Question

Calculate the (i) mass defect, (ii) binding energy and (iii) the binding energy per nucleon for a ${}_6{C}^{12}$ nucleus. Nuclear mass of ${}_6{C}^{12}=12.000000a.m.u.$, mass of hydrogen nucleus = 1.007825 a.m.u and mass of neutron = 1.008665 a.m.u.

Answer 1

Hint: When the protons and neutrons fuse (combine) to form a nucleus, some amount of mass is converted into energy called binding energy. The mass that is converted into energy is called mass defect. The constituent particles of the nucleus are called nucleons (protons and neutrons).
Formula used:
$\mathrm{\Delta }m=(n\times mas{s}_{neutron}+n\times mas{s}_{proton})-mas{s}_{nucleus}$
$c^2=931.5{\displaystyle \frac{MeV}{a.m.u}}$

Complete step-by-step answer:

(i) Mass defect: A nucleus consists of neutrons and protons. The protons and neutrons combine together to form a nucleus. Theoretically, the mass of a nucleus should be equal to the sum of the masses of all individual neutrons and protons. However, it is found that the mass of a nucleus is less than the sum of all the masses of neutrons and protons.
This means that when the protons and neutrons combine to form a nucleus, some amount of mass is lost (some may say it is destroyed). However, we know that a mass can neither be created nor destroyed. Therefore, we say that when the neutrons and protons fuse, some amount of mass ($\mathrm{\Delta }m$) is converted into energy. This mass ($\mathrm{\Delta }m$) is called mass defect. The energy in which the mass gets converted is called binding energy of the nucleus.
To express mass defect mathematically than it is simple the difference between the mass of the all the neutrons and protons in the nucleus and the actual mass of the nucleus i.e. $\mathrm{\Delta }m=(n\times mas{s}_{neutron}+n\times mas{s}_{proton})-mas{s}_{nucleus}$ ……..(i).
Let us calculate the mass defect for the given nucleus.
Here,
$mas{s}_{neutron}=1.008665a.m.u.$
$mas{s}_{proton}=1.007825a.m.u.$ (A hydrogen nucleus consists of a single proton and no neutron. Therefore, mass of proton is equal to mass of proton).
$mas{s}_{nucleus}=12a.m.u.$ (The mass number will be the mass of the given nucleus).
n= 6. (n is the number of protons or neutrons. It is equal to the atomic number of the nucleus).
Substitute all the respective values in equation (i).
$\Rightarrow \mathrm{\Delta }m=(6\times 1.008665+6\times 1.007825)-12$
$\Rightarrow \mathrm{\Delta }m=12.09894-12=0.09894a.m.u.$.
The mass defect in the nucleus of ${}_6{C}^{12}$ is 0.09894 a.m.u.

(ii) Let now calculate the binding energy of the nucleus. Binding energy (E) of a nucleus is equal to $E=\mathrm{\Delta }m{c}^2$, where c is the speed of light.
Generally, binding energy is calculated in the units of mega electron-volts (MeV). The value of $c^2$ is equal to $931.5{\displaystyle \frac{MeV}{a.m.u}}$.
Substitute the value of $c^2$ and $\mathrm{\Delta }m$ in the equation of binding energy.
Therefore, $E=0.09894a.m.u.\times 931.5{\displaystyle \frac{MeV}{a.m.u}}=92.15MeV$.
Hence the binding energy of the given nucleus is 92.15 MeV.
(iii) Binding energy per nucleon (the constituent particles of a nucleus are called nucleons) is simply the ratio of total binding energy to the total number of nucleons i.e. ${\displaystyle \frac{E}{2n}}$ .
Therefore, the binding energy per nucleon for the given nucleus is ${\displaystyle \frac{92.15MeV}{12}}=7.68MeV$.

Note: It is really surprising that the protons hold up together inside a nucleus. We know from our knowledge of electrostatics, particles of equal charge repel each and other. But it is completely opposite here. The protons come together to make up the nucleus. Therefore, it is said that there exists one more type of force in nature that holds up the protons together and this force is called nuclear force.