(a) Explain the processes of nuclear fission and nuclear fusion by using the plot of binding energy per nucleon (BE/A) versus the mass number A.

(b) A radioactive isotope has a half-life of 10 years. How long will it take for the activity
to reduce to 3.125% ?

(a) Plot of binding energy per nucleon mass number :


1. When we move from the heavy nuclei region to the middle region of the plot, we find that
there will be a gain in the overall binding energy and hence results in release of energy.
This indicates that energy can be released when a heavy nucleus (A ~ 240) breaks into two
roughly equal fragments. This process is called nuclear fission.

2. Similarly, when we move from lighter nuclei to heavier nuclei, we again find that there will
be gain in the overall binding energy and hence release of energy takes place.
This indicates that energy can be released when two or more lighter nuclei fuse together to
form a heavy nucleus. This process is called nuclear fusion.

(b) We know that, $\frac{\mathrm{R}}{{\mathrm{R}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}$

Given, $\frac{\mathrm{R}}{{\mathrm{R}}_0}=3.125\mathrm{\%}=\frac{3.125}{100}$
$\begin{array}{r}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{3.125}{100}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{1}{32}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\left(\frac{1}{2}\right)}^5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{n}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=5\end{array}$
Given T = 10 years
As      $\mathrm{n}=\frac{\mathrm{t}}{\mathrm{T}}$
$\therefore \frac{\mathrm{t}}{\mathrm{T}}=5[\because \mathrm{T}=10\text{ year }]$
or t = 5 x 10 = 50 years