Selected Geochemical Processes - Contaminant Geochemistry - page 36

against time yields a linear relation with a slope equal to -k that intersects the
origin.
In second-order reactions, the rate-determining step involves a transformation
where two reactants interact to give one product. The simplest case of such a
reaction is 2A ? B, and in such a case, we can write
d½A
dt ¼ k½A 2 ;

ð 2 : 20 Þ

½A 0
1 þ kt½A 0

½A t ¼ ; ð 2 : 21 Þ

where here k is the second-order rate constant with dimensions [mol -1 volume
time -1 ]. As a consequence, the decay period depends on the initial concentration
[A] 0 . This result has implications for environmental pollutants that decompose by
second-order reactions; in such cases, pollutants may persist for longer times at
low concentrations, compared to first-order reactions, because decay times become
longer as the concentration decreases.
When the reaction rate is not dependent on the reactant concentration, the
reaction is zero order:

d½A
dt ¼ k :

ð 2 : 22 Þ

In nature, the reaction rate depends on the reactant concentration. However,
practically speaking, when a reactant exists at a very high concentration, it is
essentially unchanged due to the reaction, and the reaction is called pseudo-zero
order.
In geochemistry, interest is focused on open systems, in which mass is added or
removed over observable time periods. A simple example of such a case is that of
steady fluid flow in a system, with constant inflow of species A. Then, for constant
recharge of species A at concentration [A] 0 (and discharge at concentration A), at
rate k*, with loss A in reaction with species B (recall Eq. 2.16 ), we have
d½A
dt ¼ k½A þ k ½A 0 k ½A

ð 2 : 23 Þ

so that

d½A
dt ¼ k½A þ k ð ½A 0 ½A Þ ¼ ð k þ k Þ ½A þ k ½A 0 :

ð 2 : 24 Þ

The solution of this first-order differential equation is

½A ¼ k þ ke ð k þ k Þ t
k þ k

½A 0 : ð 2 : 25 Þ