapor. We recognize that the area of the film must, in some sense, play a role analogous to volume of the vapor. Both are extensive coordinates, i.e. are generalized displacements x. In the case of the volume V, the conjugate generalized force is the pressure P; for the film, the force conjugate to the area A is called the spreading pressure and conventionally symbolized as ~bor n. Now, with these definitions, the two pressures can be expressed as partial derivatives o f the total free energy of the composite system: from (2.18), we obtain

P =-

~-V r.n

r.v"

(2.19)

The spreading pressure is, thus, unequivocally defined in terms of definite thermodynamic variables, even though there is no practical method for measuring ~b of a film on a solid surface (although it can be measured in films floating on liquids). As we will see, the impracticability of measuring ~b does not reduce its fundamental role as a thermodynamic coordinate, and, even though it might not be gauged by any direct measurements, it can be deduced from quantities with which it is related by quite general thermodynamics. Equation (2.19) implies, at least, a partial separation of the adsorbate into film and vapor fractions. We can certainly expect that the principal contribution to P will be due to the vapor; we also expect that the major term in ~b is due to the film. Thus, if we express F in terms of its

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

125

parts, (2.19) yields P

=

-

\OVJT.,

-- \ O V / T . ,

--

(2.2o)

<~ =

- \cgA /r,v - \c~A )r,v -

\ O/~/'';'7"'r,v"

These expressions, while correct in principle, would make subsequent calculations quite tedious if they had to be carried along in their entirety. While it must, indeed, be true that the three-dimensional pressure P, for example, reflects some of the elastic responses of the substrate and film, these contributions are, in all cases of interest., much smaller than that due to the vapor. Similarly, it is the film term that is dominant in 4). In both cases, we would wish to eliminate the minor terms completely, at least for all but the most exacting calculations. These reductions can be accomplished in two ways. The more elegant alternative is to define film and vapor phases so as to effect a perfect separation of terms. Thus, since the division of the partition function into film, vapor and substrate factors has been completely arbitrary to this point, we may here define the factors in whatever manner required to yield pure partial derivatives. In this way, the film would correspond to tha t portion of the system containing all of the variations on A, but no dependence on V, and the vapor would depend on V, but not at all on A. This method of separating film and vapor is effectively that of the dividing surface of Gibbs. Much of the subsequent theoretical developments in adsorption thermodynamics is based on this concept. But, in spi~ of its long tradition, this artifice brings with it a number of serious problems. It effects too complete a division between film and vapor, so that a number of inconvenient constraints arc placed on the interpretation of measurements and the construction of realistic theoretical models. For example, although there must be certain modifications produced in the equation of state of a vapor in the vicinity of a surface, these changes would have to be attributed to the film, even though the interaction region actually migh t extend quite far into the vapor phase. A less formal or exact division of the adsorbate into film and vapor portions is to accept some initial approximations concerning their

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J . G . DASH

physical natures. As will be seen, these approximations will, in their limiting cases, turn out to create precisely the same separation as the dividing surface of Gibbs. But the advantage of the approximate approach is that, just as in real adsorption systems, the separation between film and vapor is not exact, and it holds open the opportunity for some future, more faithful imitation of the actual situations. The assumptions are as follows: (a) The portion of vapor volume within the range of surface forces is a very small fraction of the total V. (b) The volume occupied by film is a very small fraction of V. (c) Any distortions or deformations of the substrate caused by adsorption are reversible single-valued functions of the quantity of film adsorbed. As already noted, the assumptions (a) and (b), in their extreme forms, yield pure expressions for q~ and P, whereas they are, in principle, always composed of three terms, as in (2.20). In the following sections, we usually adopt the extreme forms, but will also inspect the conditions under which they are satisfactory approximations. Assumption (c) is actually a matter of definition, and need not be inexact, depending upon subsequent models. It is a less restrictive and more physically reasonable assumption than that typically made concerning the nature of adsorbents. In conventional treatments, it is assumed that the substrate is inert, i.e. unaffected by adsorption. This assumption is, in principle, unphysical. Furthermore, there are several known mechanisms by which the properties of a surface are appreciably modified in the presence of an adatom, tla) Fortunately, one need not require the inert substrate assumption in order to proceed with the theory. It is only necessary that the substrate distortions be caused on an adatom-by-adatom basis, i.e. to be a single-valued function of the quantity of gas adsorbed on the surface. For, if this is the case, then the perturbation is associated with the adsorption state of the adatom and "belongs" to the adatom just as intimately as its binding energy, dipole moment or vibration amplitude. In fact, a proper calculation of the binding energy and other properties of the adsorption state must take into account the substrate distortions: as long as they can be related to the quantity adsorbed they may be treated as attributes of the adatoms themselves.

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

3. F I L M - V A P O R

127

EQUILIBRIUM

The establishment of equilibrium in an adsorption system involves the flow of energy between substrate and adsorbate and an exchange of particles between film and vapor. Both of these processes can be analyzed in a variety of ways, depending on wha t parameters of the system are imagined to be held constant. These different situations, corresponding to the microcanonical, canonical and grand canonical ensembles, all yield the same results as to equilibrium properties: they differ only in their fluctuations. Let us first consider the microcanonical ensemble, i.e. an isolated adsorption system having fixed total energy, volume, adsorption area and number of particles. Equilibrium corresponds to the state of maximum entropy of the total system with respect tO redistributions of energy and particles. Concerning energy flow, we first consider the system as composed of two parts; substrate (s) and adsorbate (a): S = S + S, = constant,

(3.1a)

E = E s + E° = constant.

