Marc 0. Baleiko and H. Ted Davis
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Thermal Diffusion in Binary Dense Gas Mixtures of Loaded Spheres and Rough Spheres arc 0 . Baleiko and H. Ted Davis" Deparhnents of Chemical Engineering and Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 23, 1973; Revised Manuscript Received November 73, 7973)
(Received January
The results of fairly extensive numerical calculations of the diffusion and thermal diffusion coefficients obtained from Enskog's theory are presented for binary dense gas mixtures of loaded spheres and oE rough spheres. The common result of both model theories in the limit of vanishing polyatomic parameters is the smooth hard-sphere value of the transport coefficients. The second-order Chapman-Enskog binary diffusion coefficients for rough spheres mixtures and for loaded sphere mixtures are well approximated over the entire density range by the first-order Chapman-Enskog formulas. Effects due to differences in masses and differences in eccentricity parameters in loaded sphere molecules are of equal importance in determining the sign and order of magnitude of the thermal diffusion coefficient. The effect due to different molecular diameters is nearly an order of magnitude smaller. In the rough sphere model, the effects of different masses and molecular sizes generally outweigh the effects due to different moments of inertia and usually determine the sign and order of magnitude of the thermal diffusion coefficient. Calculations for the dense gas Dz-HT system for both polyatomic models show strong composition and density dependence of the thermal diffusion factor. The form of the trial function has a significant effect upon the value of the thermal diffusion factor for rough spheres and the effect of spin anistropy, in particular, becomes very important as the density increases. The orientational anisotropy has a negligihie effect upon the thermal diffusion factor for loaded spheres, whereas the spin anisotropy accounts for about 10% of the thermal diffusion factor for this model.
I. Introduction The formal resuits of the Enskog transport theory of dense gas mixtures of loaded spheres1 and of rough spheres2 (samtl mistakes in ref 2 have been corrected in ref 1) are used herein to obtain numerical estimates of the diffusion and thermal diffusion coefficients for these model fluids. Since the details and results of the formal theory are exceedingly lengthy, we present here only the barest outline of the theory and concentrate instead on assessing by numerical computation the role of rotational motions in diffusive processes. The formulas used in the computations are available in ref 1 and 2 and on microfilm.3 It is hoped that the many tables and figures given in the following sections will be helpful in predicting qualitative trends %r real fluids at high temperatures (where the attractive interactions are not important). The loaded sphere molecules are characterized by loaded sphere parameters or eccentricity parameters, which are reduced moments of inertia defined as a, = m&2/ r Vandu, = mu,,(42/Tplwhere m,,, [E m,m,,/(m, + m,)] is the reduced mass for the species v and p , ( is the eccentricity or displacement of mass center from geometric center, and Tu is t he principal moment of inertia for a molecule of species i t . For real molecules that we hope to be adequately described as loaded spheres, the eccentricity $, i s quite small. The eccentricity parameter having a value of zero corresponds to the case of a degenerate loaded sphere (actually a smooth hard sphere where the centers of mass and geometry coincide). Perhaps the largest physically realistic value of the eccentricity parameter is 0.167 for the HT molecule in the I&-H'rbinary system. The rough sphere molecules are characterized by rough sphere parameters, which are reduced moments of inertia d e h e d as K \ = 4 1 x / r n \ i r h 2 for X = v , p . I A is the moment of The Jourriai o i Physicai Chemistry. Vol. 78, No. 7 5 . 1974
