Physical Chemistry Third Edition

10.2 Transport Processes 445

at a steady speed. Since Newton’s second law relates force and the rate of change of

velocity (and thus the rate of change of momentum) the same information is contained

in either choice.

Driving Forces and Linear Laws

To a first approximation each rate variable depends on a single state variable that we

call thedriving forcefor that variable. The temperature gradient is the driving force

for heat conduction, the concentration gradient of substanceiis the driving force for

diffusion of that substance, and the rate of shear is the driving force for viscous flow. The

empirical formulas that describe these phenomena specify that the rate of the process is

directly proportional to the driving force, so that these formulas are calledlinear laws.

In a thorough treatment of irreversible thermodynamics, more carefully defined driving

forces are used and the possibility is included that the driving force for one transport

process can make a contribution to the rate of another transport process. An example

is thermal diffusion, in which a diffusion flow is driven by a temperature gradient. We

will not discuss this and othercross-effects, although there is a considerable literature

involving them.

Fourier’s Law of Heat Conduction

The flow of heat in a solid, liquid, or gas system is described by a linear law known as

Fourier’s law of heat conduction:

q−κ∇T (Fourier’s law) (10.2-1)

whereκis a coefficient called thethermal conductivity. If the temperature varies only

in thezdirection, this equation is

qz−κ

∂T

∂z

(10.2-2)

Fourier’s law is named for Jean Baptiste
Joseph Fourier, 1768–1830, a famous
French mathematician and physicist
who also invented Fourier series and
Fourier transforms.

Fourier’s law is called alinear lawbecause the rate is directly proportional to

(linearly dependent on) the gradient of the temperature,∇T, which is the driving

force. The thermal conductivityκcan depend on the composition, the temperature,

the pressure, and on the identities of the substances present, but does not depend on

the temperature gradient. Fourier’s law is also called aphenomenological law, which

means that it is not derived theoretically but simply describes a phenomenon. Fourier’s

law holds quite accurately for gases, liquids, and solids. Table A.16 in Appendix A

gives the values of the thermal conductivity for several substances.

EXAMPLE10.2

A cubical cell 0.100 m on a side is filled with benzene. The top surface is maintained at

25.0◦C and the bottom surface is maintained at 15.0◦C. After some time, the system will

reach asteady statein which the state of the system does not change with time although heat

is flowing. Calculate the amount of heat flowing through the benzene per hour after a steady

state is achieved, neglecting convection.