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RALPHK. BIRDWHISTEU
University of West Florida Pensacola,FL 32504
Confusion in the Expressions for Transport Coefficients Boyd L. Earl University of Nevada, Las Vegas, Las Vegas, NV 89154 The transport coefficients for a gas of hard spheres are usuallv nresented in nhvsical chemistry texts. The diffusion coeffi&&, D, the thermal conductivity, K , and the viscosity, v. can convenientlv be ex~ressedin terms of simple kinetic theory quantities:. D = o,EL
where E is the average speed, L is the mean free path, n is the number density, c, is the constant-volume heat capacity per molecule, and m is the molecular mass. The constants a,,az, and as are numerical multipliers, the values of which depend on the level of rigor in the development of these expressions. Physical chemistry texts generally use rough physical arguments to predict the forms of these equations, which then yield values of 113 or 112 for the multipliers, depending on the approach. Since these results are clearly stated to be approximate, the discrepancies are not of concern. Many texts go on to state expressions that are results of the rigorous kinetic theory of hard spheres. I t is here that confusion arises. Expressions given by various texts correspond to different values for the multipliers, which are summarized in the table for 13 texts surveyed. What is the source of these discrepancies? More advanced works! "state expressions resulting from the rigorous theory whichcorresnond t o a ? = 3 ~ 1 1 6= 0.589. a? = 25~164= 1.227. and aa = 5 s h 2 = 0.49i; these are, indeed,frequently cited as the rigorous results. Based on the statements made concerning these results in physical chemistry texts, the innocent student (or teacher) might will be led to believe that these are exact results; in fact, in some texts, they are explicitly stated to be exact. Therein lies the source of confusion. Rigorous exact. In fact the rigorous kinetic theory, based on the Boltzmann equation, does not yield exact results. Rather, results are obtained at various levels of approxima-
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McQuarrle. D. A. Statisfical Mechanics; Harper and Row: New York. 1976; Chapter 19. Curtiss. C. F.; Bird. R. B. Molecular Theory of Hirschfelder,J. 0.; Gases andLiqulds; Wiley: New York, 1954; Chapter 8. $Chaoman.S.: Cowlino. T. G. The Mathematical Theory of Non~niforrn'Gases.3rd ed.: Gmbridae Universitv: London. 19%:. Chao. ters 9 and 10. 'Mason, E. A. J. Chem. Phys. 1954,22,169-186. =Chapman and Cowling3 present results on convergence of successive approximations forthe case of a "Lorentzian gas", which has two components, one being much heavier than the other. Here, additional approximationsmay be made to simpllfy calculations.For a Lorentzian gas of hard spheres, the first aDnroximations for all three coefficienti are much worse and succ&ive approximations converge less rapidly, than for a pure gas.
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Varlous Values lor the Nurnerlcal Multipilers in the Expressions for the Transport Coefflclentsa
Values Given by Texts 3d16 = 0.5S9(4)b 0.599(3) 3&/64 = 0.20S(1)5 no value (5)
25d64 =
1.227(6) 1.261(2)
no value (5)
5d32 = 0.491(5) 0.499(4) % (2) no value (2)
V a l ~ from s the Rlgwous Themy First
ApprOX. 3 ~ / 1 6
25~164
5r/32
1.255 0.498 Approx. 0.599 mird 1.258 0.499 " ,. , m A 0.600 -The top pan gives values cited by tens as the results of the rigomus theory. The number01 tens citingavalue is inpareMh-*.The bonompangivestheactval r e a u k 01 Second
the rigorous theory. bone of these tens had 3/18. whlch wan assumed to be a misprint. 'Presumably, mio is just an unacooumable error.
tion. The values just cited are the first approximations. The second and higher approximations include small, but significant, corrections to these values. Values from the first three approximations are given in the table. For thermal conductivity and viscosity, the third approximations have been shown to be within 0.1% of the true v a l ~ e . The ~ . ~ diffusion coefficient is more problematical. An authoritative theoretical treatment of this subject is given by Chapman and Cowli r ~ gwho , ~ decline to discuss the third approximation to this coeffic~ent,as i t is "extremely complicated". However, Mason' has worked out the formula, and the calculated results for rigid spheres are shown. I t appears that the successive a~nroximationsfor the diffusion coefficient are convereina abbut as rapidly as those for the other two transport coeff; cient~.~ A value from any level of approximation of the rigorous theory may be considered "correct," particularly if the level of approximation is stated. Nevertheless, finding different results for what is presented as the same quantity is different textbooks is somewhat disconcerting. I t would seem to be appropriate for physical chemistry texts to state the first approximations. (They are more than adequate for calculating average collision diameters, which are the only interaction parameters for hard spheres.) But they should be presented as approximations, with an indication that refinements in the values are possible. Some values from higher approximations might then be given, to illustrate the magnitudes involved. Volume 66 Number 2
February 1989
147
