A radioactive sample decays by two modes by a | vedclass

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Question

Let decay constants for two modes are $\lambda_{\alpha}$ and $\lambda_{\mathrm{b}}$ respectively.

$\frac{\mathrm{d} \mathrm{N}_{\alpha}}{\mathrm{dt

Vedclass · Accepted Answer

$90$

Vedclass · Answer

Let decay constants for two modes are $\lambda_{\alpha}$ and $\lambda_{\mathrm{b}}$ respectively.

$\frac{\mathrm{d} \mathrm{N}_{\alpha}}{\mathrm{dt}}=-\lambda_{\alpha} \mathrm{N}$

$\frac{\mathrm{d} \mathrm{N}_{\beta}}{\mathrm{dt}}=-\lambda_{\beta} \mathrm{N}$

$\frac{\mathrm{d} \mathrm{N}_{\alpha} / \mathrm{dt}}{\mathrm{dN}_{\beta} / \mathrm{dt}}=2 \Rightarrow \frac{\lambda_{\alpha}}{\lambda_{\beta}}=2$      $(i)$

Also $\frac{\ln 2}{\lambda_{\alpha}+\lambda_{\beta}}=60$

$\frac{\ln 2}{\lambda_{\alpha}+\frac{\lambda_{\alpha}}{2}}=60$

$\frac{\ln 2}{\lambda_{\alpha}}=90$ years.

column $I$	column $II$
$(A.)$Nuclear fusion	$(P.)$ Absorption of thermal neutrons by ${ }_{92}^{213} U$
$(B.)$Fission in a nuclear reactor	$(Q.)$ ${ }_{27}^{60} Co$ nucleus
$(C.)$ $\beta$-decay	$(R.)$ Energy production in stars via hydrogen conversion to helium
$(D.)$ $\gamma$-ray emission	$(S.)$ Heavy water
	$(T.)$ Neutrino emission

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