Class 12 Physics Nuclei | Nuclear binding energy |

**Nuclear binding energy**

- Nuclear binding energy is the energy required to hold an atom’s protons and neutrons together in the nucleus.
- Energy required holding neutrons and protons together therefore keeps the nucleus intact.
- It can also be defined as the energyneeded to separate the nucleons from each other.

- Importance of nuclear binding energy describes how strongly nucleons are bound to each other. By determining its value we will come to know whether the neutrons and protons are tightly or loosely bound to each other.
- If nuclear binding energy is high -> high amount of energy is needed to separate the nucleons this means nucleus is very stable.
- If nuclear binding energy is low -> low amount of energy is needed to separate the nucleons this means nucleus is not very stable.
__Mass defect__:-- Mass defect is the difference in the mass of nucleus and its constituents(neutrons and protons).
- It is denoted by ΔM.
- Mathematically :-
**ΔM = [Z m**_{p}+ (A-Z) m_{n}]- M- Where m
_{p}=mass of 1 proton, Z=number of protons,(A-Z)= mass of neutrons, m_{N}= mass of 1 neutronand M =nuclearmass of the atom.

- Where m
- For example: -(
^{16}_{8}O)àOxygen atom has 8 protons and 8 neutrons. - Mass of 8 protons à (8x1.00866) u and Mass of 8 neutrons à(8x1.00727) u.
- Therefore Oxygen nucleus à(8p+8n) à8(1.00866 + 1.00727) = 16.12744u.
- From spectroscopy ->Atomic mass of (
^{16}_{8}O) =15.9949u. - Mass of 8 electrons =(8x0.00055) u.
- Therefore Nuclear mass of (
^{16}_{8}O) = (15.9949 – (8x0.00055)) =15.99053u. - Nuclear mass is less than sum of the masses of its constituents.
- This difference in mass is known as
__mass defect__. - It is also known as excess mass.

- Relation between Mass defect and Nuclear binding energy:-
- Nuclear binding energy is denoted by E
_{b}. **E**_{b}= ΔMc^{2}- Where E
_{b}= nuclear binding energy, ΔM=mass defect.

- Where E
- As there is difference in the mass so there is energy associated with it. This energy is known as nuclear binding energy.
- Nuclear binding energy is a measure of how well a nucleus is held together.

- Nuclear binding energy is denoted by E

** Problem:- **Find the energy equivalent of one atomic mass unit,first in Joules and then in MeV. Using this, express the mass defectof (

** Answer:- **1u = 1.6605 × 10

To convert it into energy units, we multiply it by c^{2} and find thatenergy equivalent

= 1.6605 × 10^{–27} × (2.9979 × 10^{8})^{2} kg m^{2}/s^{2}

= 1.4924 × 10^{–10} J

= (1.4924 × 10^{–10})/ (1.602x10^{-19}) eV

= 0.9315 × 10^{9} eV

= 931.5 MeV

Or, 1u = 931.5 MeV/c^{2}

For (_{8}^{16}O), ΔM = 0.13691 u = 0.13691×931.5 MeV/c^{2}

= 127.5 MeV/c^{2}

The energy needed to separate (_{8}^{16}O) into its constituents is thus127.5 MeV/c^{2}.

** Problem:- **Find the energy equivalent of one atomic mass unit,first in Joules and then in MeV. Using this, express the mass defectof

** Answer:- **1u = 1.6605 × 10

To convert it into energy units, we multiply it by c^{2} and find thatenergy equivalent

= 1.6605 × 10^{–27} × (2.9979 × 10^{8})^{2} kg m^{2}/s^{2}

= 1.4924 × 10^{–10} J

= (1.4924 × 10^{–10} J)/ (1.602x10^{-19}) eV

= 0.9315 × 10^{9} eV

= 931.5 MeV

Or, 1u = 931.5 MeV/c^{2}

For ^{16}_{8}O, ΔM = 0.13691 u = 0.13691×931.5 MeV/c^{2}

= 127.5 MeV/c^{2}

The energy needed to separate ^{16}_{8}O into its constituents is thus127.5 MeV/c^{2}.

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