Page 38 - Bird R.B. Transport phenomena

§1.4 Molecular Theory of the Viscosity of Gases at Low Density 23

unless there are chemically dissimilar substances in the mixture or the critical properties
of the components differ greatly.
There are many variants on the above method, as well as a number of other empiri-
cisms. These can be found in the extensive compilation of Reid, Prausnitz, and Poling. 5

EXAMPLE 1.3-1 Estimate the viscosity of N at 50°C and 854 atm, given M = 28.0 g/g-mole, p = 33.5 atm, and
c
2
T = 126.2 K.
Estimation of Viscosity c
from Critical Properties SOLUTION
Using Eq. 1.3-lb, we get
2/3
1/2
ix = 7.70(2.80) (33.5) (126.2Г 1/6
c
= 189 micropoises = 189 X 10~ 6 poise (1.3-3)
The reduced temperature and pressure are
1J 2 56; 2 5 5 (13 4ab)
W^=- *=Ш= - - '
From Fig. 1.3-1, we obtain /x = /JL/IJL = 2.39. Hence, the predicted value of the viscosity is
r C
6
6
/л = fi (fi/fi ) = (189 X 1(T )(2.39) = 452 X 10~ poise (1.3-5)
c c
6
6
The measured value is 455 X 10~ poise. This is unusually good agreement.
§1.4 MOLECULAR THEORY OF THE VISCOSITY
OF GASES AT LOW DENSITY
To get a better appreciation of the concept of molecular momentum transport, we exam-
ine this transport mechanism from the point of view of an elementary kinetic theory of
gases.
We consider a pure gas composed of rigid, nonattracting spherical molecules of di-
ameter d and mass m, and the number density (number of molecules per unit volume) is
taken to be n. The concentration of gas molecules is presumed to be sufficiently small
that the average distance between molecules is many times their diameter d. In such a
gas it is known 1 that, at equilibrium, the molecular velocities are randomly directed and
have an average magnitude given by (see Problem 1C.1)

«=

in which к is the Boltzmann constant (see Appendix F). The frequency of molecular
bombardment per unit area on one side of any stationary surface exposed to the gas is

Z = \пп (1.4-2)

5
R. C. Reid, J. M. Prausnitz, and В. Е. Poling, The Properties of Gases and Liquids, McGraw-Hill, New
York, 4th edition (1987), Chapter 9.
6 A. M. J. F. Michels and R. E. Gibson, Proc. Roy. Soc. (London), A134, 288-307 (1931).
1
The first four equations in this section are given without proof. Detailed justifications are given in
books on kinetic energy—for example, E. H. Kennard, Kinetic Theory of Gases, McGraw-Hill, New York
(1938), Chapters II and III. Also E. A. Guggenheim, Elements of the Kinetic Theory of Gases, Pergamon
Press, New York (1960), Chapter 7, has given a short account of the elementary theory of viscosity. For
readable summaries of the kinetic theory of gases, see R. J. Silbey and R. A. Alberty, Physical Chemistry,
Wiley, New York, 3rd edition (2001), Chapter 17, or R. S. Berry, S. A. Rice, and J. Ross, Physical Chemistry,
Oxford University Press, 2nd edition (2000), Chapter 28.