=
V f / V ,L13/L13 V J V , L,3/L,3 € $ / E / , L,3/Lf3 h,/w, L/3/L,3
=
( T , - T ) / ( T - T ) , dimensionless
= = =
0
0
= =
= =
P/ P.
=
w
=
=
Literature Cited
(1) Louisell, W. H., “Coupled Mode and Parametric Electronics,” Wiley, New York, 1960. 0
1
1
R . H. W I L H E L M A. W. R I C E A. R. BENDELIUS‘
0
( T , - , T ) / ( T - T ) , adsorbent temperature, dimensionless C,/C,, concentration in fluid, dimensionless C,/COj concentration in adsorbent, LJ3/L,3 C*/Co, concentration in fluid that would be in equilibrium with &, dimensionless density of interparticle fluid, M:/Lj3 density of adsorbent particles, M I L S 3 time scale, 1/7
Princeton Unioersity Princeton, N . J .
RECEIVED for review August 16, 1965 ACCEPTED December 6, 1965 1
N.
Present address, Pfister Chemical LVorks, Inc., Ridgefield,
.r.
CORRESPONDENCE BINARY DIFFUSION COEFFICIENTS IN NONPOLAR GASES SIR: The paper by Holsen and Strunk (2) presents new diffusion data for some binary systems near room temperature as well as an interpretation of the difficulties encountered in attempting to predict diffusivities from Lennard-Jones correlations of viscosity measurements. We find the paper misleading in several respects, and we would like to try to clarify these points. In one step of their procedure Holsen and Strunk attempt, without success, to find “unique” Lennard- Jones parameters for their data. Because the Lennard-Jones potential is itself not physically unique and because experimental values are not unique numbers, one cannot expect to find a unique set of parameters. There is, instead, a region of values of u and e / k which will reproduce data within the experimental error. The regions peculiar to diffusivity data and to viscosity data may or may not intersect. When they do not intersect, there is indeed difficulty in reconciling correlations of the two properties. The regions of correlation are not ordinarily revealed accurately by graphical procedures such as the Buckingham plot illustrated in Holsen and Strunk’s Figure 1. Rather one needs to use a careful numerical search procedure, based on a least-error criterion. M’e have carried out such a procedure (3) using Holsen and Strunk’s data as well as some others, with the results listed in Table I. I t is evident from the mean
Table 1. System
N o . of Points and Reference
He-Ar
3 (2) 14 ( 4 )
He-air He-COZ Air-COz COz-Ar
a
144
Parameters from aiscosity data.
l&EC FUNDAMENTALS
deviations shown that effective correlation has been achieved. Holsen and Strunk‘s data appear to be somewhat more precise than most results for diffusion, which commonly exhibit correlative deviations of 1 to 374. We have also shown in Table I our correlation of the results of Saxena and Mason (4)for the helium-argon system. T h e two sets of data are in good agreement. There is apparently no reason for discarding any of their own He-Ar data, as Holsen and Strunk suggest may be necessary. The parameters e l k = 35.6’ K. and u = 2.987 A. provide a n excellent fit of available viscosity measurements with helium-argon. Table I illustrates that if uncertainties in the measurement of diffusivities are of the order of 1.7 to 3.0%, a realistic range, viscosity, and diffusivity can be successfully correlated by the same set of L-J parameters for the temperatures a t which Holsen and Strunk worked. M’hen there is a t least one polyatomic species in the mixture, predictions of diffusivities from viscosities will be considerably in error. This is illustrated for the COrargon system in Table I. For the self-diffusion coefficient of C O P even larger discrepancies, of the order of 3074, are observed (7). Contrary to the assumptions of Holsen and Strunk, however, these discrepancies increase monotonically with temperature above about 300’ K. Walker and b’estenberg ( 5 ) observe that this is also true for the helium-argon system. They conclude their
Force Constants for Lennard-Jones ( 1 2-6) Potential
Temp. Range, ’ K. 276-346 251-418 276-346 276-346 251-418 276-346 276-346 276-317 276-317 276-317 276-1798 276-1 798
elk, ’ K. 25 69 69 35.65 35. ba 40 45 25 21 277 277 158a
u, A .