(3.1b)

The entropy is maximized with respect to energy changes, so that at equilibrium =0

(3.2)

N, V, A

Using the relations (3.1), we obtain

Es/lN, V, A

__ (0so kf~Ea,]N,V, A"

(3.3)

These partial derivatives, as can be seen from (2.17), are just the reciprocals of the absolute temperatures of substrate and adsorbate: thus, (3.3) implies the equality of temperatures: T~ = T

(3.4)

and the result can obviously be extended to any two portions of the system: T is then uniform throughout. We now consider the exchange of particles between film and vapor:

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J. 6. DASH

at equilibrium the entropy is a maximum, so that

If we include all three parts of the total entropy, S = S f.~t- Sv --~- S s

the partial derivative in (3.5) becomes

t os

(

Nf/IE, v, A + ~ N f / ] E , V , A + ~dNv]F.,V.A

(3.6)

The second term would be awkward to carry through subsequent manipulations, but fortunately it vanishes according to assumption (c), since all substrate changes caused by adsorption are to be associated with the adatoms: such changes are actually embedded in the first term, (aSf/aNf). Thus we have the fundamental equation of vaporfilm equilibrium,

N filE, v, A

~t~Nv/JE, V, A

The quantities in (3.7) involve the chemical potentials of the film and vapor. To bring this out, we note that the total differentials dE and dF in (2.17) and (2.18) must actually include terms proportional to the quantity of material: in addition to the products of conjugate forces and displacements ckdA and PdV, the sums Y.X~ dX~ also include products involving dN. Now, if we introduce this explicit dependence in (2.17), and also write out the separate contributions to the energy, entropy, etc., we obtain dE = dEy + dE v + dE s = T ( d S f + dS + dSs)

--dpdA - P dV + g f d N f + la,,dN,, + lasdN .

(3.8)

The #'s are introduced here simply as the generalized coordinates conjugate to particle number. They are directly related to certain partial derivatives of the derivatives of the energies and entropies: differentiating with appropriate quantities being held constant, we

STATISTICAL TItER.MODYNAbflCS OF PI-IYSISORPTION

129

obtain I~"t"= \ON.r.]s, a, v = T \ O N t . ] s , a, v'

(

,' os x

t~v = \ O N J s ,

a, v = T ~ - ~ o ) E , a , v ,

(oE

(3.9)

.__.,'osx

Thus, (3.7) is just the standard condition for thermodynamic equilibrium between two phases, but specifically applied to adsorption: t~s = #2.

(3.10)

It is, in a sense, a trivial result, except for two points. Firstly, we emphasize that the derivation does not depend at all upon the sharpness of the division between film and vapor. The equilibrium condition is, in fact, a special case of the universal condition that the chemical potential of a substance is uniform in thermodynamic equilibrium, even in the presence of arbitrary fields and boundaries. The second point is that it does not require that the substrate be inert, but may, in fact, participate in the adsorption process in a way that affects its properties rather profoundly. So far, we have shown that the chemical potential is related to certain partial derivatives of the entropy and energy. There are additional connections to other thermodynamic functions, which we now discuss in the context of physical adsorption. The Helmholtz free energy F, Gibbs free energy G, enthalpy H and Landau potential 12 for arbitrary systems are: F = E -

TS,

(3.11)

G = F + ~ X,, x~,,

(3.12)

H = E + ~X, x,,

(3.13)

~t

12 = F - G.

(3.14)

Differentiating each state function and substituting for d E in the form

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J . G . DASH

(3.8), we obtain dE = - S d T dG = - S d T

dpdA - P d V + ~,#~dN i,

(3:15)

+ A d O + V d P + ~,#idN~,

(3.16)

where the index i refers to film, vapor, or substrate. d H = TdS + Addp + V dP + ~,#~dN~,

(3.17)

dr) = - A d ¢ - q~dA -

(3.18)

V dP - P dV.

Thus, the chemical potentials can be related to the different functions:

#i= ~l-~ii)T,A,V,N

(3.19) Now we can show that the chemical potential is equal to the Gibbs free energy per particle, just as for ordinary bulk matter. According to the division of the composite system into film, vapor and substrate, we can write (3.16) as three component equations (noting that according to (2.20) even P and ¢ can be divided into separate contributions] all of the form dG i =

- S i d T + Addpi + V dP i + # i d N i

(3.20)

and thus the implicit functional form of the Gibbs free energy is, for each of the components, G = G(T, ¢, P, N).

But G, just as all of the other functions of state, is an extensive quantity, i.e. proportional to the total number N. Therefore, since all of the other variables T, ¢, P in its implicit relation are intensive, G must have the functional form G = Na(T, ¢, P),

(3.21)

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

131

and, therefore, g(T, d?, P) = G / N = (dG/ON)r .,. p.

(3.22)

But, comparing to (3.19), we see that g(T, ~b, P) is just equal to the chemical potential: in other words, for each component Gi = Ni/~i.

4. H O M O G E N E I T Y

(3.23)

OF FILM V A P O R

Equilibrium in the mixed film-vapor system is characterized by the uniformity of T and ~t, but not of P, t# or the other intensive variables. Interactions between the adsorbate and the substrate fall off as some function of their separation, hence P, ~b and density vary in the neighborhood of the adsorbing surface. However, the chemical potential of the adsorbate is uniform throughout the equilibrium system: it is in fact the uniformity of/~ that makes possible the calculation of local properties. The proof that p is uniform in the presence of substrate fields is identical, in form, with the preceding proof that #I = Pv: one could just as well consider an arbitrary division of the adsorbate into two regions having different energies of interaction with the surface. Thus, it is not necessary to assume that the film and vapor are homogeneous phases, although it is often convenient to ignore the variations. The gradients in q~ can be particularly large, due to the proximity of the film to the surface. It is clear that an imagined barrier process, such as illustrated in Fig. 1, would measure an effective spreading pressure that is not actually a two-dimensional ~b, but some ~b averaged over the thickness of the edge of the barrier. Variations along the direction normal to the plane of the film are, of course, present near the filmsubstrate and film-vapor surfaces, even if the substrate were a smooth hard wall with no long-range forces. The actual gradients in real films depend upon the specific interactions between the adsorbate molecules themselves and between adsorbate and substrate. The variations are particularly important in the physics of films of a few molecules thickness. We will have more to say on this point in § 7, but for the present we suppress the need for investigating gradients in the film by treating

132

J.G. DASH

the film as a thin region enclosed between the a d s o r b e n t surface a n d G i b b s ' s fictitious d i v i d i n g surface. This device creates a s h a r p division b e t w e e n film a n d v a p o r , such t h a t the p r o p e r t i e s of the film a r e extensive with surface area, a n d v a p o r p r o p e r t i e s are extensive with volume. W e n o w wish to explore, in terms of simple p h y s i c a l models, to w h a t extent one m a y actually ignore v o l u m e terms in the film a n d surface t e r m s in the vapor. H e r e we will show that the two cross t e r m s are interrelated.

