Prove that the instantaneous rate of change of activity of a radioactive substance is inversely proportional to the square of half life.

Explanation

If a radioactive substance cotains N atoms at any time t, its activity is
`A=(dN)/(dt)=lambdaN`
Differentiating w.r.t. t
`(dA)/(dt)=d/(dt)(lambdaN)=lambda(dN)/(dt).......(i)`
`=lambdad/(dt)(N_(0)e^(-lambdat))`
`=lambdad N_(0)(-lambda) e^(-lambdat)`
`=lambdad^(2) N_(0)e^(-lambdat)`
`(dA)/(dt)=lambda^(2)N` (leaving neg. sign)
As `lambda=0.693/T`
`:. (dA)/(dt)=(0.693/T)^(2)N=((0.693)^(2)N)/(T^(2))`
Hence, `(dA)/(dt) prop 1/(T^(2))` which was to be proved.