Chemistry Reference
In-Depth Information
If the standard free energy is the same in the initial and final states and the
energy barrier is symmetrical, the free energy of activation will be the same in
the forward and backward directions. The specific rate constant k is, therefore,
the same for the flow in either direction. By definition, the concentrations of the
solute molecules in the initial and final states of diffusion are c and c
d
dx
,
respectively. The number of molecules moving in the forward direction (i.e., from
left to right) through a unit cross-sectional area is given by
1λ
k
molecules
m
2
s
v
forward
5
λ
N
a
c
(6-58)
where N
a
is the Avogadro number and k is the specific reaction rate for diffusion
(i.e., the number of times a molecule moves from one position to the next per sec-
ond). Similarly, the number of molecules moving in the backward direction (i.e.,
from right to left) is given by
dc
dx
k
molecules
m
2
s
v
backward
5
1λ
λ
N
a
c
(6-59)
Owing to the concentration gradient, the resultant flow is from left to right
and its quantity is given by
2
k
dc
dx
molecules
m
2
s
v
5
v
forward
2
v
backward
52
N
a
λ
(6-60)
Equation (6-1)
can be written in the following form:
DN
a
dc
dx
molecules
m
2
s
v
52
(6-61)
Equating
Eqs. (6-60) and (6-61)
yields
2
k
dc
DN
a
dc
dx
2
N
a
λ
dx
52
(6-62)
or
2
k
D
5λ
(6-63)
Replacing k in
Eq. (6-63)
by
Eq. (6-57)
yields an equation that relates the
self-diffusion coefficient and viscosity of the liquid. Obviously, by doing so, it is
assumed that all
λ
i
values in the viscous flow and those in the diffusion process
are comparable.
5
λ
1
k
B
T
λ
2
λ
3
η
D
(6-64)
Here, it is interesting to compare
Eq. (6-64)
with the Stokes
Einstein equation
[i.e.,
Eq. (6-42)
]. As mentioned, the Stokes
Einstein equation relates the diffu-
sion of Brownian particles with the viscosity of solvent that is made up of mole-
cules that are much smaller than the Brownian particles. As a result, the liquid
