Geology Reference
In-Depth Information
Box 3.2 Kinetics of radioactive decay I: the Rb-Sr system
the radioactive isotope of rubidium,
87
rb (Box 10.1),
decays to the strontium isotope
87
Sr. the kinetics of this
nuclear reaction can be treated in the same way as a
chemical reaction:
where
n
0
is the number of parent nuclei initially present
(when
t
= 0). the decay profile for a short-lived isotope is
shown in Figure 3.2.1a. If we transform both sides of
equation 3.2.3 into natural logarithms and rearrange
things a bit, we get:
87
87
β
Rb
→++
Sr
v
(3.2.1)
0
n
n
'
parent
isotope
'
'
daughter
isotope
'
p
ln
=λ
t
(3.2.4)
p
where the
β
-particle (
β
-
) and the antineutrino (
v
) are
released by the reaction.
the decay rate of any radioisotope is proportional to the
number of the radioisotope ('parent') nuclei present in the
sample (
n
p
) at the moment in question. this can be written
as a rate equation:
this equation is more useful because it is linear in relation
to time (Figure 3.2.1b). For radioisotopes that decay rap-
idly, the decay constant can be determined by measuring
the gradient of this graph (see exercise 3.1).
the half-life
t
2
of a radioisotope is the time it takes for
n
p
to decay to half of its original value (
nn
p
=
2
0
). thus
p
d
d
n
t
p
decay rate
=−
=
λ
n
(3.2.2)
ln(
21
2
/
=λ
t
)
p
λ
(3.2.5)
therefore
=
.
0 6931
/
t
2
Because there is only
one
concentration term (
n
p
) on the
right-hand side (unlike equation 3.3), this is called a
first-
order
reaction.
λ
is the rate constant analogous to
k
in
equation 3.3, but in this context it is called the
decay con-
stant
(same algebra, different names).
the rate equation can be integrated to show how
n
p
varies with time:
Because the decay of
87
rb is very slow, the numerical
value of
λ
is extremely small: 1.42 × 10
−11
year
−1
. During
one year only about 14 out of every million million
87
rb
nuclei are likely to decay. the
87
rb remaining in the earth
today has survived from element-forming processes
(Chapter 11) that took place before the solar system was
formed 4.6 billion years ago.
nn
t
p
0
=
e
−
λ
(3.2.3)
p
(a)
(b)
n
0
n
p
n
p
In
n
0
2.0
Readings of
n
p
taken at
different times
0.5
n
0
1. 0
Straight line graph
indicates first-order
kinetics
t
1/2
0
50
50
t
/ hours
Figure 3.2.1
(a) the falling number
n
p
of radionuclei in a sample plotted against time. (b) the function ln[
n
0
/
n
p
] plots as
a straight line against time; the gradient equals
λ
.
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