10.2 Transport Processes 449
Fick’s Second Law of Diffusion
We substitute Eq. (10.2-4) into Eq. (10.2-9) to obtain Fick’s second law of diffusion
for the one-dimensional case:∂ci
∂t∂
∂z(
Di∂ci
∂z)
(Fick’s second law of diffusion
in one dimension)(10.2-11)
IfDiis independent of position we obtain thediffusion equationfor one dimension:∂ci
∂tDi∂^2 ci
∂z^2(the diffusion equation
in one dimension)(10.2-12)
If the concentration depends on all three coordinates and ifDiis constant, the diffusion
equation is∂ci
∂tDi[
∂^2 ci
∂x^2+
∂^2 ci
∂y^2+
∂^2 ci
∂z^2]
Di∇^2 ci(the diffusion
equation in three
dimensions)(10.2-13)
The Laplacian is named for Pierre The operator∇^2 (“del squared”) is called theLaplacian operator.
Simon, Marquis de Laplace,
1749–1827, a great French
mathematician and astronomer who
proposed that the solar system
condensed from a rotating gas cloud.
The one-dimensional diffusion equation in Eq. (10.2-12) contains partial derivatives
and is called apartial differential equation. The solution of such an equation requires
not only the equation, but also specification ofinitial conditions. For example, if a
solution initially containing a solute (substance 2) at concentrationc 0 in a solvent
(substance 1) is placed in the bottom half of a cell and pure solvent is carefully layered
above it in the top half of the cell, the initial condition isc 2 (z,0){
c 0 ifz< 0
0ifz> 0wherez0 is the center of the cell. The solution of Eq. (10.2-2) for this initial condition
in an infinitely long cell is^1c 2 (z,t)c 0
2[
1 −erf(
z
2√
D 2 t)]
(10.2-14)
where erf(···) denotes theerror function, introduced in Chapter 9 and described in
Appendix C. This solution is shown in Figure 10.3 for three values oftand for a value(^1) See D. P. Shoemaker, C. W. Garland, and J. W. Nibler,Experiments in Physical Chemistry, 6th ed.,
McGraw-Hill, New York, 1996, for a solution pertaining to a cell of finite length.