Question

Using the equation $N=N_0{e}^{-\lambda T}$, obtain the relation between half-life $\left(\lambda \right)$ and decay, constant (A) of a radioactive substance.

Answer 1

Let $N_0$ be the number of atoms present in a radioactive substance at time t = 0 and N the number at a later time t. Then, by Rutherford-Soddy law,

We have $N=N_0{e}^{\lambda T}$

where $\lambda$ is the decay constant for the substance. If the half-life of the substance be T, then at time $t=T$ the number of atoms left will be $N_o/2$. So putting t = T and $N=N_o/2$ in the above equation, we have

$\frac{N_0}{2}=N_0{e}^{-\lambda T}$

$\frac{1}{2}=e^{-\lambda T}$

$e^{\lambda T}=2$

$\lambda T=log{e}^2$ [on taking natural log both sides]

$T=\frac{log{e}^2}{\lambda }$

$=\frac{0.6931}{\lambda }$

This is the relation between half time and decay constant.