Physical Chemistry Third Edition

446 10 Transport Processes

Solution

Since the top of the vessel is at a higher temperature than the bottom, we assume that

convection can be neglected. The thermal conductivity of pure benzene at 20◦C and 1.00

atm pressure is equal to 0.151 J m−^1 s−^1 K−^1. Assuming that the temperature depends only

onz, the vertical coordinate, the temperature gradient is

dT

dz

≈

∆T

∆z



10 .0K

0 .100 m

100 K m−^1

We have used an average value for the gradient by replacingdT /dzby the quotient of finite

differences∆T/∆z. Fourier’s law now gives the result

qz−κ

(

∆T

∆z

)

−(0.151 J m−^1 s−^1 K−^1 )(100 K m−^1 )− 15 .1Jm−^2 s−^1

The cross-sectional area is 0.0100 m^2 , so the total amount of heat flowing in 1.00 hour is

q

∣∣

∣−

(

15 .1Jm−^2 s−^1

)(

0 .0100 m^2

)

(3600 s)

∣∣

∣544 J

Exercise 10.1

A bar of aluminum with length 0.400 m and uniform cross-sectional area 1. 00 × 10 −^4 m^2 is

placed between two large objects as in Figure 3.9. One object is maintained at 30.0◦C and the

other is maintained at 20.0◦C. Calculate the amount of heat in joules that is conducted through

the bar in 15.0 minutes.

Fick’s Law of Diffusion

Diffusion in solids, liquids, and gases is described by a second linear law, Fick’s law

of diffusion:

Ji−Di∇ci (Fick’s law) (10.2-3)

whereciis the concentration of substanceiand whereDiis a coefficient called the

diffusion coefficientof substancei. It depends on temperature, pressure, composition,

and on the identities of all substances that are present, but not on the concentration

gradient. If the concentration varies only in thezdirection, Fick’s law is

Jiz−Di

(

∂ci

∂z

)

(10.2-4)

Fick’s law is named for Adolf Fick,
1829–1901, a German physiologist.

According to Fick’s law, the driving force for diffusion is the concentration gradient,

∇ci. The thermodynamically correct driving force for diffusion is the chemical potential

gradient. Larger values of the chemical potential correspond to larger values of the

concentration, so the gradient of the concentration is related to the gradient of the

chemical potential. Fick’s law is a good approximation for most gaseous, liquid and

solid systems.

Table A.17 in the appendix gives the values of several diffusion coefficients. Many

liquid substances with “ordinary size” molecules have diffusion coefficients roughly