Chemistry Reference
In-Depth Information
,
t
{
m
),
Eq. (6-38)
can be simplified to
Eq. (6-39)
by doing a Taylor series expansion of the exponential term.
First, at short
times (i.e., 0
k
B
T
x
2
m
t
2
h
ð
t
Þi 5
(6-39)
This result reveals that the particle at short times moves with a constant ther-
1
=
2
). However, at long times, the exponential term in
kT
m
mal velocity (i.e., v
5
Eq. (6-38)
diminishes. As a result,
2
k
B
T
γ
h
x
2
ð
t
Þi 5
t
(6-40)
The above equation simply shows the effects of the environment (i.e., temper-
ature and friction due to the solvent molecules) on the motion of the particle.
Comparing
Eq. (6-40)
with
Eq. (6-29)
derived from the random walk model
yields the following equation.
k
B
T
γ
D
5
(6-41)
, is related to the
viscosity of the solvent and the size of the Brownian particle. Combining
Eqs. (6-32) and (6-41)
yields the famous Stokes
As mentioned earlier (
Eq. 6-32
), the friction coefficient,
γ
Einstein equation for the self-
diffusion of spherical Brownian particles:
k
B
T
6
D
5
(6-42)
πη
r
EXAMPLE 6-4
Consider a small spherical Brownian particle with a radius of 3 nm in water.
a) Calculate the self-diffusion coefficient of this particle in water at 25
C.
b) Given that there are many Brownian particles initially located at the origin of the x-axis
diffusing outward in both positive and negative directions on the x-axis, what is the aver-
age distance that these particles travel in a day?
Solution
a) At 25
C, the viscosity of water is 10
23
kg
ms
, and the friction coefficient is:
10
212
kg
s
10
23
10
29
γ 5
6
πη
r
5
6
3π3
3
3
3
5
56
:
5
3
The self-diffusion coefficient of a particle is:
10
223
k
B
T
γ
1
:
38
3
3
298
:
15
57:28310
211
m
2
s
D 5
5
56:5310
212
b) From
Eq. (6-32)
, we have
