Chemistry Reference
In-Depth Information
diffuse as a result of their Brownian motions. The corresponding time dependency of the con-
centration profile along the x-axis is governed by Fick's second law as shown in the following
equation:
2
c(t ; x)
@x
2
@
c(t
;
x)
@t
5D
@
(6-22)
The initial conditions are that the point source has a positive concentration of the parti-
cles, C
0
,atx 50 and that there is zero concentration elsewhere. Solving the above partial
differential equation (see Problem 6-3) yields
Eq. (6-23)
:
C
0
4
x
2
4Dt
C(t ; x) 5
p
e2
(6-23)
π
Dt
At a given moment in time,
ð
1
N
C(t ; x)dx 5C
0
(6-24)
2
N
Therefore, one can normalize the particle concentration distribution found in
Eq. (6-23)
using
Eq. (6-24)
. Doing so leads to the probability density distribution function that
describes the spatial distribution of the particles undergoing one-dimensional Brownian
motion:
C(t ; x)
Ð
1
N
2
N
C(t ; x)
C
0
P(t
;
x)dx
5
C(t ; x)dx
dx
5
dx
(6-25)
The evolution of the probability density distribution function is depicted in
Fig. 6.4
.
And the mean square displacement of the particles can be calculated as follows:
ð
N
hx
2
x
2
P(t ; x)dx
i 5
(6-26)
2
N
0
@
1
A
x
2
4Dt
dx
ð
N
ð
N
x
2
1
4πDt
4Dt
dx 5
2
2Dt
x
2
x
2Dt
e
2
hx
2
x
2
i 5
p
e2
p
4πDt
(6-27)
2
N
2
N
Performing the integration by parts yields:
N
2
N
2
ð
N
4Dt
dx
ð
N
x
2
4Dt
x
2
x
2
4Dt
dx
i 5
2
2Dt
2Dt
4πDt
hx
2
p
4πDt
xe2
e2
5
p
e2
(6-28)
2
N
2
N
By changing variables,
p
4Dt
p
4Dt
x
2
4Dt
5.
y
2
5
x
5
y
5.
dx
5
dy
One obtains
ð
N
ð
N
p
4Dt
2Dt
4
2
e
2y
2
e
2y
2
dy 52Dt erf (
N
) 52Dt
hx
2
i 5
p
dy 52Dt
p
(6-29)
π
Dt
2
N
2
N
Comparing
Eq. (6-29)
with
Eq. (6-21)
shows that the self-diffusion coefficient of a one-
. All the above analyses can be applied
L
2
2τ
dimensional random walker is simply equal to