IVAPOR

]

(0)

FILM I ~377~P7; ; 7 / / / / / / / / / , 4 ,V/////SU BST RAT E~ ' / ~ >q V / / / / / /~" / Z / ~

...!:,:-~!~VA P o R

////

r'~C~

...... A v f

FIG. 1. (a) Components of an adsorption system, illustrating a volume compression of the vapor by an ordinary three-dimensional piston and an areal compression of the film by a surface barrier. To a first approximation, a thin film is assumed to be strictly two-dimensional, but diagram (b) shows that the actual situation can be much more complicated. GradienLs m Hmhilayer films are discussed in § 7.

W e a s s u m e that the v a p o r beha;ees as an ideal gas, if s u b s t r a t e interactions are absent. T h e ideality of the v a p o r is assured in m o s t a d s o r p tion w o r k b y the t y p i c a l l y low pressures. F u r t h e r m o r e , d e v i a t i o n s from ideality, due to m o l e c u l a r i n t e r a c t i o n s or statistical effects, are a l m o s t always c o m p l e t e l y m a s k e d by a d s o r p t i o n effects. In the absence of s u b s t r a t e fields, we have the usual free energy of an ideal classical gas (12) /e~V\ F v = - N k T In ~ - ~ )

(4.1)

where 2 = h ( 2 n r n k T ) - ~ is the t h e r m a l de Broglie wavelength. (We assume, for simplicity, that the m o l e c u l e s have no i n t e r n a l degrees

133

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

of freedom.) From (4.1) we obtain the standard relations:

(~Fo~ / #V \ So= - \ O T / s , , v = N o k l n L ~ 2 a ) ,

P

=

(4.2) (4.3)

\TVIT, No --V-' =

I£v = ~ x - ~ v , J T .

V =

--

k T In

= - k T In

.

(4.4)

Equations (4.1) to (4.4) correspond to a hard-wall container and no film present: the dividing surface coincides with the solid boundary of V and there are no long-range substrate fields. If both film and long-range fields arc present, the fields being due to both vapor-substrate and vapor-film interactions, the free energy of the vapor fraction is =

In ~

e -pu(')

(4.5)

where N o = N - N f and u(r) is the potential energy of a vapor molecule at r from the solid wall. F'o is to be compared with the free energy of an ideal gas of N o molecules in the volume V. This can be done in the style of the theory of imperfect gases39~ We define a surface second virial coefficient B by

Bs =- S [1 - e -~u(')] dar.

(4.6)

O

Then we can write (4.5) in the form

-- #V

F'~ = - N o k T In | ~ - - f f (1 - B,/V)

]

= Fv(No) - NokT In (1 - BJV) N°kT B -~ F°(Nv) + ---V-- "

(4.7)

if Bs/V ~. 1. F,,(No) has the ideal gas form (4.1) for N o molecules. From (4.7), we get the perturbed equation of state of the vapor

p, = _ (OF'~ \OV/N~, r

= N°kT (1 + BJV) = P(1 + BJV) v

(4.8)

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J.G. DASH

where P is the pressure of an ideal gas of N v molecules. In adsorption work, one is often interested in a particular temperature derivative of pressure known as the isosteric heat of adsorption qsc This quantity, which is discussed later, is defined as (~?ln P~

kT2\, OF INl. a

qst

(4.9)

From (4.8), we find that the perturbed heat of adsorption is

q' ~-st

qst

+

kT2(dBs]. V \dT]

(4.10)

Equations (4.8) and (4.10) show that the departures from ideality, caused by vapor-film and vapor-substrate interactions, can be reduced to arbitrarily small values by making V arbitrarily large. But this would also have the effect of making all film properties of negligible importance. Thus, it becomes a matter of practical necessity to effect some compromise between sensitivity to film effects and a simple equation of state for the vapor. The effects can be put on a quantitative basis, if we adopt a few simplifying assumptions. We approximate the substrate and film as planar and homogeneous, attracting gas atoms with characteristic energies varying as the inverse cube of the distance, t14' 15) Such a variation is appropriate over intermediate distances ranging from several to perhaps a hundred atomic diameters: for simplicity, we will assume it to hold all the way down to contact, where repulsion will be approximated as a hard wall. Thus, for an atom at a normal distance z from a substrate, the interaction with the wall is represented by u(z)

=----

us

Z>ff

Z3 '

= oo, z < a. (4.11) If the surface is covered by a homogeneous film of thickness d, the total interaction is u(z) -

=oo,

(us - us) ul z3 - (z - d) 3' z > d +

O
(4.12)

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

135

We now calculate B, for this interaction. The repulsive region can be evaluated immediately, but for the attractive interaction we assume a hioh temperature approximation flu(z), ~ 1,which then allows evaluation of the exponential by the leading term in the power series expansion. This approximation is less stringent than for the adsorbate in general, since it refers to the vapor fraction only, and its density is low just because of the smallness of the exponential factor. Thus, carrying out the calculation in the manner described, one obtains

dz +

B "" A d+a~,j !

i

Aft

0

u(z) dz

d+o

A_~V_u_£ (u~ - uf)-] = A(d + 0.) - 2kT L0.2 + (d + 0.)2j.

(4.13)

w e are now interested in the relative importance of the surface virial coefficient, as, for example, in the extent of its influence on the heat of adsorption. We can put these changes on an interesting comparative basis as follows. The quantity A(d + 0.) is approximately equal to Vy, the volume occupied by the film, and Aa = V1, the volume occupied by a single layer. Then the perturbed heat of adsorption (3.10), with B given by (4.13),can be written as qst -'

qst =

I 0 .3 ]

Lv

L(d

F(u~-u't)l + 0.)a_]"

+

(4.14)

The bracketed factors in both terms are roughly equal to certain heats of adsorption. For example, if there is very little adsorption, d = 0 and V: = VI. Then, since us~0.a ~- qst of the bare substrate, we obtain the fractional correction q,st _ qst ~ V l (4.15) qst -- 2V" In the case of a thick film, where d ,> 0., qS

--

st

qst

%

V_Vz -- 2V'

(4.16)

where % refers to the heat of adsorption (or heat of evaporation) of the bulk liquid. Thus, in both examples, the fractional correction in the

136

J . n . DASH

heat of adsorption is comparable to the fractional volume occupied by film. In other words, the deviations from ideality of both film and vapor are coupled: the importance of surface terms in the vapor equation of state scales with the volume terms due to the film. Therefore, one may reduce both correction terms to extremely small values simply by increasing the vapor volume, but this would cause a loss of ability to detect any of the properties of the film. As a practical matter, it is then a compromise between the two trends, and some substrateassociated non-ideality of vapor will have to be accepted as a necessary concomitant of sensitivity to film characteristics.

