For a radioactive material, its activity A and rate of change of its activity R are defined as , 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 activities at t = 2τ are RP and RQ, respectively. If , then the value of n is

Engineering

Physics

Radioactivity New

Question

For a radioactive material, its activity A and rate of change of its activity R are defined as $A = - \frac{dN}{dt} and R = - \frac{dA}{dt}$ , 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 activities at t = 2τ 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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N = N₀e^–λt

$A = - \frac{dN}{dt}$ = λN₀e^–λt = λN

$R = - \frac{dA}{dt}$ = +λ²N₀e^–λt = λA = λ²N = λ²N₀e^–λt

Now, λ_PN_P0 = λ_QN₀

$\frac{R_{P}}{R_{Q}} = \frac{λ_{P}}{λ_{Q}} \frac{A_{P}}{A_{v}} = 2$

$= 2 \frac{e^{- 2}}{e^{- 1}} = \frac{2}{e}$

n = 2