Two radioactive nuclei X and Y initially contain equal number of atoms. The half life is 1 hour and 2 hours respectively. Calculate the ratio of their rates of disintegration after two hours.

Solution

Given, nucleii X and Y contain equal number of atoms.
Half-life of X , T₁ = 1 hr
Half-life of Y , T₂ = 2 hr

Ratio of radioactive sample of X left after 2 hours,

N_{1} = {(\frac{1}{2})}^{2 / 1} N_{0} = \frac{1}{4} N_{0}

Ratio of radioactive sample of Y left after 2 hours,

N_{2} = {(\frac{1}{2})}^{2 / 2} N_{0} = \frac{1}{2} N_{0}

Now, deacay rate is given by,

R = λN = \frac{0.693 N}{T_{1 / 2}}

\Rightarrow

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

i.e.,

\frac{R_{1}}{R_{2}} = \frac{(T_{1 / 2})_{2}}{(T_{1 / 2})_{1}} \times \frac{N_{1}}{N_{2}} = \frac{2}{1} \times \frac{\frac{N_{0}}{4}}{\frac{N_{0}}{2}} = 1

Hence, the activity of the two samples will be equal after 2 hours.

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Wave Optics

Two radioactive nuclei X and Y initially contain equal number of atoms. The half life is 1 hour and 2 hours respectively. Calculate the ratio of their rates of disintegration after two hours.

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