Define half life of a radioactive element and deduce the expression for the same .

Explanation

Half life of a radioactive sample is the time taken for half the number of nuclei to disintegrate. It is denoted by `T_((1)/(2))`. From the decay equiation `N_(1)=N_(0)e^(-lambdat)`, it follows that for `t rarr infty, N_(1) rarr 0`.
Thus it takes an infinitely long time for all the nuclei of any radioactive sample to disintegrate. But the time taken for half the number nuclei in a given sample to disintegrate is finite and is characteristic of the given sample. It is called the half life of the radioactive substance .
When , `t=T_((1)/(2)),N_(t)=(N)/(2)`
`therefore` From the equation `N_(t)=N_(0)e^(-lambda t)`
We have `(N_(0))/(2)=N_(0)e^(-lambdat) or e^(-lambdaT_((1)/(2)))=2`
or `lambdaT_((1)/(2))=log_(e)2=2.303 log_(10)2=0.693`
`therefore T_((1)/(2))=(0.693)/(lambda)`
Thus half life of a radioactive substance is inversely proportional to its decay constant .