Kinetics of nuclear reactions or nuclear chemistry
Question

# The nuclei of a certain radioactive element are produced at a constant rate of  $\alpha$ per second. This element has a decay constant $\lambda$ . Initially ${\mathrm{N}}_{0}$  nuclei of the element are present.If in the above question $\alpha =2\text{\hspace{0.17em}}\lambda \text{\hspace{0.17em}}{N}_{0}$  calculate the number of nuclei left in the element after one half life period of the element.

Difficult
Solution

## $\lambda t=\mathcal{l}n\text{\hspace{0.17em}}\left(\frac{2\lambda {N}_{0}-\lambda {N}_{0}}{2\lambda {N}_{0}-\lambda N}\right)$$\lambda \frac{\mathcal{l}n\text{\hspace{0.17em}}2}{\lambda }=\mathcal{l}n\text{\hspace{0.17em}}\left(\frac{\lambda {N}_{0}}{2\lambda {N}_{0}-\lambda N}\right)$$2=\frac{\lambda {N}_{0}}{2\lambda {N}_{0}-\lambda N}$$4\lambda {N}_{0}-2\lambda N=\lambda {N}_{0}$$N=\frac{3}{2}{N}_{0}$

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