Geology Reference
In-Depth Information
An analogy is with a bath; the inflow from a tap (S) is equal to the outflow
(R) when the bath is full. An increase in S is balanced by an increase in R.
If the total amount of substance A in the reservoir (analogy
¼
mass of
water in the bath) is A, then the lifetime, t is defined by
A
ð
kg
Þ
S
ð
kg s
1
Þ
t
¼
ð
7
:
1
Þ
In practical terms, the lifetime is equal to the time taken for the
concentration to fall to 1/e (where e is the base of natural logarithms)
of its initial concentration, if the source is turned off. If the removal
mechanism is a chemical reaction, its rate may be described as follows
R
0
¼
d
½
A
dt
¼
k
½
A
ð
7
:
2
Þ
(In this case d[A]/dt describes the rate of loss of A if the source is
switched off; obviously with the source on, at equilibrium d[A]/dt
¼
0.)
The latter part of Equation (7.2) assumes first order decay kinetics, i.e.
the rate of decay is equal to the concentration of A, termed [A],
multiplied by a rate constant, k. As discussed later this is often a
reasonable approximation.
Taking Equation (7.1) and dividing both numerator and denominator
by the volume of the reservoir, allows it to be rewritten in terms of
concentration. Thus
½
A
ð
kg m
3
Þ
S
0
ð
kg m
3
s
1
Þ
t
¼
ð
7
:
3
Þ
since S
0
¼
R
0
t
¼
½
A
k
½
A
¼
k
1
ð
7
:
4
Þ
Thus the lifetime of a constituent with a first order removal process is
equal to the inverse of the first order rate constant for its removal.
Taking an example from atmospheric chemistry, the major removal
mechanism for many trace gases is reaction with hydroxyl radical, OH.
Considering two substances with very different rate constants
5
for this
reaction, methane and nitrogen dioxide
CH
4
þ
OH
-
CH
3
þ
H
2
O
(7.5)
d
dt
½
CH
4
¼
k
2
½
CH
4
½
OH
k
2
¼
6
:
2
10
15
cm
3
molec s
1
ð
7
:
6
Þ
