Step 1: Nuclear Density Definition
Nuclear density \( \rho \) is defined as: \[ \rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} \] Given that nuclear mass is proportional to the mass number \( A \), and nuclear volume is proportional to \( R^3 \), the density can be expressed as: \[ \rho = \frac{m \times A}{\frac{4}{3} \pi R^3} \] where \( R \) represents the nuclear radius.
Step 2: Nuclear Radius Dependence on Mass Number
The nuclear radius \( R \) is empirically related to the mass number \( A \) by: \[ R = R_0 A^{1/3} \] with \( R_0 \) being a constant.
Substituting this relationship into the volume formula yields: \[ V \propto (A^{1/3})^3 = A \] Consequently, the nuclear density simplifies to: \[ \rho \propto \frac{A}{A} = \text{constant} \]
Step 3: Ratio of Nuclear Densities
As nuclear density is independent of the mass number \( A \), the ratio of nuclear densities for two nuclei with mass numbers in the ratio \( 4:3 \) is: \[ \frac{\rho_1}{\rho_2} = 1:1 \]Final Answer: The ratio of the nuclear densities of the two nuclei is \( 1:1 \).