2118
P. Ripa and 0. Zgrablich
We have made some attempts to observe the infrared and Raman spectra of the isotropic solid; thus far we were not successful. In the infrared there is a problem arising from the nonzero vapor pressure of the ethane; it sublimes to a slightly colder region of the conventional lpw-temperature cell. We have designed a special infrared cell that incorporates two cold windows with provision for introducing ethane between them and isolating the sample from the insulating vacuum. For Raman work the sample was contained in a quartz tube connected to an expansion bulb; cooling was done by a stream of cold nitrogen gas. It was not possible to hold the sample temperature in the 0.4O range of stability for the isotropic phase for more than a few minutes. A different cell, with provision for much better temperature control, has been designed; we plan to report the infrared and Raman spectra of the isotropic solid phase in a separate paper. In some work on condensed phases of ethane at high pressures, Webster and Hoch' reported a solid-solid phase transition; solid I, which lies between the liquid and solid 11, has a range of stability of about 5' at a pressure of 1 kbar. These workers suggested that there must be a solid I-solid 11-liquid triple point between 0 and 1 kbar; they were unable to conduct measurements in this region. We believe that the isotropic solid phase that we have observed
is the same as their solid I phase, and that there is no triple point involving these phases in the range of pressures from 0 to 5 kbars, the upper limit of their measurements. It would be most interesting to have additional measurements of the heat capacity of highly purified ethane in the vicinity of the solid-solid transition temperature as well as near the melting point. We suspect that the enthalpy change of the solid-solid transition may be substantially larger than the enthalpy change of melting.
Acknowledgment. We are grateful to Professor B. J. Zwolinski for suggestions concerning the literature on ethane. We are also grateful to Mr. J. Van Zee for suggesting the use of a certain silicon diode for measurement of temperature, and to Mr. D. Hinman for assistance in comparing our diode with the platinum resistance thermometer. References and Notes (1) H.Mark and E. Pohland. 2.Krlstallogr.,82, 103 (1925). (2)R. K. Witt and J. D. Kemp, J. Am. Chem. Soc., 59,273(1937). (3)K. Clusius and K. Weigand, 2.Phys. Chem., 846, 1 (1940). (4) L. J. Burnett and E. H. Muller, J. Chem. Eng. Data, 15, 154 (1970). ( 5 ) W. Wahl, 2.Phys. Chem., 88, 133 (1914).
( 6 ) S.B. Tejada and D. F. Eggers, Jr., to be submitted for publication. (7) D. S.Webster and M. J. R. Hoch, J. Phys. Chem. Solids, 32, 2663 (1971). We are grateful to Professor W. A. Steele for pointing out the relation of this work to ours.
Effect of the Potential Correlation Function on the Physical Adsorption on Heterogeneous Substrates P. Rlpa and 0. Zgrabllch* Departamento de Fhica, Universidad de San Luis, San Luis, Argentina (Received April 2 1, 1975)
A model for heterogeneous substrates with a multivariate gaussian distribution for the adsorption potential is proposed. The evaluation of second and third virial coefficients shows a considerable dependence on the correlation length, in the range of the other parameters where heterogeneity plays an important role. The model has as limiting cases the homogeneous and large patches models; but the behavior at finite correlation lengths is by no means intermediate between that corresponding to those extremes.
1. Introduction
The most commonly used model for the adsorption of gases on heterogeneous substrates depicts the adsorption potential ( Uad) as being constant through large 2atches of the surface.1 Another model was also proposed in which the surface is partitioned in an infinity of single adsorbing sites and U,d is randomly distributed on them with no correlation between the values of Uad for neighboring sites.2 For both mdels, the best distribution for the values of Uad was found to be g a ~ s s i a n . ~ . ~ We propose here a model in which Uad is considered to have a chaotic (stochastic) structure with a finite correlation length. The potential has a multivariate gaussian distribution and, therefore, all its statistical properties are given in terms of its mean value and covariance function. The usual formulae for the evaluation of the two-dimensional virial coefficients (reviewed in section 2) are used in The Journal of Physical Chemistry, Vol. 79, No. 20, 1975
section 3 with the distribution of the present model to obtain a general expression for the nth coefficient. The second and third coefficients are evaluated in section 4 for a square well parametrization of the interparticle potential, while the conclusions are discussed in section 5. We include an Appendix with the deduction of the multivariate gaussian distribution for the sake of completeness. 2. Virial Expansion of the Adsorption-Isotherm Equation We are interested in those adsorption processes that can be pictured in terms of an ideal nonadsorbed phase (NAP), a two-dimensional adsorbed phase (AP), and an inert solid substrate, responsible for the adsorption force. For simplicity we take both phases to have one component. The NAP is macroscopically described by means of its absolute temperature T, the volume particle density PO,
2119
Physical Adsorption on Heterogeneous Substrates and the pressure p (equal to kTpo in our case), while the AP is described by T, the surface adsorbed particle density p, and the two-dimensional or spreading pressure 4. The latter can be expanded in powers of p giving
4 = kTp(1
+ C B,(T)p"-l)
(1)
n22
where B,( T) is the nth two-dimensional virial coefficient. Using this expansion in Gibbs equation po d 4 = p dp, which gives the condition for statistical equilibrium between the NAP and the AP, yields
where K(T) is an integration constant. We further assume that the potential energy of the system of adsorbed particles can be written as the sum over all pairs of the interparticle potentials Ugg(Jf, - j z j l ) , plus the sum over all particles of the particle-solid potential4 1 Uad(Zi,%i)= - kz(zi - z , ) ~ v(fi) (3) 2 In the above equation f = (x,y) denotes the two-dimensional position vector. Under these assumptions, the coefficients K(T) and B, (T) can be evaluated as
+
K ( T) = ( k T k z / 2 ~ ) ' / exp( ~ ( - V/kT) )-l B,(T) = -Z,/(n(n
- 2)!11")
(4)
with I, =
SA.. 1 .
dP1.. . d3,h(f1
+.
