Nuclear Density

Question

If the density of a nucleus with mass number A is D, then what is the density of a nucleus having mass number 2A?

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Solution

The Mass number, !!A!! of a nucleus represents the total number of nucleons (protons and neutrons) inside the nucleus.

The protons and neutrons have nearly the same mass. Therefore,

!!m_p = m_n = m!!

Therefore, total mass of the nucleus is:

!!M = m * A!!

Since the nucleus is spherical in shape, its volume is given by:

!!V = (4pi)/3 R\^3!!

where, !!R = R_o A\^(1/3)!!

where, !!R\o = 1.2 * 10\^-15 m = 1.2fm!!

Substituting !!R!! in the expression for volume, we get:

!!V = (4pi)/3 (R_o)\^3 A!!

Therefore, density is:

!!D = (mass)/(volume)!!

!!= (mA)/((4pi)/3 (R_o)\^3 A)!!

!!= 3/(4pi(R_o)\^3)!!

Thus, !!D = 2.3 * 10\^17 (kg)/(m\^3)!!

which is a constant.

Therefore, the density of all nuclei is same ans is independent of their mass number.

Application

The nuclear density is a very huge number! This means that a very large mass is confined to a very narrow space inside the nucleus. How is this possible? Well, that is because of the extremely strong nuclear force which acts between the nucleons and keeps them together. Moreover, the fact that nuclear density is constant for all nuclei also tells us another amazing thing about this nuclear force and this is that it is very short range!

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