A three-step decay chain is stable

PHYSICS · Subjective · AoPS · Mathematics · Exponents and Logarithms · Interest-ing Problems

Problem

A three-step decay chain is

A\rightarrow B\rightarrow C\rightarrow

stable

D

. The half-lives are

T_{1/2}(A)=3\,\text{h}

T_{1/2}(B)=30\,\text{min}

and

T_{1/2}(C)=5\,\text{min}

. At the instant

t=0

there are

1.0\times10^{18}

atoms of

A

and none of

B

C

. (a) Calculate the number of atoms of

A

B

and

C

present after

4\,\text{h}

. (b) Hence find the activity of

C

t=4\,\text{h}

. (c) Determine the ratio of the activities of

C

B

at this time and comment on what this indicates about the stage reached in the decay sequence.