inertia, mh is the mass, and (TA is the diameter of a molecule of species A. The value of K A may vary from a value of zero when all the mass is concentrated a t the center of the sphere to a value of 2h when all the mass is evenly distributed on the surface of the sphere. The limit K~ = K,, 0 corresponds to the smooth hard-sphere limit. The general theory of the dense gas loaded sphere (LS) mixture has been applied to a binary mixture. Likewise, the general theory of the rough sphere (RS) dense gas mixture has been applied to the case of a binary system. For these two models the expression for the relative diffusion velocity in a two-component system of species v and p has the form n? V, - V, = - __ DTV In TI (1.1) nunu[a>,,d,,, where 3& is the binary diffusion coefficient and UT is the thermal diffusion coefficient. d,, is a generalized diffusion force which for loaded spheres is given by -+
+
dyII(Ls)
I
(PnW-'(P,VP" - P"VP,) n-lbyuguP(n.,Vn,, - n,Vn,)
PYPU
pnhT (m,-'xY - m,-'x,,) -" cru
+ n-lbuu---- 20,, ITu
n,n,Vg,
+
(1.2)
For rough spheres we have d,,'RS'
(PnkT)-l(p,VP, - p Y V P k )
= P"P,
-1
(mu-!xu- m,,
x,) I-
n-'b,,g,&,Vvcp - npvrd (1.3) In these expressions n, is the number density of species v , n is the total number density, pu is the mass density of
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Thermal Diffusion in Binary Dense Gas Mixtures species v, p is the total mass density, xu is the external force on a particle of species u , P , is the hydrostatic pressure due to species u , u, is the diameter of a particle of b,, = ( 2 / 3 ) ~ ( r , , ,and ~ g,, is species v, uy,, = (u, -ia,)/2, the contact value of the equilibrium pair correlation function for a pair of particles of species u and p . The generalized diffusion force for loaded spheres has the additional term proportional to Vg,,. This term arises from a particular evaluation of the pair correlation function in the molecular chaos assumption used to close the hierarchy of equations for the singlet distribution function. In the development of the kinetic equations in the formal theory it is assumed that the doublet distribution function for a pair of particles, say molecule 1 of species u with center of mass located at position r characterized by the dynamical variables T , and molecule 2 of species p with center of mass located a t r -t 6,l- 6 , ~characterized by dynamicall variables 7 2 , may be written as the product of a correlation function which is independent of momenta, and two1 singlet distribution functions. This assumption has the form
+ 6,1
fLk(r.T,,r
cients are associated with the particular forms of the driving forces given here. For comparison of theory and experiment some care must be exercised so that the mass flux is expressed in terms of the same driving forces. The theoretical and experimental values of the transport coeffi. cients may then be properly compared as coefficients of the same driving forces. The formulas for the binary diffusion coefficient and the thermal diffusion coefficient in terms of the scalar coefficients arising in the Chapman-Enskog solution technique to the order of approximations made are for the loaded sphere model
- 6 1 r . ~ , ;t ) =
-+ h o J S u ( r ,T ~t)f,(r ; For loaded spheres g,,(p
ii
= [Del
+ 6”1 - ~ , L , T ~t ); (1.4)
- Xcvk
where k is the unit vector in the direction from the geometric center of molecule 2 to the geometric center of molecule 1 a t the moment of impact, and e is the unit vector in the direction from the center of mass to the center of geometry with t the distance the center of mass is displaced from the cen;er of geometry. For rough spheres whose centers of mass and geometry coincide, E , = [, 0, so that 6”l
=
- Xa,k
6 p ~ =’/,a,k
In writing the molecular chaos assumption it seems consistent to evaluate g V ! a,, is identically zero for a , = u p , and is positive and increases with bn for a < a,, . The magnitude of the thermal diffusion coefficient for the case of different eccentricity parameters is nearly the same as that for different masses. The effect is greatest when the molecules exhibit the greatest dissimilarity, i.e., when 1~ - a,] is maximal. The Journal of Physical Chemistry. Vol. 78. No. 15. 1974
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Marc 0. Baleiko and H. Ted Davis
l2
t
-121
KP
bn i)TiRS’ in units of ( k T / ~ m ) l I ~ n - 1 ( 2 ~ ) - 2 vs. reduced density bn: K ” = 0.1, m,,/m, = u V / u , = 1, x , = x,, = 0.5 (dashed curves); K , = 0 . 3 , mv/m,, = u,/u, = 1, x u = x,, = 0.5 (solid curves).
