Sometimes a radioactive nucleus decays


Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is

Sulphur $\stackrel{38}{\stackrel{\text { half-life }}{=2.48 h}} C l^{38} \stackrel{\text { half-life }}{=0.62 h} A r^{38}$ (stable)

Assume that we start with 1000 38S nuclei at time t = 0. The number of 38Cl is of count zero at t = 0 and will again be zero at t = ∞ . At what value of t, would the number of counts be a maximum?


Let λ1 and λ2 be the disintegration constants for S38 and Cl38

dN1/dt = – λN1

dN2/dt = rate of decay of Cl38 + rate of formation of Gl38

dN2/dt = -λ2N2 + λ1N1

λ2tdN2 + λ2N2e λ2t dt = λ1Noe(λ2 – λ1)dt

Integrating the above equation we get,

t = 1.65 h

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