Question

Why nuclear density is not dependent on the mass number?

Answer 1

Nuclear density:

• Nuclear density is the density of the nucleus of an atom.
• It is the mass-to-volume ratio inside the nucleus.

Relationship between radius and mass number

• The radius (size) R of nucleus is related to its mass number (A) as
• $\mathrm{R}={\mathrm{R}}_0{\mathrm{A}}^{1/3}$where ${\mathrm{R}}_0=1.1\times {10}^{-15}\mathrm{m}$
• If m is the average mass of a nucleon, then
• mass of nucleus = mA, where A is the mass number

Find the volume of the nucleus

$$\begin{gathered} \mathrm{Volume}\mathrm{of}\mathrm{nucleus}=\frac{4}{3}{\mathrm{\pi R}}^3 \\ =\frac{4}{3}\mathrm{\pi }{\left({\mathrm{R}}_0{\mathrm{A}}^{1/3}\right)}^3 \\ =\frac{4}{3}{\mathrm{\pi R}}_0^3\mathrm{A} \end{gathered}$$

Finding the density of the nucleus

Let nuclear Density be ${\mathrm{\rho }}_{\mathrm{N}}$

$$\begin{gathered} {\mathrm{\rho }}_{\mathrm{N}}=\frac{\mathrm{mass}}{\mathrm{volume}} \\ {\mathrm{\rho }}_{\mathrm{N}}=\frac{\mathrm{mA}}{{\displaystyle \frac{4}{3}}{\mathrm{\pi R}}_0^3\mathrm{A}} \end{gathered}$$

${\mathrm{\rho }}_{\mathrm{N}}=\frac{\mathrm{m}}{{\displaystyle \frac{4}{3}{\mathrm{\pi R}}_0^3}}$

Since the formula of nuclear density is free of A(mass number).

Hence we can conclude that nuclear density is not dependent on the mass number.