Callister - An introduction - 8th edition

5.3 Steady-State Diffusion • 127
Figure 5.4
(a) Steady-state
diffusion across a
thin plate. (b) A
linear concentration
profile for the
diffusion situation
in (a).
Gas at
pressure P A
P A > P B
and constant
Thin metal plate
Gas at
pressure P B
Direction of
diffusion of
gaseous species
Concentration of diffusing species, C
C A
C B
x A
Position, x
x B
Area, A
(b)
(a)
Fick’s first law—
diffusion flux for
steady-state diffusion
(in one direction)
diffusion coefficient
Fick’s first law
driving force
In the present treatment, the concentration profile is assumed to be linear, as depicted
in Figure 5.4b, and
concentration gradient ¢C
(5.2b)
¢x C A C B
x A x B
For diffusion problems, it is sometimes convenient to express concentration in terms
of mass of diffusing species per unit volume of solid (kg/m 3 or g/cm 3 ). 1
The mathematics of steady-state diffusion in a single (x) direction is relatively
simple, in that the flux is proportional to the concentration gradient through the
expression
J D dC
dx
(5.3)
The constant of proportionality D is called the diffusion coefficient, which is expressed
in square meters per second. The negative sign in this expression indicates
that the direction of diffusion is down the concentration gradient, from a high to a
low concentration. Equation 5.3 is sometimes called Fick’s first law.
Sometimes the term driving force is used in the context of what compels a reaction
to occur. For diffusion reactions, several such forces are possible; but when
diffusion is according to Equation 5.3, the concentration gradient is the driving force.
One practical example of steady-state diffusion is found in the purification of
hydrogen gas. One side of a thin sheet of palladium metal is exposed to the impure
gas composed of hydrogen and other gaseous species such as nitrogen, oxygen, and
water vapor. The hydrogen selectively diffuses through the sheet to the opposite
side, which is maintained at a constant and lower hydrogen pressure.
1
Conversion of concentration from weight percent to mass per unit volume (in kg/m 3 ) is
possible using Equation 4.9.