Geology Reference
In-Depth Information
compared to RT
:
The value of k may, however, be found to depend on the actual
value of the potential / if measurements are made over a wide range of /.
Where the transport of discrete entities is envisaged, the flux density can be
expressed as j
¼
c
N
v where v is the velocity of the entities and c
N
their concentration
(number per unit volume) at the reference cross-section. We can now define the
mobility M of the entities as their velocity when unit force is acting on them, so that
v
¼
MX
ð
3
:
15
Þ
Then we have k
¼
c
N
M and (
3.14
) becomes
j
¼
c
N
MX
ð
3
:
16
Þ
In the case of a diffusing substance, the thermodynamically defined mobility is
the velocity of transport in unit gradient of chemical potential; this mobility can be
related to the diffusion coefficient D in the empirical treatment of diffusion by the
Einstein formula
M
¼
D
RT
ð
3
:
17
Þ
in the case of ideal solutions, as will be shown in
Sect. 3.5
, where non-ideality is
also discussed. Eq. (
3.17
) thus relates a thermodynamically defined quantity M to
an empirically defined quantity D. Formally, one could also take dc
N
=
dx as a
''force'' and obtain an empirically defined ''mobility'' that would be equal to
D
=
c
N
; however, in practice the concept of mobility is normally only used in
relation to thermodynamically defined forces.
In the practical analysis of transport processes, a distinction has to be made
between transient or evolving situations and steady states. The relationship (
3.14
),
which in terms of the potential / can be written as
j
¼
k
d/
dx
;
ð
3
:
18
Þ
serves to fully describe the steady state, in which the flux density and potential
gradient at any given point are unchanged with time. However, steady-state
measurements do not reveal all the properties relevant to transport in more general
situations; in particular, they give no information about the ''capacity'' of the
medium for the transported property, as defined by dZ
=
d/. Thus, in a transient
situation the amount per unit volume, qz
;
of the property Z at any given point is a
function of time involving this generalized capacity (z is the amount of Z per unit
mass and q the density). Consideration of the fluxes in and out of an elementary
volume leads to the continuity or conservation equation
o
q
ðÞ
ot
¼
o
j
ð
3
:
19
Þ
ox
