MARCH 2024 — Physics STD 12 Science — Question

→ The nucleus is made up of neutrons and protons. Therefore, it may be expected that the mass of the nucleus is equal to the total mass of its individual protons and neutrons.
→ But the nuclear mass $M$ is found to be always less than the total mass of its individual protons and neutrons.
→ For example :
${ }_8 O ^{16}$, a nucleus which has 8 neutrons and 8 protons.
Mass of 8 neutrons $=8 \cdot 1.00866 u$

Mass of 8 protons $=8 \cdot 1.00727 u$

Mass of 8 electrons $=8 \cdot 0.00055 u$
→ Therefore, the expected mass of ${ }_8 O ^{16}$ nucleus
$\begin{array}{l}=(8 \cdot 1.00866+8 \cdot 1.00727) \\ =8(1.00866+1.00727) \\ =8 \cdot 2.01593 u \\ =16.12744 u\end{array}$
→ The atomic mass of ${ }_8 O ^{16}$ found from mass spectroscopy experiments is seen to be $15.99493 u$.
→ Subtracting the mass of 8 electrons
$(8 \cdot 0.00055 u=0.0044 u)$ from this we get the experimental mass of ${ }_8 O ^{16}$ nucleus to be $15.99053 u$.
→ Thus, the mass of the ${ }_8 O ^{16}$ nucleus is less than the total mass of its constituents by
$
(16.12744-15.99053)=0.13691 u
$
→ "The difference in mass of a nucleus and its constituents, $\Delta M$ is called the mass defect" and is given by
$
\Delta M=\left[Z m_p+(A-Z) m_n\right]-M
$
Where, $Z =$ number of protons
$A - Z = N =$ neutron number
$m_p$ - mass of proton
$m_n$ - mass of neutron
M - mass of a nucleus
→ The energy equivalent to this mass defect is called the binding energy of nucleus.
$\therefore$ Binding energy $E _b=\Delta M c^2$
→ Binding energy per nucleon is the binding energy divided by the total number of nucleons.
$
\therefore E_{b n}=\frac{E_b}{A}
$
→ The binding energy per nucleon gives a measure of the stability of the nucleus.
→ A nucleus for which the value of $E _{b n}$ is comparatively higher is said to be more stable and for a nucleus for which the value of $E _{b n}$ is comparatively less is said to be less stable.