Physical Chemistry Third Edition

1192 28 The Structure of Solids, Liquids, and Polymers

The motion of a single molecule into a molecule-size hole is not the only simple

molecular diffusion process that can be treated in a simple model theory. A variety of

processes, including the exchange in position of two adjacent molecules, have been

considered.^31 If there is a vacancy adjacent to the pair of molecules, this process might

make a significant contribution. The exchange in position of two molecules of different

solute species moves a molecule of one species in one direction and a molecule of

another species in the opposite direction. Such process can lead to “cross-effects,” so

that Fick’s law must be written in an extended form, in which the concentration gradient

of one species makes a contribution to the diffusion flux of another species:

Jiz−

∑c

j 1

Dij

∂cj

∂z

(28.6-12)

In the theory of nonequilibrium thermodynamics, such cross-effects are systematically

studied, and thermodynamic theorems relating the cross-coefficients are proved.^32

Viscosity

In viscous (shearing) flow in a liquid, one layer of molecules flows past an adjacent

layer. Newton’s law of viscous flow is

Pzyη

(

∂uy

∂z

)

(Newton’s law) (28.6-13)

Whereuyis the velocity component of the fluid in theydirection and∂uy/∂zis

called therate of shear. The force per unit area required to maintain the shearing flow

is denoted byPzy, andηis the viscosity coefficient.

If we use a moving coordinate system, one layer of molecules is stationary and the

adjacent layer moves relative to it. In our simple model, this motion is accomplished

by the motion of individual molecules into holes in the liquid, rather than by concerted

motion of a whole sheet of molecules at once. Therefore the rate of shear is propor-

tional to the rate constant in Eq. (28.6-2). By Eq. (28.6-13), the viscosity is inversely

proportional to the rate of shear, so that the viscosity should obey a formula of the

Arrhenius type,

ηAηe+Eaη/RT (28.6-14)

whereEaηis an activation energy andAηis a preexponential factor. This activation

energy should roughly equal that for diffusion in the same liquid.

Exercise 28.12

The viscosity of carbon tetrachloride at 20◦C is equal to 9. 69 × 10 −^4 kg m−^1 s−^1 , and is equal

to 6. 51 × 10 −^4 kg m−^1 s−^1 at 50◦C. Calculate the Arrhenius activation energy for viscosity.

(^31) R. G. Mortimer and N. H. Clark,Ind. Eng. Chem. Fundam., 10 , 604 (1971).
(^32) S. R. DeGroot and P. Mazur,Nonequilibrium Thermodynamics, North Holland Publishing Co.,
Amsterdam, 1962.