Figure 4.
Figure 5. D T ( ~ in~ )units of ( k T / a m ) 1 / 2 n - 1 ( 2 a ) - 2 K,,: bn = a, m,/m,, = uy/u,, = 1, x u = x , = 0.5.
I - my /m,=O 2- m, /m,=O
The rough sphere thermal diffusion coefficient was calculated for different values of the reduced moments of in. species of equal mass and molecular ertia K , arid K ~ For radius, dissimilar but symmetric mass distributions in the spheres give rise t o different moments of inertia, and hence. provide a driving force that creates a thermal diffusion effect in such systems. Figure 4 shows plots of us. reduced density bn for combinations of K~~ and K,, for equal masses and equal radii. First we note that the effect of thermal diffusion due to differences in moments of inertia in the rough sphere model is generally one or two orders o l magnitude smaller than the effect due to different eccentricity parameters in the loaded sphere model. At higher densities for moderately large values of K , and K ~ the , qualitative behavior of DT(Lb)is the same as D1(Ls). That is, for reduced densities greater than ap) proximately one, ffor a fixed value of K,, D T ( ~is~negative for K , < K,, and increases in a positive sense with increasing K ~ is , identically zero for K , = K , (for all densities), . low and is positive for K~ > K , and increases with K ~ At densities the qualitative behavior of DT(RS)does not appear the same. Figure 5 is a plot of DTIRS)us K~ for the dilute gas limit bn = 0 . It is not true that a maximum thermal diffusion efrect corresponds to the greatest dissimilarity in the species. Qualitatively speaking, our results are in agreement with the results of Trubenbachero who studied the dilute gas rough sphere binary mixture. It was not possible to compare our numerical values with his since he studied thermal diffusion in the limit that the concentration of one species vanishes and for small differences in K and h, At this point we wish to examine the effect that loads and roughness have on the thermal diffusion effect when the species present have different masses or radii. For this purpose we define the reduced thermal diffusion coefficlents ~
DT(R\‘
I
D , R”(K, =~
K,,
b,,n, mu/m,. a,/a,.
D41(SHs’(byun, m,lm,,, a,/a,, The J o u r n a l o f Physicai Chemistry. Voi. 78. No. 15. 1974
n,, n,,) nu,n,)
(3.1)
0=0.05,
vs.
5 667
3-m,/rrF=09
v=
0.90
0 85 )(LS)’ T
0.80
0.75
0.70
0
Figure 6. DTiLs)* vs.
=i,~,=x,,=a~5
I
bn
2
reduced density bn. a, --- a, = a,
.,/up
and tLS,* = DT
DT(LS’(a,,a , b,,n, mu/mu, - 0,/ c p ,nu.n,> D,’SHS’(b,pn,m,lm,,, c u / c #nu, , n,)
(3.2)
The loaded sphere thermal diffusion coefficient was calculated for equal eccentricity parameters for case i. Figure 6 is a plot of us. bn for case i. The addition of equal loads to molecules of different masses results in reducing the thermal diffusion effect. Very little density dependence is exhibited. For a fixed value of a,DTiLS)* decreases as m,/m,, tends toward unity. For a fixed value of m,/m,,, the addition of equal loads to the molecules decreases as a increases. The thermal diffusion coefficient for a particular mass ratio may be decreased only as much as 15-20% if the molecules have the maximum physically realistic load parameters. Rough sphere thermal diffusion coefficients were calcu-
Thermal Diffusion in Diriary Dense Gas Mixtures
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6
4
-4
c I
2- m, /m.= i 0.664 3- m,/mp=0.5 0
2
l
3
bn Figure 7 . DT(Rs)* YS. f , x u = x, = 0.5.
