Question

An unstable element is produced in a nuclear reaction of a constant rate $ \frac{Mgx}{3} $ . Its disintegration constant is $ \eta $ . Find the number of nuclei after time t, if initially it was zero.

• A. $ 2mg\,[1+{{(\eta /L)}^{2}}] $
• B. $ mg\,[1+{{(\eta /L)}^{2}}] $
• C. $ \frac{mg}{2}\,[1+{{(\eta /L)}^{2}}] $
• D. $ 4mg\,[1+{{(\eta /L)}^{2}}] $

Answer 1

Answer: (D)

For radioactive decay, we can write $ \frac{dN}{dt}\eta -\lambda N $ or $ \int\limits_{0}^{N}{\frac{dN}{\eta -\lambda N}}=\int\limits_{0}^{t}{dt} $ or, In $ \left( \frac{\eta -\lambda N}{\eta } \right)=-\lambda t $ or $ \frac{\eta -\lambda N}{\eta }={{e}^{-\lambda t}} $ or, $ N=\frac{\eta }{\lambda }[1-{{e}^{-\lambda t}}] $