Question

Derive the expression for the half-life of a radio active nuclide.

Answer 1

At time $t$ the number of atoms present in a radioactive sample is given by the equation:
$N=N_0{e}^{-\lambda t}$
At $t=T_{\frac{1}{2}}$, we have $N=\frac{N_0}{2}$
$\Rightarrow \frac{N_0}{2}=N_0{e}^{-\lambda \left(T_{\frac{1}{2}}\right)}$
$\Rightarrow e^{\lambda T_{\frac{1}{2}}}=2$
$\Rightarrow \lambda T_{\frac{1}{2}}={\log }_{e}2$
$\Rightarrow \lambda T_{\frac{1}{2}}=2.303{\log }_{10}2$
$\Rightarrow \lambda \left(T_{\frac{1}{2}}\right)=2.303\times 0.3030=0.693$
$\Rightarrow T_{\frac{1}{2}}=\frac{0.693}{\lambda }$