reduced density bn: K,, = K , = K , cr,/cr, =
lated for the case K~ = K~ = K for systems in which the species have different masses. Figure 7 is a plot of DT(RS)* us. hn for case i. The curves exhibit a definite ordering at high densities where the collisional mode of transport dominates. At low densities D T ( R s ) * does not vary monotonically with K for a fixed mass ratio, but exhibits a minimum with respect to K (a trait that seems to characterize the low density pig-a-back transport with rough sphere molecules). Generally speaking, D T ( R S ) is reduced when roughness is added to molecules of different masses. The effect varies in magnitude but is only as much as 15% in a dense system with m,/m,, = 0.5 and K at its largest physically realistic .value. Claypoollo gives an expression for the thermal diffusion coefficient, derived from the Longuet-Higgins and Poplefl scheme, for an isotopic mixture of rough spheres. He considered species of equal molecular size but different masses and moments of inertia. The theory gives the following result for the thermal diffusion coefficient
This expression predicts that thermal diffusion vanishes when the masses are identical even though the species have different mioments of inertia. It also predicts that the reduced thermal diffusion coefficient is independent of density and is given by (K,,, + 1 ) / ( 2 ~ ”+~ 1).This behavior is in disagreement with what we have predicted above with Enskog’s theory. The thermal diffusion coefficients for loaded spheres with a, -= a, = a and for rough spheres with K,, = K , = K were calculated for the case of unequal radii, i.e., case ii. In Figure 8, which is a typical plot for one value of .,/a, in case ii, the thermal diffusion coefficients pass from negative to positive values as density increases as did Drr(SM5].This is because the difference in molecular radii is still the only driving force for thermal diffusion. The presence of equal eccentricity parameters in loaded
-80 0.1
0.3
0.5
09
0,K
Figure 8. L+(LS) vs. a, DT(RS) vs. K , in units of ( k T / l r m ) ’ / * n - ’ o,,,-~ for crv/ap = 0.667: a , = a, = a, K, = K, = K , rn, = rnp = rn, x u = x, = 0.5.
spheres that have different radii tends to decrease the value of Drr(Ls)in a negative sense for all physically realistic values of the eccentricity parameters. The presence of roughness and equal rough sphere parameters in molecules that have different radii tends to increase D,r‘KSJin the positive sense for all but the highest densities. The polyatomic effects are most important at values of u U / u p near unity since the thermal diffusion coefficient is small and the polyatomic effects represent a large per cent deviation from the SHS results. The polyatomic effects are also of most importance for reduced densities in the range 1.0 < b,,n < 1.5, where the thermal diffusion coefficient vanishes due to competing transport modes. To determine whether the effects of different masses and different loaded sphere parameters are of equal imwas plotted us. a, in Figure 3. The plot portance, Drr(LS)* displays two different mass ratios for high and low densities for the case a, = 0, which will provide the greatest loaded sphere effect. The plot demonstrates a qualitative additivity of the effects of different masses and loaded sphere parameters. In a system where the masses are quite different, e.g., m,/m, = 0.5, the polyatomic effect of dissimilar loaded spheres can reduce the thermal diffusion effect by 40-50%. In a system where the mass difference is small, e.g., m,/m, = 0.9, the effect of dissimilar loaded sphere parameters dominates, and the sign of the thermal diffusion coefficient can be reversed. This is not surprising, as earlier we saw that the effects of dissimilar loaded spheres was of the same order of magnitude as the effects due to dissimilar masses. Though not explicitly depicted in a figure, the effect of introducing loaded sphere character to molecules that have different radii is substantial. The effects of different loaded sphere parameters are dominating in determining the sign and order of magnitude of the thermal diffusion effect in situations where the species exhibit a size difference. This again is not surprising in light of the fact that differences in loaded sphere parameters were found to produce a thermal diffusion effect on order of magnitude greater than simple differences in molecular size To determine the importance of considering both mass The Journal of Physical Chemistry. Voi. 76, No. 75 1974
Mare 0.Baleiko and H. Ted Davis
1570
1
-1.00
I
0.I
0.2
2
0.3
% Figure 9.
