Physics

Problem

Isotope A decays to isotope B, which subsequently decays to stable isotope C. The half-life of A is

20\,\text{min}

, and the half-life of B is

5.0\,\text{min}

. At

t = 0

the sample consists of

N_{0} = 1.2 \times 10^{20}

atoms of A and no atoms of B or C. a. Part 1: (a) Derive an algebraic expression for the time

t_{\text{peak}}

at which the activity of isotope B attains its maximum value. Express your result solely in terms of the decay constants of A and B. b. Part 2: (b) Calculate the numerical value of

t_{\text{peak}}

using the given half-lives. c. Part 3: (c) Determine the activity of isotope B at

t_{\text{peak}}

and express it as a fraction of the initial activity of isotope A at

t = 0

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