TERRELL rA.HILL
456
Vol. 63
acknowledge the valuable assistance of his many eo- lowship a t the Mellon Institute; and direct and workers and collaborators with whom during the indirect support of his work at the Johns Hopkins past 35 years he has been studying the properties University by W. R. Grace Company and the Daviof catalysts and the nature of catalytic processes. son Chemical Company. Finally, he wishes t o He is glad also to acknowledge gratefully the thank the Kendall Company for the award, this year support of his work by the Gulf Oil Company in and for its continued support of the Kendall Awa.rd the form of funds for the Multiple Petroleum Fel- in colloid chemistry.
RELATIONS BETWEEN DIFFERENT DEFINITIONS OF PHYSICAL ADSORPTION1*2 BYTERRELL L. HILL Department of Chemistry, University of Oregon, Eugene, Oregon Received October 7, 1968
The conventional two dimensional definition, a mechanical (or “bound” molecule) definition, and the correct Gibbs (surface excess) definition of physical adsorption are compared in the Henry’s law region in terms of “effective” solid-gas molecule potentials. The diff’erences are considerable. Also, the Gibbs definition is put in the form of an exact “two dimensional” virial expansion (surface pressure expanded in powers of excess surface density) and compared with the approximate virial expansion according to the two dimensional definition. The exact “two dimensional” treatment does not lead to a law of corresponding states or a pair-wise additive intermolecular potential as usually assumed.
I. Introduction
I n Fig. 1, we obtain the number of adsorbed moleThe object of the present paper is to compare cules by subtracting the number of molecules in V three different definitions of physical adsorption in Fig. la from the number in Vin Fig. lb. We shall for an idealized system consisting of a classical mon- call this the Gibbs definition. From a rigid thermodynamic point of view,4none atomic gas in contact with a solid possessing a of the above definitions can be made both exact and mathematically uniform surface (Fig. lb), the 5, The centers of the atoms of the solids are operational for a real system. However, the Gibbs r] plane. assumed to be distributed with uniform number definition resembles in principle the conventional density n in the region ( 6 0. We consider the low operational definition (measurement of “dead gas pressure region only, where the first monolayer space” by helium adsorption). Certainly, €or the idealized system being considered and for theoretis just beginning to form. ical purposes, the Gibbs definition should be reThe three definitions, which we consider in detail garded as correct. Incidentally, unlike the Gibbs below, are the following. (1) The actual interaction between a gas mole- definition, definitions (1) and (2), without modificule and the solid is replaced by a parabolic poten- cation, become obviously unrealistic in multilayer tial function having the correct potential energy adsorption. minimum (at ( = lo)and curvature at the mini11. Two Dimensional Definition of Adsorption mum. A molecule trapped in this parabolic potenThe canonical ensemble partition function for N tial well is “adsorbed.” The molecule carries out molecules can be written here as simple harmonic (-motion. Treatment of the motion in the t, q directions is simplified by assuming that it is independent of or separable from the (motion and restricted to the plane ( = bo; thus, as where fax as the tJ q directions are concerned, the adh = h/(2nmkT)’/r (2) sorbed molecules are treated as a strictly two dimenand sional gas. This is the conventional definition of or model for mobile monolayer adsorption. We shall refer to it as the two dimensional definition. (2) If, at any instant, the p contribution to the kinetic energy, pr2!2m, of a given gas molecule is The superscript (2) refers to the two dimensions E less than the negative of the potential energy of in- and q({ = bo plane). U d 2 )is the interaction poteraction between the molecule and the solid, then tential energy between N gas molecules in the plane the molecule is considered “adsorbed” (or, the {- ( = ( 0 . Q. is the surface area. To obtain qV, we motion is “bound”). This is the mechanical defini- have to discuss first u((), the interaction potential between a gas molecule and the solid. If the potention of adsorption.8 (3) Finally, the number of adsorbed molecules tial energy of interaction with a single atom of the can be defined as a surface excess, following Gibbs. solid a t a distance T is taken to be (1) Tbi, work was supported in part by a grant from the National Science Foundation. (2) Preaented a t the San Francisco meeting of the American Chemia d Society, April, 1958. (3) T. L. Hill, J. Chsm. Phys., 16, 181 (1948).
