(3.14)
By integration, Eq. (3.14) becomes
(3.15)
Equation (3.15) gives the activity of the daughter nuclide dat time tas
a result of growth from the parent nuclide pand also due to the decay of
the daughter itself.
Transient Equilibrium
If ld>lp, that is, (t1/2)d<(t1/2)p, then e−ldtin Eq. (3.15) is negligible compared
to e−lptwhen tis sufficiently long. Then Eq. (3.15) becomes
(3.16)
(3.17)
This relationship is called the transient equilibrium. This equilibrium
holds good when (t1/2)pand (t1/2)ddiffer by a factor of about 10 to 50. The
semilogarithmic plot of this equilibrium equation is shown in Figure 3.4.
The daughter nuclide initially builds up as a result of the decay of the parent
nuclide, reaches a maximum, and then achieves the transient equilibrium
decaying with an apparent half-life of the parent nuclide. At equilibrium,
the ratio of the daughter to parent activity is constant. It can be seen from
Eq. (3.17) that the daughter activity is always greater than the parent activ-
ity, because (t1/2)p/([t1/2]p−[t1/2]d) is always greater than 1. The time to reach
maximum daughter activity is given by the formula:
(3.18)
A typical example of transient equilibrium is^99 Mo (t1/2=66 hr) decaying
to 99mTc (t1/2 = 6 hr). Because 87% of^99 Mo decays to 99mTc, and the
remaining 13% to the ground state, Eqs. (3.15), (3.16), and (3.17) must be
multiplied by a factor of 0.87. Therefore, in the time–activity plot, the 99mTc
daughter activity will be lower than the^99 Mo parent activity (Fig. 3.5).
Also, the 99mTc activity reaches maximum in about 23 hr, i.e., 4(t1/2)d
[Eq. (3.18)].
ttt tt
ttpd pd
pdmax.ln
=×()×()× []()()
[]()−()
(^14412121212)
12 12
=
()( )
()−()
tA
ttp p t
pd12
12 12=
()
−
l
lldpt
dpA
A
A
d t edp
dp()= ()pt
−l −
ll0 lAN
A
dt dd dpee
dp()==() pt dt
−l ()− − −l
ll0 l ldN
dtd=−llppNNd d30 3. Kinetics of Radioactive Decay