Nuclear Binding Energy

 

Nuclear Binding Energy :- We have seen in the article “Composition of nucleus” that 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. However, the nuclear mass M is found to be always less than this, i.e., the rest mass of the nucleus is smaller than the sum of the rest masses of nucleons constituting it. This is due to the fact that when nucleons combine to form a nucleus, some energy (binding energy) is liberated. Hence we may define :-

“Binding energy of a nucleus is the energy with which nucleons are bound in the nucleus. It is measured by the work required to be done to separate the nucleons an infinite distance apart from each other, so that they may not interact with each other.”

Expression for Nuclear Binding Energy

Consider a nucleus _ZX^A, here

Z = atomic number = number of protons in the nucleus

A = mass number = total number of nucleons in the nucleus(protons + neutrons)

Let

m_p = mass of a proton

m_n = mass of a neutron

m_N = mass of a nucleus of _ZX^A

Mass Defect(Δm) :- The difference between the total mass of the nucleons and mass of the nucleus is called the mass defect of the nucleus.

∴ Δm = [Zm_p + (A-Z)m_n – m_N]

From Einstein’s mass energy equivalence,

Nuclear Binding Energy(B.E.) = Δmc²

B.E. = [Zm_p + (A-Z)m_n – m_N] c²     …..(1)

where c is the velocity of light in vacuum.

If the masses are taken in atomic mass unit, then nuclear binding energy is given by

B.E. = [Zm_p + (A-Z)m_n – m_N] × 931.5 MeV     …..(2)

Expression (1) can also be written as,

B.E. = [Zm_p+ Zm_e + (A-Z)m_n – m_N – Zm_e] c²

B.E. = [Z(m_p+ m_e) + (A-Z)m_n – (m_N + Zm_e)] c²

B.E. = [Zm_H + (A-Z)m_n – m(_ZX^A)] c²     …..(3)

Let us consider example of oxygen nucleus. It contains 8 protons and 8 neutrons.

Mass of 8 neutrons = 8 × 1.00866 u
Mass of 8 protons = 8 × 1.00727 u
Mass of 8 electrons = 8 × 0.00055 u

Therefore the expected mass of ₈O¹⁶ nucleus = 8 × (1.00727 + 1.00866) u = 8 × (2.01593) = 16.12744 u.

Now atomic mass of ₈O¹⁶ = 15.99493 u (found from mass spectroscopy experiments)

Subtracting the mass of 8 electrons (8 × 0.00055 u) from atomic mass of ₈O¹⁶

Experimental mass of ₈O¹⁶ nucleus = (15.99493 – 8 × 0.00055) u = 15.99053 u.

Mass defect = 16.12744 u – 15.99053 u = 0.13691 u

Thus, we find that the mass of the ₈O¹⁶ nucleus is less than the total mass of its constituents.

 

 

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