Environmental Engineering Reference
In-Depth Information
force
F
acting on the particle from an external field is the particle mobility. Thus,
the definition of the mobility
b
of a particle is
w
D
b
F
.
(4.35)
We now assume that the test particles in a gas are in thermodynamic equilibri-
um with the gas subjected to the external field. According to the Boltzmann for-
mula (1.43), the distribution for the number density of the test particles is
N
D
N
0
exp(
U
/
T
), where
U
is the potential due to the external field and
T
is the tem-
perature of the gas. The diffusive flux of the test particles according to (4.26) is
j
D
U
is the force acting on the test particle.
Because thermodynamic equilibrium exists, the diffusive flux is compensated for
by the hydrodynamic particle flux,
j
D
w
N
D
N
D
D
F
N
/
T
,where
F
D
r
r
D
b
F
N
. Equating these fluxes, we find
that the kinetic coefficients are related as
D
T
b
D
.
(4.36)
This expression is known as the Einstein relation in accordance with Einstein's
study of Brownian motion [13, 15, 16]. In reality, this relation was obtained earlier
by Nernst [17] and Townsend and Bailey [18, 19], but Einstein used this relation
for the analysis of Brownian motion (see [20, 21]). The Einstein relation is valid
for small fields that do not disturb the thermodynamic equilibrium between the
test and gas particles. On the basis of the Einstein relation (4.36) and the estima-
tion (4.29) for the diffusion coefficient of a test particle in a gas, one can estimate
the particle mobility as
1
σ
p
mT
.
Note that the mobility of a charged particle is introduced as the proportionality
coefficient for the relationship between the drift velocity of a test charged particle
w
and the electric field strength
E
as
b
N
w
D
K
E
.
(4.37)
Correspondingly, the Einstein relation (4.36) between the mobility
K
and the diffu-
sion coefficient
D
for a charged particle has the form
eD
T
K
D
.
(4.38)
4.2.4
Heat Transport
Heat transport can be treated in a manner analogous to that employed for particle
transport. The heat flux is defined as
Z
2
m
v
q
D
v
fd
v
,
(4.39)
2
