Physics Problem: isobars and binding energy..?

Question

Two Nuclei having the same mass number are known as isobars. Calculate the difference in binding energy per nucleon for the isobars 23/11 Na (23 being the mass number and 11 being atomic number) and 23/12 Mg. How do you account for the difference?

Question in my physics book which i do not understand!

the answer that my book gave me was: 0.210 MeV/nucelon greater for 23/11 Na, attributable to less proton repulsion.

Help! Explain how the answer was solved. Thanks so much!

Answer 1

Well, just from the numbers given in the question, you cannot account for the difference in binding energy. Depending on the information available, there are various ways to get the result:

1. check a nuclear data table. Usually the binding energy is listed. For example this would give (186564 keV - 181724 keV) / 23 nucleons = 210 keV/nucleon

2. use a nuclear model and calculate the binding energies from primary principles. I don't try this one. Maybe the liquid drop model gives some usable result.

3. use the masses from a nuclear chart to calculate the binding energies.

E.g. the binding energy of Na-23 is c(squared) times the sum of the masses of the constituents minus the mass of Na-23: 11 protons + 12 neutrons + 11 electrons - Na-23

(11 * 1.007276 + 12 * 1.008665 + 11 * 0.000549 - 22.98977) amu*c^2 = 186564keV

or in energy units:

11 * 938272 keV + 12 * 939565 keV + 11 * 511keV - 22.98977amu*c^2 = 186560 keV

(rounded)

Do the same for Mg-23 and then procede as in section 1.

4. Instead of calculating the binding energies as in section 3, you may calculate the difference in binding energy from the decay energy and the mass of an electron a neutron and a proton.

The decay energy is known to be 4.057MeV. Na-23 has one electron less, one proton less and one neutron more than Mg-23. So we get:

(4.057MeV - 511keV + 939565keV - 938272keV) / 23nucleons = 210keV/nucleon

Note that this is probably much more accurate than the solution in section 3!

N.B.: In the answers 3 and 4 I neglect the electon binding energies. I assume that the difference in electron binding energies in Na and Mg is less than 1keV. (Ionization potential of Mg being some eV)