Question:

A researcher studying nuclear physics designs an experiment to investigate the binding energy per nucleon for different elements. The researcher uses a particle accelerator to increase the kinetic energy of protons and collides them with a target nucleus. By analyzing the characteristics of the resulting nuclear reactions, the binding energy per nucleon for each element is determined.

a) Explain how the nuclear binding energy per nucleon is related to the stability of a nucleus.

b) The following data represents the binding energies per nucleon for four different elements:

(i) Determine the element with the highest average binding energy per nucleon.

(ii) State and explain whether this element is more or less stable compared to the other elements given.

Answer: a) The nuclear binding energy per nucleon is the energy required to separate the nucleons (protons and neutrons) within a nucleus. The stability of a nucleus is related to its binding energy per nucleon. A higher binding energy per nucleon indicates a more stable nucleus. This is because a stable nucleus has a higher energy state compared to the combined energy of all its individual nucleons. To separate the nucleons, energy must be supplied, resulting in an increase in the potential energy of the system. This potential energy increase is equal to the binding energy per nucleon.

b) (i) To determine the element with the highest average binding energy per nucleon, we need to calculate the average binding energy per nucleon for each element and compare the values.

Average binding energy per nucleon = Total binding energy / Number of nucleons

For Carbon: Carbon has 12 nucleons. Total binding energy of Carbon = 7.68 MeV * 12 = 92.16 MeV Average binding energy per nucleon = 92.16 MeV / 12 = 7.68 MeV

For Oxygen: Oxygen has 16 nucleons. Total binding energy of Oxygen = 7.98 MeV * 16 = 127.68 MeV Average binding energy per nucleon = 127.68 MeV / 16 = 7.98 MeV

For Iron: Iron has 56 nucleons. Total binding energy of Iron = 8.80 MeV * 56 = 492.80 MeV Average binding energy per nucleon = 492.80 MeV / 56 = 8.80 MeV

For Uranium: Uranium has 238 nucleons. Total binding energy of Uranium = 7.59 MeV * 238 = 1807.42 MeV Average binding energy per nucleon = 1807.42 MeV / 238 = 7.59 MeV

Comparing the values, we find that Iron has the highest average binding energy per nucleon (8.80 MeV).

(ii) Iron is more stable compared to the other elements given. A higher average binding energy per nucleon indicates that the nucleons in Iron's nucleus are bound together more strongly, requiring greater energy to separate them. The higher binding energy per nucleon suggests a higher degree of stability, as more energy is needed to disturb the nucleus. Therefore, Iron is more stable compared to Carbon, Oxygen, and Uranium.