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Question: The binding energy per nucleon of \({}_5{B^{10}}\) is 8.0 MeV and that of \({}_5{B^{11}}\) is 7.5 Me...

The binding energy per nucleon of 5B10{}_5{B^{10}} is 8.0 MeV and that of 5B11{}_5{B^{11}} is 7.5 MeV. The energy required to remove a neutron from 5B11{}_5{B^{11}} is (mass of electron and proton are 9.11×10−319.11 \times {10^{ - 31}} kg and 1.67×10−271.67 \times {10^{ - 27}} kg).
(A) 2.5 MeV
(B) 8.0 MeV
(C) 0.5 MeV
(D) 7.5 MeV

Explanation

Solution

1. We will first calculate the binding energy of 5B10{}_5{B^{10}} and 5B11{}_5{B^{11}} , and then find their difference to calculate the energy required to remove a neutron from 5B11{}_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 5B10={}_5{B^{10}} = 8.0 MeV
Binding energy per nucleon of 5B11={}_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, 5B10{}_5{B^{10}} has 10 nucleons the binding energy for 5B10(E1){}_5{B^{10}}({E_1}) becomes:
E1=8.0×10{E_1} = 8.0 \times 10
E1=80MeV{E_1} = 80MeV
Similarly, 5B11{}_5{B^{11}} has 11 nucleons so the binding energy for 5B11(E2){}_5{B^{11}}({E_2}) becomes:
E2=7.5×11{E_2} = 7.5 \times 11
E2=82.5MeV{E_2} = 82.5MeV
Now, we observe that 5B10{}_5{B^{10}} has 10 total nucleons and 5B11{}_5{B^{11}} has 11 total nucleons, that is, 5B10{}_5{B^{10}} has one less nucleon than 5B11{}_5{B^{11}} . This means that if we subtract the binding energy of 5B10{}_5{B^{10}} from the binding energy of 5B11{}_5{B^{11}} , we will get the energy required to remove a neutron from 5B11{}_5{B^{11}} .Thus, the difference is written as follows:
ΔE=E2−E1\Delta E = {E_2} - {E_1}
ΔE=82.5−80\Delta E = 82.5 - 80
⇒ΔE=2.5MeV\Rightarrow \Delta E = 2.5MeV
Thus, the energy needed to remove a neutron from 5B11{}_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.