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13.Nuclei
hard
જો પદાર્થનો અર્ધ-આયુ $20$ મિનીટ હોય તો $33\,\%$ ક્ષય અને $67\,\%$ ક્ષય વચ્ચેનો સમય અંતરાલ ગણો : (મિનીટ માં)
A
$60$
B
$20$
C
$40$
D
$13$
(JEE MAIN-2021)
Solution
$N _{1}= N _{0} e ^{-\lambda t_{1}}$
$\frac{ N _{1}}{ N _{0}}= e ^{-\lambda t _{1}}$
$0.67= e ^{-\lambda t _{1}}$
$\ln (0.67)=-\lambda t _{1}$
$N _{2}= N _{0} e ^{-\lambda t _{2}}$
$\frac{ N _{2}}{ N _{0}}= e ^{-\lambda t _{2}}$
$0.33= e ^{-\lambda t _{2}}$
$\ln (0.33)=-\lambda t _{2}$
$\ln (0.67)-\ln (0.33)=\lambda t _{1}-\lambda t _{2}$
$\lambda\left( t _{1}- t _{2}\right)=\ln \left(\frac{0.67}{0.33}\right)$
$\lambda\left( t _{1}- t _{2}\right) \cong \ln 2$
$t _{1}- t _{2} \simeq \frac{\ln 2}{\lambda}= t _{1 / 2}$
Half life $= t _{1 / 2}=20\,minutes.$
Standard 12
Physics
