Question

Write the formula relating the radius and mass number of a nucleus and show that nuclear density does not depend on the mass number.

Hint:

Use \( R = R_0 A^{1/3} \); then \( V \propto A \) and \( M \propto A \), so density \( M/V \) has \( A \) cancel out.

Answer 1

Step 1: Radius formula.
Experiments show that the radius \( R \) of a nucleus of mass number \( A \) is
\[ R = R_0 A^{1/3} \]
where \( R_0 = 1.2\times10^{-15}\ \text{m} = 1.2\ \text{fm} \) is a constant.

Step 2: Mass of the nucleus.
A nucleus of mass number \( A \) contains \( A \) nucleons, each of mass \( m \) (about \( 1.67\times10^{-27}\ \text{kg} \)). So the nuclear mass is
\[ M = A\,m \]

Step 3: Volume of the nucleus.
Treating the nucleus as a sphere of radius \( R \),
\[ V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi \left(R_0 A^{1/3}\right)^3 = \frac{4}{3}\pi R_0^3 A \]

Step 4: Density.
\[ \rho = \frac{M}{V} = \frac{A\,m}{\dfrac{4}{3}\pi R_0^3 A} \]
The mass number \( A \) cancels:
\[ \rho = \frac{3m}{4\pi R_0^3} \]

Step 5: Conclusion.
The right side contains only constants \( m \) and \( R_0 \), with no \( A \). Hence nuclear density is the same for all nuclei and is independent of the mass number.

\[\boxed{\rho = \frac{3m}{4\pi R_0^3}\ \text{(independent of } A)}\]