YS.
a , for a, = 0: u, = u,, = 6,xu = x p = 0.5.
differences and the polyatomic effect of rough spheres differing in moments of inertia, DT(RS)was calculated for case i for different values of K ~ ,K ~ .Figure 10 depicts D T ( R S ) * us bn. for m,/m, = 0.5 for K , = 0.1 and various . in rough sphere parameters do values of K ~ Differences not have a large effect on the thermal diffusion coefficient for this large s mass difference. In fact, even for small differences in masses we have found (but not reported here) that the maximum effect of different moments of inertia is a decrease in the thermal diffusion coefficient by 25%. The thermal diffusion coefficient was also calculated for rough spheres with different rough sphere parameters for case in. Our calculations show that the effects of different radii and different rough sphere parameters are not strongly coupled. Different molecular radii were found to produce a thermal diffusion effect of an order of magnitude greater than the effect of different moments of inertia. Except in the region of moderate densities where the thermal diffusion effect from unequal radii may vanish due to competing pig-a-back and collisional transfer modes, or for small values of the radius ratio, the sign and order of magnitude of the thermal diffusion effect is determined by the value of the radius ratio.
1IV. Dense (GasMixtures of D2 and HT At this point we have made no attempt to model an actual binary system by either the rough sphere or loaded sphere model. Mere we examine a dense gas mixture of D2 and ZIT for both models. The gaseous system exhibits a thermal diffusion effect,, and both the dilute gas rough sphere model and the dilute gas loaded sphere model have been employed in attempts to satisfactorily predict the phenomenon. TrubenbacherQ considered the molecules to be rough spheres with different moments of inertia and calculated the thermal diffusion factor a =-. DT/x,x,3JC),,,for the binary mixture. With v designating the species D2 and p designating the species FIT, he postulated and molecules to be of equal mass and equal molecular size with K , = 0.090 and K~ = 0.068. His prediction of IYT = 1.9 X 10-3 is smaller than the experimental value12 of 0.028 by a factor of about 15. The Journal of Physical Chemistry. Vol. 78. No. 75. 1974
Figure IO. D T ( ~ ~vs. ) *reduced density bn: 0.5, u, = u p = u , xu = xu = 0.5.
K,
= 0.1, rn,/m, =
Sandler and Dahler,l3 realizing the system lends itself to description by the loaded sphere model, postulated the D2 molecule as a smooth hard sphere and the H T molecule as a loaded sphere. Masses and rnoleculttr radii were assumed equal. Using a calculated value of the eccentricit y parameter for H T of a, = 0.167, they calculated a thermal diffusion factor approximately twice the experimentally observed value. When they used a, as an adjustable parameter to predict a proper rotational relaxation time for HT, a “semiempirical” value of a,, was determined to be a, = 0.092. This produces a value of CYT= 0.026, which is in excellent agreement with experiment. Our purpose in examining the dense gas rough sphere mixture model and the dense gas loaded sphere mixture model for the D2-HT system is not to determine the success or failure of the models in predicting the dilute gas experimental value of the thermal diffusion factor; this question was answered quite well by Trubenbacher and Sandler and Dahler. We performed the calculations to give a concrete basis on which to compare the results of the dense gas theories, and also as a further check on our calculations in the dilute gas limit. We were unable to compare our calculations for the dense D2--HT system to any experimental data. The molecules D2 and HT have a very small mass differences (mD, - mHT = 4.0294 - 4.0250 = 0.0044 au) which by itself is incapable of adequately explaining the thermal diffusion effect. We calculated the diffusion coefficient and thermal diffusion factor for the dense gas D2-HT system for the models: (i) D2 and HT are smooth hard spheres with cr,/gp = I, m,/m, = 4.0294/4.0250. (ii) D2 and H T are rough spheres with K , = 0.090, K~ = 0.068, and m v / m , = crD/a, = 1. (iii) Dz and IIT are rough spheres with K , = 0.090, K~ = 0.068, m,/m, = 4.02941 4.0250, and cr,/a, = 1. (iv) DP and HT are loaded spheres with a, = 0, a,, = 0.167, m u / m p= cr,/a,, = 1. (v) Dz and HT are loaded spheres with a, = 0, a, = 0.092, m,lm, = , . / , a = 1. The thermal diffusion factor is given in Table I1 (microfilm, see ref 3) for cases i-v for reduced densities in the range 0 < bn < 3 for various compositions. For case i CYT varies only with density, that is, it is composition inde-