-2a
(i)” +
2 I )e: (
(4) E. A. Guggenbeim, reviewed by T. L. Hill, Advances in Catalysis, 1, 211 (1952). pp. 261-265.
RELATIONS BETWEEN DIFFERENT DEFINITIONS OF PHYSICAL ADSORPTION
April, 19.59
then the interaction with the entire solid isav6
0
457
0
loo O
0
where (5)
The minimum in eq. 4 occurs a t u/kT = uo/kT, j . / ~ * = TO/T* = 5-’/6 = 0.7647
o
If we expand eq. 4 about 1: = to, we find
o
0 0
0
0
0
O
0
0
O
00
O
where U H O ( { ) is the harmonic oscillator approximation to u ( { ) . Then qv = (~*/A)ZHO
where ZHO =
(7) 0
cm
e-UHo,‘kTd(r/r*)
0 0
7 1 0 O
(8)
o
0
0
0
00
Whether one writes the lower limit in eq. 8 as 0 or - 03 is immaterial. The well-known expression
0 0
relates the vibrational frequency v to the parameters used above. The grand partition function is
I Fig. 1.-(a)
(9)
-
where cp is the two dimensional pressyre, and 8 is an activity defined so that 8 4 = N / a as r + 0. That is, in the Henry’s law region (I’ 0), the amount of adsorption is given by r = 8 = T*zHoeP/kT
(10)
A8
Of course p is determined by the fugacity of the gas in equilibrium with the adsorbed phase. By standard methods* we find the virial expansion 2 = r + ~ ~ ( r22 )+ B p r a + . . . (11) kT &(a’
E
- 22(Z*“ - as) a
(12)
etc. If u12(p) is the intermolecular pair potential, then B2(W =
-
sd”
We will assume that u&) form
[e-UdP)/kT
0
0
0 0
kT qv = e-w/kT hv
-
llpdp
T
.
O
o
O
o
0
0 0 0
0
Reference system-no adsorbing surface: (b) “real” system with adsorbing surface.
ory. At higher surface densities, approximate van der Waals7 and Lennard-Jones-DevonshireR theories are available. A two dimensional law of corresponding states is of course obeyed if U N ( ~is) pairwise additive and u12has only two parameters as in eq. 14. 111. Mechanical Definition of Adsorption We limit ourselves in this section to the Henry’s law region (adsorbed molecules independent of each other). The partition function for a single molecule in the volume V (Fig. lb) is (after integrating over E, 7,PE and P,l> Now if we definea an adsorbed molecule as one whose {-motion is bound pr2/2m
< - u(t)
then the partition function q~ for a single adsorbed molecule is, instead of eq. 15
(13)
has the Lennard-Jones
Equations 9-13 represent the beginnings of a twodimensional version of Mayer’s imperfect gas the( 5 ) T. L. Hill, Advances in Catalysis, 4, 211 (1952). (6) See, for example, T. L. Hill, “Statietical Mechanics,” MoGrawHill Book Co., Inc., New York, N . Y., 1956, pp. 134-136,143-144.
If we define an effective potential U M ( ~ by ) e-uu(l)/kr = e-u(fYk7‘ei-f d - u ( f ) / k T = 0 u(S)> 0
u(c) < 0
(17)
(7) T. L. Hill, J . Chsrn. Phys., 14, 441 (1946). ( 8 ) A. F. Devonshire, Proc. Roy. SOC.(London), 1686, 132 (1837).
