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Q.

For a radioactive material, is activity A and rate of change of its activity R are defined as $A = - \frac{d N}{d t}$ and $R = - \frac{d A}{d t}$ . Two radioactive sources P (mean life = $τ$ ) and Q (mean life = $2 τ$ ) have the same activity at t = 0. Their rates of change of activities at t = $2 τ$ are $R_{P}$ and $R_{Q}$ respectively. If $\frac{R_{p}}{R_{Q}} = \frac{n}{e},$ then the value of n is

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answer is 2.

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Detailed Solution

$\begin{array}{l} N = N_{o} e^{- λ t} \\ A = \frac{- d N}{d t} \\ R = \frac{- d A}{d t} = \frac{d^{2} N}{d t^{2}} \Rightarrow R = \frac{d^{2}}{d t} (N_{o} e^{- λ t}) \\ \Rightarrow R = λ A_{o} e^{- λ t} \Rightarrow \frac{R_{p}}{R_{g}} = \frac{λ_{p} A_{o} e^{- λ_{0} t}}{λ_{Q} A_{0} e^{- λ_{0} t}} (λ = \frac{1}{t}) \\ \frac{R_{P}}{R_{Q}} = \frac{2}{e} = \frac{n}{e} \Rightarrow n = 2 \end{array}$

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For a radioactive material, is activity A and rate of change of its activity R are defined as A=-dNdt and R=-dAdt . Two radioactive sources P (mean life = τ ) and Q (mean life = 2τ ) have the same activity at t = 0. Their rates of change of activities at t = 2τ areRP and RQ respectively. If RpRQ=ne, then the value of n is