Question

As the mass number A increases, which of the following quantities related to a nucleus do not change?
(a) Mass
(b) Volume
(c) Density
(d) Binding energy

Answer 1

(c) Density

Radius of a nucleus with mass number A is given as
$R=R_o{A}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$
Here, R_o = 1.2 fm
∴ Volume of the nucleus = $\frac{4\pi R^3}{3}=\frac{4\pi {R_o}^{3}A}{3}$

This depends on A. With an increase in A, V increases proportionally.
Mass of the nucleus $\simeq$ Am_N
Here, m_N is the mass of a nucleon.
Therefore, mass of the nucleus also increases with the increasing mass number.
Binding energy also depends on mass number (number of nucleons) as it is the difference between the total mass of the constituent nucleons and the nucleus. Therefore, it also varies with the changing mass number.

On the other hand,
Density = $\frac{\mathrm{Mass}}{\mathrm{Volume}}=\frac{A{m}_N}{{\displaystyle \raisebox{1ex}{$4\pi R^3$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}=\frac{A{m}_N}{{\displaystyle \raisebox{1ex}{$4\pi {R_o}^{3}A$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}=\frac{m_N}{{\displaystyle \raisebox{1ex}{$4\pi {R_o}^3$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}=\frac{3m_N}{{\displaystyle 4\pi {R_o}^3}}$
This is independent of A and hence does not change as mass number increases.