The electrostatic energy of protons uniformly distributed throughout a spherical nucleus of radius is given by The measured masses of the neutron, , and are , , and , respectively. Given that the radii of both the and nuclei are same, ( is the s

Question

The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by $E = \frac{3}{5} \frac{Z (Z - 1) e^{2}}{4 π ε_{0} R}$

The measured masses of the neutron, ${}_{1}^{1}H$ , ${}_{7}^{15}N$ and ${}_{8}^{15}O$ are $1.008665 u$ , $1.007825 u$ , $15.000109 u$ and $15.003065 u$ , respectively. Given that the radii of both the ${}_{7}^{15}N$ and ${}_{8}^{15}O$ nuclei are same, $1 u = 931.5 MeV / c^{2}$ ( $c$ is the speed of light) and $e^{2} / (4 π ε_{0}) = 1.44 MeV fm$ . Assuming that the difference between the binding energies of ${}_{7}^{15}N$ and ${}_{8}^{15}O$ is purely due to the electrostatic energy, the radius of either of the nuclei is ( $1 fm = 10^{- 15} m$ ). [2016]

Smart Achievers · Accepted Answer

(3)

Binding energy of nitrogen atom

$[8 M_{n} + 7 M_{p} - M_{N}] \times 931$

$= [8 \times 1.008665 + 7 \times 1.007825 - 15.000109] \times 931$

Binding energy of oxygen atom

$[8 M_{n} + 8 M_{p} - M_{O}] \times 931$

$= [8 \times 1.008665 + 8 \times 1.007825 - 15.003065] \times 931$

$∴$ $Difference = 0.0037960 \times 931 MeV$ $\dots (i)$

$Also$ $E_{O} = \frac{3}{5} \times \frac{8 \times 7}{R} \times \frac{e^{2}}{4 π ε_{0}} = \frac{3}{5} \times \frac{56}{R} \times 1.44 MeV$

$E_{N} = \frac{3}{5} \times \frac{7 \times 6}{R} \times \frac{e^{2}}{4 π ε_{0}} = \frac{3}{5} \times \frac{42}{R} \times 1.44 MeV$

$∴$ $E_{O} - E_{N} = \frac{3}{5} \times \frac{14}{R} \times 1.44 MeV$ $\dots (i i)$

$From eq. (i) & (ii)$

$\frac{3}{5} \times \frac{14}{R} \times 1.44 = 0.0037960 \times 931$

$∴$ $R = 3.42 fm$

Solution

1 2.85 fm

2 3.03 fm

3 3.42 fm

4 3.80 fim

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