SOLID SURFACES and the Gas-Solid Interface

---

A General Theory of Monolayer Physical Adsorption WILLIAM A. STEELE Department of Chemistry, The Pennsylvania State University, University Park, Pa.

A general theoretical approach to monolayer phys­ ical adsorption is discussed. In this theory, the isotherms and heats of adsorption at given μ, Τ are given as functions of the interaction energies of the adsorbed atoms with the solid and with each other. The general equations reduce to localized and mobile adsorption when the potential varia­ tions over the surface are either very large or very small. Intermediate cases are also included. Gas atom-solid interaction energy functions are computed from the known pair interaction poten­ tials for several rare gas systems, and it is shown that a considerable amount of information can be obtained about the adsorption properties of such systems from these potential functions.

T h e relationship between the properties of a physical adsorption system and the energies of the adsorbed particles with each other and with the solid has been the subject of numerous investigations. In particular, reasonably good agreement has been obtained between theoretically calculated energies of adsorption and experimental heats at very low coverages on reasonably uniform surfaces (16). Halsey et al. (7, 22) have interpreted measurements of the high temperature in­ teraction of dilute gases with solids as gas imperfection caused by the potential energy of gas atoms in the vicinity of the gas-solid interface. More recently, the formal equations necessary to express the properties of an adsorption system at any Τ, μ as functions of the interaction energies have been given (12, 23). In these treatments, it is assumed that the properties of the adsorbent surface remain essentially unperturbed by the presence of the adsorbed gas. This approximation allows one to identify the adsorption problem with that of a fluid in the external potential field which exists near the surface of the adsorbent. This approach was then used in a treatment of the specific problem of monolayer adsorption (11, 24) and it was shown that a general theory of monolayer adsorption could be obtained in which many of the assumptions made in previous theories were no longer neces­ sary. In particular, the general theory contains both localized and mobile adsorp­ tion as special cases. In this paper, we bring together some of the results of the previous papers which are pertinent to the monolayer adsorption problem. The application of the theory to several simple adsorbate-adsorbent pairs is discussed, 269

270

ADVANCES IN CHEMISTRY SERIES

and it is shown that one can obtain a great deal of information about the nature of the gas-solid potential energy function from an analysis of the experimental data for suitable systems. Theoretical Most experimental investigations consist in measurements of the isotherms, the heats of adsorption, or both. Therefore, we shall direct our attention to these two properties. From the point of view of a statistical mechanical calculation, the most convenient definition of the amount adsorbed, N , is : a

J

"

(i)

ri) -

(2.1)

po]dn

where p is the gas density in the adsorption volume, v, in the absence of gassurface interactions, and ( Ο is the density in the presence of the interactions. p^ ) (r ) is, of course, a function of the position of atom 1 (denoted by r ) , and is expected to be large compared to p when atom 1 is in the vicinity of the surface, and to decrease rapidly to p as the distance from the surface increases. Since adsorption systems are generally at known Τ, μ (the chemical potential of the adsorbed gas is obtained from the adsorption pressure), it is clear that it is most convenient to use the grand canonical ensemble in computing the properties of the system. In general, p^ ) (rj) is defined as the probability that a molecule will be found at r irrespective of the positions of the other molecules. Therefore, the expression for pW (r ) in a canonical ensemble of Ν particles is given by a Boltzmann factor which includes the interaction energies of all Ν atoms with each other and with the surface, integrated over all possible positions of all atoms except atom 1, and divided by the proper normalizing factor. In the grand canonical en­ semble, Ν is unrestricted, and, therefore, one has a sum of Boltzmann factors, one for each possible value of N. When the proper normalizing and weighting factors are included, the classical statistical mechanical equation for pW (r ) is written: 0

1

x

x

0

0

1

l9

x

x

(i) P

(ri) = Î ^ T j

(]y^T"ï)j J

J

e x

P l-U (ri, N

ζ = εγρ(μ/*Γ)/Λ

r )/kT]dr N

2

(2.2)

dr

N

(2.3)

8

Ν U (ri,. N

. . .r ) N

= Σ

« s(ri) +

]Γ

u

^

(

s-Zâi** Ζ

•/ ί

Ν

=

I .... Ι

exp [ — U (n,. N

2

(2

. . .r )/kT]dn. N

'

·

4

)

5)

(2.6)