Thermal Diffusion in Binary Dense Gas Mixtures pendent. The thermal diffusion factor for this case is positive and increases quite rapidly with density. At very low densities aT for cases ii and iii has little dependence upon the composition. varyirg only about 7% for bn = 0, but is a strong function of composition at higher densities. In case ii for rough spheres of equal mass, O ~ decreases T from small positive values at low densities to large negative values at high densities. This density dependence may very well be the effec: of the different transport modes characterizing the iow and high density ranges. Case iii includes the effect of the small mass difference on the rough spheres. We see that for this case of a very small mass difference and small value of K ” and K~ the individual effects due to the -ass difference and different moments of inertia super mpose qualitatively and add in a nearly quantitative manner to produce a net thermal diffusion effect. Previously we saw that this was not the case for large values of K , and K ~ , ,where a coupling of the mass difference and different moments of inertia produced a net thermal diffusion effect quite different than would be expected if the two effects were independent. At high densities the very small mass difference determines the sign of the thermal diffusion factor. varies only about 10% over the For cases i v and v, entire range of compositions at low densities and less than 5% at high densities. L Y ~is positive and increases quite rapidly with density for ail compositions. The agreement of our calculaitians at zero density and the results of Sandler and Dahlerl3 i s excellent. All our reported rough sphere calculations were performed in the $,pin anisotropic approximation which inin the expansion of A and C in cludes the vector W - Q Q the Chapman-Enskog scheme. A lower order approximation does not include this term, but only retains terms of the form a.W, b-W(5/2 - WZ),and h.W(3/2 - Q2) in the expansions of A and C, These two approximations are referred to, respectively, as the Kagan-Afanas’ev and Pidduck approximations.14 For cases ii and iii the binary diffusion coefficient and thermal diffusion factor were calculated for equal mole fractions in the Pidduck approximation so we could compare the results without Kagan-Afanas’ev (KA) calculation and determine the sensitivity of the transport coefficients to the choice of the trial functions. As reported in section 11, we found the value of the binary diffusion coefficient was virtually the same in both alpproximations. The thermal diffusion factor was quite sensitive to the effect of spin anisotropy. In cases ii and ~ in the KA approximation was decreased iii, 0 1 calculated in the negative sense from the values given by the Pidduck approximation. Bit zero density in case ii the Pidduck approximation gave a value of (YT = 1.54 x Our calculation in the KA approximation is 1.25 x about 19% smaller, For case iii at zero density the Pidduck approximation gave a value of CYT = 2.03 x Our calculation in the KA approximation is = 1.75 x 10F3, nearly 14% smaller. At higher densities the effect of spin anisotropy is much larger. For case ii at bn = 3, the Pidduck approximation gave a value of KT = -11.54 x while our ECA calculation is KT = -31.78 x For case (iii) at bn = 3 the values of aT were found to be 45.29 x and 25.88 x for the Pidduck and KA approximations, respectively. These results illustrate the fact that the effects of spin anisotropy are very important in the dense rough sphere fluid, and perhaps warrant an extension of these calculations to test the effect of still higher approximations on the thermal diffusion coefficient.
