Nuclear - Binding energy

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The sum of the individual masses of various particle in the nucleus must be equal to the nuclear mass. But this is not so in actual practice. The nuclear mass is somewhat less than the sum of the individual masses of various nuclear particles. The difference between the actual nuclear mass and the expected nuclear mass (sum of the individual masses of nuclear particles) is referred to as mass defect. The mass defect can be converted into equivalent energy by means of Einstein equation (DE = Dmc²).

DE = Energy liberated

Dm = Loss of mass

c is the speed of light.

The energy equivalent to mass-defect is responsible for holding the nucleus together and is called binding energy of the nucleus.

The mass of the hydrogen atom is equal to the sum of the masses of a proton and an electron.

The binding energy of helium nucleus is thus calculated. (He nucleus has 2 protons + 2 neutrons)

Mass of 2 free neutrons (2 x 1.00867 u) = 2.01734 u

Mass of 2 free protons (2 x 1.00728 u) = 2.07456 u

Sum of masses of 2 free protons _________

+ 2 free protons 4.03190 u

Observed mass of 2 protons and

2 neutrons merged in a helium = 4.00150 u

nucleus _________

Mass defect (Dm) 0.03040 u

= 28.3 MeV.

(MeV stands for million electron volt or mega electron volt)

The binding energy in MeV can be directly found by multiplying the mass defect (Dm) in u (amu) with (931.48) a conversion factor as explained below:

Using E = mc²

= 931.48 MeV

Thus binding energy (MeV) = Dm (u) x 931.48

Binding energy (J) = Dm (u) x 1.4924 x 10^-10

In comparing the binding energies of different nuclei, it is more useful to consider the binding energy per nucleon.

For e.g., helium nucleus with 4 nucleons,

Binding energies of the nuclei of other atoms can be calculated in a similar manner. The binding energy per nuclei particle is a measure of the stability of nucleus. Binding energy may also be considered as the energy required to separate the individual particles of the nucleus.

fig.11.2 - A plot of nuclear binding energy per nucleon against the mass number for naturally occurring nuclides.

We observe the following features from the figure shown above:

i) Nuclei with mass number around 60 have the highest binding energy per nucleon.

ii) Species of mass numbers 4, 12 and 16 have high binding energy per nucleon implying that the nuclei ⁴H, ¹²C, and ¹⁶O are particularly stable.

iii) The binding energy per nucleon decreases appreciably above mass number 100.