Chemistry Reference
In-Depth Information
0
@
1
A
exp
2 ðω
1
3
1
:
158
1 ð
1
2ω
1
Þ 3
0
:
43
3
1
:
005
Þ
0:621ω
1
10:0648
0
@
1
A
29950
1:9863323:15
D
1
5
73
:
2 exp
0
@
1
A
ðcm
2
51:848310
25
exp
2
0
:
726
ω
1
2
0
:
432
0:621ω
1
10:0648
=sÞ
Figure 6.12
shows the resulting concentration dependence of D
1
over the concentration
range of 0
2 wt% of n-pentane.
PROBLEMS
6-1
(a) Using Sterling's approximation show that
in the case of a large
number of steps, N,
the probability in
Eq. (6-17)
will
reduce to
1
N
n
2
2N
e
2
P
ð
n
Þ 5
π
(b) Normalize the probability found in part (a) and show that the result is a
Gaussian distribution given in
Eq. (6-19)
.
6-2
Using the normal distribution in
Eq. (6-19)
and time dependency of random
walker [
Eq. (6-20)
], show that the mean square displacement of the random
walker after N steps is given by
Eq. (6-21)
.
6-3
Show that
the solution of
Eq.
(6-22)
shown in Example 6-2 yields
Eq. (6-23)
.
6-4
(a) Solve the differential equation in
Eq. (6-37)
and show that the final
result is
h
i
kT
γ
2
m
t
h
x
ð
t
Þ
x
ð
t
Þi 5
1
2
exp
(b) And then use the fact that
1
2
d
dt
h
x
2
h
x
ð
t
Þ _
x
ð
t
Þi 5
ð
t
Þi
to show that
Eq. (6-38)
is obtained.
6-5
Derive an analytical model for the time-dependent position of a harmonic
oscillator (
Fig. 6.8
) in a solvent. Using this result, calculate the mean square
displacement of the oscillator.
