Question

A counter rate meter is used to measure the activity of a radioactive sample. At a certain instant, the count rate was recorded as $475$ counters per minute. Five minutes later, the count rate recorded was $270$ counts per minute. Calculate the decay constant and half life period of the sample.

Answer 1

Let $N_0$ and $N$ be the number of atoms of the radioactive substance present at the start and after $5$ minutes respectively.

Rate of disintegration at the start $=\lambda N_0=475$

and rate of disintegration after $5$ minutes $=\lambda N=270$

Dividing both, $\frac{\lambda N_0}{\lambda N}=\frac{475}{270}$

or $\frac{N_0}{N}=1.76$

We know that, $\lambda =\frac{2.303}{t}{\log }_{10}\frac{N_0}{N}$

$\lambda =\frac{2.303}{5}{\log }_{10}1.76=0.113{minute}^{-1}$

Half life period $=\frac{0.693}{\lambda }=\frac{0.693}{0.113}=6.1$ minutes