V A P O R PRESSURE The equilibrium vapor pressure is determined by the equations of film and vapor and the uniformity of chemical potential. If the condition for overall ideality of vapor is satisfied, then the equation for P is given by equating p, in (4.4) to/~I' leading to 2nm P = (-~--)(kT)'exp(t~f/kT).

(5.1)

Specific equations for P, in terms of the physical parameters of the system, require detailed microscopic models of the film. However, in this discussion, we refrain from the adoption of particular models and instead explore the general interrelationships among the several thermodynamic quantities. A model-independent connection between film and vapor properties can be obtained at this stage. If the area and volume cross terms can be neglected in the free energies of film and vapor, then the differential Gibbs functions are, according to (3.19): dGy = - S r d T + A dqb + P.r d N I ,

(5.2)

dG v = - S v d T + V d P + i~ d N r .

(5.3)

With the relations (3.23) Gf = Nfl~j. and Go = Nv/~~ in differential

STATISTICAL THERMODYNAMICS OF PHYSISORIrrION

137

form, we obtain dltf = - s I d T + a ddp,

(5.4)

dp v = - s v d T + v d(a,

(5.5)

where sj, = S I / N I , a = A / N I , etc. Equation (5.5) is the ordinary G i b b s - D u h e m relation for simple bulk matter; (5.4) is sometimes referred to as the Gibbs-Duhem relation for a film. If we now consider an isothermal change in the equilibrium system, i.e. equating (5.4) and (5.5) and setting d T = 0, then a ddp = v dP. (5.6) Equation (5.6) provides, in principle, a means for deducing film properties from the behavior of its equilibrium vapor. If the molecular area a is independently known as a function of vapor pressure, then ~b can be obtained for any P by the integral p'

,,7, 0

where it is assumed that ~b (0) = 0. If the vapor behaves as an ideal gas, v = k T / P , and we obtain p'

ck(P') = k~T A f Nf(P)d(lnP).

(5.8)

0

Equation (5.7) is known as Gibbs' adsorption theorem, and (5.8) is sometimes called the Gibbs isotherm. It is a most simple and direct relation between film and vapor quantities, but we note that, just as for most of the explicit thermodynamic connections between the phases, it neglects cross-terms and assumes that the vapor is an ideal gas. If the condition for overall ideality of the vapor is satisfied, this does not automatically imply that there are no gradients in the vapor; it merely means that the inhomogeneous region is quite small. In fact, an argument based upon local ideality, explicitly taking account of spatial variations, leads to a useful gauge of film thickness. Suppose,

138

J. (3. DASH

for simplicity, that the intrinsic thermodynamic state of the film is the same as that of bulk adsorbate at the same temperature, i.e. that the film is a thin slice of the bulk liquid or solid. In this case, the chemical potential of a vapor atom above a film of some thickness d on a substrate differs from the chemical potential of saturated vapor (i.e. in equilibrium with the bulk phase) only due to the energy change caused by the presence of the substrate instead of the deeper regions of the bulk. Thus, if #vo(T) is the chemical potential of saturated vapor in equilibrium with the bulk phase, the chemical potential at the surface of a film of thickness d is #v(T, d) = i~ o(T ) + Au(d) (5.9) where Au(d) is the change in energy caused by the substitution of substrate for bulk adsorbate below the depth d. If the vapor is locally ideal, its chemical potential is in the form given by (4.4), and we obtain In (P/Po) = flAu(d)

(5.10)

where P is the vapor pressure of the film and Po is the saturated vapor pressure of the bulk. If the interactions are of the type given by (4.1 I) and (4.12), then I-'z

x

7

J

(5.11)

For thick films, d ,> e, the thickness dependence tends to exp ( - const/d3). The exponent of d may have other values (as, for example, at long range, where retardation effects tend to produce a d -4 variation). With slight variations, the simple power law version was proposed independently by Frenkel, um Halsey t6~ and Hill t17~ (FHH). We emphasize that (5.6) and the F H H isotherm are based upon the assumption that the film is a uniform slice of the bulk phase. This condition must be approached asymptotically as d increases, but it can only be an approximate relation for films of a few layers' thickness.

6. H E A T C A P A C I T Y A N D

ENTROPY

Direct measurement of the total heat capacity of the system yields contributions from the film, vapor and substrate. For fixed N, F and A,

STATISTICALTHERMODYNAMICSOF PHY$ISORPTION

139

the total heat capacity

" ~ ( S f + S v + Ss) . JN, V,A

(6.1)

The substrate contribution can be measured in the absence of adsorbate and subtracted from the total. We are then left with the adsorbate contribution, which we denote as Cc~. This equilibrium heat capacity C,q is composed of three terms: parts due to fixed quantities of film and vapor, and a contribution due to film-vapor conversion along the equilibrium curve. Explicitly, we have (d_~S'~

(a_~S'~

(a_~S"~

(d_~N'~

OT /N,A = \OT ,/N,,A + ~ONf/IT, A ~ d r

(6.2)

Jeq

and a corresponding expression for the vapor. We define Cf and Cv as the heat capacities for fixed quantities: C, =- T , - - -

\ cgT /N,,A

,

Cv -

\ c3T..IN~,V"

(6.3)

Then the equilibrium heat capacity for the combined film and vapor is Ceq: T {~(Sf

+ Sv) t 3N, V,A

--C +C~.+

rr as A TL~)T,V--kONf}T,A_Jk-~-~},q. (6.4)

The difference in partial entropies (the term in square brackets) in (6.4) is related to the heat of adsorption introduced earlier. Using the expression (4.2) for the entropy of the vapor, we find (aS~'~ = S_z~- k. (~Nv/T, r N,,

(6.5)

For the partial entropy of the vapor we write

ONfiT, A = -- ONf~T -~ -- k OT JN,,A"

(6.6)

140

J. G. DASH

Now, with (4.9) for qst in terms of pressure and by substituting the general relation (5.1) for P,

With {If = {lv' and once again using (4.2) for Sv' the first two terms on the right-hand side of (6.7) can be written as (6.8) Combining (6.6) and (6.8) in (6.7), we get a general relation between qst and the vapor and film entropies: qst =

T[~v - (~)T,J·

(6.9)

Equation (6.9) provides a thermodynamic basis for determining the partial entropy of a film from vapor pressure measurements. The same quantity qst also enters into the heat capacity of the mixed film-vapor system. Substituting (6.9) in (6.4), we obtain Ceq

= Cv + Cf

+ (qst

-

kT)(ddi)eq'

(6.10)

We now explore the interrelationships among heat capacity, vapor pressure and heat of adsorption, in order to extract some expressions of practical use. In (6.10), the conversion term (dN jdTJ e can itself be related to qst' Differentiating the ideal gas law (4.3), q

dN) ( dT

eq

=

d (PV) dT kT

= -

PV pV(dlnp) kT2 + kT -n

eq'

(6.11)

The pressure derivative is the sum of two terms: In P) (ddT

= eq

(a In P) aT

(~).

