Page 40 - Bird R.B. Transport phenomena

§1.4 Molecular Theory of the Viscosity of Gases at Low Density 25

or, by combining Eqs. 1.4-1, 3, and 8

(1.4-9)

This expression for the viscosity was obtained by Maxwell 2 in 1860. The quantity ird is
2
called the collision cross section (see Fig. 1.4-2).
The above derivation, which gives a qualitatively correct picture of momentum
transfer in a gas at low density, makes it clear why we wished to introduce the term
"momentum flux" for r yx in §1.1.
The prediction of Eq. 1.4-9 that /x is independent of pressure agrees with experimen-
tal data up to about 10 atm at temperatures above the critical temperature (see Fig. 1.3-1).
The predicted temperature dependence is less satisfactory; data for various gases indi-
cate that /A increases more rapidly than V T . TO better describe the temperature depen-
dence of /JL, it is necessary to replace the rigid-sphere model by one that portrays the
attractive and repulsive forces more accurately. It is also necessary to abandon the mean
free path theories and use the Boltzmann equation to obtain the molecular velocity dis-
tribution in nonequilibrium systems more accurately. Relegating the details to Appendix
D, we present here the main results. ' '
3 4 5

Circle of area ird 2

Fig. 1.4-2 When two rigid spheres of diameter d approach
each other, the center of one sphere (at O') "sees" a circle of
2
area ird about the center of the other sphere (at O), on
2
which a collision can occur. The area ird is referred to as the
"collision cross section."

2
James Clerk Maxwell (1831-1879) was one of the greatest physicists of all time; he is particularly
famous for his development of the field of electromagnetism and his contributions to the kinetic theory
of gases. In connection with the latter, see J. C. Maxwell, Phil. Mag., 19,19, Prop. XIII (1860); S. G. Brush,
Am. J. Phys, 30,269-281 (1962). There is some controversy concerning Eqs. 1.4-4 and 1.4-9 (see S. Chapman
and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, 3rd
edition 1970), p. 98; R. E. Cunningham and R. J. J. Williams, Diffusion in Gases and Porous Media, Plenum
Press, New York (1980), §6.4.
3 Sydney Chapman (1888-1970) taught at Imperial College in London, and thereafter was at the
High Altitude Observatory in Boulder, Colorado; in addition to his seminal work on gas kinetic theory,
he contributed to kinetic theory of plasmas and the theory of flames and detonations. David Enskog
(1884-1947) (pronounced, roughly, "Ayn-skohg") is famous for his work on kinetic theories of low- and
high-density gases. The standard reference on the Chapman-Enskog kinetic theory of dilute gases is
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University
Press, 3rd edition (1970); pp. 407-409 give a historical summary of the kinetic theory. See also D. Enskog,
Inaugural Dissertation, Uppsala (1917). In addition J. H. Ferziger and H. G. Kaper, Mathematical Theory of
Transport Processes in Gases, North-Holland, Amsterdam (1972), is a very readable account of molecular
theory.
4 5
The Curtiss-Hirschfelder extension of the Chapman-Enskog theory to multicomponent gas
mixtures, as well as the development of useful tables for computation, can be found in J. O. Hirschfelder,
C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 2nd corrected printing
(1964). See also С F. Curtiss, /. Chem. Phys., 49, 2917-2919 (1968), as well as references given in Appendix
E. Joseph Oakland Hirschfelder (1911-1990), founding director of the Theoretical Chemistry Institute at
the University of Wisconsin, specialized in intermolecular forces and applications of kinetic theory.
5
C. F. Curtiss and J. O. Hirschfelder, /. Chem. Phys., 17, 550-555 (1949).