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The decay constant of a radio isotope is $\lambda$. If $A_1$ and $A_2$ are its activities at times $t_1$ and $t_2$ respectively, the number of nuclei which have decayed during the time $(t_1 - t_2)$ is
A
$A_1t_1 - A_2t_2$
B
$A_1 - A_2$
C
$(A_1 - A_2)/\lambda$
D
$\lambda(A_1 - A_2)$
Detailed Solution
Activity of a sample: $A = \lambda N$, where N is the number of undecayed nuclei present.
At time $t_1$: $A_1 = \lambda N_1$, so $N_1 = \frac{A_1}{\lambda}$
At time $t_2$: $A_2 = \lambda N_2$, so $N_2 = \frac{A_2}{\lambda}$
Number of nuclei decayed between the two instants = $N_1 - N_2$
$N_1 - N_2 = \frac{A_1 - A_2}{\lambda}$
At time $t_1$: $A_1 = \lambda N_1$, so $N_1 = \frac{A_1}{\lambda}$
At time $t_2$: $A_2 = \lambda N_2$, so $N_2 = \frac{A_2}{\lambda}$
Number of nuclei decayed between the two instants = $N_1 - N_2$
$N_1 - N_2 = \frac{A_1 - A_2}{\lambda}$