. . .dr

N

where u ( r ) is the potential energy of interaction of atom 1 at position r with the solid, and ι/(ι#) is the mutual interaction of atoms i and / separated by a dis­ s

x

T

tance ry. Methods of computing p^ ) (r ) from Equations 2.2 to 2.6 have been dis­ cussed (23); however, in the case of monolayer adsorption, it is often more convenient to deal directly with the grand partition function of the system. When Equation 2.2 is substituted in Equation 2.1 and the integrations are carried out, the amount adsorbed is given by 1

x

STEELE

271

Monolayer Physical Adsorption

"•-ΐΣίΛ^ϊτ.^-"·

( 2

·

7 )

where N is the total number of atoms which would be held in the adsorption container if there were no gas-surface interactions. It is now easy to show that the total number of gas atoms in the container (N + N ) can be obtained directly from the grand partition function in the usual manner: 0

a

* +

0

Γ,Κ

(

2

·

8

)

However, the expression given in Equation 2.5 for the grand partition function can be written in a form which is better adapted to adsorption calculations. This is accomplished by dividing the volume of integration into two parts: a relatively small volume in the immediate vicinity of the adsorbent surface (denoted by V ) , and the remainder of the gas space volume (denoted by V^). The dividing line is chosen to be at the point where the gas-surface interactions become negligibly small. [In most monolayer problems, the exact location of this boundary is un­ important, since all the adsorbed atoms will be found in a small region of maximum attractive energy; however, at sufficiently high temperatures, one must treat this problem somewhat more carefully (22).] For convenience, it will be assumed that the gas in the volume V is perfect. Then one can write 8

g

Z

= Zs

N

N

+ NV Zs _r g

N

+ (~ N

N

X

+

V*Zs _2

)

N

(2.9)

where Zs

N

= j

. _ . . . J*exp [-U (r . N

. . .T )/kT}dn. . . .dr

u

N

N

(2.10)

When Equation 2.9 is substituted in 2.5, and terms are collected, it emerges that Z

Σ, W 1^

Ξ =

Z

>Γ-Λ

N

N

Zsn

Ν £ 0

"

W

V

G

(2· ) 11

N

N^O

z is the absolute activity of the adsorption system, and is equal to the absolute activity of the gas in equilibrium with the system. As is usual in treatments of ideal gases, we set ζ = p/kT (10). Then the second sum in Equation 2.11 is just the grand partition function for an ideal gas in volume V . We define g

A

LJ

=

N >

Z

0

S

(2-12)

N

'

It is now easy to show that N

a

(2.13)

= z(à\nZ*/àz) ,

T v

where it has been assumed that V ^ v. The energy of the adsorbed phase E can also be obtained from the grand partition function in the usual manner: E = kT\b In Έ*/οΤ) , + Νμ (2.14) g

a

a

ν μ

= kT\b In S * / o r )

α

F ( Z

+

^kinetic

(2.15)

Up to this point, the theory has been presented in a form suitable only for adsorption systems which can be treated by classical statistical mechanics. When the quantization of the motion of the atoms is important, one must replace the Boltzmann factors in the integrals for Z by the appropriate Slater sums. After integrating over the coordinates, one obtains N

272

ADVANCES IN CHEMISTRY SERIES

= A"**

(Zs \ N

luantum

exp [-e(jN)/kT]

(2.16)

Λ = (hyiirmkTyi*

(2.17)

e(/V) is the ;th eigen value of the energy of a collection of Ν atoms in the adsorp­ tion volume V . Equation 2.12 is in the form of the grand partition function for an imperfect gas, and the adsorption isotherm can be obtained by the usual methods of im­ perfect gas theory (10) : g

N

+ ...

= Zs,(p/kT) + (Zs - Z *)(p/kTf

a

2

Sl

(2.18)

If the surface of the adsorbent is homogeneous in the sense that it is made of N elements of area A all of which are identical, Equation 2.18 can be put in a con­ venient form for the analysis of experimental data by making the substitutions:

8

89

kT/p

= Γ Γ exp [-u (ri)/kT]dn J zJ A

= Z /N

0

Sl

s

(2.19)

s

s

s

β = N -

Zs N /Zsi

s

/i expj 2

2

2

s

u (T )]/kT}dridr

— [Κ«(ΓΙ) +

s

2

2

(2.20)

f(((

exp {— [u (ri) +

u (r )]/kT}dndr

s

s

2

2

%J Z %) A nJZ %) A S

S

S

fu

s

= exp [—u(m)/kT] — 1

The isotherm equation becomes θ = Ν /Ν α

= (p/p )[\ - β(ρ/ρ ) + · · ·]

