Question

The binding energy of \({ }_{2}^{3} \mathrm{He}\) is lower than that of \({ }_{1}^{3} \mathrm{H}\). Provide a plausible explanation, taking into consideration the Coulomb interaction between two protons in \({ }_{2}^{3} \mathrm{He}\).

Short answer

Based on the above analysis, provide a plausible explanation for why the binding energy of Helium-3 (\({ }_{2}^{3}\mathrm{He}\)) is lower than that of Tritium (\({ }_{1}^{3}\mathrm{H}\)). The binding energy of Helium-3 (\({ }_{2}^{3}\mathrm{He}\)) is lower than that of Tritium (\({ }_{1}^{3}\mathrm{H}\)) mainly due to the repulsive Coulomb interaction between the two protons in the Helium-3 nucleus. While the nuclear force contributions to the binding energies of both nuclei are more or less the same, the presence of two protons in Helium-3 results in a decrease in binding energy due to their repulsion. In contrast, Tritium has only one proton and hence does not experience repulsion from the Coulomb force, leading to a higher binding energy than Helium-3.

Step-by-step solution

State the composition of \({ }_{2}^{3}\mathrm{He}\) and \({ }_{1}^{3}\mathrm{H}\) nuclei

The Helium-3 nucleus consists of 2 protons and 1 neutron (\({ }_{2}^{3}\mathrm{He}\)), while Tritium consists of 1 proton and 2 neutrons (\({ }_{1}^{3}\mathrm{H}\)).

Compare the nuclear forces in \({ }_{2}^{3}\mathrm{He}\) and \({ }_{1}^{3}\mathrm{H}\) nuclei

In both nuclei, there are two nuclear force interactions: one between the proton and neutron, and one between the neutrons. Since the strength of the nuclear force is more or less the same for both isotopes, there is no significant difference in the nuclear force contributions to the binding energies.

Discuss the effect of the Coulomb force in the \({ }_{2}^{3}\mathrm{He}\) nucleus

In the \({ }_{2}^{3}\mathrm{He}\) nucleus, there are two protons which will repel each other due to the Coulomb force. This repulsion results in a reduction of the binding energy in \({ }_{2}^{3}\mathrm{He}\).

Discuss the effect of the Coulomb force in the \({ }_{1}^{3}\mathrm{H}\) nucleus

In the \({ }_{1}^{3}\mathrm{H}\) nucleus, there is only one proton, so there is no repulsion due to the Coulomb force.

Conclude the reason for the lower binding energy of \({ }_{2}^{3}\mathrm{He}\) compared to \({ }_{1}^{3}\mathrm{H}\)

Since the nuclear forces for both nuclei are more or less the same, the main difference comes from the Coulomb interaction between the protons in the Helium-3 nucleus. The repulsion between the two protons reduces the binding energy in the \({ }_{2}^{3}\mathrm{He}\) nucleus, making it lower than the binding energy of the \({ }_{1}^{3}\mathrm{H}\) nucleus, which has only one proton and thus does not experience repulsion from the Coulomb force.

Key concepts

Coulomb Force

The Coulomb force, also known as the electrostatic force, is a fundamental interaction between charged particles. In the context of atomic and nuclear physics, it specifically refers to the force between protons, which are positively charged.

According to Coulomb's law, the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as: \[ F = k \frac{{|q_1q_2|}}{{r^2}} \] where \( F \) is the force, \( q_1 \) and \( q_2 \) are the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant.

In a nucleus with more than one proton, such as the Helium-3 nucleus, the Coulomb force acts as a repulsive interaction, counteracting the attractive strong nuclear force and thus reducing the nucleus's overall binding energy. This is why Helium-3, with two protons, has a lower binding energy than Tritium, which has only one proton and therefore experiences less electrostatic repulsion.

Helium-3 Nucleus

The Helium-3 (\( {}_{2}^{3}\text{He} \) nucleus is composed of two protons and one neutron. The presence of two protons means that the electrostatic repulsion within this nucleus will be significantly greater than in a nucleus with only one proton.

Despite the strong nuclear force trying to hold the nuclear particles together, the additional repulsion due to the Coulomb force in the Helium-3 nucleus must be overcome to maintain stability. This repulsion contributes to a lower binding energy for Helium-3, as energy must be expended to counteract the proton-proton repulsion.

The lower binding energy can make Helium-3 less stable compared to other isotopes with a stronger binding force between the nuclear components. However, Helium-3 is still a stable isotope due to the balance of forces within the nucleus.

Tritium Nucleus

Tritium, with its nuclear composition represented as (\( {}_{1}^{3}\text{H} \) or Hydrogen-3, contains one proton and two neutrons. The single proton means that the Coulomb repulsion within the Tritium nucleus is significantly less than in the Helium-3 nucleus.

This lower repulsive force results in a higher binding energy for the Tritium nucleus. As a result, it is more tightly bound and more stable against decay than Helium-3. While Tritium is a radioactive isotope and thus unstable in that regard, its instability comes from the weak force and not from Coulomb repulsion which is the focus in the context of stability due to binding energy.

Nuclear Forces

Nuclear forces, primarily the strong nuclear force, are the interactions that hold the nucleons (protons and neutrons) together within the nucleus. The strong force is one of the four fundamental forces of nature and is the strongest force over short distances, such as those within the atomic nucleus.

It acts equally between protons and neutrons, irrespective of their charge, and has a very short range. Outside of this range, the force drops off exponentially, making it a purely nuclear force, with no long-range effects like the Coulomb force.

In the context of the Helium-3 and Tritium nuclei, while the strong nuclear force acts to maintain the integrity of the nucleus, its strength is consistent across both types of nuclei. Therefore, the main differentiator in binding energy is a result of the Coulomb force, as the strong force is relatively uniform for both isotopes.