Geoscience Reference
In-Depth Information
When k* k, that is, the recharge is very slow, we return to a simple first-
order reaction. When the recharge is very fast, compared to the reaction rate, that
is, when k* k, then [A] % [A] 0 and the concentration of A remains constant in
the system during the period of observation.
Many reactions use the Arrhenius parameter; for example, plotting ln k against
1/T gives a linear relationship that can be written in the form
ln k ¼ ln A Ea
RT ;
ð 2 : 26 Þ
where Ea is the activation energy, and ln A corresponds to the interception point
when 1/T = 0; A is called the pre-exponential factor or frequency factor. The
higher the Ea, the stronger the dependence of the rate constant on temperature. The
activation energy Ea is the minimum kinetic energy that reactants must have to
form a product. The pre-exponential factor is a measure of the rate at which
transformation of a reactant to products occurs, irrespective of their energy.
An excellent example combining thermodynamics, kinetics, and equilibrium
considerations was presented by O'Day ( 1999 ), who considered the precipitation
reaction of solid lead carbonate, in the form of the mineral cerussite (PbCO 3(s) )
according to the reaction
Pb 2 þ þ CO 2 3 PbCO 3 ð s Þ :
ð 2 : 27 Þ
The equilibrium constant is governed by a function of temperature and pressure
and can be expressed using the standard state equilibrium constant (K eq ) and the
change in standard free energy of the reaction:
a PbCO 3 ð s Þ
a Pb 2 þ a CO 3 2
K eq ¼
ð 2 : 28 Þ
DG r ¼ RT ln K eq ;
ð 2 : 29 Þ
where a is the activity of each species, and DG t denotes the change in free energy
of a reaction at standard state. By convention, the activities of chemically pure
solids are set equal to 1. The description of cerussite precipitation given by
Eq. ( 2.27 ) is an example of a thermodynamic equilibrium. However, when
examining the reaction system at the molecular level, before equilibrium is
achieved, the reaction is much more complicated. For example, aqueous species
take part in pH-dependent reactions that determine the form of carbon in solution
and thus affect the precipitation of cerussite:
CO 2 ð aq Þ þ H 2 O H 2 CO 3
ð 2 : 30 Þ
H 2 CO 3 HCO 3
ð 2 : 31 Þ
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