Question 10.6.Q2: In Prob. 220 we determined general expressions for the numbe......

In Prob. 220 the Bateman equation and Bateman constants were consolidated into one general equation which was used to determine general expressions for the number of nuclei and activity for the first four generations of a radioactive decay series: P → D → G → GG. The general Bateman equation is written as follows

N_n(t)=\sum_{m=1}^n C_m e^{-\lambda_m t}=N_1(0) \sum_{m=1}^n\left[\frac{\prod_{i=1}^{n-1} \lambda_i}{\prod_{\substack{i=1 \\ i \neq m}}^n\left(\lambda_i-\lambda_m\right)}\right] e^{-\lambda_m t} .         (10.156)

(1) The general expressions for the number of nuclei N_1(t),\ N_2(t),\ N_3(t),\ and\ N_4(t) were given as follows with N_1(0) the initial number of parent nuclei and λ_i the decay constant of generation i.

N_1(t)=N_1(0) e^{-\lambda_1 t},          (10.157)

N_2(t)=N_1(0) \frac{\lambda_1}{\lambda_2-\lambda_1}\left[e^{-\lambda_1 t}-e^{-\lambda_2 t}\right]          (10.158)

(2) The general expressions for activities \mathcal{A}_1(t),\ \mathcal{A}_2(t),\ \mathcal{A}_3(t),\ and\ \mathcal{A}_4(t) were given as follows with \mathcal{A}_1(0), the initial activity of the parent radionuclide

\mathcal{A}_1(t)=\lambda_1 N_1(t)=\mathcal{A}(0) e^{-\lambda_1 t}          (10.161)

\mathcal{A}_2(t)=\lambda_2 N_2(t)=\mathcal{A}_1(0) \frac{\lambda_2}{\lambda_2-\lambda_1}\left[e^{-\lambda_1 t}-e^{-\lambda_2 t}\right],         (10.162)

(3) Decay constant λ_i is related to half-life (t_{1/2})_i through the standard expression

\lambda_i=\frac{\ln 2}{\left(t_{1 / 2}\right)_i}             (10.165)

(4) Initial number of parent nuclei N_1(0) is related to initial parent activity \mathcal{A}_1(0) as follows

(5) Table 10.5 provides a summary of relevant specific data for the first four generations of the radioactive decay series analyzed in this problem.

(a) Expressions for the number of parent N_1(t),\ daughter\ N_2(t), granddaughter N_3(t), and great granddaughter N_4(t) nuclei as a function of time t is obtained by inserting (10.165) and appropriate decay constants into (10.157), (10.158), (10.159), and (10.160), respectively

(1)

N_1(t)=N_1(0) e^{-\lambda_1 t}=1.73 \times 10^{13} \times e^{-\left(0.462 \mathrm{~d}^{-1}\right) t},          (10.167)

(2)

(3)

(4)

(b) Expressions for the activity of the parent \mathcal{A}_1(t), daughter \mathcal{A}_2(t), granddaughter \mathcal{A}_3(t), and great granddaughter \mathcal{A}_4(t) as a function of time t is obtained by inserting \mathcal{A}_1(0) = 2.5 mCi and appropriate decay constants into (10.161), (10.162), (10.163), and (10.164), respectively

(1)

\mathcal{A}_1(t)=\lambda_1 N_1(t)=\mathcal{A}_1(0) e^{-\lambda_1 t}=\left[2.5 \times e^{-\left(0.462 \mathrm{~d}^{-1}\right) \times t}\right] \mathrm{mCi},          (10.171)

(2)

A_2 (t) = λ_2 N_2 (t) A_1 (0) \frac{λ_2}{λ_2  –  λ_1}[e^{-λ_1 t }  –  e^{-λ_2 t}] \\ \quad = 2.88 \times [e^{-(0.462  d^{-1}) \times t}  –  e^{-(3.466  d^{-1}) \times t}] mCi,     (10.172)

(3)

(4)

(c) Numbers of nuclei N_1(t),\ N_2(t),\ N_3(t),\ and\ N_4(t) present at time t = 2.5 d are calculated inserting t = 2.5 d into (10.167), (10.168), (10.169), and (10.170), respectively

(1)

N_1(t)=1.73 \times 10^{13} \times e^{-0.462 \times 2.5}=5.45 \times 10^{12}        (10.175)

(2)

N_2(t)=2.66 \times 10^{12} \times\left[e^{-0.462 \times 2.5}-e^{-3.466 \times 2.5}\right]=0.838 \times 10^{12} \text {, }            (10.176)

(3)

(4)

(d) Activities \mathcal{A}_1(t),\ \mathcal{A}_2(t),\ \mathcal{A}_3(t),\ and\ \mathcal{A}_4(t) present at time t = 2.5 d are calculated by multiplication of the number of nuclei N_1(t),\ N_2(t),\ N_3(t),\ and\ N_4(t), respectively, with appropriate decay constant λ

(1)

(2)

(3)

(4)

Results of (c) and (d) are summarized in Table 10.6.

(e) Data calculated in (c) for the numbers of nuclei N_1(t),\ N_2(t),\ N_3(t),\ and\ N_4(t) present at time t = 2.5 d are superimposed onto curves plotting number of nuclei N(t) against time t, as shown in Fig. 10.10.

(f) Data calculated in (d) for activities \mathcal{A}_1(t),\ \mathcal{A}_2(t),\ \mathcal{A}_3(t),\ and\ \mathcal{A}_4(t) present at time t = 2.5 d are superimposed onto curves plotting activities \mathcal{A}_1(t),\ \mathcal{A}_2(t),\ \mathcal{A}_3(t),\ and\ \mathcal{A}_4(t) against time t, as shown in Fig. 10.11.