State and derive the law of radioactive decay.

(N/A) The law of radioactive decay states that in any radioactive sample,the number of nuclei undergoing decay per unit time is proportional to the total number of nuclei present in the sample.
Let $N$ be the number of nuclei in the sample at time $t$,and let $\Delta N$ be the number of nuclei that decay in a small time interval $\Delta t$. According to the law:
$\frac{\Delta N}{\Delta t} \propto N$
Since the number of nuclei $N$ decreases over time,the rate of change of $N$ is negative. Thus,we write:
$-\frac{dN}{dt} = \lambda N$
where $\lambda$ is the decay constant or disintegration constant.
Rearranging the terms to integrate:
$\frac{dN}{N} = -\lambda dt$
Integrating both sides:
$\int_{N_0}^{N} \frac{dN}{N} = -\int_{0}^{t} \lambda dt$
$\ln(N) - \ln(N_0) = -\lambda t$
$\ln\left(\frac{N}{N_0}\right) = -\lambda t$
Taking the exponential of both sides:
$N(t) = N_0 e^{-\lambda t}$
This is the law of radioactive decay,where $N_0$ is the initial number of nuclei at $t = 0$.