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

Explanation

Radioactive law : The rate of disintegration of radio active atoms present in the sample of an element is directly proportional to the number of radioactive atoms present at that instant.

i.e., `(dN)/(dt) prop N`
or `(dv)/(dt)=-lambdaN`
where `lambda` is known as the disintegration constant . The -ve sign indicates that the number of radioactive nuclei/atoms decreases with the passage of time.
Hence, `(dN)/N=-lambdadt`
Integrating both sides we get
`log_e=-lambda t +C`
Applying the initial condition for `t=0 , N=N_0`
We get , `C=log_e N_0`
i.e., `log_e (N/N_0)=-lambdat`
or `N/N_0=e^(-lambdat)`
Hence, `N=N_0e^(-lambdat)`
A graph .N. v/s time .t. gives the exponential curve as shown in the figure.