Two radioactive isotopes, A and B

PHYSICS · Multi-part · A Level · Physics · Nuclear physics · Nuclear physics

Problem

Two radioactive isotopes, A and B, decay independently according to the laws

A(t)=500e^{k_At}

and

B(t)=800e^{k_Bt}

, where

t

is measured in days and masses are in milligrams. The half-life of isotope A is

14\,\text{days}

, and the half-life of isotope B is

21\,\text{days}

. (a) Determine the exact values of

k_A

and

k_B

. (b) Derive an equation whose positive solution gives the time

t_1

at which the masses of the two isotopes are equal. (c) Solve the equation from part (b) to find

t_1

correct to the nearest tenth of a day. (d) Evaluate

\displaystyle\lim_{t\to\infty}\dfrac{A(t)}{B(t)}

and interpret your result in the context of the two isotopes.