Question

11. Use a one dimensional particle in a box model for the nucleus to answer the following questions. a. Explain why a nucleus with two neutrons and two protons has less kinetic energy than a nucleus with 3 neutrons and one proton. b. Why do nuclei with large numbers of nucleons have an excess of neutrons over protons?

Answer 1

(a) From the particle in a box model we know that for the ground state energy, the expression is,

T E = 2ma

From this we can see that the ground state energies are inversely proportional to the mass of the particles. Hence in the first case 2 protons and 2 neutrons have lesser mass as compared to 3 neutrons and one proton. But still 2 protons and 2 neutrons have lesser kinetic energy, this is due to the repulsive interaction between the two protons, a part of energy goes to the coulombic potential created, whereas in the case of 3 neutrons and 1 proton, there is no coloumbic interaction as there is only one proton, so no energy is used up for creating the potential energy

(b) This coloumbic interaction between the protons is the reason why neutrons are more in number in heavier nuclei. This is because as the number of protons increase the repulsive forces between them also increases and the nucleas would become unstable. But when the number of neutrons are increased, there is no repulsion, and the strong nuclear force helps to stabilise the nucleas more as compared to addition of proton. So nucleas is more stable if neutron number is more, hence the number of neutrons increases for heavier elements.