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 $\underset{0}{\overset{N}{\int }}\frac{dN}{\eta -\lambda N}=\underset{0}{\overset{t}{\int }}dt$ or, In $\left(\frac{\eta -\lambda N}{\eta }\right)=-\lambda t$ or $\frac{\eta -\lambda N}{\eta }=e^{-\lambda t}$ or, $N=\frac{\eta }{\lambda }[1-e^{-\lambda t}]$