A) \[1\,s\]
B) \[2\ln \left( \frac{4}{3} \right)s\]
C) \[\ln \,2\,s\]
D) \[\ln \left( \frac{4}{3} \right)s\]
Correct Answer: B
Solution :
[b] Let N be the number of nuclei at any time t then, \[\frac{dN}{dt}=100-\lambda N\] or \[\int\limits_{0}^{N}{\frac{dN}{(100-\lambda N)}=\int\limits_{0}^{t}{dt}}\]\[-\frac{1}{\lambda }[\log (100-\lambda N)]_{0}^{N}=t\] \[\log (100-\lambda N)-\log 100=-\lambda t\] \[\log \frac{100-\lambda N}{100}=-\lambda t\] \[\frac{100-\lambda N}{100}={{e}^{-\lambda t}}\] \[1-\frac{\lambda N}{100}={{e}^{-\lambda t}}\] \[N=\frac{100}{\lambda }(1-{{e}^{-\lambda }}t)\] As, \[N=50\]and \[\lambda =0.5/\sec \] \[\therefore \] \[50=\frac{100}{0.5}(1-{{e}^{-0.5t}})\] Solving we get, \[t=2\,\,ln\,\left( \frac{4}{3} \right)\,\sec \]
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