8

0

(2.21 )

0

At small p/p , higher terms can be omitted and Equation 2.21 can be written in a form which is particularly useful in the comparison of experiment with theory: 0

NJN

(2.22)

= (p/p )/[\ + β(ρΙρ )\

S

0

0

If N /ρ is plotted as a function of N . the slope at small N gives —β/p , and the intercept gives N /p . If an independent estimate of N is available—from multi­ layer data, for instance—one can obtain both β and p . The partial molar heat of adsorption can be derived either from Equation 2.15 or from the well known relation (l

Q9

8

(l

0

0

s

0

q

st

(2.23)

= R[d\np/d(\/T)]

Q

In either case, one obtains q

8t

= R[b\np /o(\/T)] 0

+ [RB/(\ - βθ)][άβ/ά(\/Τ)]

+ ...

(2.24)

Equation 2.24 predicts that, unless β is constant, q will change with coverage even in systems which are topographically uniform—i.e., no imperfections in the lattice structure of the surface. st

Calculation of Parameters To carry out a complete and unambiguous computation of the parameters p and β, a detailed knowledge of the gas-surface potential function is necessary, as well as the gas-gas potential function. The computation of the gas-surface potential function for all possible positions of the gas atom relative to the surface 0

STEELE

273

Monolayer Physical Adsorption

is extremely difficult, and it is much more likely that experimental results will be used to obtain information about this potential, than the converse. It is possible to simplify this process if one can approximate the gas-surface potential interaction as a relatively simple function of the position coordinates of the gas atom. This procedure is feasible only if it is assumed that monolayer adsorption is adsorption under conditions where all the adsorbed atoms are at distances from the surface which are very nearly equal to the distance corresponding to the minimum poten­ tial energy. This distance is denoted by z , and the gas-solid interaction energy at that point by e . Also, let be a two-dimensional vector which is parallel to the plane of the adsorbent surface. In general, both z and e will vary with τ (measured relative to some convenient origin). We define e (τ*) to be the variation in e due to changes in , e„, (0) to be the potential energy at = 0 and ζ = z , and e ( ζ — z ) to be the change in the interaction energy as the ζ coordinate of the atom is changed from z . Then the gas-surface potential func­ tion in the neighborhood of the minimum can be written as m

τ

m

m

m

r

X i

m

m

s

τ

m

m

u.(n) = e (0) + 6 (τ») + € (z - z ) m

2

Τ

(3.1)

m

where it has been assumed that e has no appreciable dependence upon τ· Equation 2.19 for p can now be written z

0

\/po = (\/kT)cxp

[-e (0)/kT] m

I exp [-e (z - z )/kT]dz Jz z

I exp J A

m

s

[-e (xi)/kT]dXi T

s

(3.2) In many systems, the variation of e~ with ζ is sufficiently rapid that one cannot suc­ cessfully use the classical integral with respect to ζ which appears in Equation 3.2; if this degree of freedom is considered to be quantized, the equation for p becomes

0

MPo = ( M r ^ e x p [-e (0)/kT]

^

n

η

exp [-e (n)/fcT]

exp [ - € ( τ ι ) Α Γ ] ^ τ ι (3.3)

|

z

Τ

J A

s

where z (n) is the nth energy level for motion in the ζ direction. It is generally most convenient to choose = 0 to be the point of maximum attractive energy; in this case, e will always be greater than or equal to zero, and = 0 will correspond to the central point of an adsorption site. The assumption that all atoms are at or very near to ζ — z allows one to simplify the expression for β. The integrations over ζ (or sums over quantum states) in the numerator and denominator of Equation 2.20 are identical in this case, and cancel out. The resulting equation for β involves only the variable: z

