Question

Two radioactive sources A and B initially contain equal number of radioactive atoms. Source A has a half-life of 1 hour and source B has a half-life of 2 hours. At the end of 2 hours, the ratio of the rate of disintegration of A to that of B is :

Answer 1

Two radioactive nuclei X and Y initially contain equal number of atoms. The half life is 1 hour and 2 hours respectively.

Calculate the radio of their rates of disintegration after two hours.

Given, nucleii X and Y contain equal number of atoms.

Half-life of X, $T_1=1hr$

Half-life of Y, $T_2=2hr$

Radio of radioactive sample of X left after 2 hours,

$N_1=(\frac{1}{2}{)}^{2/1}{N}_0=\frac{1}{4}{N}_0$

Radio of radioactive sample of Y left after 2 hours,

$N_2=(\frac{1}{2}{)}^{2/2}{N}_0=\frac{1}{2}{N}_0$

Now, decay rate is given by,

$R=\lambda N=\frac{0.693N}{T_{1/2}}$

$\Rightarrow R\propto \frac{N}{T_{1/2}}$

$\frac{R_1}{R_2}=\frac{(T_{1/2}{)}_2}{(T_{1/2}{)}_1}\times \frac{N_1}{N_2}$

i.e., $=\frac{2}{1}\times \frac{\frac{N_0}{4}}{\frac{N_0}{2}}$

$=1$