Question

The binding energies per nucleon are 5.3MeV, 6.2MeV and 7.4MeV for nuclei with mass numbers 3, 4 and 5 respectively. If one nucleus of mass number 3 combines with one nucleus of mass number 5 to give 2 nuclei of mass number 4, then:
a) 0.3MeV energy is absorbed
b) 0.3MeV energy is released
c) 28.1MeV energy is absorbed
d) 3.3MeV energy is absorbed

Answer 1

In the above question it is the binding energies of three different nucleons are given to us with their respective mass number. It is also given to us when the nucleus of mass number 3 combines with one nucleus of mass number 5 to give 2 nuclei of mass number 4. Hence will obtain the difference in the binding energy of the reactant and the products to determine whether the energy is released or absorbed and how much.

Complete answer:
Let us denote the element X with mass number 3 and binding energy 5.3MeV, Y with mass number 4 and binding energy 6.2MeV and Z with mass number 5 and binding energy 7.4MeV. Hence as per the information given to us in the question we can write the following reaction i.e.
${{X}^{3}}+{{Z}^{5}}\to 2{{Y}^{4}}$
Mass number represents the total number of nucleons of the element. Hence the mass number times the binding energy per nucleon will give us the total binding energy. Since the total binding energy of the reactants should be equal to the total binding energy of products should be the same, let us first determine whether it is true or not.
Let us say the binding energy of X,Y and Z be ${{E}_{X}},{{E}_{Y}}\text{ and }{{E}_{Z}}$ respectively. Hence these are numerically equal to,
$\begin{aligned} & {{E}_{X}}=3\times 5.3MeV \\\ & \Rightarrow {{E}_{X}}=15.9MeV \\\ \end{aligned}$
$\begin{aligned} & {{E}_{Y}}=4\times 6.2MeV \\\ & \Rightarrow {{E}_{Y}}=24.8MeV \\\ \end{aligned}$
$\begin{aligned} & {{E}_{Z}}=5\times 7.4MeV \\\ & \Rightarrow {{E}_{Z}}=37MeV \\\ \end{aligned}$
As per the law of conservation of energy the total energy on the reactant side should be equal to total energy on the product side. Hence we can write,
$\begin{aligned} & {{E}_{X}}+{{E}_{Z}}=2{{E}_{Y}} \\\ & \Rightarrow 2{{E}_{Y}}-({{E}_{X}}+{{E}_{Z}})=2(24.8)-(15.9+37) \\\ & \Rightarrow 2{{E}_{Y}}-({{E}_{X}}+{{E}_{Z}})=49.6-52.9 \\\ & \Rightarrow 2{{E}_{Y}}-({{E}_{X}}+{{E}_{Z}})=49.6-52.9=-3.3MeV \\\ \end{aligned}$
The minus sign indicates that some of the energy is absorbed.

Hence we can conclude that the correct answer of the above question is option d.

Note:
If the difference between the final and the initial energy is positive that means some energy is released when the reaction occurs. Such reactions do not make any sense if there is no mass defect observed. The value can only be positive if there is mass conversion to energy.