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Question:

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by E = (3/5)Z(Z-1)e²/4πε₀R. The measured masses of the neutron, ¹H, ¹⁵N, and ¹⁶O are 1.008665 u, 1.007825 u, 15.000109 u, and 15.003065 u, respectively. Given that the radii of both the ¹⁵N and ¹⁶O nuclei are the same, 1 u = 931.5 MeV/c² (c is the speed of light), and e²/(4πε₀) = 1.44 MeV fm. Assuming that the difference between the binding energies of ¹⁵N and ¹⁶O is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10⁻¹⁵ m): 3.03 fm, 2.85 fm, 3.42 fm, 3.80 fm

3.42 fm

3.80 fm

2.85 fm

3.03 fm

Solution:

For ¹⁵N: Z₁ = 7, N₁ = 8
For ¹⁶O: Z₂ = 8, N₂ = 8
Difference in the electrostatic energy, ΔE = E₂ - E₁
ΔE = (3/5)[(8(8-1)/R) - (7(7-1)/R)] × 1.44 = 12.096/R MeV fm
Mass defect, ΔM = ZmH + Nmn - matom
∴ For ¹⁵N, ΔM₁ = 7(1.007825) + 8(1.008665) - 15.000109 = 0.12399 u
For ¹⁶O, ΔM₂ = 8(1.007825) + 8(1.008665) - 15.003065 = 0.12019 u
Difference in binding energy, ΔB.E = [ΔM₁ - ΔM₂] × 931.5 MeV
⇒ ΔB.E = [0.12399 - 0.12019] × 931.5 = 3.54 MeV
According to the question,
12.096/R = 3.54
⇒ R = 3.42 fm