Question

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

Answer 1

Activity or decay rate at any instant of time is

A= -N λ
differentiating w.r.t time we get instantaneous rate of change of activity of radioactive substance..
dA / dt= (-dN /dt) λ
Again dN / dt= -N λ
thus dA/ dt= N λ2

dA / dt= ( 0.693)2N /(T1/2)2 (since λ =0.693/ T1/2)
thus...
dA /dt is directly proportional to T1/2

Answer 2

at any instant of time is A= -N λdifferentiating w.r.t time we get instantaneous rate of change of activity of radioactive substance..dA / dt= (-dN /dt) λAgain dN / dt= -N λthus dA/ dt= N λ2dA / dt= ( 0.693)2N /(T1/2)2 (since λ =0.693/ T1/2)thus...dA /dt is directly proportional to T1/2