A nucleus has density 2.29 × 10¹⁷ kg·m⁻³ and mass 19.926 × 10⁻²⁷ kg. Estimate its mass number A, given nuclear radius R = R₀A¹ᐟ³ with R₀ = 1.2 × 10⁻¹⁵ m and 4π = 12.56.
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Question

A nucleus has density 2.29 × 10¹⁷ kg·m⁻³ and mass 19.926 × 10⁻²⁷ kg. Estimate its mass number A, given nuclear radius R = R₀A¹ᐟ³ with R₀ = 1.2 × 10⁻¹⁵ m and 4π = 12.56.

G1 Physics · Accepted Answer

Step 1: Density = mass/volume. For a spherical nucleus, M = (4π/3)R³ρ. Step 2: Nuclear radius obeys R = R₀A¹ᐟ³. Hence R³ = R₀³A. Step 3: M = (4π/3)R₀³Aρ. Rearranging: A = 3M/(4πR₀³ρ). Step 4: Substitute: A = [3 × 19.926 × 10⁻²⁷]/[12.56 × (1.2 × 10⁻¹⁵)³ × 2.29 × 10¹⁷]. Step 5: Evaluate numerator = 5.9778 × 10⁻²⁶; denominator ≈ 4.97 × 10⁻²⁷; A ≈ 12.0. Final answer: A ≈ 12 (Option C).