For a radioactive material, its activity A and rate of change of its activity R are defined as $$A=$$−dNdt and $$R=$$−dAdt, where Nt is the number of nuclei at time 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 RP and RQ, respectively. If $$RPRQ=ne$$, then the value of n is

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For a radioactive material, its activity  A and rate of change of its activity  R are defined as $$A=$$−dNdt  and  $$R=$$−dAdt, where Nt  is the number of nuclei at time  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 RP  and  RQ, respectively.  If  $$RPRQ=ne$$, then the value of n is

Infinity Learn · Accepted Answer

$$N=N0e-$$λt $$A=$$−dNdt $$=$$λ$$N0e-$$λt       $$R=$$−$$dAdt=$$λ$$2N0e-$$λt         P   Mean life τ    Q   Mean life  $$2$$τ→ initially At time ‘t’$$=0$$             $$AP=AQ$$⇒λ$$1N1e$$−λ$$0=$$λ$$2N2e$$−λ$$0$$⇒λ$$1N1=$$λ$$2N2$$ →  After  $$t=2$$τRatio of rate of change of activities gives,$$RPRQ=$$λ$$12N1e$$−λ$$1t$$λ$$22N2e$$−λ$$2t$$⇒$$RPRQ=$$λ$$1$$λ$$2$$×e−$$1T$$×$$2Te$$−$$12T$$×$$2T$$⇒$$RPRQ=$$τ$$2$$τ$$1e$$−$$2e$$−$$1=2$$ττ×e−$$1$$⇒$$RPRQ=2e$$

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