J. Phys. Chem. 1988, 92, 4165-4171 and hole barriers in the Hg12-acetonitrile system, and (2) the barriers are expected to be large and the infinite barrier approximation is not expected to produce gross differences in the calculated results, as borne out by the above described case. The similarity between the spectra of ionic Hg12Cl- and the colloid spectra is consistent with multiple transitions from the colloidal system that approach the behavior of molecular or ionic species. This observation is also consistent with that of Wang and H e r r 0 n ' ~ 3in~ their study of PdS and CdS encapsulated in zeolite; clusters containing 12-14 molecules were observed to have properties that approach those of molecular species. Our estimated colloid size of 36 Hg12 molecules may well fall into this regime. Finally, we note that our results do not prove conclusively that the clusters in the Hg12colloidal sol are truly crystalline. Mercuric halides are known to form clusters that can involve solvent molecule^.^^^^^*^^ Further work is in progress to establish the detailed nature of the particles in more direct and unequivocal experiments, such as small-angle X-ray scattering and EXAFS. Con cIusion
We conclude from a series of optical measurements together with physical separation techniques involving ultracentrifugation, ultrafiltration, and treatment with ion-exchange resins that the reaction of 1 X lo4 M HgClz with 2 X lo-" M NaI in acetonitrile produces colloidal particles of Hg12and the ionic complex Hg12Cl-. About 60% of the mercury is present as HgI, clusters. These clusters are charged and contain C1- ions in normally unfilled tetrahedral coordination sites at the edges of the cluster; the empirical formula is Hg12Clo,60~6.Other possible molecular and ionic species, such as molecular Hg12, HgC12, HgICl, HgClzI-, Hg13-, Hg12C122-,HgC131-, HgI,Cl-, 13-, and I2 have been elim-
4165
inated as components of the initial colloidal solution. Absorption difference spectra from the ultracentrifugation and ultrafiltration of the initial colloidal solution, as well as the resultant spectrum obtained by treating the initial colloidal solution with the ionexchange resin Amberlite, are identical and this spectrum is attributed to small HgIz particles that are less than about 25 A. The spectrum of colloidal Hg12 exhibits three peaks at 4.26, 4.94, and 6.04 eV, which are attributed to quantization effects in the small Hg12 particles. Using a simple particle-in-a-box model with infinite potential barriers and the appropriate selection rules for optical transitions, the three peaks are consistent with either (1) the first three transitions in a Hg12 particle with the usual tetrahedral structure containing 36 Hg12 molecules arranged in four layers (total thickness of 26.1 A) with nine molecules per layer (total lateral dimension of 13.3 A), or (2) particles with magic numbers containing from one to four layers (7.6-26.1 A), depending upon the s ecific optical transition, and lateral dimensions The results of our physical separation exfrom 9.6 to 15.8 periments favor the former model of multiple transitions from a single dominant particle size. Model calculations using finite potential barriers of 4 eV did not seriously affect the results or our conclusions. Additional work is required to unequivocally establish that the 25-A colloidal particles are crystalline with the same hexagonal, layered crystal structure as red HgI,.
1.
Acknowledgment. This work was funded by the U S . Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, and the SERI Director's Development Fund. O.I.M. was supported by the US.-Yugoslavia Joint Research Fund. Registry No. Hg12, 7774-29-0.
Second Virial Coefficients of Molecules Absorbed in Spherical Cavities and Slltlike Micropores Perla B. Balbuenat and Donald A. McQuarrie* Department of Chemistry, University of California, Davis, California 95616 (Received: August 3, 1987; I n Final Form: January 20, 1988)
Virial expansions for the properties of a dilute gas within a spherical cavity and a parallel-plate micropore are developed, and calculations are presented for the second virial coefficients of a Lennard-Jones gas that interacts with the walls of a spherical cavity and a slitlike pore by a Lennard-Jones potential. The second virial coefficients of molecules within spherical cavities and parallel-plate micropores are expressed as single integrals involving Fourier transforms.
We start with the grand partition function, E , which is given
Introduction
Recently Rowlinson' has developed a formalism for the virial expansion of inhomogeneous systems. This formalism is based upon the grand potential, Q, which is taken to be a function of the temperature, T, the chemical potential, K , and a function of an external one-body potential, $, which replaces the conventional fixed geometric boundaries of a system. The types of systems for which this formalism is directly applicable are fluids within a system of capillary pores, or in the interstices of a chlathrate structure or a zeolite. In this paper we shall model a zeolite as a regular structure consisting of more or less separate spherical cavities and shall present a formal analysis of the absorption properties of a model zeolite and then present explicit calculations of a virial expansion of the isosteric heat of absorption. 'Permanent address: INTEC,Giiemes 3450,3000-Santa
Fe, Argentina.