τ

T

τ

m

τ

/i exp | - [ € ( τ ι ) + IAJAS 2

β =

Τ

€ (τ )]/Α7Ίί/τι the diameter of an atom of the solid. The fractional change in energy, —E/e (0), was obtained by linear extrapolation of the theoretically computed change in e near the site center θΰ

sa

m

T

The two curves shown are for an atom over the 100 face of a simple cubic solid, and over the 100 face of a face-centered cubic solid the site center [the site centers ( = 0) for a surface such as this occur at points equidistant from four surface atoms]. This potential was computed by assuming that the total interaction was equal to a sum of pairwise interactions between the gas atom and the atoms of the solid. The pair interactions were calculated from a Lennard-Jones potential function and the sums were carried out on an IBM 650 computer. Since extensive tabulations of these functions have been published (24), we merely point out here that the data show that the angular dependence of e for such a lattice is reasonably small near the site center. In this region, can thus be represented by τ

T

6 r

er = 2Er/a

(3.5)

where a is the edge length of a site, r is the radial distance from the site center, and Ε is the change in the minimum of the energy between the site center and the mid­ point of a site edge, calculated by linear extrapolation of the variation near the center. Ε was found to depend strongly upon the relative sizes of the gas atom and the solid atom. — E/e (0), the extrapolated fractional change in energy between r = 0 and r = a/2, is shown in Figure 1 as a function of the ratio of the atomic diameters. The extrapolated fractional change in energy between the site center and the midpoint of a site edge for an atom adsorbed on the 100 race of a face-centered cubic crystal is also shown—the details of this calculation are discussed in the following section. As E/e (0) approaches zero, the monom

m

STEELE

275

Monolayer Physical Adsorption

layer properties approach those of a completely mobile monolayer, and p and β become 0

\/po = (AkT)~ A cxp l

8

[-e (0)/kT] m

exp [ - € , ( « ) / * Γ ]

Σ

(3.6)

η β =

/,*/τ

-1/A,

(3.7)

As expected, Equation 3.7 for β is similar to the surface second virial coefficient which occurs in the usual treatment of the two-dimensional gas approach to mono­ layer adsorption (9). For E/kT » l,p becomes 0

\/p = (AkT)-\Ta /2)(kT/E)* 0

2

exp [-e (0)/kT] m

^

exp [-e (n)/kT] z

(3.8)

η Furthermore, the rapid variation in e when E/kT » 1 means that there is negligible probability of the adsorbed atoms being anywhere other than at their site centers. In this case, the integrand in the denominator of Equation 3.4 is finite only when both atoms are at the centers of their sites. Thus, T

0 = 1 - c(exp[ -u( )/kT]

- 1) + · · ·

aij

(3.9)

where u (a^) is the interaction of a pair of atoms on one of the c pairs of nearest neighbor sites whose centers are separated by a distance a^. Higher terms in the series include similar contributions to β from next nearest neighbor sites, etc. β — 1 if interactions with atoms on neighboring sites are negligible; in this case, Equa­ tion 2.22 is identical to the Langmuir isotherm. The calculation of β and p for intermediate values of E/kT requires consider­ able computational labor, and is discussed in detail elsewhere (I J, 24). 0

Application It is desirable to compare the predictions of the theory presented here with experimental results obtained on some systems in which an independent com­ putation of the gas-surface potential function can be carried out. A calculation of the potential functions for the adsorption of rare gases on solid rare gases involves the least number of unknown parameters. The rare gases crystallize into facecentered cubic solids with known lattice constants. Furthermore, the parameters appearing in the Lennard-Jones potential functions for the gas-gas and the gassolid atom interaction can be estimated to a good degree of accuracy from experi­ ments on the gas properties as well as from the empirical combining laws for potential parameters. Furthermore, some experimental results have already been reported for these adsorption systems (18, 20). In particular, the potential functions for helium adsorbed on argon and for neon, argon, and xenon adsorbed on xenon are shown here, and the theoretically predicted adsorption properties are compared with experiment, wherever possible. u (ri) for a Lennard-Jones atom interacting with a crystalline solid is given by the equation: s

u (n) = 4e*p*«[SM s

ρ* = σ/α,

+

P*«SU(P)]

ρ = ζ/α

(4.1) (4.2)

where €* and σ are the usual parameters in the Lennard-Jones equation for the interaction between a pair of atoms (one gas atom and one solid atom) and a is

276

ADVANCES IN CHEMISTRY SERIES -220

-260

-300

-340

-380

Figure 2. The minimum potential energy, t of a helium atom interact­ ing with the 100 face of an argon crystal is plotted as a function of the position of the helium atom relative to the surface lattice m>

