The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by $$E=35ZZ$$−$$1e24$$πε$$0R$$ The measured masses of the neutron, $$11H$$, $$715N$$ and $$815O$$ are $$1.008665$$ u, $$1.007825$$ u, $$15.000109$$ u and $$15.003065$$ u, respectively. Given that the radii of both the $$715N$$ and $$815O$$ nuclei are same, $$1u=931.5$$ $$MeV/c2$$ (c$$ is the speed of $$light) and $$e2/4$$πε$$0=1.44$$ MeV fm. Assuming that the difference between the binding energies of $$715N$$ and $$815O$$ is purely due to the electrostatic energy, the radius of either of the nuclei is $$1fm=10$$−$$15$$

Question

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by  $$E=35ZZ$$−$$1e24$$πε$$0R$$ The measured masses of the neutron,    $$11H$$,  $$715N$$  and  $$815O$$  are $$1.008665$$ u, $$1.007825$$ u, $$15.000109$$ u and $$15.003065$$ u, respectively. Given that the radii of both the  $$715N$$  and   $$815O$$ nuclei are same,  $$1u=931.5$$ $$MeV/c2$$ (c$$ is the speed of $$light) and $$e2/4$$πε$$0=1.44$$ MeV fm. Assuming that the difference between the binding energies of   $$715N$$ and  $$815O$$  is purely due to the electrostatic energy, the radius of either of the nuclei is  $$1fm=10$$−$$15$$

Infinity Learn · Accepted Answer

Binding energy of nitogen $$atom=8$$×$$1.008665+7$$×$$1.007825$$−$$15.000109931.5$$​Binding energy of oxygen $$atom=7$$×$$1.008665+8$$×$$1.007825$$−$$15.003065931.5$$​∴Difference in $$BE=0.0037960$$×$$931.5=3.54$$ MeVNow, ​Difference in electrostatic $$energy=EO$$−$$EN=358$$×$$7R$$−$$7$$×$$6R1.44MeV$$ $$fm=12.096RMeV$$ fmBoth differences are equal, ​$$12.096RMeV$$ $$fm=3.54$$ MeV​⇒$$R=3.42fm$$

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