Question

A and b are two radioactive substances whose half life are 2 and 4 years respectively. Initially 10g of A and5g of B are taken . The time after which they will have same quantity?

Answer 1

half life of radioactive substance is given by, $T_{1/2}=\frac{ln2}{\lambda}$

where, $\lambda$ is radioactive decay constant.

for radioactive substance A ; half life = 2 years

$\lambda_A=\frac{ln2}{2}$

for radioactive substance B; half life = 4 years

$\lambda_B=\frac{ln2}{4}$

Let after t years, both radioactive substances will have same quantity.

for A ; initial amount of substance = 10g

so, $N_A=10e^{-\lambda_A t}$

for B ; initial amount of substance = 5g

so, $N_B=5e^{-\lambda_B t}$

a/c to question,

$N_A=N_B$

or, $10e^{-\lambda_A t}=5e^{-\lambda_B t}$

or, $2e^{-\lambda_A t}=e^{-\lambda_B t}$

or, $e^{(\lambda_A-\lambda_B)t}=2$

or, $(\lambda_A-\lambda_B)t = ln2$

or, $ln2[\frac{1}{2}-\frac{1}{4}]t=ln2$

or, t = 4 years

hence, after 4 years they will have same quantity.