Prove that the instantaneous rate of change of activity of a radio-active substance is inversely proportional to the square of - Physics - Nuclei

Question

Prove that the instantaneous rate of change of activity of a
radio-active substance is inversely proportional to the square of
its half life

Sodhi · Answer

Activity or decay rate 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
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
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Prove that the instantaneous rate of change of activity of a radio-active substance is inversely proportional to the square of its half life.