The origin of the χ coordinate is at the center of a site, and the orientation of the χ axis is parallel to the site edge. The horizontal line intersecting the potential curve shows the location of the ground state energy level of a helium atom vibrating along the χ axis. ψ * is the probability amplitude in the χ direction for an adsorbed helium atom in its ground state, plotted in arbitrary units χ

the lattice parameter of the solid. S and S are the sums (over the entire solid) of the attractive and repulsive terms in the Lennard-Jones expression. These quantities depend upon the coordinates of the gas atom, as well as the symmetry of the lattice and the crystallographic indices of the exposed face. If it is assumed that the 100 face is exposed, S and S for a face-centered cubic lattice can easily be obtained from the tabulated sums for a simple cubic lattice. The results show that the edge length of a repeating unit in the surface (or site) is a/V2, the dis­ tance of closest approach of two solid atoms in this lattice. Two of the corners of a site coincide with two atoms on an edge of a unit cell, and the other two corners coincide with atoms in the centers of the faces of two adjacent unit cells. The parameters used in these calculations were calculated from the combin­ ing rules: 6

6

e* =

Λ/€*ΙΙ€*

1 2

1 2

2 2

.σ = ( σ η + σ ) / 2 22

(4.3)

The values used for c * , a n , etc., were taken from Dobbs and Jones (5); these authors also list the requisite lattice parameters for argon and xenon at 0 ° K. The values of the parameters used in the calculation are listed in Table I. The poten­ tial functions which were obtained when these parameters were substituted in Equation 4.1 are shown in Figures 2 and 3. The potential energy at ζ = z is plotted as a function of χ (or y), the displacement of an adsorbed atom from the site center parallel to a site edge. It was found that the best simple representation of e in the neighborhood of a site center on a face-centered cubic crystal is n

m

T

STEELE

Monolayer Physical Adsorption

277

e = (2V2E/a)(\x\ + \y\)

(4.4)

T

where Ε is, as before the extrapolated difference in energy between the site center and the midpoint of a site edge. Theoretical values of the quantities pertinent to computations of adsorption data are shown in Table II. c (n) are shown for η — z

' 400 I

•

J

X (A.)

Figure S. The minimum po­ tential energies of several rare gases interacting with the 100 face of a xenon crystal are plotted here The definition of the χ coordi­ nate is identical to that given in Figure 2 Table I. Interaction Parameters for Adsorbate-Adsorbent Pairs AdsorbateAdsorbent

e*/k, ° Κ.

He-A Ne-Xe A-Xe Xe-Xe

35.6 89.5 166.0 225.0

σ, A.

a, A. at 0 K.

3.00 3.41 3.74 4.07

5.31 6.1 6.1 6.1

0,1. These values were computed by fitting the calculated u (z) curves in the neighborhood of z to perturbed harmonic oscillator functions [incidentally, these H

m

278

ADVANCES I N CHEMISTRY SERIES

Table II. Theoretical Minimum Interaction Energies and Energy Levels for Adsorbed Gases (All energies given in units of cal./mole) AdsorbateAdsorbent

He-A Ne-Xe A-Xe Xe-Xe

e (0)

e,(7) 172 184 214 175

**(0)

m

-370 -937 -1862 -2756

66 66 73 59

Ε

-E/e (0)

u(aij)

e (0)

m

250 660 875 1150

x

60 60 59 47

0.69 0.71 0.52 0.42

-4.1 -18.0 -177.0 -370.0

140 140 135 108

calculations indicated that e (n) has a negligible dependence upon for the systems considered]. Also shown is the value of the interaction energy of a pair of gas atoms which are assumed to be completely localized at the centers of two nearest neighbor sites [w(a^) ]. In view of the rapid variation of potential energy with distance shown in Figures 1 and 2, it seemed worthwhile to calculate the energy levels for motion of the adsorbed atoms in the χ or y directions. The Schroedinger equation has been solved elsewhere for the potential function of Equation 4.4 (20); the results can be summarized as : τ

z

e (m)

=

x

HmCL(2^/iE/a)

