Chemistry Reference
In-Depth Information
A
2
Δ
x
y
z
X
−Δ
x
X
+Δ
x
x
FIGURE 6.1
A volume element in a diffusion region.
Applying the Taylor series expansion of J around point x, J(x
2Δ
x) and J
(x
1Δ
x) can be expressed as follows:
Δ
dJ
dx
J
ð
x
2Δ
x
Þ 5
J
ð
x
Þ 2
x
1
high-order terms
(6-4)
Δ
dJ
dx
ð
1Δ
Þ 5
ð
Þ 2
1
J
x
x
J
x
x
high-order terms
(6-5)
Δ
x)
2
,(
Δ
x)
3
,
If
Δ
x approaches zero, the higher order terms, including (
...
,
vanish.
Eq. (6-3)
can be written as:
d
dx
dx
d
dx
dx
J
ð
x
Þ 2
2
J
ð
x
Þ 2
dC
dt
5
(6-6)
2dx
or
dJ
dx
5
dC
dt
2
(6-7)
If the diffusion coefficient is assumed to be constant (i.e., independent of con-
centration and time), combining
Eqs. (6-1)
and
(6-7)
yields:
2
C
@
@
C
@
D
@
t
5
(6-8)
x
2
Equation (6-8)
is known as Fick's second law of mass transfer (or the diffu-
sion equation). Solution of this equation is dependent on the boundary conditions
as well as the initial condition of the system of interest. Obviously, when the dif-
fusion coefficient is not a constant (i.e., dependent on concentration and/or time),