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The fact that our result for case ii at zero density is somewhat lower than the result given by TrubenbackerDis not surprising in view of the fact that he used a lower approximation than the Pidduck, namely, a.W + b.W(5/2 -
W). All our reported loaded sphere calculations include the effects of both an orientational anisotropy term W-ee, and a spin anisotropy term W.QQ as expansion vectors for the functions A and C determining the diffusion and thermal diffusion coefficients. When orientational anisotropy is neglected the expansions for A and C have the KA form. When both orientational and spin anisotropy are neglected we have the Pidduck approximation. We performed the calculations for case iv and v for equal mole fractions in both the spin isotropic or Pidduck approximation and the orientation isotropic or KA approximation. The expansion vector W-ee was not included in the dilute gas theory of Sandler and DahleP3 as previous evidence indicated it contributes a small amount to the value of transport coefficients. This was substantiated by our calculations in the dilute gas limit. Inclusion of the term leaves the binary diffusion coefficient essentially unchanged and lowers the thermal diffusion factor by less than 1% over the entire density range. The effect of spin anisotropy in the loaded sphere diffusion coefficient is also negligible. The effect on the thermal diffusion factor is greater in case iv than in case v, indicating that the effect increases with eccentricity parameter. The effect of spin anisotropy decreases with increasing density. For case iv at zero density the Pjdduck approximation gives a value of iyT = 0.0474 and the KA approximation a value of 0.0556, some 17% higher. For the ~ 6.32 whereas the case bn = 0, the Pidduck value of 0 1 is KA value is 6.81, an increase of less than 8%. The differences between the Pidduck and KA approximations for case v are somewhat smaller, being 12.2% at zero density and only 5% at bn = 3. We observe the thermal diffusion factor to increase rapidly with increasing density which suggests that at densities characteristic of the liquid phase we may expect the establishment of a concentration gradient one or two orders of magnitude larger than in very dilute gas systems. The tendency to establish a larger concentration gradient in the liquid system could then be employed to perhaps effect better separations while enjoying the advahtage of the small volume in the liquid state.
Acknowledgments. The authors express their gratitude for financial support of this research furnished them by the National Science Foundation and the U. S. Army Research Office-Durham. Supplementary Material Available. Listings of vector and matrix elements from which are obtained the scalar coefficients determining the mutual and thermal diffusion coefficients will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 x 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JP6-74-1564. ~
References and Nates (1) M Baleiko, Ph D Thesis, Universityof Minnesota. 1971
The Journal of Physical Chemistry. Vol. 78. No. 15. 1974
1572 (2) R. A. Stenrei, M . S. Thesis, University of Minnesota, 1967. (3) See paragraph at end of paper regarding supplementary material. (4) J. 0 . Hirschfelder, C . F. Curtiss, and R . B. Bird, "The Molecular Theory of Gases and Liquids," Wiley, New York, N. Y., 1954. (5) J. L. Lebowitz and J. S . Rowlinson, J. Chem. Phys., 41, 133 (1964), (6) S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases," Cambridge University Press, New York, N, Y , , 1953. (7) R. Brown and H.T. IDavis, J . Phys. Chem., 75, 1970 (1971). (8) I. L. McLaughlin and H, T. Davis, J. Chem. Phys., 45, 2020 (1966).
The Journal of Physicai Chemistry. Voi. 78. No. 15. 7974
Marc 0.Baleiko and H.Ted Davis (9) E.Trubenbacher, Z. Naturforsch. A, 17, 539 ('1962). (10) J. L. Claypool, M.S. Thesis, Universityof Minnesota, 1968. (11) H. C. Longuet-Higgins and J. A. Pople, J. Chem. Phys., 25, 884 (1956), (12) J. Schirdewahn, A . Klemm, and L. Waldmann, Z. Naturforsch. A , 16,133 (1961). (13) S. I . Sandler and J. S. Dahler, J. Chem. Phys., 47, 2621 (1967). (14) For references to the work of F. 8. Pidduck and of Yu. Kagan and A. M. Afanas'ev see D. W. Condiff, W. K. Lu, and J, S. Dahler. J. Chem. Phys., 42, 3445 (1965) and B. J. McCoy, S. I. Sandier, and J, S. Dahler, ibid., 45, 3485 (1966).