(4.5) a =

(9αη /64χ/2ττ ΜΕγι* 2

2

where M is the mass of the adsorbed atom, and £ is a numerical factor having the values 0.778, 1.875, 3.480, for m = 0,1,2 The ground and first excited states for motion in the χ direction were calculated from Equation 4.5 for the various adsorption systems, and are listed in Table II. The horizontal line in Figure 2 shows the net energy of an adsorbed helium atom after e (0) is added to the total potential energy; the probability amplitude in the χ direction for an atom in the ground state (ψ ) is also plotted in arbitrary units. It is clear that the zero point energy and zero point motion of an adsorbed atom are large in this system, and in fact, the total zero point energy (summed over all three degrees of free­ dom) amounts to 190 cal. per mole. Therefore, helium atoms adsorbed on an argon surface must be treated as completely quantized at experimental tempera­ tures ( ^ 2 0 ° Κ or less). The quantum expressions for p and β are: m

x

2

χ

0

1/Po = ( Λ ^ Γ ) - ι exp [-e (0)/kT]

exp [-e(ji)/kT]

m

(4.6)

η

where j is the quantum number for the energy levels of an isolated atom on a site; 1

exp

Σ

[-e(j )/kT] 2

-

β = Ν* N{ s

Σ

2

(4.7)

exp[-e(j\)/kT]}

ii

where ; is the quantum number for the eigen states of a pair of interacting atoms. For helium adsorbed on argon, the values of p computed from Equation 4.6 using the theoretical values of the energy states agree very well with the experi­ mental measurements over the range 1 4 ° to 2 0 ° Κ (20). The experimental β were found to range from ^ 2 at 1 4 ° Κ to ^ 1 at 2 0 ° K; qualitative considera­ tions, using Equation 4.7, lead to the tentative conclusion that these results are indicative of a net repulsive interaction between atoms on nearest neighbor sites. The classical calculation (see Table II) indicates a weak attractive energy; how­ ever, the net interaction will most likely be repulsive when the large zero point 2

0

STEELE

279

Monolayer Physical Adsorption

motion of the adsorbed atoms is taken into account. Figure 2 shows that dis­ placements of 1 A. from the site center are reasonably probable; if two helium atoms on neighboring sites are displaced toward each other by 1 Α., their mutual interaction will be repulsive b y ^ l O O O cal. per mole. A detailed calculation of the eigen states of a pair of interacting atoms on a surface is now being attempted. The calculated ground and first excited states shown in Table II for the Ne-Xe, Ar-Xe, and Xe-Xe systems hardly change from one pair to another. This constancy is accidental: As the atomic number of the adsorbate is increased, the increase in mass tends to decrease the spacing of the levels; however, this decrease is can­ celled by the increase in spacing due to the larger gradients in potential energy present in the systems with larger atomic numbers. The separation of the energy levels is roughly the same in all three dimensions; this indicates that one must use either the completely quantum equations for p and β (Equations 4.6 and 4.7) or the completely classical equations (Equations 2.19 and 3.4). Since monolayer experiments are generally carried out at temperatures higher than the freezing points of the adsorbates, it may be surmised that the quantum equations may be required in the Ne-Xe systems, the classical equations in the Xe-Xe case, and pos­ sibly a semiclassical (13) treatment for Ar on Xe. The Ar-Xe system has been studied experimentally (18). Unfortunately for the purpose of this paper, these measurements are primarily concerned with the multilayer properties. An analysis of the low coverage portion of these data is now being attempted. There is, however, one interesting point of comparison. The experiments show that the heat of adsorption of argon on xenon at zero coverage is about 250 cal. per mole greater than the heat of sublimation of solid argon at 6 0 ° K. The theoretically calculated heat of adsorption at 0 ° K, zero coverage, is -1862 + 2(59) + 73 = -1671 cal. per mole. This value is 180 cal. per mole greater than the experimental heat of sublimation of Ar at 0 ° Κ (5). Since the enthalpy change of argon between 0 ° and 6 0 ° Κ would be expected to be about the same for the adsorbed and the solid states, one may conclude that the theoret­ ical and experimental estimates are in good agreement. 0

Discussion It is widely accepted that localized adsorption is a result of large variation in potential energy from point to point on the adsorbent surface, and that the lateral interaction in such systems must be affected to some extent by the existence of these variations (3). This paper consists in a more quantitative formulation of these concepts than has been available previously. Furthermore, it is shown that, for adsorption on crystallographically perfect surfaces, a considerable amount of in­ formation about the nature of the potential field at the surface of the solid can be obtained from a proper treatment of the experimental data. This approach to the adsorption problem can also be extended to give the formal equations ap­ plicable to adsorption on a heterogeneous surface; however, it seems unlikely that much practical use could be made of these equations because of the very large number of unknown parameters which appear. The calculations of the potential functions for the various rare gas pairs in­ dicate that monolayer adsorption in these systems will be highly localized at temperatures near the boiling points of the adsorbates. From the magnitudes of the values of —E/e (0) shown in Figure 1, it may be surmised that localized ad­ sorption will generally be found if the adsorbate atoms are smaller than the atoms m

