Physical Chemistry Third Edition

452 10 Transport Processes

a model process such that an object repeatedly takes steps of fixed length in randomly

chosen directions. The displacement is proportional to the square root of the elapsed

time, not to the elapsed time, since as time passes there are more chances for the object

to reverse direction.

Exercise 10.5

Look up the integral and show that Eq. (10.2-19) is correct.

Although we obtained the root-mean-square distance traveled by a diffusing mole-

cule by considering a special case in which all of the molecules started out at the same

value ofz, the molecules diffuse in the same way for other initial conditions, and we

can use Eq. (10.2-19) for any kind of initial conditions and for thexandydirections

as well.

EXAMPLE10.4

Find the root-mean-square distance in three dimensions diffused by glucose molecules in

30.0 minutes in water at 25◦C.

Solution

The root-mean-square distance traveled in three dimensions is

rrms

〈

x^2

〉

+

〈

y^2

〉

+

〈

z^2

〉 1 / 2

 3

〈

z^2

〉 1 / 2



√

6 D 2 t (10.2-20)

where we use the fact that all three directions are equivalent. From Table A.17 of Appendix

A, the value of the diffusion coefficient of glucose at 25◦Cis0. 673 × 10 −^9 m^2 s−^1.

rrms 6 D 2 t^1 /^2 

√

6

(

0. 673 × 10 −^9 m^2 s−^1

)

(30.0 min)(60 s min−^1 )

 2. 70 × 10 −^3 m 0 .270 cm 2 .70 mm

Newton’s Law of Viscous Flow

Consider the arrangement that is shown in Figure 10.1. There is a force per unit area
denoted byPzythat is exerted on the upper plate to keep it moving at a constant speed.
The first subscript onPzyindicates that the plate is perpendicular to thezdirection,
and the second subscript indicates that the force is in theydirection. Theycom-
ponent of the velocity depends onzas indicated in Figure 10.1, corresponding to
shearing flow.
Newton’s law of viscous flow is named
for Sir Isaac Newton, 1642–1727, the
great British mathematician and
physicist who is famous for Newton’s
laws of motion and for being one of the
inventors of calculus.

Newton’s law of viscous flow is the linear law for viscous flow:

Pzyη

(

∂uy

∂z

)

(Newton’s law of viscous flow) (10.2-21)

The coefficientηis called theviscosity coefficientor theviscosity. It depends on the

temperature and the identity of the substance but does not depend on the rate of shear

if Newton’s law is obeyed. Table A.18 of Appendix A gives values for viscosity coef-

ficients for a few liquids and gases. Equation (10.2-21) holds not only for the force