Question

A radioactive nucleus is being produced at a constant rate $100\text{per sec}$. Its decay constant is $50\text{per sec}$. If $N_0$ are the number of nuclei at time $t=0,$ then maximum number of nuclei possible are _____.

Answer 1

The accumulation rate of the nucleus is given by,

$\frac{dN}{dt}=\text{(Rate of production) - (Rate of disintegration)}$

Let $\alpha$ be the rate of production and $\lambda$ be the decay constant.

$\therefore$ Rate of disintegration $=\lambda N$

So, at any given time $t$, the governing equation will be,

$\frac{dN}{dt}=\alpha -\lambda N$

When $N$ reaches a maxima,

$\frac{dN}{dt}=0$

$\therefore N=\frac{\alpha }{\lambda }$

Here, $\alpha =100{\text{s}}^{-1};\lambda =50{\text{s}}^{-1}$

$\therefore N=\frac{100}{50}=2$

Hence, the correct answer will be $2$.