280

ADVANCES IN CHEMISTRY SERIES

of the adsorbent. In view of the fact that the numerous "lattice gas" treatments of adsorbed monolayers (2, 6,10,14) apply only to localized systems, a knowledge of the degree of localization is of great value in the interpretation of experimental data in terms of these theories. For instance, many investigations of the proper­ ties of adsorbed films on graphitized carbon black have been reported (1, 4, 15, 17, 19, 21). These data should be suitable for theoretical interpretation, since the surface of the adsorbent can be made reasonably homogeneous. The lattice spacing of the exposed face of the graphite lattice is small compared to the atomic sizes of most adsorbates, and thus one might suppose that the potential variation from point to point over a carbon surface would be small (except possibly for helium). Calculations show this to be correct for the argon-carbon system (4). The parallel variations should thus be negligible for all adsorbates with molecular sizes larger than argon. If this is the case, one ought to be able to compare the monolayer adsorption data for such systems with the two-dimensional imperfect gas model. Such comparisons would be of particular interest in view of the state­ ments that some adsorbed films made up of molecules larger than argon form localized monolayers on graphitized carbon (15,19). Many adsorption systems can be shown to correspond either to a localized adsorption model or to a two-dimensional gas. However, there must also be a large number of adsorbate-adsorbent pairs for which the potential energy func­ tions will have variations intermediate between the extremes required for the limiting models. The formal equations applicable to these systems are included in the theory presented here, and furthermore, it appears to be feasible to carry out specific computations of the isotherms and heats for such systems up to moderate coverages (11, 24). Literature Cited (1) Aston, J. G., Greyson, J., J. Phys. Chem. 61, 613 (1957). (2) Bumble, S., Honig, J. M., J. Chem. Phys. 33, 424 (1960). (3) Champion, W. M., Halsey, G. D., Jr., J. Phys. Chem. 57, 646 (1953). (4) Crowell, A. D., Young, D. M., Trans. Faraday Soc. 49, 1080 (1953). (5) Dobbs, E. R., Jones, G. O., Repts. Progr. Phys. 20, 516 (1957). (6) Fowler, R., Guggenheim, Ε. Α., "Statistical Thermodynamics," p. 437, Cambridge University Press, Cambridge, England, 1952. (7) Freeman, Μ. Ρ., Halsey, G. D., Jr., J. Phys. Chem. 59, 181 (1955). (8) Hill, T. L., J. Chem. Phys. 17, 762 (1949); 14, 441 (1946). (9) Hill, T. L., J. Phys. Chem. 63, 456 (1959). (10) Hill, T. L., "Statistical Mechanics," McGraw-Hill, New York, 1956. (11) Hill, T. L., Greenschlag, S., J. Chem. Phys. 34, 1538 (1961). (12) Hill, T. L., Saito, N., Ibid., 34, 1543 (1961). (13) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., "Molecular Theory of Gases and Liquids," p. 419, Wiley, New York, 1954. (14)

(15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

Honig, J. M., ADVANCES IN C H E M . SER., NO. 33, 239 (1961).

Isirikyan, Α. Α., Kiselev, Α. V., J. Phys. Chem. 65, 601 (1961). Kiselev, Α. V., Quart. Revs. 15, 99 (1961). Pace, E. L., Siebert, A. R., J. Phys. Chem. 63, 1398 (1959). Prenzlow, C., Halsey, G. D., Jr., Ibid., 61, 1158 (1957). Ross, J. W., Good, R. J., Ibid., 60, 1167 (1956). Ross, M., Steele, W. Α., J. Chem. Phys. 35, 0000 (1961). Spencer, W. B., Amberg, C. H., Beebe, R. Α., J. Phys. Chem. 62, 613 (1957). Steele, W. Α., Halsey, G. D., Jr., J. Chem. Phys. 22, 979 (1954). Steele, W. Α., Ross, M., Ibid., 33, 464 (1960). Ibid., in press.

RECEIVED June 12, 1961.