State the law of radioactive decay. Define the term 'decay constant' for a radioactive substance. How is it related to half life ?

Explanation

Law of radioactive decay. The number of atoms disintegrated per second is directly proportional to the number of radioactive atom actually present at that instant.
Decay constant i.e. `-(dN)/(dt) prop N_(0)`
Decay constant is defined as the time after which the number of radioactive atoms reduce to 1/e times the original number of atoms.
Relation between half life T and `lambda`.
Half-life period
The half-life of a radioactive substance is defined as the time during which half of the atoms of radioactive substance will disintegrate.
Let us denote the half-time of a substance by T. Then, by definition, after time T, number of atoms left behind will be `(N_(0))/(2)`. ...(1)
Setting this condition i.e. when `t=T, N=(N_(0))/(2)` in the given equation (1), we get
`(N_(0))/(2)=N_(0)e^(-lambda T)`
or `e^(-lambdaT)=1/2 ` or `e^(lambdaT)=2`
or `lambda T=log_(e)2=2.303 log_(10)2`
`=2.303 xx 0.3010=0.693`
Therefore, `T=(0.693)/(lambda)` ...(2)
Thus, half life of radioactive substance is inversely proportional to its decay constant and is a characteristic property of its nucleus and cannot be altered by any known method.