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(D) The activity $R$ of a radioactive sample is given by $R = \lambda N$,where $\lambda$ is the decay constant and $N$ is the number of undecayed nucl

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$(R_1 - R_2)T$

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(D) The activity $R$ of a radioactive sample is given by $R = \lambda N$,where $\lambda$ is the decay constant and $N$ is the number of undecayed nuclei.Given the mean life $T = \frac{1}{\lambda}$,we have $\lambda = \frac{1}{T}$.At time $T_1$,the number of nuclei present is $N_1 = \frac{R_1}{\lambda} = R_1 T$.At time $T_2$,the number of nuclei present is $N_2 = \frac{R_2}{\lambda} = R_2 T$.The number of nuclei disintegrated between time $T_1$ and $T_2$ is the difference in the number of nuclei present at these times:$\Delta N = N_1 - N_2 = R_1 T - R_2 T = (R_1 - R_2)T$.

The radioactivity of a sample at time $T_1$ is $R_1$ and at time $T_2$ is $R_2$. If the mean life of the sample is $T$,then the number of nuclei disintegrated in the time interval $(T_2 - T_1)$ is:

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