A count rate meter is used to measure the activity of a given sample. At one instant the meter shows 4750 counts per minute. Five minutes later it shows 2700 counts per minute. Find the half – life of the sample in minute.( Take ${\log }_{10}1.760=0.2455$)

Initial velocity  ${\mathrm{A}}_{\mathrm{i}}={\frac{\mathrm{dN}}{\mathrm{dt}}|}_{\mathrm{t}=0}=\lambda {\mathrm{N}}_0=4750$.....(i)

Final velocity  ${\mathrm{A}}_{\mathrm{f}}={\frac{\mathrm{dN}}{\mathrm{dt}}|}_{\mathrm{t}=5}=\lambda \mathrm{N}=2700$...(2)

Dividing (1) by (2) we get $\frac{4750}{2700}=\frac{N_0}{N_1}\dots$...(3)

The decay constant is given by  

$\lambda =\frac{2.303}{\mathrm{t}}\log \left(\frac{{\mathrm{N}}_0}{{\mathrm{N}}_1}\right)\Rightarrow \lambda =\frac{2.303}{5}\log \left(\frac{4750}{2700}\right)=0.113{\min }^{-1}$

Half life of the sample is  $\mathrm{T}=\frac{0.693}{\lambda }=\frac{0.693}{0.113}=6.13\min$