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The mass of $_{ 1 }{ H^{ 2 } }$ atoms is 2.014102 amu, where n and p have their weight 2.016490 amu. Neglect the mass of the electron. The binding energy for $_{ 1 }{ H^{ 2 } }$ atom is :
(a) 4.4464 MeV
(b) 2.2232 MeV
(c) 1.116 MeV
(d) None of the above

Last updated date: 20th Jun 2024
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Hint: Mass cannot be created nor destroyed. Therefore, if the theoretical mass is greater than the experimentally determined mass, then it implies that the difference of mass has been converted into energy and released.

Complete step by step answer:
In order to solve this question, we have to look into some of the concepts of nuclear chemistry. Of this, two terms, the binding energy and mass defect are the most important.

Binding Energy
It is the energy that is required in order to separate the different components i.e. protons and neutrons of a nucleus. This energy is always positive.

Mass Defect
There exists a difference between the observed mass of an atom’s nucleus and its theoretical mass based on the sum of the masses of its protons and neutrons.

The energy (E) and mass (M) are related by the following equation:
$ E=M{ c }^{ 2 } $
Where c is the speed of light and M is the mass in Kg. For the nuclei, the observed mass is always less than the sum of the individual masses of the protons and neutrons since the formation of nuclei releases energy. This energy which is equivalent to the binding energy accounts for a mass according to the above mentioned equation which is lost. This mass is known as the mass defect.

Mass defect can be calculated as the difference between the observed atomic mass and the theoretical mass (calculated by adding the individual masses of the protons and the neutrons of the nucleus). Mass of the proton is around 1.00728 amu and that of neutrons is 1.00867 amu.

Once this mass defect is known, we can easily find out the nuclear binding energy by converting that mass to energy by using the $ E=M{ c }^{ 2 } $.

For solving the above question, we should know the following relations:
$ 1amu=1.6605\times { 10 }^{ -27 }Kg$
$ 1eV=1.602\times { 10 }^{ -19 }J$

In the question the sum of neutron and proton for $_{ 1 }{ H^{ 2 } }$ is 2.016490 amu while the actual mass of $_{ 1 }{ H^{ 2 } }$ is 2.014102 amu. Therefore mass defect will be:
Mass defect=$2.016490\quad amu-2.014102\quad amu=0.002388\quad amu$

Using the relation $ 1amu=1.6605\times { 10 }^{ -27 }Kg$, we will convert the mass defect in Kg:
Mass defect=$\cfrac { 0.002388\quad amu }{ 1\quad amu } \times 1.6605\times { 10 }^{ -27 }Kg=3.9653\times { 10 }^{ -30 }Kg$

Using the relation $ E=M{ c }^{ 2 } $, we will convert this mass defect in energy:
Energy=$3.9653\times { 10 }^{ -30 }Kg\times (3\times { 10 }^{ 8 }\quad m/s{ ) }^{ 2 }=3.569\times { 10 }^{ -13 }J$

Now we will convert this energy into MeV using the relation $ 1eV=1.602\times { 10 }^{ -19 }J$ :
Energy= $\cfrac { 1\quad eV }{ 1.602\times { 10 }^{ -19 }j } \times 3.569\times { 10 }^{ -13 }J=2.2232\quad MeV$

Hence the correct answer is (b) 2.2232 MeV.

Note: The energy releases will actually have a negative sign since this energy is being released when the nucleons are forming the nucleus whereas in the case of binding energy. It will be required to supply this energy in order to separate the nucleons.