Question

Obtain the relation between the decay constant and half life of a radioactive sample.

Answer 1

The number of atoms at any instant in a radioactive sample is given by

N=N0e^−λt

where

N=total number of atoms at any instant

N₀=number of atoms in radioactive substance at t=0

λ=decay constant

t=time

When t=T (Where T is the half life of the sample)

${\displaystyle N=\frac{N_0}{2}}$

${\displaystyle \Rightarrow \frac{N_0}{2}=N_0{e}^{-\lambda t}}$

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

${\displaystyle \Rightarrow e^{\lambda T}=2}$

Taking log on both the sides, we get

${\displaystyle \lambda T={\log }_{e}2=2.303lo{d}_{10}2}$

${\displaystyle \Rightarrow T=\frac{2.303lo{d}_{10}2}{\lambda }}$

${\displaystyle \Rightarrow T=\frac{2.303\times 0.3010}{\lambda }}$

${\displaystyle \Rightarrow T=\frac{0.6931}{\lambda }}$

Thus, half life of a radioactive substance is inversely propotional to decay constant.