Question

State three properties of nuclear forces. Show that the density of nuclear matter is independent of mass number A.

Answer 1

(a) Strcture of nucleus: Rutherford's experiments on the scattering of α-particles by atoms led to the discovery of a positively charged heavy central core, called nucleus, of radius of the order of 10^-15 m.

1. The entire positive charge and almost the entire mass of the atom is concentrated in the nucleus.

2. Nucleus contains protons and neutrons called nucleons. A proton carries a unit positive charge and mass 1836 times that of an electron. The neutron is electrically neutral and has mass 1836.6 times that of an electron. The number of protons in the nucleus is called atomic number (Z) whereas the total number of neutrons and protons is called mass number (A).

3. Negatively charged electrons round the nucleus revolve in various orbits. The negative charge on the electron is of the same magnitude as the positive charge on the proton. The number of electrons revolving round the nucleus is equal to that of protons so that an atom is normally electrically neutral.

(b) Nuclear charge: The positive charge of the nucleus is due to the presence of protons. The charge on a proton is equal to 1.6 x 10^-19 C. If Z be the atomic number, then nuclear charge = Z x 1.6 x 10^-19 C.

The mass of the nucleus is equal to the sum of the masses of all protons and neutrons minus the mass defect.

Let Z = atomic number

A = mass number

Mn = mass of neutron

and ∆m = mass defect

Then the number of neutrons = (A - Z)

∴ Mass of the nucleus = ZM_p + (A - Z) M_n - ∆m.

Nuclear size: The distance of the closest approach of α-particle from. the nucleus is the measure of radius of nucleus or size of nucleus. For metals its value is about 10^-14 m.
The dimensiohs of nucleus have been measured by several experiments involving the scattering of n,p,e, etc. Every time it is found that the volume of nucleus is directly proportional to the number of nucleons (A) in the nucleus.

If R is the radius of the nucleus having mass number A, then

\(\frac{4}{3}\) π R³ ∝ A

or R³ ∝ A

or R ∝ A^1/3

or R = R₀A^1/3

where R₀ = 1.2 x 10^-15 m = 1.2 fm. It shows that greater the mass number, greater the nuclear radius.

Density of nucleus (ρ) It is defined as the nuclear mass per unit volume.

Thus the nuclear density is of the order of 10¹⁷ kgm^-3 and is independent of its mass number. Therefore, all nuclei have the same approximate density.

Mass defect: It is found that the mass of a stable nucleus is always less than the sum of masses of its constituent protons and neutrons in their free state. The difference between the sum of the masses of nucleons constituting it and the rest mass of the nucleus is called mass defect.

Let

M_p = mass of protons in the nucleus

M_n = mass of neutrons in the nucleus

M = mass of nucleus

N = number of neutrons

Z = number of protons

A = total number of nucleous

∴ The mass defect

∆m = ZM_p + (A - Z) M_n - M.

Packing fraction (f): Though the atomic masses are considered to be whole numbers, yet they differ from the integers by small amount. This variation of atomic masses from whole numbers is considered to be due to some deviations of the mass of the nucleus from the mass of the constituents of the nucleus (i.e. protons and neutrons). The term packing fraction is defined as the mass defect per nucleon.

∴ f = \(\frac{\text {mass defect}}{\text {mass number}} = \frac{\bigtriangleup m}{A} = \frac{M - A}{A}\)

where M is the atomic mass, A is the mass number (i.e. number of protons and neutrons in a.m.u.)

Thus f can be +ve, -ve or zero accordingly as M > A, M < A, M = A.

Packing fraction is a fundamental characteristic of the nucleus and is directly related to the availability of the energy of the nucleus and its stability. A negative packing fraction which will also correspond to a negative mass defect will mean that the atomic mass of the nucleus is less than the nearest whole number which will correspond to some conversion of mass into energy during the formation of the nucleus. Also, negative packing fraction will imply that the nucleus is exceptionally stable because we have to supply more energy to break the nucleus into its constituent nucleons while a positive packing fraction will indicate the nucleus to be less stable.