Define radioactive decay constant and half-life and derive a relation between them.

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.