Question

A radioactive nucleus is being produced at a constant rate $\alpha$ per second. Its decay constant is $\lambda$. If $N_0$ is the number of nuclei at the $t=0$, then maximum number of nuclei possible are:

• A. $N_0+\frac{\alpha }{\lambda }$
• B. $N_0$
• C. $\frac{\lambda }{\alpha }+N_0$
• D. $\frac{\alpha }{\lambda }$

Answer 1

The correct option is D $\frac{\alpha }{\lambda }$

The rate of production of the nuclei is $\frac{dN}{dt}=\alpha -\lambda N$

When the number of nuclei are maximum, i.e. $N$ is maximum, $\frac{dN}{dt}=0$

$⟹\alpha -\lambda N_{max}=0$

$⟹N_{max}=\frac{\alpha }{\lambda }$