For a nucleus of mass number \(A\) and radius \(R\), the mass density of nucleus can be represented as how?

The mass density ( \(\rho\)) of a nucleus is defined as the ratio of its mass to its volume:
\(\rho=\) Mass of nucleus / Volume of nucleus
For a spherical nucleus with radius R , the volume is:
\(\text { Volume }=(4 / 3) \pi R^3\)
The mass of the nucleus is related to the mass number A :
\(\text { Mass }-\mathrm{A} \times \mathrm{m} \text {u}\)
(where m_u is the atomic mass unit, approximately \(1.66 \times 10^{-27} \mathrm{~kg}\))
Therefore:
\(\rho=\left(\mathrm{A} \times \mathrm{m} \_\mathrm{u}\right) /\left[(4 / 3) \pi \mathrm{R}^3\right]\)
Now, there's an important relationship between the mass number A and nuclear radius R. Empirically, it has been found that:
\(\mathrm{R}=\mathrm{Ro} * \mathrm{~A}^{\wedge}(1 / 3)\)
Where Ro is a constant approximately equal to \(1.2 \times 10^{-15} \mathrm{~m}(1.2\) fermi).
Substituting this into the density equation:
\(\begin{aligned}
& \rho=\left(\mathrm{A} \times \mathrm{m} \ \mathrm{u}\right) /\left[(4 / 3) \pi\left(\mathrm{Ro} \times \mathrm{A}^{\wedge}(1 / 3)\right)^3\right] \\
& \rho=(\mathrm{A} \times \mathrm{m} \text { u }) /\left[(4 / 3) \pi \mathrm{Ro}^3 \times \mathrm{A}\right] \\
& \rho=\mathrm{m} \_\mathrm{u} /\left[(4 / 3) \pi \mathrm{Ro}^3\right]
\end{aligned}\)
This shows that the nuclear density is approximately constant for all nuclei, regardless of their mass number. This is a fundamental property of nuclear matter known as nuclear saturation density.
The numerical value is:
\(\rho=2.3 \times 10^{17} \mathrm{~kg} / \mathrm{m}^3\)