Geochemical systems - Geochemistry: An Introduction - page 123

Let us now consider a radioactive element with a decay constant λ i . The radioactive

λ i MC i , can be subtracted from (6.1) , modifying (6.2) as:
τ H

decay

d C i
d t

C in

C i + = (6.13)

i i

1 + β + λ i τ H 1 + β + λ i τ H

This equation applies, in particular, when determining the mixing time of the ocean,
which is approximated by the residence time of carbon in the deep ocean. Because the deep
ocean is isolated from the atmosphere, the radioactive carbon-14 it acquired during its time
at the surface is no longer renewed and so decays within a closed system. Oceanographic
measurements indicate that the 14 C/ 12 C ratio of deep water makes up only 84% of the
same ratio in the surface water. Suppose that the ocean composition has reached steady
state (d C i 0). Let us arbitrarily divide the ocean into two reservoirs, one of deep

water and one of surface water, which exchange water by vertical mixing. The inflow into
the deep reservoir is therefore the downwelling of surface water. Dividing (6.13) for 14 C
by (6.2) for the stable 12 C, we get:
14 C
12 C

/ d t =

deep =

14 C
12 C

C

1 + β

(6.14)

C

1 + β + λ 14C τ H

surface

C , which is identical for both carbon isotopes, is small
enough compared with unity to be ignored. The reader will check that by introducing the
relative values of isotope ratios given above and the value of

The term for chemical reactivity β

λ 14C from Table 3.1 , the

renewal time for deep seawater τ H represents

the time between two successive transits of sea water through the deep reservoir, this value
is often referred to as the mixing time of the ocean.

τ H can be calculated at 1600 years. Because

6.2 Interaction of multiple reservoirs and geochemical cycles

The dynamics of a multiple-reservoir system differ from those of a single reservoir in
different respects. Collective readjustment is invariably faster than it would be if the reser-
voirs were considered separately. We will see that ignoring this principle would lead to
very serious errors.
Let us consider the case of a change in the number of atoms of an element in two reser-
voirs that exchange matter with one another, for example the set of the “surface water”
and “deep water” oceanic reservoirs exchanging material by vertical advection, or of the
mantle-continental crust system. Let us call p 1 → 2 the probability for an atom to be trans-
ferred from reservoir 1 to reservoir 2 in the unit time, with a similar definition for p 2 → 1 .If
n 1 and n 2 stand for the number of moles of the element in question in each reservoir, we
get the two equations:

d n 1
d t =−

p 1 → 2 n 1 + p 2 → 1 n 2 (6.15)

d n 2
d t =

p 1 → 2 n 1 − p 2 → 1 n 2 (6.16)