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Question

For a radioactive material, its activity A and rate of change of its activity R are defined as A=dNdt and R=dAdt, where N(t) is the number of nuclei at time t. Two radioactive sources P(mean life τ) and Q (mean life 2τ) have the same activity at t = 0. Their rates of change of activitites at t = 2τ are RP and RQ, respectively. If RPRQ=ne, then the value of n is

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Solution

N=Noeλt
A=dNdtR=dAdt=+ddt(dNdt)=d2Ndt2

R=ddt[ddt(N0eλt)]

=N0(λ)ddt(¯eλ+)
=λ(λN0)eλt
=λA0eλt----(1)

From (1)
RpRQ=λpAoeλptλQAoeλQt
(where λ=1T)
=1τe1τ2τ12τe12τ2τ
=2e1
So,value of n is 2

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