Question

The binding energy per nucleon of ${}_5{B^{10}}$ is 8.0 MeV and that of ${}_5{B^{11}}$ is 7.5 MeV. The energy required to remove a neutron from ${}_5{B^{11}}$ is (mass of electron and proton are $9.11 \times {10^{ - 31}}$ kg and $1.67 \times {10^{ - 27}}$ kg).
(A) 2.5 MeV
(B) 8.0 MeV
(C) 0.5 MeV
(D) 7.5 MeV

Answer 1

Hint:
1. We will first calculate the binding energy of ${}_5{B^{10}}$ and ${}_5{B^{11}}$ , and then find their difference to calculate the energy required to remove a neutron from ${}_5{B^{11}}$ .
2. Binding Energy is calculated using the following formula: -
3. Binding Energy = Binding Energy per nucleon x No. of nucleons

Complete step-by-step solution
Binding energy of the system is defined as the energy released out when its constituents are brought from infinity to form the system.
From the question we know: -
Binding energy per nucleon of ${}_5{B^{10}} = $ 8.0 MeV
Binding energy per nucleon of ${}_5{B^{11}} = $ 7.5 MeV
Now, the binding energy of an element can be formulated as below: -
E= Binding Energy per nucleon x No. of nucleons
Since, ${}_5{B^{10}}$ has 10 nucleons the binding energy for ${}_5{B^{10}}({E_1})$ becomes:
 \[{E_1} = 8.0 \times 10\]
 \[{E_1} = 80MeV\]
Similarly, ${}_5{B^{11}}$ has 11 nucleons so the binding energy for ${}_5{B^{11}}({E_2})$ becomes:
 \[{E_2} = 7.5 \times 11\]
 \[{E_2} = 82.5MeV\]
Now, we observe that ${}_5{B^{10}}$ has 10 total nucleons and ${}_5{B^{11}}$ has 11 total nucleons, that is, ${}_5{B^{10}}$ has one less nucleon than ${}_5{B^{11}}$ . This means that if we subtract the binding energy of ${}_5{B^{10}}$ from the binding energy of ${}_5{B^{11}}$ , we will get the energy required to remove a neutron from ${}_5{B^{11}}$ .Thus, the difference is written as follows:
 \[\Delta E = {E_2} - {E_1}\]
 \[\Delta E = 82.5 - 80\]
 \[ \Rightarrow \Delta E = 2.5MeV\]
Thus, the energy needed to remove a neutron from ${}_5{B^{11}}$ is 2.5 MeV.

Hence, the option (a) is the correct solution.

Note
1. The mass of the electron and proton which is given in the question is just extra information and has nothing to do with the solution for this question.
2. It is important to remember that the value written in the superscript of the element symbol represents the number of nucleons present in it and the value written in the subscript of the element symbol represents the number of protons present in it.