Geology Reference
In-Depth Information
function of the concentrations c
i
of the components involved. The form of the
function fc
ðÞ
is determined empirically. Usually, it is found possible to write it as
a product of the concentrations of the reactants only, each raised to a suitable
power (0, 1, 2…). In this case, the kinetics of the reactions are said to be of first
order if the corresponding power is unity, and so forth, the sum of the powers of
the terms involved being called the overall order. The concept of order is par-
ticularly used in connection with chemical reactions (Atkins
1986
, Chap. 28), but
it
can
also
be
usefully
applied
to
processes
such
as
recovery
and
phase
transformation.
If the reaction rate is found to depend on the concentration of only one reactant
and to be first order in it, that is, fc
ðÞ¼
c
;
where c is the concentration of that
component, and if the rate of reaction is specified by dc
=
dt
;
then we have the
particularly simple form
1
c
dc
dt
¼
k
ð
3
:
6
Þ
where the rate coefficient k is identical to the specific rate of the process and has
the dimensions s
-1
. In the more general form (
3.5
), the dimensions of k depend on
the form of fc
ðÞ:
The temperature and pressure dependence shown explicitly in
(
3.1
) and (
3.2
) are normally incorporated in the rate coefficient k as
k
¼
k
0
e
RT
ð
3
:
7
Þ
where the pre-exponential factor k
0
has the same dimensions as k
;
being again a
frequency in the simple case (
3.6
) or whenever fc
ðÞ
has the same dimensions as n
:
3.2.3 Statistical Approach to Thermal Activation
From a statistical thermodynamics point of view, the nature of a thermally acti-
vated reaction is often depicted as in Fig.
3.2
, showing a change from an
assemblage of reactants 1 to an assemblage of products 2 via what is assumed in
transition state theory to be a thermodynamically definable intermediate state
corresponding to the energy peak. It is assumed that temperature and pressure are
the independent variables, so that Gibbs energies are used to define the states. The
forward change involves an activation barrier G
m
and it is often useful to picture it,
in a simple-minded way, as being governed by a relation
n
þ
¼
f
þ
c
ðÞ
m
þ
e
G
RT
ð
3
:
8
Þ
there being at the same time a corresponding tendency for the reverse change to
occur at the rate
n
¼
f
c
ðÞ
m
e
G
m
þ
DG
m
ð
3
:
9
Þ
RT