0022-3654/88/2092-4165$01.50/0
by
where the activity X = ePikT, the deBroglie wavelength A = (h2/2amkT)1/2and
vN =
s ...
sdr,
... drN exp{-pU(r, ,...,rN) - @$(rl,...,rN)] (2)
where 8 = l / k T is the configuration integral. Following Rowlinson, we write
= VN/vIN (1) Rowlinson, J. S. Proc. R . Soc. London 1985, A402, 67.
0 1988 American Chemical Society
(3)
4166
Balbuena and McQuarrie
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988
so that each molecule over which the integration in V, is taken has with it a weighting factor of VI-l exp(-p$(r)), where $(r) is defined by N
W l , . . . , r d = Z$(rj)
(4)
J=I
In this notation eq 1 becomes
z = cZN(VN/N!)
and VI, is a free volume Vla
= j d r ~XP(-W(~))
(17)
Because the two systems are in equilibrium with each other, A / A 3 is the same in both systems. By eliminating A / A 3 between the series for the intensive functions na/Vlaand ppg, we obtain
(5)
'$20
where z = AV1/A3. We shall be using the potential $(rj) to limit the j t h molecule to lie within a spherical cavity. We now assume that our model zeolite consists of M independent, spherical cavities, each of which contains a maximum of m absorbed molecules. If we let nJ be the number of cavities that are occupied by exactly j absorbed molecules, then2
where B2& T ) is simply the ordinary second virial coefficient of a homogeneous gas B2g(T) = -I/21m[e-@u(r) - 1]4?rr2d r
(20)
and B,(T) is the second virial coefficient of the molecules absorbed in a cavity and is given by
where uj.' = (1 / j ! ) U j
(7)
and where the asterisk on the summation indicates the constraint on the set of nj that
B2a(T ) =
j 1dr, dr,
e-@~(rl)e-@~(r2)[e-@U(riz)-
11 (21)
The isosteric heat of absorption is given by
m
cjnj = N j= 1
The grand potential and the grand partition function for this model are given by
dn = ,"(M,k,T) = (1
+ ul'z + u2'z2 + ... + u , ' Z ~ ) ~(9)
where p: is a standard pressure. Applying eq 22 to eq 19 we obtain
The average number of absorbed molecules is given by
and the average occupancy of a cavity is
n
8=-=
+ 2u2/z2 + ... + mum'zm 1 + U 1 f Z+ u2fz2 + ... + (;,'Zm u1'z
(1 1)
($)B,,(T)
In anticipation of a virial expansion to be developed below, we introduce the reducible cluster integrals l!bl = u1 = u i / 2!b2 = V ? - VI'
= 2 ~ 2' (
so that
(13)
J>
To apply these equations to the absorption of gaseous molecules by a model zeolite of spherical cavities or by a slitlike pore and to derive an expression for the isosteric heat of absorption, we consider two systems in equilibrium with one another. The first system is a macroscopic volume Va of the absorbing solid (zeolite) and its absorbed molecules and the second system is a macroscopic volume V, of the gas. In the first system we have
fla = c j b , , z i J b1
and in the second system we have PP, = Cb,& J>
I
-
dB2g -
1
2 5 + O(@P,)~)(24) a
dT
(12)
n in eq 10 is given by IV = cjbJzJ 1
B,a(T)
where
~ 1 ' ) ~
3!b3 = u3 - 3UzC.1 + 2uI3
+ "*','-
(14)
(15)
If we ignore the terms involving B2&7') or its derivative in eq 23, then we obtain the same expression obtained by Brauer et al.3 by quite a different method. The Second Virial Coefficient in a Square-Well Spherical Cavity
It remains for us to evaluate B2a(T ) for given potentials $(r) and u(r) in eq 21. This expression is a second virial coefficient of two molecules within a spherical cavity. This quantity was evaluated by Janssens and Prigogine4 by a geometric argument for square-well potentials and was estimated by Pople5 for a more general potential. Both of these studies involved the contribution of double occupancy to the communal entropy of the LennardJones Devonshire cell theory of liquids. Brauer et aL3 also have evaluated (with misprints) B2a(r ) for a square-well potential in a study of the absorption of gases by zeolites. Recently McQuarrie
In eq 14 and 15 Z,
= VlaA/A3
zg = V,A/A3
(16)
(2) See, for example. Hill, T. L. Statistical Thermodynamics; AddisonWesley. Reading, MA, 1960; Chapter 7
(3) Brauer, P.; Lopatkin, A. A,; Stepanez, G. PH. Molecular Sieve Zeolites II; Advances in Chemistry Series No. 101; American Chemical Society: Washington, DC, 1971. (4) Janssens, P.; Prigogine, I. Physica 1950, 16, 895. (5) Pople, J. Proc. R. SOC.A 1951, 42, 212.
