Page 547 - Bird R.B. Transport phenomena

§17.3 Theory of Diffusion in Gases at Low Density 527

The dimensionless quantity П а лв —the "collisional integral" for diffusion—is a func-
tion of the dimensionless temperature KT/S . The parameters a AB and s AB are those ap-
AB
pearing in the Lennard-Jones potential between one molecule of A and one of В (cf. Eq.
1.4-10):
(17.3-13)

This function £lc l/AB is given in Table E.2 and Eq. E.2-2. From these results one can com-
pute that ЯЬ increases roughly as the 2.0 power of Г at low temperatures and as the 1.65
АВ
power of T at very high temperatures; see the p y —> 0 curve in Fig. 17.2-1. For rigid
spheres, П^ would be unity at all temperatures and a result analogous to Eq. 17.3-10
/ЛВ
would be obtained.
The parameters cr and s AB could, in principle, be determined directly from accurate
AB
measurements of ЯЬ over a wide range of temperatures. Suitable data are not yet avail-
АВ
able for many gas pairs, and one may have to resort to using some other measurable
property, such as the viscosity of a binary mixture of A and B. In the event that there are
4
no such data, then we can estimate a and e from the following combining rules: 5
AB
AB
AB I^PA °*B)/ AB — vs As B (17.3-14,15)
a = + £
for nonpolar gas pairs. Use of these combining rules enables us to predict values of 3) ЛБ
within about 6% by use of viscosity data on the pure species A and B, or within about
10% if the Lennard-Jones parameters for A and В are estimated from boiling point data
by use of Eq. 1.4-12. 6
For isotopic pairs, cr * = a A = cr * and s * = s A = s *; that is, the intermolecular
AA
A
A4
A
force fields for the various pairs A-A*, A*-A*, and A-A are virtually identical, and the
parameters a and s A may be obtained from viscosity data on pure A. If, in addition, M A
A
is large, Eq. 17.3-11 simplifies to
сЯЬ АА * = 3.2027 X 10" 5 / T 7 - - = - ^ (17.3-16)
M
V A <г^ *
и л
The corresponding equation for the rigid-sphere model is given in Eq. 17.3-9.
Comparison of Eq. 17.3-16 with Eq. 1.4-14 shows that the self-diffusivity 2) * and
дл
the viscosity /JL (or kinematic viscosity v) are related as follows for heavy isotopic gas
pairs at low density:
M _ „ _ 5 %АА*
П
in which 11^ ~ l.lfty;,AA* over a wide of кТ/е А/ as may be seen in Table E.2. Thus ЯЬ^* ~
1.32^ for the self-diffusivity. The relation between v and the binary diffusivity ЯЬ is not so
АВ
simple, because v may vary considerably with the composition. The Schmidt number
Sc = \x/p4b is in the range from 0.2 to 5.0 for most gas pairs.
AB
Equations 17.3-11, 12, 16, and 17 were derived for monatomic nonpolar gases but
have been found useful for polyatomic nonpolar gases as well. In addition, these equa-
tions may be used to predict % for interdiffusion of a polar gas and a nonpolar gas by
AB
using combining laws different 7 from those given in Eq. 17.3-14 and 15.

4 S. Weissman and E. A. Mason, /. Chem. Phys., 37,1289-1300 (1962); S. Weissman, /. Chem. Phys., 40,
3397-3406 (1964).
5 J. O. Hirschfelder, R. B. Bird, and E. L. Spotz, Chem. Revs., 44, 205-231 (1949); S. Gotoh, M. Manner,
J. P. Stfrensen, and W. E. Stewart, /. Chem. Eng. Data, 19,169-171 (1974).
6 R. C. Reid, J. M. Prausnitz, and В. Е. Poling, The Properties of Gases and Liquids, 4th edition,
McGraw-Hill, New York (1987).
7 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, 2nd corrected
printing, Wiley, New York (1964), §8.6b and p. 1201. Polar gases and gas mixtures are discussed by E. A.
Mason and L. Monchick, /. Chem. Phys. 36, 2746-2757 (1962).