TERRELL L. HILL
458
Vol. 63
we wish to emphasize here is that this basically three dimensional treatment can be recast in two dimensional form, and can thus be compared directly with Section 11. For example, cp can be expanded in powers of l7 with exa,ct "two dimensional" virial coefficients. I n the absence of a surface (Fig. la), we have'"
i i i
l! 'VbP = 2 1 0 = V , b10 = 1 2! Vb20 = z 1 0 220
=
v*
lV
e-m/kT
drl drz
z = er/kT/Aa
Fig. 2.-Relationship
between UI*(p ) and UIZ*(P,TI (schematic).
etc. The average number of molecules in the system is In the presence of a surface (Fig. lb)
l ! Vb, = 2! Vbz = z a
Z1 = Zz =
lV
2 1
-
212
e-u(fi'l)/kTdrl
$y e - [ u ( h )+ ~ f . b ) + u i h a )I / k T d r l d r I
etc. The average number of molecules in the system is Vjbjzf
=
Fig. 3.-Comparison of actual van der Waals potential with three "effective" potentials for uo/kT = -2.
(23)
j> 1
We subtract eq. 20 from 22 and 21 from 23 to find
then We have QN
where IM=
1
= mqmN
lom e-uM/kTd(
f/r*)
(19)
Equations 18 and 19 are formally similar to eq. 10 and 8, respectively. This treatment resembles a discussiong of cluster formation in an imperfect gas using a "mechanical" definition of a cluster. IV. The Gibbs Dekition of Adsorption Various aspects of this problem already have been discussed by Wheeler, Ono, Freeman and Halsey and Hill.'" The amount of adsorption (surface excess) can be expressed as a power series in the gas pressure or fugacity, etc. The new feature which (9) Reference 13, pp. 152-164. (10) See, for example, pp. 424-425 of reference 6. ences are given there.
The other refer-
Equations 28 and 30 are analogs of eq. 10 and 8, and of 18 and 19, respectively. When u > 0, e - w I k T < 0 and U G is complex. (and cp) can be
RELATIONS BETWEEN DIFFERENT DEFINITIONS OF PHYSICAL ADSORPTION
April, 1959
negative with the Gibbs definition of adsorption, but not according to the other two definitions. It should be noted that Gibbs’ definition of adsorption is closely related to Mayer’s definition of a cluster.” In view of eq. 26 and 27, we define a “two-dimensional” activity 8 by
*7
459
I
so that 8 + I’ as I’ -+ 0 (as in Section 11). If we define q, by the relation 8 = q v e r / k T / A 2 = zq,A (32) as in eq. 9, then V(bi
- l)/aA
We use eq. 32 to.replace z by 8,and eq. 24 becomes (34)
where (35)
Fig. 4.-Comparison of actual van der Waals potential with three ‘effective” potentials for uo/kT = - 5 .
“h 2
1
(37)
0
The “two dimensional” virial expansions according to the Gibbs definition of adsorption is then kT =
r
r*+ B P r*+ . . .
+
(38)
with given by eq. 12, etc. Let us define an efective ‘(two dimensional” intermolecular potential ulz*by the equation
- a2 =
Z2(2)
la
(e-Uip*/kT
-
-
-2
l)drl(*)d r P (39)
Then from eq. 37 we find that e-urr*b,T)/kT
-1
-3
1=
-4
where r2 = P2
=
(€2
+ (b - Td2 - €P+ pa
(tl2
11)2
Thus, in an exact treatment, u&) in eq. 13must be replaced byulz*(p, 5”). Sinceulz*foragivengas will be different for different adsorbents, a two dimensional law of corresponding states cannot be strictly obeyed. Also, consideration of Z(z)shows, as might be expected, that the efective potential u12*is not strictly pair-wise additive (assuming that the actual intermolecular potential is pair-wise additive), To get an approximate idea of the difference between ulz*(p, T) and U&J) in eq. 40,we make the (11) See p. 162,reference 6.
-5 Fig. 5.-Gibbs and mechanical “effective” potentials aa functions of van der Waals potential.
following first-order calculation (which will underestimate the difference). First, we use the parabolic approximation, eq. 6, to u(p),neglect unity compared to e - u / k T , and expand e--zLip/hTabout T = p in powers of x = pz - rl: e-urr(r)/kT = e - w z ( P ) / k T
b =
+ bx2 + cx4 + . . .