3.100 2.774 2.774 2,987a 2.987a ~
3.060 3.240 4. . 4411 . ._ 4. . 448 . ._
3.208 3.208 3. 665a
Mean Deviation, % 0.7 1.9 2.6 1.7 3.0 1.1 0.6 0.4 0.5 2.5 2.8 13.6
paper with the statement: “High-temperature transport properties computed with a Lennard-Jones potential based upon low-temperature gas properties and lacking other corroboration must therefore be viewed with extreme skepticism.” \Ve are a t a loss to see how Holsen and Strunk, having read the paper of Walker and Westenberg, could make the statement: ”Likewise, experimental diffusivities obtained a t elevated temperatures appear to correlate well with theoretical predictions, yielding intermolecular parameters which correspond to similar parameters obtained from viscosity data.” Apparently on the basis (of this statement, which is simply not correct, Holsen and Strunk assumed a n asymmetry parameter with reciprocal temperature dependence. The use of such a n asymmetry parameter is inconsistent with empirical evidence and probably with theoretical evidence ( 7 ) ; such use undoubtedly gives misleading results.
SIR: Although sratistical or numerical search procedures can produce most probable values for the pair of Lennard-Jones potential parameters, the results are hardly satisfying when parameters determined from diffusion data for polyatomic molecules are compared with the corresponding parameters from viscosity data. This is evident from Pakurar and Ferron’s (5) Table I, where a series of parameters is tabulated for the COZ-Ar system. Yet relatively close agreement is found in the corresponding pairs of parameters from P-V- T and viscosity data. as noted by Hirschfelder. Curtiss, and Bird ( 3 ) . I n attempting to correlate iheir room temperature diffusion coefficients with Lennard-Jones parameters, the writers found that the temperature dependence of the data was inconsistent 1% ith the diffusion coefficimt equation when used with LennardJones parameters. Thus they were led to seek a n explanation which mas temperature dependent. T h e asymmetry parameter defined by Holsen and Strunk ( 4 ) was empirical, and was so noted. T h e form cited was suggested in part by a theoretical derivation of De Boer (7) which considered the effects of asymmetric polarizabilities on the field between two hydrogen molecules. For the effrctive attractive field de Boer obtained a n expression of the form
Literature Cited
(1) Ember, G., Ferron, J. R., Wohl, K., J . Chem. Phys. 37, 891-7 (1962).
(4) Saxena, S. C., Mason, E. .4., Mol. Phys. 2, 379 (1959). (5) Walker, R. E., IVestenberg, A. A., J . Chem. Phys. 31, 519 (1959). Thomas A . Pakurar’ J o h n R. Ferron University of Delaware :Yewark, Del. 1
Present address, E. I. du Pont de Nemours & Co., Richmond,
Va.
T h e effective repulsive field was characterized by a negative inverse temperature dependence related to the fact that “at low temperatures the molecules prefer orientations of small potential energy, which contribute only little to the repulsive field.” A reciprocal temperature dependence also appears in analyses of the effects of strong electric fields on the polarizability. T h e reciprocal temperature dependence of the anisotropic polarizability parameter was suggested by the above arguments and not by reference to other high temperature diffusion data. T h e remarks made should have been limited to the data cited in the paper and not extended to the interpretation of diffusion data a t higher temperatures. T h e correlation of experimental data presented in Figure 3 of (4) shows a consistent deviation from the temperature dependence found with Lennard- Jones viscosity parameters which would be evident regardless of the specific function of temperature selected. T h e fact that these deviations may differ in nature from the high temperature deviations noted by Ember, Ferron, and Ll’ohl (2) may indicate that the nonspherical effects in the two regions differ in nature.
literature Cited
T h e nature of C is discussed by De Boer (7). mean polarizability defined by
Here 5 is the
where Q and aI. denote polarizabilities along axes parallel and perpendicular, respectively, to the bond, and
(1) De Boer, J., Physica 9, 363 (1942). (2) ..Ember, G., Ferron, J. R., Wohl, K., J . Chem. Phys. 37, 891 (1962).
(3) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” Wiley, Xew York, 1954. (4) Holsen, J. N., Strunk, M. R., IND.ENG.CHEM.FUNDAMEKTALS 3, 143 (1964). (5) Pakurar, T. A , , Ferron, J. R., Ibid., 5 , 144 (1966). James X. Holsen Tt’ashington University, St. Louis, M o . M a i l a n d R. Strunk University of Missouri, Rolla, Mo.
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FEBRUARY 1966
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