9.20 Diffusion
379
of particles into the layer:
(J x (x) − J x (x + dx)) A =
∂C
∂t
Adx .
(9.146)
The left-hand side divided by −Adx defines the derivative of the flux. Now we apply
the relation to the diffusion flux, Eq. (9.144), to find
D
∂ 2 C
∂x 2 =
∂C
∂t
.
(9.147)
This is the diffusion equation. In three dimensions, it reads
2 C =
1
D
∂C
∂t
.
(9.148)
Let us consider a particular solution C of the diffusion equation. First, the
diffusion equation in three dimensions is explicitly:
1
D
∂C
∂t
=
∂ 2 C
∂x 2 +
∂ 2 C
∂y 2 +
∂ 2 C
∂z 2 .
(9.149)
A solution to this equation for the particle concentration C initially localized near
the origin of a large volume is given by the Gaussian
C = N
1
4πD t
3/2
exp
−
x
2
+ y
2
+ z
2
/ (4D t)
.
(9.150)
The spread of these molecules after a time t, can be measured by the root-meansquare width of the concentration peak, r RMS ≡
r 2
, which is given by
√
6D t ,
so that the time of spread to a given r RMS is
t =
r 2
RMS
6D
.
(9.151)
More generally, if diffusion occurs in a space of dimension d, such as within a
cellular microtubule (d = 1) or on a organelle membrane (d = 2), then
t =
r 2
RMS
2d D
.
(9.152)