Second Virial Coefficients of Absorbed Molecules
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988 4167
TABLE I: The Reduced Free Volume VI,* for a Lennard-Jones Spherical Cavity (Eq 34) and Its Temperature Derivative dV,,*/dT as a Function of a / a and T * for c = 10 a/a = 0.100 u / a = 0.200 ala = 0.333 a/a = 0.500 T* VI.* d v,.* I d T* VI.* dVi.*/dT* VI.* d v,* /d T* VI,* dVl.*/dT 0.600 0.800 1.ooo 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3 .OOO
0.2689 0.2696 0.2702 0.2708 0.2714 0.2718 0.2723 0.2727 0.273 1 0.2735 0.2738 0.2741 0.2744
0.002 97 0.003 22 0.003 05 0.002 8 1 0.002 56 0.002 35 0.002 16 0.001 99 0.001 85 0.001 72 0.001 61 0.001 51 0.001 42
0.2168 0.2139 0.2126 0.2121 0.2118 0.2118 0.2120 0.2122 0.2124 0.2127 0.2129 0.2132 0.2135
-0.021 99 -0.009 36 -0.004 19 -0.001 69 -0.000 36 0.000 39 0.000 84 0.001 11 0.001 27 0.001 37 0,001 42 0.001 44 0.001 45
0.1885 0.1711 0.1623 0.1572 0.1539 0.1518 0.1502 0.1492 0.1484 0.1479 0.1475 0.1473 0.1471
-0.1263 -0.0588 -0.0327 -0.0201 -0.0 13 1 -0.0089 -0.0062 -0.0045 -0.0031 -0.0022 -0.0015 -0.001 1 -0.00067
0.1658 0.1280 0.1 107 0.1010 0.0950 0.0910 0.0882 0.0861 0.0846 0.0835 0.0825 0.08 19 0.0814
-0.2947 -0.1198 -0.0623 -0.0371 -0.0241 -0.0 166 -0.0119 -0,008 7 -0.0066 -0.00 5 0 3 -0.00387 -0.00300 -0.00233
TABLE 11: The Reduced Second Virial Coefficient B2.* for a Lennard-Jones Spherical Cavity (Eq 39) and Its Temperature Derivative dBt*/dT* as a Function of T * and u / a and for c = 10 a/a = 0.100 a l a = 0.200 a l a = 0.333 a l a = 0.500 r B2a* dB2,*/dP B2a’ dBZa*/dT* B2a* dB2,*/dT* B2a* dB2,*/dT* 0.600 0.800 1.ooo 1.100 1.200 1.300 1.400 1so0 1.600 1.700 1.800 1.900 2.000 2.200 2.400 2.600 2.800 3.000
-4.624 -2.756 -1.854 -1.560 -1.326 -1.137 -0.9803 -0.8488 -0.7369 -0.6406 -0.5 568 -0.4832 -0.4182 -0.308 5 -0.2 196 -0.1462 -0.0847 -0.0325
14.02 5.928 3.223 2.515 2.016 1.656 1.377 1.166 1.000 0.8664 0.7581 0.6688 0.5943 0.4782 0.3930 0.3286 0.2787 0.2393
-3.416 -1.944 -1.263 -1.046 -0.87 5 5 -0.7 3 8 7 -0.6264 -0.5 328 -0.45 34 -0.38 5 5 -0.3266 -0.2751 -0.2297 -0.1535 -0.09196 -0.04155 +0.000968 +0.03685
1 1.20 4.687 2.446 1.895 1.509 1.229 1.019 0.8585 0.7326 0.6321 0.5507 0.4838 0.4283 0.3420 0.2790 0.2317 0.1951 0.1664
and Rowlinson6 in a study of the virial expansion of the grand potential at spherical and planar walls have presented a general method for evaluating B2a(T)by recognizing it to be mathematically equivalent to the third virial coefficient of a homogeneous gaseous mixture. The higher cavity virial coefficients can also be mapped on to some of the graphs of the virial coefficients of mixtures. It can be shown that the nth cavity virial coefficient maps on to some of the graphs of the ( n + 1)th ordinary virial coefficient with one molecule of species a and n of species b. Unfortunately these graphs are the more highly connected, and consequently the more difficult to evaluate. For example the connected graphs of the third cavity virial coefficient are
-2.762 -1.293 -0.7327 -0.5709 -0.4503 -0.3 5 74 -0.28 39 -0.2245 -0.1755 -0.1345 -0.09972 -0.06990 -0.04407 -0.001579 +0.03 185 +0.05879 +0.08093 +0.09944
9.851 3.414 1.656 1.241 0.9613 0.7652 0.6225 0.5156 0.4336 0.3694 0.3182 0.2768 0.2428 0.1908 0.1536 0.1261 0.1052 0.0889
-1.186 -0.5059 -0.1668 -0.09 104 -0.03957 -0.00459 +0.02054 +0.03920 +0.05342 +0.06451 +0.073 33 +0.08046 +0.08633 +0.09533 +0.1018 +0.1067 +0.1106 +0.1136
and transform B2a(7‘) to B2a(79 = 4 ~ ( 2 a ) ~ / t2y2(t) ~ ~ ~4(t) d t
Following Katsura,s we introduce the Fourier transforms
(28)
For two hard spheres of diameter u within a hard spherical cavity of radius a g(r)=l O