1 uIp”( P) - 2P e-uiz(p)/kT kT -.-
460
J. T. KUMMER
Vol. 63
V. Discussion The solid-gas molecule potentials,uHo,UM and U G are compared in Figs. 3 and 4 with u (eq. 4), for uo/kT = -2 and -5, respectively. It is clear that the Gibbs and mechanical definitions provide effective "bound" potentials just as does the harmonic oscillator model. Figure 5 shows U M and U G plotted against -u. = e-uirb)/bT + ( b / a ) + (3c/a*) (41) In the Henry's law region, the amount of adsorpwhere tion, according to the three definitions, is given by 5 1 = f l - fo, 5 2 = h - r o eq. 10, 18 and 28. The three values of I' differ only a = - -27Qz -(-$) through the integral I (eq. 8, 19 and 30). IHO 2r*2 can be evaluated analytically; we also have calculated values of IM and IG by numerical integration If we substitute the expressions for b and c in eq. 41, (and analytical integration for large for uo/kT = factor out e-ulr@)/kT, take the logarithm of both -2 and - 5. These results are shown in Table I. sides, expand each of the terms on the right-hand side about p = TO up to ( p TO)^, and finally locate TABLE I the minimum in the resulting expression for u12*(p, RELATIVEAMOUNTSOF ADSORPTION T ) ,we find -uo/kT
r)
-
pmin =
rg
(1 - Y-2+
2
...
) (42)
U*lZ(min)
=
- EO
+. " (-"- 2 1
.,
IHO Ir io IEO/IO Iaa/IO
5
1.93
24.5
4.83
34.3 30.2 0.81
1.81 1.07 2.67
1.13
eo
03
W
W
1 .oo 1.00
IM is much larger than IG for u&T
where
The function uI2*(p,T ) is compared schematically with u l ~ ( p )in Fig. 2. The general relationship shown in this figure results from the fact that oscillations, perpendicular to the surface, of two molecules a distance p apart in the E, 7 directions will necessarily make their average separation somewhat greater than p. Also, an average over u12will necessarily lead to a minimum in ulz*above the minimum in u12 itself. If, for simplicity, a strictly two dimensional treatment (virial expansion, two dimensional condensation, critical properties, etc.) as in Section I1 is used, the theory will be improved by the use of the adjusted parameters of eq. 42 rather than TOand eo themselves.
= -2. The excess arise? as follows: 0.6 in the interval 0 to 0.64 (in which IG has a negative contribution and IM none); 0.9 in the interval 0.64 to 2.75; and 1.5 in the interval 2.75 to a. The negative contribution to IG just referred to becomes relatively less important as -uo/kT increases. This effect is responsible for the reversal in relative magnitude of IHO and IG seen in the table. It is apparent from Table I that significant errors can be made in estimating theoretically the amount of adsorption in the Henry's law region by not using the correct (Gibbs) definition of adsorption. Finally, it should be mentioned again that classical rather than quantum mechanics has been used throughout. The author is indebted to Dr. Dirk Stigter for his interest and assistance.
THE CHEMISORPTION OF OXYGEN ON SILVER BY J. T. KUMMER The Dow Chemical Company, Midland, Michigan Received October 7, 1968
The silver-oxygen system has been studied by measuring the change in surface potential of a silver surface produced by oxygen chemisorption, and by measuring the paramagnetism of a silver surface after oxygen chemisorption. The change in surface potential was obtained by measuring the current flow into or out of a condenser consisting of gold and silver plates at -220' when oxygen is admitted to the system. The change in surface potential is negative -0.2 volt. The surface paramagnetism wa8 measured by ortho-para hydrogen conversion, and the surface after oxygen chemisorption is slightly paramagnetic, corresponding to one unpaired electron per 150 A.2
The catalytic oxidation of ethylene to ethylene oxide over a silver surface occurs by the reaction between ga.s phase or weakly adsorbed ethylene and oxygen chemisorbed on the surface. In order to (1)
0.H. Twigg, Trans. Faraday
Soc., 42, 284 (19463.
understand the catalytic reaction more thoroughly, we have measured the surface potential of a silverpxygen surface relative to a silver surface and have measured its paramagnetism. The silver-oxygen bond in Ag20is largely covalent in character and the
4
