Assume that protons and neutrons have equal masses. Mass of a nucleon is 1.6×10⁻²⁷ kg and radius of nucleus is 1.5×10⁻¹⁵ A^(1/3) m. The approximate ratio of the nuclear density and water density is n×10¹³. The value of n is __________.

Solution

Physics sir · 4 steps

Mass and volume both grow as A, so A cancels and every nucleus has the same density.

1. Mass and volume. A nucleus with A nucleons has mass Am. Its radius is R = R₀ A^(1/3), so its volume is 4/3π R₀³ A. Both carry exactly one factor of A.

   ρ_(nucleus) = Am/(4/3π R₀³ A) = m/(4/3π R₀³)

2. Put in numbers. R₀³ = (1.5 × 10⁻¹⁵)³ = 3.375 × 10⁻⁴⁵ m³, and 4/3π × 3.375 × 10⁻⁴⁵ = 1.414 × 10⁻⁴⁴ m³. So ρ_(nucleus) = (1.6 × 10⁻²⁷)/(1.414 × 10⁻⁴⁴) = 1.13 × 10¹⁷ kg m⁻³.
3. Compare with water. Water has density 10³ kg m⁻³. The ratio is (1.13 × 10¹⁷)/10³ = 1.13 × 10¹⁴ = 11.3 × 10¹³.
4. So. Written as n × 10¹³, the ratio has n ≈ 11.

The idea to keep. Since R ∝ A^(1/3), nuclear volume is proportional to A, just like the mass. Nuclear density is the same for all nuclei, about 10¹⁷ kg m⁻³.

Answer from the NTA final answer key. Solution by PrepWiser.