Question

Assume that protons and neutrons have equal masses Mass of a mucleon is $16 \times 10^{-27} kg$ and radius of nucleus is $15 \times 10^{-15} A ^{1 / 3} m$ The approximate ratio of the nuclear density and water density is $n \times 10^{13}$ The value of $n$ is ___

Answer 1

Correct Answer: 11

The density of a nucleus is: \[ \rho = \frac{\text{mass of nucleus}}{\text{volume of nucleus}}. \] 1. Mass of the nucleus: The mass of a single nucleon is: \[ m = 1.6 \times 10^{-27} \, \text{kg}. \]
2. Volume of the nucleus: The volume of a nucleus is given by: \[ V = \frac{4}{3} \pi R^3, \] where \(R = 1.5 \times 10^{-15} \, \text{m}\). Substituting: \[ V = \frac{4}{3} \pi (1.5 \times 10^{-15})^3 = \frac{4}{3} \pi \cdot 3.375 \times 10^{-45}. \] Approximate: \[ V \approx 14.14 \times 10^{-45} \, \text{m}^3. \]
3. Nuclear density (\(\rho\)): \[ \rho = \frac{m}{V} = \frac{1.6 \times 10^{-27}}{14.14 \times 10^{-45}} \approx 0.113 \times 10^{18} \, \text{kg/m}^3. \]
4. Water density (\(\rho_w\)): \[ \rho_w = 10^3 \, \text{kg/m}^3. \]
5. Ratio of nuclear density to water density: \[ \frac{\rho}{\rho_w} = \frac{0.113 \times 10^{18}}{10^3} = 0.113 \times 10^{15} = 11.31 \times 10^{13}. \] Thus, the approximate ratio is: \[ n = 11. \]