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The half life...

Question

The half life of a radioactive sample is $T .$ If the activities of the sample at time $t_{1}$ and $t_{2}$ $(t_{1} < t_{2})$ are $R_{1}$ and $R_{2}$ respectively, then the number of atoms disintegrated in time $t_{2} - t_{1}$ is proportional to-

(R_{1} + R_{2}) T

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(R_{1} - R_{2}) T

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\frac{R_{1} + R_{2}}{T}

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\frac{R_{1} R_{2}}{R_{1} + R_{2}} T

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Solution

The correct option is B $(R_{1} - R_{2}) T$
As we know, the activity of a sample is,

$R = λ N$

$\Rightarrow R_{1} = λ N_{1}; R_{2} = λ N_{2}$

$\Rightarrow R_{1} - R_{2} = λ [N_{1} - N_{2}]$

$\Rightarrow R_{1} - R_{2} = \frac{0.693}{T} [N_{1} - N_{2}]$

$\Rightarrow N_{1} - N_{2} = \frac{1}{0.693} (R_{1} - R_{2}) T$

$\Rightarrow (N_{1} - N_{2}) \propto (R_{1} - R_{2}) T$

 Hence, $(B)$ is the correct answer.

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