Chemistry Reference
In-Depth Information
very low particle concentrations. According to the equipartition theorem
[15]
, the
mean square velocity of the particle in the one-dimensional case satisfies the rela-
tion that
mx
2
k
B
T
2
2
5
, which yields the following equation:
k
B
T
m
x
2
h
_
i 5
(6-36)
Thus,
Eq. (6-35)
can be rewritten in the following form:
dt
h
x x
i52
m
h
x x
i 1
d
k
B
T
m
(6-37)
0, solution of the above equation
(see Problem 6-4) yields the mean square displacement of the particle.
Given the initial condition that
h
x
ð
0
Þ _
x
ð
0
Þi 5
h
i
2
k
B
T
γ
1
m
exp
2
m
t
x
2
h
ð
t
Þi 5
t
(6-38)
This solution provides two important
insights into Brownian motion as
depicted in
Fig. 6.6
.
2
(
t
)
x
Linear term
Exponential term
Time
FIGURE 6.6
Plot of mean square displacement against time for a Brownian particle.
