A nucleus with mass number 242 and binding energy per nucleon as $7.6~ \mathrm{MeV}$ breaks into two fragment each with mass number 121. If each fragment nucleus has binding energy per nucleon as $8.1 ~\mathrm{MeV}$, the total gain in binding energy is _________ $\mathrm{MeV}$.

<p>The total binding energy of a nucleus is the binding energy per nucleon multiplied by the number of nucleons (protons and neutrons), which is the mass number.</p> <p>The initial total binding energy of the nucleus is $242 \times 7.6 \, \text{MeV}$.</p> <p>After the break, each fragment has a total binding energy of $121 \times 8.1 \, \text{MeV}$. <br/><br/>Since there are two such fragments, the final total binding energy is $2 \times 121 \times 8.1 \, \text{MeV}$.</p> <p>The gain in binding energy is the final total binding energy minus the initial total binding energy. Therefore, the gain in binding energy is:</p> <p>$2 \times 121 \times 8.1 \, \text{MeV} - 242 \times 7.6 \, \text{MeV} = 1960.2 \, \text{MeV} - 1839.2 \, \text{MeV} = 121 \, \text{MeV}$.</p> <p>Therefore, the total gain in binding energy is $121 \, \text{MeV}$.</p>