Expression For Nuclear Binding Energy With Example - Class 12 Physics Chapter 11 Nuclei

Defination nuclear binding energy

The total energy required to disintegrate the nucleus into its constituent particles (i.e. nucleons) is called nuclear binding energy or binding energy of the nucleus.

Or

Binding energy is basically the energy required to hold the nucleons in a nucleus. 

Thus, the energy equivalent to mass defect is the binding energy of the nucleus.

i.e.

$\text{Binding energy, } E_b=\Delta mc^2$

where $ \Delta m$ is mass defect and c is velocity of light in vacuum.

Since

$\Delta m=\left[\{m_p Z+m_n (A-Z)\}-M\right]$

where $m_p$ is mass of proton, $m_n$ is mass of neutron, Z is atomic number and A is mass number.

Therefore,

$E_b=\left[\{m_pZ+m_n(A-Z)\}-M\right]c^2$

Binding energy is expressed in joule, if mass defect is measured in kg.

When mass defect $\Delta m$ is expressed in a.m.u., then the binding energy is written as,

$E_b=\left[\{m_pZ+m_n(A-Z)\}-M\right]\text{ a.m.u.}$

Since

$1\ \text{a.m.u.}=931\ \text{MeV}$

$E_b=\left[\{m_pZ+m_n(A-Z)\}-M\right]\times931\ \text{MeV}$

or

$E_b=\Delta m\times931\ \text{MeV}$

Numerical Example 1 

Calculate the binding energy of a helium nucleus $_2He^4$. Given mass of $_2He^4$ = 4.0026 a.m.u. Mass of a proton = 1.0073 a.m.u. and mass of a neutron = 1.0087 a.m.u.

Solution

STEP 1. nucleus contains 2 protons and 2 neutrons.

Mass of 2 protons=$2\times1.0073=2.0146\ \text{a.m.u.}$

Mass of 2 neutrons=$2\times1.0087=2.0174\ \text{a.m.u.}$

Mass of all nucleons =$2.0146+2.0174=4.0320\ \text{a.m.u.}$

STEP 2.

Mass of nucleus=$4.0026\ \text{a.m.u.}$

Mass defect

$=\text{mass of nucleons}-\text{mass of nucleus}$

$=4.0320-4.0026$

$=0.0294\ \text{a.m.u.}$

Since

$1\ \text{a.m.u.}=931\ \text{MeV}$

$\therefore \text{Binding energy of }{}_2He^4\text{ nucleus}$

$=\text{mass defect}\times931\ \text{MeV}$

$=0.0294\times931\ \text{MeV}$

$=27.3714\ \text{MeV}$

Example 13.3: 

Energy Equivalent of One Atomic Mass Unit and Mass Defect of Oxygen-16

Find the energy equivalent of one atomic mass unit (u), first in Joules and then in MeV. Using this, express the mass defect of in MeV/.

Given

$1u = 1.6605 \times 10^{-27}\ \text{kg}$

$c = 2.9979 \times 10^8\ \text{m/s}$

Step 1: Energy Equivalent of 1 u in Joules

Using Einstein's mass-energy relation:

$E=mc^2$

$E = (1.6605 \times 10^{-27})(2.9979 \times 10^8)^2$

$E = 1.4924 \times 10^{-10}\ \text{J}$

Therefore,

$\boxed{1u = 1.4924 \times 10^{-10}\ \text{J}}$

Step 2: Convert into Electron Volts

Since

$1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$

$E = \frac{1.4924 \times 10^{-10}}{1.602 \times 10^{-19}}$

$= 0.9315 \times 10^9\ \text{eV}$

$= 931.5\ \text{MeV}$

Hence,

$\boxed{1u = 931.5\ \text{MeV}/c^2}$

Mass Defect of For Oxygen-16:

Z = 8 and N = 8

Mass of a hydrogen atom:

$m_H = 1.007825\,u$

Mass of a neutron:

$m_n = 1.008665\,u$

Atomic mass of oxygen:

$M(^{16}_{8}\mathrm{O}) = 15.994915\,u$

Step 1: Calculate Mass of Free Nucleons

Mass of 8 hydrogen atoms:

$8m_H = 8 \times 1.007825$

$= 8.062600\,u$

Mass of 8 neutrons:

$8m_n = 8 \times 1.008665$

$= 8.069320\,u$

Total mass of free nucleons:

$M_{\text{free}} = 8.062600 + 8.069320$

$= 16.131920\,u$

Step 2: Calculate Mass Defect

$\Delta M= M_{\text{free}} - M(^{16}_{8}\mathrm{O})$

$= 16.131920 - 15.994915$

$= 0.137005\,u$

$\boxed{\Delta M \approx 0.137\,u}$

Step 3: Express Mass Defect in MeV/

$\Delta M = 0.137005 \times 931.5$

$= 127.6\ \text{MeV}/c^2$

$\boxed{\Delta M \approx 127.6\ \text{MeV}/c^2}$

Final Answer

$\boxed{1u = 1.4924 \times 10^{-10}\ \text{J}}$

$\boxed{1u = 931.5\ \text{MeV}/c^2}$

For Oxygen-16:

$\boxed{\Delta M = 0.137\,u}$

$\boxed{\Delta M = 127.6\ \text{MeV}/c^2}$

Therefore, the binding energy of the nucleus is approximately

$\boxed{127.6\ \text{MeV}}$

which is the energy required to separate the nucleus into its constituent nucleons.