Introduction to Health Physics: Fourth Edition - Ruang Baca FMIPA UB

126 CHAPTER 4
the daughter. The time after isolation of the parent that the daughter reaches its
maximum activity, tmd, may be computed by differentiating Eq. (4.53), the daughter
activity, setting the derivative equal to zero, and then solving for the time at which
the daughter activity is at its maximum, tmd.
λB NB = λBλANA0
(e
λB − λA
−λAt −λBt
− e )
d(λB NB)
dt
= λBλANA0
(−λAe
λB − λA
−λAt
+ λBe −λBt
) = 0,
λAe −λAt = λBe −λB t
λB
λA
ln λB
λA
= e −λAt
e −λB t
= (λB − λA)t
t = tmd = ln(λB/λA)
λB − λA
= e (λB−λA)t
= 2.3 log(λB/λA)
. (4.57)
λB − λA
For the case illustrated in Figure 4-15,
and
λA = 0.693
10.6 = 0.065 h−1 ,
λB = 0.693
1.01 = 0.686 h−1 .
The time at which the daughter ThC will reach its maximum activity is
tmd =
ln (0.686/0.065)
0.686 − 0.065
= 3.8 hours.
The time when the total activity is at its peak may be determined in a similar
manner. In this case, the total activity A(t) must be maximized. The total activity at
any time is given by
A(t) = λANA + λB NB. (4.58)
Substituting Eqs. (4.43) and (4.52) for NA and NB, respectively, in Eq. (4.58), we
have
A(t) = λANA0 e −λAt
λBλANA0
+
λB − λA
Differentiating Eq. (4.59) yields
dA(t)
dt =−λ2 A NA0 e −λAt + λBλANA0
λB − λA
(e −λAt − e −λBt ). (4.59)
(−λAe −λAt + λBe −λB t ) = 0. (4.60)