,
.+3,) x
Figure 1.
Interparticle interaction contribution for the integrand of In
in eq 4, for n = 2,3,and 4. Each dotted line represents a factor fp
scopic portion of the total surface has all the meaningful information. Now, all statistical information about the surface is given by the multivariate potential distribution 9, for all values of n, where CP, dV1 . . . dV, is the fraction of the surface such as the potential at the point fi is between V, and Vi + dVi, fori = 1 , 2 , . . . , n . Equation 4 shows that to evaluate K(T) we need only 91, whereas to calculate B,(T) we must know the interparticle interaction and 9,. B,(T), however, does not depend on the mean value of V(f). For the distribution of V(2) over the surface it has been common to use a gaussian form
where -kTa and kT, are the mean value and standard deviation of V(f ), respectively ( V ( f ) ) = -kTa;
. . ,~ n - 1 -
-3/2
[ ( 2 ~ ) det " HI-l exp
where 5'1.2'
= f 1 2 = exp(-Ug,()fi s1,2,3'
- ?,I
=
5 (V(Ei) + kTa) X
i,j=l
where H,, = ((V(3,) + kTa)( V(3,)
6f12f13f14f23f34 -I-
1 A
R, . . . ,f k + R )
(5)
Hill6 used eq 4 and 5 to evaluate B2 with V(f) a periodic function of 2. The case of a heterogeneous potential is treated in next section. 3. Adsorption on a Heterogeneous "Gaussian"Surface In order to use eq 4 when the adsorption potential has a chaotic structure we must postulate the validity of the statistical homogeneity hypothesis
(F(fi + 8 , . . . ,f k + 5)) = (F(f1,.. . ,f k ) ) (6) i.e., averages must be functions of the differences of coordinates. Physically this hypothesis means that any macro-
(9)
+ kTa)) = (kT,)2C(Z, -
f,), is the covariance matrix.
Thus CP, can be evaluated for any set of positions f,, in terms of Ta, T,, and a single function, namely, the correlation function C(?). The latter satisfies C(0) = 1, C ( m ) = 0 andlC(?)I < 1 forl?l + 0. Using (9) in (4)gives B,(T) = -
-x
S, d R F ( f 1
5,)
~-1)L,(v(f,) + kTa)J
3fl2f23f34f14 and in general SI, ', is the sum over all completely connected diagrams (see Figure 1). In the above formulae we denote by ( F ( f 1 ,. . . , 3k) ) the average of F over the surface ( F ( 3 1 , .. . , f k ) ) = limA-,
(8)
1
)/kT) - 1
= fl2f13f23
= fi2fi3fi4f23f24f34
+ kT,)2) = (kT,)2
This distribution gives K ( T ) = (kTkz/2r)lI2exp[-Ta/T 1k(T,/T)21 For the evaluation of B,(T) we propose for 9, a multivariate gaussian form (see Appendix) CPn(V1,. . . , Vn;f1 -32,.
si,2,3,4'
((V(f)
n(n - 2)!
s...
I d 3l . . . d P , x
A
6(X1 + . . . + f n ) SI,
fi
exp[(T,/T)2C(3, - Z,)]
(10)
1>,=1
The above equation can be diagrammatically represented as in Figure 2. One could pose the question whether it is correct or not to average the integrand of I,, which is not a directly measurable quantity. If we call Z, =
1
d f 1 . . . df,
h(fl+
. . . + f n )x Si,
,n'
fi exp[-V(f,)/kT]
r=l
it can be easily proved that the vanishing limit off(?) and C(?) as IF1 (short range of interparticle potential and +
Q)
The Journal of Physical Chemistry, Vol. 79.No. 20, 1975
2120
P. Ripa
and G. Zgrablich
of T,/T and T,,/T without increasing considerably the computation time. The limiting values when T/T, 03 (or ro 0) are -+
J1(2)
= 1; J,(Z) = R2
-1
-
Flgure 2. 6, for heterogeneous substrates, for n = 2, 3. and 4 (eq
10). A solid line represents a factor fij exp(k$,//(kT)2) and a wiggly line denotes a factor exp(Hil/(kT)2).
2 -sin
Ic/1 R(R2
ir
adsorptive potential correlation) leads to (ZnZ,) = IJ,, Le., the 2, are deterministic variables (standard deviation zero) with mean value I,. We have then constructed a model in which all the necessary information about the adsorption potential is given by its mean value, -kT,, and the covariance function (kTs)2C(F).In the spirit of this model, indeed, this is all the physically meaningful information, i.e., the adsorptive potential of two substrates with identical macroscopic properties (for instance, two carbon blacks graphitized a t the same temperature) will very likely have different detailed structures, but identical mean values and covariances. In next section we calculate B2 and B3 using
C(P) = exp[-lh(rlr~)~]
Ugg(r)=
for r < a -kT,, for a 0 for b < r
1
