Physics

Problem

A radioactive nuclide

A

decays to the daughter nuclide

B

which, in turn, decays to a stable product

C

. At

t=0

only

N_{A0}

atoms of

A

are present. (i) Write the differential equations governing

N_A(t)

and

N_B(t)

and hence derive an expression for

N_B(t)

in terms of

N_{A0}

\lambda_A

and

\lambda_B

. (ii) Obtain the condition for the activity of

B

to attain its maximum value and show that the corresponding time is

t_{\text{max}}=\frac{1}{\lambda_B-\lambda_A}\ln\!\left(\frac{\lambda_B}{\lambda_A}\right).

(iii) The half-life of

A

10\,\text{h}

and that of

B

2\,\text{h}

. If

N_{A0}=2\times10^{20}

, calculate: (a) the time (in hours, correct to two significant figures) at which the activity of

B

is maximum, (b) the maximum activity of

B

(in Bq).