BINARY DIFFUSION COEFFICIENTS IN NONPOLAR GASES J A M E S N . H O L S E N A N D M A I L A N D R . S T R U N K School of Engineering, Washington C'niuersity, St. Louts, Mo.
Experimental measurements by the Loschmidt method of binary diffusion coefficients of the gas pairs He-A, He-air, He-C02, air-COz, and A X 0 2 are reported for the temperature range from 276" to 346" K. Within this range, the temperature dependence of the coefficients is found to b e inconsistent with the theoretical value predicted with Lennard-Jones potential parameters derived from viscosity data when used with the usual combining rules for unlike molecules. Similar deviations are found in previously published data.
VUMBER
of investigators have found it difficult to interpret
A experimental diffurion coefficients of nonpolar gas systems in terms of the molecular diameters and intermolecular potential energy functions defined in kinetic theory. The reported results of Bunde (2) for the systems H 2 - x ~ and H2-He and of Rumpel ( 6 ) for the system He-N2 may be cited as typi cal. \Vorking with experimental diffusivities measured over the temperature range of 25' to 85" C,, both authors were unable to obtain unique values of the intermolecular parameters ivhich would satisfy the systems over the \*(
TAB*)
[
.MA
+ M , '"
~-
2MA,M,
]
(2)
where the subscripts A and B refer to the t\vo diffusing species. 'rhe integral W1 I # * is a function of a reduced temperature 7',,* = k 7 - C A B . T h e u A B and c A B are potential parameters appropriate to the assumed form of the intermolecular potential function \vhich is required for the evaluation of the (1-integral. 'The potential parameters corresponding to ,an assumed potential function may in principle be obtained by applying Equation 2 to the exprrimental data a t each of two or more 144
I&EC FUNDAMENTALS
GQJ Pair He-.A
298
Comparison with Published Data Idue Computed Published from Value, D Tabif I I , DifferRej Sq C m lSec Sq Cm Sec ence, 0 633 - 1 25 0 688 -1 29 (8) 0 696 0 689 -1 00 (9) 0 979 0 955 -2 45 (70) 0 725 ( A 0 -2'
298
(70)
Temp., K.
273 28'
9 288 2 254 2
';
;1
;2;
trace)
0 754 ( H e -
0
72'
trace)
Air-CO? X-CO2
He-COz
273 293 2 303
i':i
(7)
:;?
0 595 ( C 0 2 trace)
0 1385 0 146 0 616
0 36
3 5
tential for the A-COz system has been computed using the usual empirical combining rules I.o
m
a“ #y-
0.9
0 IV
a
LL
z
0
F 0.8 V ul
K #y 0 V
0.7
2 2.8
3.0
3.2
3.4
3.6
3.8
( I / T ) x 103,0K
Figure 3. Correction factors to be applied to the Lennard-Jones potential energy parameter (from viscosity data) to reproduce the experimental diffusion coefficients Experimental d a t a far the n2-N~ and He-N? systems obtained from d a t a of Bunde and o f Rumpel, respectively
fied Buckingham potentials without success. Figure 1 illustrates one of the graphical solutions which were obtained for the Lennard-Jones parameters. T h e pair of parameters predicted by the usual combining rules ( 4 ) is located near the upper right hand corner on the figure. T h e experience reported here is not unique; as noted earlier. Bunde (2) and Rumpel (6)\\orking with similar systems were also unable to fit potential parameters to their experimental data. Investigator< \\orking \vith monatomic gases have been most successful in obtaining potential parameters from experimental diffusion coefficients. Mirschfelder (5) has shown that at a given temperature those collisions most effective in promoting diffusion occur at closer distances of approach than do those collisions which are most effective in the transfer of momentum and energy. T h e argument is as follows. T h e derivation of Equation 2 shows that
(3) From a similar analysis for the coefficient of viscosity
where fi(2 D * is the appropriate %integral for the first approx imation to the coefficient of viscosity. and uRS and o L , ,are the effective collision diameters for the rigid sphere and LennardJones models, respectively. If the Lennard- Jones potential function is to satisfv both mechanisms, the u L J must be identical in the t\vo equations: and the effective rigid sphere diameters will differ. T h e relative values of the rigid sphere diameters are aqsumed to be related to the degree of intermolecular penetration characteristic of the two processes. These relation? are illustrated in Figure 2 where the Lennard-Jones po-
and parameters for the pure components are derived from viscosity data ( 4 ) . The rigid sphere diameters a t 276’ K . for mass diffusion and momentum transfer are located on the potential curve at the points indicated. This qualitative analysis suggests that the relatively closer distances of approach characteristic of the diffusion process may have some effects on the diffusion coefficient which would not appear in data for viscosity and thermal conductivity. \\’hen the measured values of the temperature exponent rn from Equation 1 are compared with the corresponding values predicted by Equation 2, using potential parameters from viscosity data, deviations from theory are greatest for those systems which contain large or azymmetric molecules. This relationship has been noticed before. Wilke and Lee ( 7 7 ) , for instance, have suggested that a better correlation of experimental data is obtained if the constant term in Equation 2 is modified and that a relationship may exist between the constant and the molecular weight group. Hirschfelder, Curtiss. and Bird (4)have reviewed the London theory for dispercion forces between asymmetric molecules. At large separations. the dispersion energy is proportional to the reciprocal sixth power of the distance of separation but the magnitude depends upon the relative orientation of the colliding molecules. At shorter distances of approach, the dicpersion energy deviates from the reciprocal sixth power relation a n d becomes smaller than would be predicted for large separations. For this case, the average potential energy of interaction, averaged over all orientations, will be temperature dependent. I t is suggested that the observed deviations from the theory for symmetric molecules may be related to the anisotropic nature of the polarizabilities of asymmetric molecules, Mean polarizabilities of linear molecules may be defined in terms of the polarizabilities along each of the three principal axes by
where a,l and a i denote polarizabilities along axes parallel and perpendicular, respectively, to the bond. T h e function
will be taken to be representative of the degree of anisotropy in the polarizability of a given molecular species, A . An empirical correlating parameter representative of asymmetric effects is now defined by
This particular function. although suggested bv the observation that sums of squared terms often appear in expressions for the dispersion enerqy, has no immediate justification in theory I t is reaconable that the asvmmetric effects 5hould be related to the reduced temperature. T*. and should be most significant a t the lower reduced temperatures characterictic of low velocitv collisions. For the higher velocity collisions. orientation forces can be expected to be of less importance relative to the momentum of the particles. VOL. 3
NO. 2
M A Y
1964
145
T = 273 “K I.( b b
%
\
\
$ a0 c
\
0.’
‘bd
\ \
0
\
