Environmental Engineering Reference
In-Depth Information
particle. The time dependence of this concentration, or density, of the
tagged particle follows from the mass balance:
2
∗
∗
d
ρ
d
ρ
s
=
D
,
2
dt
dz
and the initial condition is given by:
(
)
( )
ρ
zt
,
=
0
= δ
z
,
where the delta function ensures that there is one particle at
z
=
0. This
is a well-known differential equation which has as its solution:
(
)
1
()
2
s
ρ
zt
,
=
exp
−
z
/ 4
D t
s
4
π
Dt
This result looks a bit strange, as we have only one particle: how can the
density be less than one? The way to interpret this result is to suppose
that we repeat the experiment many times and at each time
t
the particle
will be in a different position. The expression for the density gives the
probability of fi nding a particle at time
t
at position
z
. With this equation
we can compute the average position of a particle at time
t
:
(
)
1
∞
∞
()
()
2
s
∫
∗
∫
zt
=
z
ρ
ztdz
,
=
z
exp
−
z
/ 4
Dt dz
=
0,
−∞
−∞
s
4
π
Dt
which is intuitively clear as there is no gradient in the concentration and
hence the particle has an equal probability of going right or left. Hence,
the average will be zero. The second moment, or mean squared dis-
placement, does not disappear:
(
)
1
∞
∞
()
()
2
2
2
2
s
s
∫
∗
∫
zt
=
z
ρ
ztdz
,
=
z
exp
−
z
/ 4
Dtdz
=
2
Dt
−∞
−∞
s
4
π
Dt
This equation shows that by measuring the mean squared displacement
of a labelled particle one can get the diffusion coeffi cient. Because the
integral in the above equation extends from -
∞
to +
∞
, the simple propor-
tionality of mean squared displacement to the diffusion coeffi cient
assumes many displacements, i.e., over long time scales:
1
(
)
2
()
( )
s
D
=
lim
z t
−
z
0
2
t
t
→∞
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