# (a) Explain the processes of nuclear fission and nuclear fusion by using the plot of binding energy per nucleon $\left ( \frac{BE}{A} \right )$ 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%?

From the graph, we note that
During nuclear fission
From the graph when we go through from heavy region to the middle region there is a given in binding energy per nucleon which means energy is released
During nuclear fusion
From the graph, when we move from lighter nuclei region to heavier nuclei region there is a gain in binding energy per nucleon. Which means energy is released

b)
Given,
$T_{\frac{1}{2}}= 10\; years$
activity after 't' time At = 3.125 %
The law of radioactivity decay is given as $A_{t}= A_{0}e^{-\lambda t}$

$\lambda= \frac{0\cdot 693}{T_{\frac{1}{2}}}$

where $\lambda =$ decay constant

$\lambda = \frac{0\cdot 693}{10}= 0\cdot 0693$
Let $A_{0}= 100\, ^{0}/_{0}$( initial activity) Then,
$3\cdot 125= 100\; e^{-0\cdot 0693t}$
$\frac{3\cdot 125}{100}= e^{-0\cdot 0693t}$
Taking logarithm on both sides
$\l_{n}\, \frac{3\cdot 125}{100}= 0\cdot 0693\, t$
$\l_{n}\, \frac{100}{3\cdot 125}= 0\cdot 0693\, t$
$t= \frac{l_{n}32}{0\cdot 0693}= 50\; years$

Therefore the time taken for the activity to reduce to 3.125 % is 50 years

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