Page 549 - Bird R.B. Transport phenomena

§17.4 Theory of Diffusion in Binary Liquids 529

in which fi B is the viscosity of the pure solvent, R A is the radius of the solute particle, and
/3 AB is the "coefficient of sliding friction" (formally the same as the \xl'£ of problem 2B.9).
The limiting cases of /3 AB = °° and [3 AB = 0 are of particular interest:

a =
- Элв °° (no-slip condition)
In this case Eq. 17.4-2 becomes Stokes' law (Eq. 2.6-15) and Eq. 17.4-1 becomes

~lk 1 = ~fr*R (17 4 3)
- "
A
which is usually called the Stokes-Einstein equation. This equation applies well to the dif-
fusion of very large spherical molecules in solvents of low molecular weight 7 and to sus-
pended particles. Analogous expressions developed for nonspherical particles have been
used for estimating the shapes of protein molecules. ' 8 9

bo p AB = 0 (complete slip condition)
In this case Eq. 17.4-1 leads to (see Eq. 4B.3-4)

(17.4-4)

If the molecules Л and В are identical (that is, for self-diffusion) and if they can be as-
sumed to form a cubic lattice with the adjacent molecules just touching, then 2R A =
U3
(V /N ) and
A A
' i/3
(17.4-5)
Equation 17.4-5 has been found 10 to agree with self-diffusion data for a number of liq-
uids, including polar and associated substances, liquid metals, and molten sulfur, to
within about 12%. The hydrodynamic model has proven less useful for binary diffusion
(that is, for Л not identical to B) although the predicted temperature and viscosity depen-
dences are approximately correct.
Keep in mind that the above formulas apply only to dilute solutions of Л in В. Some
attempts have been made, however, to extend the hydrodynamic model to solutions of
finite concentrations. 11
The Eyring activated-state theory attempts to explain transport behavior via a quasi-
12
crystalline model of the liquid state. It is assumed in this theory that there is some uni-
molecular rate process in terms of which diffusion can be described, and it is further
assumed that in this process here is some configuration that can be identified as the "ac-
tivated state." The Eyring theory of reaction rates is applied to this elementary process in
a manner analogous to that described in §1.5 for estimation of liquid viscosity. A modifi-

7
A. Poison, /. Phys. Colloid Chem., 54, 649-652 (1950).
8
H. J. V. Tyrrell, Diffusion and Heat Flow in Liquids, Butterworths, London (1961), Chapter 6.
9
Creeping motion around finite bodies in a fluid of infinite extent has been reviewed by J. Happel
and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J. (1965); see
also S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-
Heinemann, Boston (1991). G. K. Youngren and A. Acrivos, /. Chem. Phys. 63, 3846-3848 (1975) have
calculated the rotational friction coefficient for benzene, supporting the validity of the no-slip condition
at molecular dimensions.
10
J. С. М. Li and P. Chang, /. Chem. Phys., 23, 518-520 (1955).
11
C. W. Pyun and M. Fixman, /. Chem. Phys., 41, 937-944 (1964).
12 S. Glasstone, K. J. Laidler, and H. Eyring, Theory of Rate Processes, McGraw-Hill, New York (1941),
Chapter IX.