Question

Obtain an expression for the number of radioactive nuclei present at any instant in terms of the decay constant and initial number of nuclei.

Answer 1

According to law of radioactivity, $-\frac{dN}{dt}\propto N$ or $\frac{dN}{dt}=-\lambda N$

where $\frac{dN}{dt}=$ rate of decay (negative sign due to decreases nuclei with time);

$N=$ number of nuclei present at time t;

$\lambda =$ decay constant.

So, $\frac{dN}{N}=-\lambda dt$

Integrating, $lnN=-\lambda t+C$ where $C=$ integrating constant.

at $t=0,$ $N=N_0$ (initial number of nuclei), so $C=ln{N}_0$

or $lnN=-\lambda t+ln{N}_0$

or $ln\left(N/N_0\right)=-\lambda t$

or $N/N_0=e^{-\lambda t}$

or $N=N_0{e}^{-\lambda t}$