Assume that protons and neutrons have equal masses. Mass of a nucleon is kg and radius of nucleus is m. The approximate ratio of the nuclear density and water density is . The value of is __________.
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Answer: 11
Solution
Nuclear density = mass/volume = (A · m_nucleon) / ((4/3)π r₀³ A) = m_nucleon / ((4/3)π r₀³)
= (1.6 × 10⁻²⁷) / ((4/3)π (1.5 × 10⁻¹⁵)³)
Denominator: (4/3)π (1.5)³ × 10⁻⁴⁵ = (4/3) × 3.1416 × 3.375 × 10⁻⁴⁵ ≈ 4.189 × 3.375 × 10⁻⁴⁵ ≈ 14.14 × 10⁻⁴⁵ m³
Nuclear density ≈ 1.6 × 10⁻²⁷ / (14.14 × 10⁻⁴⁵) ≈ 0.1132 × 10¹⁸ ≈ 1.132 × 10¹⁷ kg/m³
Ratio with water (density 10³ kg/m³): 1.132 × 10¹⁷ / 10³ = 1.132 × 10¹⁴ ≈ 11.32 × 10¹³
So n ≈ 11.
= (1.6 × 10⁻²⁷) / ((4/3)π (1.5 × 10⁻¹⁵)³)
Denominator: (4/3)π (1.5)³ × 10⁻⁴⁵ = (4/3) × 3.1416 × 3.375 × 10⁻⁴⁵ ≈ 4.189 × 3.375 × 10⁻⁴⁵ ≈ 14.14 × 10⁻⁴⁵ m³
Nuclear density ≈ 1.6 × 10⁻²⁷ / (14.14 × 10⁻⁴⁵) ≈ 0.1132 × 10¹⁸ ≈ 1.132 × 10¹⁷ kg/m³
Ratio with water (density 10³ kg/m³): 1.132 × 10¹⁷ / 10³ = 1.132 × 10¹⁴ ≈ 11.32 × 10¹³
So n ≈ 11.
[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]