+ (a In P) NJ,A

aNf

T,A

dT.

eq

(6.12)

STATISTICALTHERMODYNAMICSOF PHYSISORPTION

With the definition for q~t and

(dNv" \-~ ~ ]¢q = k -PV ~

(qst -

141

dN I = -dN,,, (6.11) and (6.12) lead to PV(~!nP~ 1-1 (6.13) kT) [1 + kT\ t~Nf/T, AJ "

Substituting (6.13) for (dN,,/dT) in (6.10) and using the ideal gas heat capacity C v = ~PV/T, one obtains tlsJ Ceq =

Cf

PVf3 (q,,

[i PV(OIRP I-' }

"~- T / 2 + \ k g ' - 1t 2 _ + ~ - - ~ \ - - ~ - f / j

(6.14)

which shows that Cy can be extracted from measurements of the total heat capacity together with data from vapor pressure isotherms and isosteres. The contribution due to desorption Cdc, - Ccq - Cf becomes relatively unimportant at low temperatures: there is usually a range in which Cd~ may be ignored. This is because P varies approximately as exp ( - q , J k T ) , while the temperature dependence of Cf is in most cases less rapid. If Cdes is not negligible, then it is possible to deduce P from heat capacity measurements alone. This can be done by comparing total heat capacities of systems having different values of V, but identical in all other respects, as by changing the part of the experimental volume that is at room temperature. The theoretical connection between P and Cd, , is obtained by equating the expressions for (aqst/~T)Nf that can be obtained from (6.14) and (4.9). This yields an exact expression, which can be solved numerically for P. When qJkT -~ 10 the solution, to within a few percent, is ~1s) P = TCdes [ 2 + T ( 0 In Cde'~ v

_

\

!,,,j

l-'

(6.15) "

If the total heat capacity is measured down to such low temperatures that one can make a reliable extrapolation to T = 0, and if a single value of P is known at some point within the measured range of C q(N, T), then one can calculate the absolute entropy St(Nf, T) of the film. The total entropy of the adsorbate is T

S(N,T)= |C,q(N,d)a~-T=Sy(Nf, T)+ S,,(N,,,T). 0

(6.16)

142

J.G. DASH

Using (4.2) and (4.3) for the vapor, we obtain T

S:(N:, T)

=

Ceq(N ,

) y

~-

[

P

\

h2 ] j

(6.17)

6 where N : = N - PV/kT. Equation (6.17) is based on the assumption that the vapor is an ideal classical gas, but it does not depend on any specific model of the film. It does not require that the desorption contribution t o Ceq be separately measured or calculated: the effects of desorption are automatically included in Sv(Nv, T). If (6.15) is combined with (6.17), then the absolute entropy of a film could be calculated from purely thermal measurements. This would be subject, of course, to the condition that the vapor pressure is sufficiently low for (6.15) to be satisfactory. In the most fortunate circumstances, P will be so small that the entire second term on the right-hand side of (6.17) can be neglected, in which case S: will be obtained simply by integrating Ceq. The major uncertainty in this evaluation is then likely to be in the extrapolation of Ceq from the lowest experimental temperature to T = 0, and this will depend upon both instrumental factors and questions concerning the detailed theoretical model of the film. An alternative evaluation of the absolute entropy of the film can be made by vapor pressure measurements alone. Since S:(N:, T) = j \ ~ N : ] r ' A dN:, o

(6.18)

using (6.9) for the partial entropy of the film and the ideal gas relations (4.2) to (4.4), we obtain N$

Sf(N:, T) = ~N:k - ~

(Pv + qst)dN: •

(6.19)

0

All of the quantities on the right-hand side of (6.19) can be determined through vapor pressure measurements. This method of determining the absolute entropy of films, while quite direct according to the formal equations, is generally less reliable than by thermal measurements. This is because the terms ~v and q~ are generally comparable in magnitude

STATISTICAL

THERMODYNAMICSOF PHYSISORPTION

143

and of opposite sign, and their difference will, in most cases, amount to less than 1 9/0 of their absolute magnitudes.

7. S U R F A C E

PHASES

If the gradients caused by the substrate fields are ignored, and we consider the adsorbate to be composed simply of film and vapor, then there can be as many as four coexisting phases of the adsorbate. This follows from the fact that the system would be described by four pairs of conjugate variables: (T, S); (P V); (~b,A); ~, N). Such a quadruple point becomes the analog to the triple point of conventional bulk systems, and will consist of three surface phases (e.g. two-dimensional solid, liquid and vapor phases) in equilibrium with the threedimensional vapor. If the adsorption system consists of only three phases, then these can coexist along an equilibrium line P = (T, tk). If there are only two phases (i.e. homogeneous film and vapor), coexistence occurs over a finite area of the P, T, tk surface. But, since substrate fields are, in fact, present in both the film and vapor fractions, the situation is actually more complicated. The substrate attraction causes gradients in the adsorbate, particularly in those layers of film closest to the substrate, and these may be so great as to make adjacent molecular layers of the film quite distinct from each other. Properties, such as the surface density of mass and entropy of adjacent molecular layers, can be so unlike as to be different phases. For example, it is possible for the layers closest to the substrate to be two-dimensional solids, while those further away are two-dimensional liquids. But, even if the differences between layers are not so dramatic, it is clear that a complete description of a multilayer system will require four pairs of conjugate variables for each layer, over the entire range of influence of the substrate fields. Equilibrium will still be governed by the uniformity of/z and T within each layer, between layers, and between any layer and any portion of the vapor. Now the number of phases in the adsorption system has become enormous. Not only is each molecular layer in the vapor as well as in the film a different phase, but each layer can break up into as many as three homogeneous fractions.

