Question

The rate at which a particular decay process occurs in a radioactive sample, is proportional to the number of radioactive nuclei present. If N is the number of radioactive nuclei present at some instant, the rate of change of N is $\frac{dN}{dt}=-\lambda N$.
Consider radioactive decay of A to B which may further decay either to X or to Y. ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ are decay constant for A to B decay, B to X decay and B to Y decay respectively. If at $t=0$ number of nuclei of A,B,X and Y are $N_0,N_0$, zero and zero respectively and $N_1,N_2,N_3,N_4$ are number of nuclei A,B,X and Y at any instant.

Rate of accumulation of at any instant will be

• A. $N_1{\lambda }_1+N_2{\lambda }_2+N_3{\lambda }_3$
• B. $N_1{\lambda }_1-N_3{\lambda }_2-N_4{\lambda }_3$
• C. $N_1{\lambda }_1-N_2{\lambda }_2-N_2{\lambda }_3$
• D. $N_1{\lambda }_1+N_2{\lambda }_2-N_2{\lambda }_3$

Answer 1

The correct option is C $N_1{\lambda }_1-N_2{\lambda }_2-N_2{\lambda }_3$


Rate of accumulation of B will depend upon radioactive decay of A to B minus radioactive decay from B to X and Y.
$\Rightarrow \frac{dB}{dt}=N_1{\lambda }_1-N_2{\lambda }_2-N_2{\lambda }_3$