Geology Reference
In-Depth Information
where J
a
are the rates of change of local extensive parameters and X
a
are conjugate
intensive parameters, the summation being over all the component processes
contributing to the entropy production. There can be some ambiguity in the fac-
toring of the terms of (
2.7
) into the J
a
and X
a
but no serious confusion arises if
consistent rules are followed (Miller
1974
).
Where heat flow, diffusion, and chemical reactions are to be taken into account,
the dissipation function can be expressed in the following terms (Katchalsky and
þ
dn
Tr
¼
J
Q
:
T grad
1
T
þ
J
i
:
T grad
l
i
dt
A
ð
2
:
8a
Þ
T
or
Þ þ
dn
Tr
¼
J
S
:
grad
ðÞþ
J
i
:
grad
l
i
ð
dt
A
ð
2
:
8b
Þ
where J
Q
, J
S
and J
i
are vectors representing the currents or rates of flow of heat,
entropy, and substance i, respectively, dn
=
dt is a scalar representing the rate of
advancement of the chemical reaction (n is the extent of chemical reaction), and
A
¼
P
m
i
l
i
is the chemical affinity driving the reaction (m
i
are the stoichiometric
coefficients for the reaction). In cases involving viscous flow, electric and mag-
netic effects, or other dissipative processes, further terms can be added. The factors
J
Q
, J
S
, J
i
and dn
=
dt in (
2.8a
) can be clearly identified as the extensive parameters
J
a
in (
2.7
) and their multiplying factors are then the respective conjugate intensive
parameters X
a
. The seeming ambiguity regarding the intensive parameter conju-
gate to J
i
arises from the different ways in which the total dissipation is partitioned
between the different terms in (
2.8a
) and (
2.8b
). The form (
2.8a
) is appropriate
where flow of heat and diffusion of substance are being individually and simul-
taneously measured in the presence of a temperature gradient, in which case it is
evident that the intensive parameter driving the diffusion is to be taken to be
T grad l
i
=ð Þ
. The form (
2.8b
) is appropriate when the entropy changes associated
with the movement in the temperature gradient of the measurable heat and of the
substance (through its heat content) are brought together in the factor J
S
; although
the latter quantity is not directly measurable, the form (
2.8b
) is useful in indicating
that, in the absence of a temperature gradient, the intensive parameter driving the
isothermal diffusion can be taken to be grad(-l
i
).
In the entropy representation, the interest centers on r
;
the rate of entropy
production itself, instead of on Tr
:
Again r is written as a sum of terms that are the
products of extensive and intensive parameters and, somewhat confusingly, the
same symbols are often used as were used in (
2.7
) for Tr; further, r or r
=
2 is often
taken as a dissipation function or potential. In both representations, the J
a
factors
are commonly termed ''fluxes'' or ''flows'', although these terms are scarcely
appropriate for quantities such as dn
=
dt, and the X
a
are variously termed the
''thermodynamic forces'' or ''affinities'', regardless of whether being in energy or
entropy representation and in spite of the 1
=
T factor subsumed in the X
a
in the