144

J.G. DASH

These arguments seem to suggest that one can never actually produce a single homogeneous phase in any experimental system, since it is never possible to remove surface effects completely. A bulk phase can still be visualized as a series of molecular layers. While this is correct in principle, it is nevertheless true that, experimentally, a macroscopic sample of bulk matter can act as a single homogeneous phase. The answer to what might seem to be a paradox is to understand phase in an operational sense, i.e. by reference to the experimental sensitivity to resolve the heterogeneities. To the extent that the gradients are smaller than can be resolved, then it only makes sense to treat such a region as a single uniform phase. This analysis just extends the arguments originally given by Gibbs when he proposed the dividing surface as a device for distinguishing between film and vapor. In the following discussion, we consider the possible changes that can occur in a single layer, bearing in mind that simultaneous changes in other layers may also occur. In the event that there are two or more surface phases co-existing with each other in a layer, and each with the vapor at a single set of variables (P, T, ~), then a number of distinctive features will be exhibited by the thermal properties of the combined systems. These features, as will be shown, depend primarily on the fact of phase coexistence, and not on the detailed nature of the phases. The fundamental attribute of the region of surface phase coexistence is that the chemical potential is independent of N, being clampedby the fact that there is a single equilibrium line P = P(T, ~b) for the system. The chemical potentials of the two surface phases (I, 2) and the vapor are all equal: /zl(~b, T) = i/2({~, T) = /~v(P, T),

(7.1)

independent of N all along the coexistence line. Since the vapor pressure is directly related to the chemical potential by (5.1), then P must also be independent of N. Thus, the region of two-phase equilibrium on the surface must show up as a vertical portion of a vapor pressure isotherm, as illustrated schematically in Fig. 2. At the same time, the isosteric heat of adsorption qst will be independent of N. The equilibrium heat capacity will also have a distinctive signature in a two-phase region. If we consider the total differential Gibbs free

145

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

energy of the mixed system, for a constant quantity of adsorbate, then (3.16) yields (7.2) dG = - S dT + A de/> + V dP. Now consider the first and second temperature derivatives of G at w

!:i

(])

a:

o

V)

-----r

az

~­

u..a: o~ ~~

1->

i=+

z::;

-----

~...J

i5S ...J

~

I-

o

VAPOR PRESSURE

I-

P

z

o I-

a. a:

o V) a

~

'-~----.,.,

u..

-

o -:;;

I'

I-CT

I

<1

c.:

I I

J:

u a: w

I

l-

V)

o

V)

FIG.

N

2. Characteristic signatures of the vapor pressure isotherm and isosteric heat of adsorption during two-phase equilibrium in a film.

constant area and volume (the typical conditions of a of heat capacity): differentiating (7.2), we obtain

(:~) 2 d G) ( dT2

= ~

= eq

N(:~)

= -

S

~

(d2p) N dT2

~

= eq

+ A(:~) +

C T +A e

(

measur~ment

V(:~)

~

,

(7.3)

2

d2 e/» dT2 eq

P) + V (ddT2

. eq

(7.4)

146

J.G. DASH

It is the fact that there is a single equilibrium line P(T, qS)that makes the several equilibrium derivatives independent of N and, hence, universal functions of T alone throughout the phase coexistence region. Rearranging (7.4), we obtain the general condition on the total heat capacity (19) Ceq = Nf(T) + g(T).

(7.5)

Thus, the signature of Ceq during phase equilibrium has the form illustrated in Fig. 3. It must be noted, however, that (7.5) is a necessary, but not sufficient, concomitant of phase coexistence: it is possible to have homogeneous surface phases with the same functional dependence as

(7.5). ,,t I-m ¢rO

/

/

/ /

/

/

k-

O-

/11

r.) /

p-

,-" "

pO

I I I I 4 !

. . . . .

N

FIG. 3. The heat capacity of the adsorption system varies linearly with total quantity of adsorbate, when the film is undergoing two-phase equilibrium.

There are quite distinctive discontinuities at the boundaries of twophase regions, i.e. where one of the phases just appears or disappears. To see this, all one has to do is to re-examine (6.4), which was derived earlier in the discussion of film-vapor equilibrium. If there are now two surface phases, there will be two parts to CI and another factor involving a difference of partial entropies, i.e. a term

l

[kaNe,/r,A, - \aNe2/T, a2J \ dT /¢q

(7.6)

STATISTICAL THERMODYNAMICS OF PHYSISORPTION

147

which represents the progressive conversion from phase 2 to phase 1 as T changes. It is only present during phase coexistence; thus, it appears or disappears abruptly with the first and last traces of a second phase. To give this discussion a more definite context, we imagine two different examples: surface-phase evaporation and surface-phase melting.

Ceq

FIG. 4. Schematic illustration of the temperature dependence of the equilibrium heat capacity for a film composed of two-dimensionally condensed patches in equilibrium with two-dimensional vapor. The discontinuous drop occurs when the last trace of condensed phase evaporates.

For evaporation, we imagine that the film is composed of condensed (liquid or solid) patches in equilibrium with a low density two-dimensional vapor. As the combined system is warmed, the two-dimensional (2D) vapor pressure ~b will, in general, increase, and the density of the 2D vapor phase will increase accordingly. As the evaporation process continues, there will be a term of the form given in (7.6) contributing to the heat capacity. Finally, a temperature will be reached at which the last trace of condensed phase evaporates: when this happens, both the conversion contribution and Cs (condensed) terms disappear abruptly: the total heat capacity has the shape illustrated in Fig. 4, The proof of the existence of the discontinuity does not require the assumption of any definite model of the condensed phase or vapor. However, fairly general considerations predict that the overall shape of the conversion term will contain an exponential factor of the activation energy type,

148

J.G. DASH

i.e. of the form ~b ~ exp ( - const/T). This is readily obtained by first constructing the surface ClausiusClapeyron equation; equating relations of the form (5.4) for two surface phases:

(dc~/dT)eq = (Ssl - sI2) (as, - as2 )"

(7.7)

Then, if one follows the same sort of approximation as in the elementary treatment of evaporation, i.e. assuming that the vapor is an ideal gas, of much lower density than the condensed phase, and that the latent heat qvap = TAs is a constant, then (7.7) leads to the 2D vapor pressure equation of the form given earlier, i.e.

