Question

How does the size of a nucleus depend on its mass number A? Hence prove that the density of nucleus is a constant, independent of A, for all nuclei.

Answer 1

Nuclear size and density

Step 1: Nuclear radius relation
The empirical formula for nuclear radius is: \[ R = R_0 A^{1/3} \] where \(R_0 \approx 1.2 \times 10^{-15}\ \text{m}\) Thus, nuclear volume: \[ V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi R_0^3 A \] So: \[ V \propto A \]

Step 2: Mass of nucleus
Mass of nucleus is approximately: \[ M = A m_n \] where \(m_n\) is nucleon mass. So: \[ M \propto A \]

Step 3: Density of nucleus
Density is: \[ \rho = \frac{M}{V} \] Substitute: \[ \rho = \frac{A m_n}{\frac{4}{3}\pi R_0^3 A} \] Cancel \(A\): \[ \rho = \frac{m_n}{\frac{4}{3}\pi R_0^3} \]

Step 4: Final conclusion
Since all terms are constants: \[ \rho = \text{constant (independent of } A\text{)} \] Final Answer:
• Nuclear radius: \(R \propto A^{1/3}\)
• Nuclear density: constant for all nuclei