Question

State the radioactive decay law.

Answer 1

In a radioactive material, it is found that the radioactive decays per unit time are directly proportional to the total number of nuclei of radioactive compounds in the sample.

If the number of nuclei in a sample is $N$ and the number of radioactive decays per unit time $\Delta t$ is $\Delta N$ then,

$\frac{\Delta N}{\Delta t}\propto N$

or $\frac{\Delta N}{\Delta t}=\lambda N,$ where $\lambda$ is the constant of proportionality called radioactive decay constant.

Also, the number of radioactive decays ΔN is reducing the total number in the sample. Convention tells us that this should be termed negative.

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

On integration we get,

$lnN-ln{N}_0=-\lambda (t-t_0)$

Here $N_0$ represents the original number of nuclei in the sample at a time $t_0$, ie $t=0$.