~9 = ~90 e-~/kT"

(7.8)

The shape of the heat capacity around a region of surface phase

meltino has a still different shape. We imagine an initial situation in which a surlace is completely" covered with some solid film phase at a low temperature and then gradually heated. In the region of pure solid phase, there is no conversion term: such a contribution will appear at some definite temperature along with the initial appearance in liquid film. Thus, C q takes a discontinuous jump and, in the coexistence region, there are three terms due to solid, liquid and conversion. As T rises, the surface pressure q~ (and also P) will change, and the mixed system will move along the solid-liquid equilibrium curve as illustrated in Fig. 5, i.e. with a steadily changing melting temperature. Finally, with the melting of the last bit of solid, the heat capacity drops discontinuously down to a single term, the homogeneous liquid phase. The existence of discontinuities in the solid-liquid transition depends on the supposition that there can indeed be first-order melting phase changes in a film. This is not necessarily the ease, however: questions involving the order of the phase change in melting of films have been discussed in detail in a recent review. (2°~ The process just considered is at constant area. It is also possible to imagine a process at constant ~b. In this case, one is considering melting at a quadruple point, i.e. melting of solid patches which are in equilib-

STATISTICALTHERMODYNAMICSOF PHYSISORPTION

SOLID /,..~

149

FLUID

(a)

I ~T I

Ceq

T

(b}

&T I T FIG. 5. Film having liquid and solid phases might have a melting line such as illustrated in (a). If a sample is heated under constant area conditions its O(T) trajectory could resemble the dashed line in (a). Then the total heat capacity of adsorbate would have the form shown in (b), with finite discontinuities due to the first appearance of liquid and the final disappearance of solid. The discontinuities are predicated upon the existence of long-range order in the solid phase, which depends upon special conditions in the particular adsorption system.t2o)

rium with 2D vapor. Then, if the melting process i~ a first-order phase transition, there must be a finite a m o u n t of heat added while the system remains at a fixed temperature: C , is infinite, just as Cp is in a first-order process in a three-dimensional system (Fig. 6). But this is again predicated on the assumption that melting of the film is indeed a first-order process. As mentioned above, this m a y or m a y not be the case in a particular a d s o r p t i o n system. A few more properties, such as the course of the vapor pressure isotherms and isosteres, can be worked out for the various cases. F o r m o r e detailed descriptions, it is necessary to assume specific microscopic models of the surface phases. This can be readily done within the framew o r k of the general theory, using relatively standard techniques and several of the relations given in the previous section.

150

J.G.

DASH

Ceq

J

J

J T

FIG. 6. At the quadruplepoint, a film melts at constant gb. Under these conditions, the total heat capacity becomes infinite, just as for Cp in a bulk system at its triple point. The singularity shown here presumes long-range order in the solid phase, as for Fig. 5.

8

HEATS OF A D S O R P T I O N

Adsorption is an exothermic process. The energetics of the process are described in terms of a heat of adsorption. In the limiting case of of T = 0 and zero coverage, the heat of adsorption is equal to the binding energy of atoms to the substrate. But, at finite temperature and coverage, the heat of adsorption also reflects the states of both film and vapor: it is the heat that must be added to change the states of particles from one phase to the other. Therefore, assuming that the vapor state is known, and indeed we assume it to be an ideal gas in most practical cases, the heat of adsorption can serve as a probe of the state of the film. However, since for thin films the attraction to the substrate is usually the dominant term in the heat of adsorption, such measurements are primarily useful for determining film thickness changes and surface heterogeneity, and not for the generally weaker effects of lateral interactions. These are more readily examined by measurements of the heat capacity. In this section, we derive several relationships involving heats of adsorption measured under a variety of experimental conditions.

STATISTICAL THERMODYNAMICS OF PHYSlSORPTION

151

Under isothermal conditions, the heat of adsorption is a differential quantity

(0Q. q ~ \ONf/IT...

(8.1)

where dQR is the heat given to a thermal reservoir when d N f is transformed from vapor to film. Different heats of adsorption refer to specific processes in which particular sets of thermodynamic variables are held fixed. The differential heat o f adsorption qn is defined as (8.2) qd -- \ O N f / r , v , a " qd is simply related to q,c Since dQR = - d Q q~ = -- T

= -

T.V,A

= -TdS,

L\ON:/T,A

-

\ONv,/r,vJ

(8.3)

where we have used dN~ = - dNy. Substituting (6.5) for the partial entropy of the (ideal) vapor in (8.3) and comparing with (6.9), q~ = qst

-

-

kT.

(8.4)

The isosteric heat of adsorption is equal to the differential heat that would be obtained in a constant pressure process. Thus, if T, P, A are held constant,

qr,,,A= --T -~:

r,e.r

L\ONf/r,a- \ON Jr,PJ"

Substituting (4.3)in (4.2),we can express the vapor entropy in terms of

T, P, N : from which we obtain

(asvh

sv

0No/T,, = ~ "

(8.6)

Substituting (8.6) in (8.5) and comparing with (6.9), qT, P,A

= rFL N v --

l

~ O N f ] T , AJ ~- qst"

(8.7)

Neither qd nor q~ correspond to the usual experimental processes in

152

J . G . DASH

measuring heats of adsorption. In typical experiments, a finite quality of gas is added to the adsorption cell from an external bulb. After equilibration, it is found that the pressure is lower than if all of the increment had remained in the (ideal) gas phase: this is identified as AN I. If A N s / N r is small, then the finite process approximates the differential q. The actual process is irreversible: in order to relate the measured heat to the equilibrium properties of the system, it must be replaced by an idealized reversible process. The external bulb is replaced by a piston by which the volume V is changed by a differential amount. In this process, N and T remain constant, while a differential quantity of vapor is converted reversibly to the film phase. The difference between this process and the constant volume process, which yields qd, is that there is an additional contribution from the heat of compression.(2.7) For the piston process the differential heat is termed the isothermal calorimetric heat of adsorption qth" According to the constant conditions of this process, (oQ, qtla -=

= _

\aN flr, A

(8.8)

os

\aN j r , A.

Expressing S in terms of SI and S o, as before, introduces a term (aS,]ONs)r,a. Sv is a function of A only indirectly, through the equation

of state of the combined film-vapor system. The cross-connection can be obtained explicitly through (4.2) and (4.3). Thus, from (4.2),

(os'

so

Nok/o "

(8.9)

Using (4.3) and dN o = - d N I, (8.10) Substituting (7.9) and (7.10) in (7.8), we obtain qth =

IN_

\?Ny] r,Aj

+ V

7"..,1 = qst

+V

T,A"

(8.11)

The pressure term can be obtained from a vapor pressure isotherm.

STATISTICAL THERMODYNAMICS OF PHYSISORFrlON

153

Now we consider a reversible adiabatic process in which the thermally isolated system experiences a change in temperature where Ny is changed. In this case, we begin with

TdS = Td(S, + Sf + S~) = 0

(8.12)

where So is the entropy of adsorbent and container. Treating the entropies as the implicit functions Sy(T, A, N f), S~(T,P, N¢), we have for constant A:

TdS¢

=

TdSv = C~e +

CfdT +

(0s t

T ~8Nf

(8.13)

r,A d N I ,

\ S N j r , e dN + T

dP

(8.14)

where C~e is the total heat capacity of vapor at constant N~, P. The second and third partials of Sv are found from (4.2) and (4.3), and we obtain for (8.14):

(8.15)

TdSv = C~e dT - -~ dNo - VdP. Substituting in (8.12) and collecting terms, ( C . + Cs + Coe) dT + T _ ~ONs r,a

dNf- VdP=O

from which we obtain

(C + Cvl, + f ) ~ r )

s = qst + V

s"

(8.16)

The adiabatic heat of adsorption q, is defined by t7)

C .[8T'~ q, = (C,e + Cve + fe)~-~)s____

(8.17)

where the component heat capacities refer to constant P as well as to constant A and numbers of particles. Since C, and Cf are independent of P, (8.16) and (8.17) yield

qs = qst +

A) ° s"

(8.18)

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9. S U M M A R Y This review has covered all of the common ground of modern thermodynamics of physisorption, and many of the results derived here were obtained by others years ago. In addition to these standard results, several novel or lesser-known points have been discussed. Here are summarized the principal features of the review. The theory is developed straightforwardly from the elements of statistical physics, with no new concepts or special definitions. Much of the first couple of sections is quite frankly a direct transcription of corresponding passages in Statistical Physics by Landau and Lifshitz: this is done in order to make very clear how unequivocal adsorption thermodynamics really is. I must admit that there is also something of pleasure in this repetition, for it presents much of familiar textbook thermodynamics in a new guise, like meeting old friends in unexpected places. This is, I think, not a wholly personal view, for many of my students and colleagues find a similar enjoyment in surface thermodynamics. Perhaps some day we will find it a more conventional topic in textbooks. It seems that one distinct benefit of the statistical approach is that several important results can be obtained by rather more economical derivations than in typical treatments. The economies are due, in large part, to the fact that one can use the explicit free energy and other functions of the ideal gas throughout. Then there is an extensive discussion of the several approximations and cross-terms that tend to be buried under the surface in most treatments. Although in the later sections of this article these terms are usually disregarded, it has been shown to what degree they need be considered in careful work, and a framework for their inclusion is prepared, so that, if need be, they can indeed be inserted. And in partial recompense for making the various corrections more visible and, hence, more difficult to ignore, we show that one particularly unphysical assumption of conventional theory is not at all necessary: the inert substrate postulate can be replaced by a much more realistic one of adatom-associated distortions. The grounding of the macroscopic thermodynamic theory on statistical foundations provides a natural and logical framework for the introduction of different microscopic models of films and substrates. These

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models can be very simple, such as the very familiar examples of the classical two-dimensional mobile gas or localized lattice gas, or extremely complicated examples of current many-body theories. But, in dny event, the thermodynamic theory as presently outlined can accept any microscopic model and then show how the macroscopic variables, such as vapor pressure, heat of adsorption, etc., would behave in an equilibrium adsorption situation. Perhaps the most original portion concerns the discussion of the phase rule and phase equilibrium, although it is in large part a rather direct application of the principles of thermodynamic equilibrium. We must conclude that, if interactions due to the surface extend for an infinite range (i.e. never become identically zero at finite distance), then the number of possible phases in an N-particle system is, in principle, of order N. But, then inspecting the effective meaning of phase, it was a natural development to propose the operational definition: there are just as many phases as can be resolved.

REFERENCES 1. J. W. GIBnS, Trans. Conn. Acad. 3, 108, 343 (1878); Scientific Papers, Longmans, Green & Co., 1906; reprinted by Dover Publishing Co., Inc., New York, 1961. 2. A. F. H. WARD, Proc. R. Soc. Lond. A, 133, 506 (1931). 3. M. POLANYI, Z. Electrochem. 35, 431 (1929). 4. E. A. GUGGEmtE~M,Trans. Faraday Soc. 36, 397 (1940). 5. T. L. HILL, J. Chem. Phys. 17, 520 (1949). 6. G. D. HALS,, Jr., J. Chem. Phys. 16, 931 (1948). 7. G. L. KINGTON and J. G. ASTON, J. Am. Chem. Soc. 75, 1924 (1951). 8. D. H. EWI~Tr and D. M. YOUNG, Trans. Faraday Soc. 48, 1164 (1952). 9. W. A. STEELEand G. D. HALSEY,Jr., J. Chem. Phys. 23, 969 (1954). 10. For a much more extensive review of the literature, see D. M. YOUNG and A. D. CROWI~LL,Physical Adsorption of Gases, Butterworths Scientific Publishers, London, 1962, references to chapter 3. 11. R. FOWLER and E. A. GUGGE~a-mlM,Statistical Thermodynamics. Cambridge, 1949. 12. L. LANDAUand E. M. LXFsHrrz, Statistical Physics, Pergamon Press, London, 1958. 13. See, for example, R. J. TYKOD1,J. Chem. Phys. 22, 1647 (1954), and M. SCmCK and C. E. CAMPBELL,Phys. Rev. A,2, 1591 (1970). 14. E. M. LIFSmTZ, J. Exp. Theoret. Phys. USSR 29, 94 (1955) [Soy. Phys. JETP 2, 73 (1956)]. 15. I. E. DZALOSmmKn, E. M. LIFSmTZ and L. P. PrrAEvsKII, Adv. Phys. 10, 165 (1961). 16. J. FRENKEL, Kinetic Theory of Liquids, Oxford University Press, London, 1949. 17. T. L. HILL, J. Chem. Phys. 17, 590 (1949).

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18. J. G. DASH,R. E. PEnmLSand G. A. STEWART,Phys. Rev. A, 2, 932 (1970). 19. G. A. STEWARTand J. G. DASH, Phys. Rev. A, 2, 918 (1970). 20. J. G. DASH, Prog. Surf. Memb. Sci. 7, 95 (1973).