Question

The binding energy of \({ }_{2}^{3}\) He is lower than that of \({ }_{1}^{3} \mathrm{H} .\) Provide a plausible explanation, taking into consideration the Coulomb interaction between two protons in \({ }_{2}^{3}\) He. a) charge b) the number of nucleons, \(A\) c) mass-energy d) linear momentum e) angular momentum

Short answer

Answer: The binding energy is lower in ${ }_{2}^{3}He$ due to the repulsive Coulomb interaction between the two protons in the nucleus, which reduces the binding energy compared to ${ }_{1}^{3}\mathrm{H}$ which has only one proton and no Coulomb interaction.

Step-by-step solution

Understand the composition of nuclei

First, let's get familiar with the composition of the two nuclei involved. \({ }_{2}^{3}He\) has 2 protons and 1 neutron, whereas \({ }_{1}^{3}\mathrm{H}\) has only 1 proton and 2 neutrons. Keep in mind that protons have a positive charge and neutrons have no charge.

Understand the role of Coulomb interaction

Coulomb interaction is the force between charged particles, such as between the protons within a nucleus. In general, the larger the distance between two charged particles, the smaller the Coulomb force and vice versa. Thus, the repulsive Coulomb force between protons contributes to the binding energy of a nucleus.

Examine the Coulomb interaction in \({ }_{2}^{3}He\) and \({ }_{1}^{3}\mathrm{H}\)

In \({ }_{2}^{3}He\), there are two protons, so there is a repulsive Coulomb force between them, which offsets some of the attractive nuclear forces, making the binding energy lower in this nucleus compared to \({ }_{1}^{3}\mathrm{H}\). In \({ }_{1}^{3}\mathrm{H}\), there is only one proton and two neutrons, so there is no repulsive Coulomb interaction between the protons. As a result, the binding energy of \({ }_{1}^{3}\mathrm{H}\) is higher than \({ }_{2}^{3}He\).

Identify the main factor responsible for the difference in binding energy

The primary factor responsible for the difference in binding energies between \({ }_{2}^{3}He\) and \({ }_{1}^{3}\mathrm{H}\) is the Coulomb interaction between the protons in each nucleus. In \({ }_{2}^{3}He\), the repulsive Coulomb force between the two protons effectively reduces the binding energy, whereas in \({ }_{1}^{3}\mathrm{H}\), since there is only one proton, there is no Coulomb interaction to decrease its binding energy. The correct answer is (a) charge.

Key concepts

Coulomb Interaction

To understand nuclear binding energy, the concept of Coulomb interaction plays a pivotal role. This fundamental force acts between entities carrying an electric charge. In atomic nuclei, the Coulomb interaction refers to the repulsive force between protons, each of which possesses a positive charge.

Given that like charges repel each other, in a helium-3 ({ }_{2}^{3}He) nucleus with two protons, this repulsion is significant. The energy required to bind the protons together in the face of this repulsion contributes to the nuclear binding energy. Conversely, in a tritium ({ }_{1}^{3}H) nucleus, with only one proton, there is no such proton-proton repulsion.

It is the strength of this Coulomb repulsion in helium-3 that leads to a lower binding energy compared to tritium, since more energy is needed to overcome the repulsion and hold the nucleus together.

Nucleon Composition

Nuclei are composed of nucleons—protons and neutrons. The composition and arrangement of these subatomic particles are crucial for the stability of a nucleus. Each proton and neutron contribute to the overall mass and energy of an atomic nucleus, but they play different roles due to their distinct properties.

Protons, for example, contribute to both the strong nuclear force that binds nucleons together and the Coulomb repulsive force that tends to push them apart due to their positive charge. Neutrons, being neutral, do not engage in Coulomb interactions, but they add to the attractive nuclear forces that help stabilize the nucleus. The overall balance between these attractive and repulsive forces determines the nuclear binding energy, which is why helium-3, with two protons, has a lower binding energy than tritium, with only one.

Nuclear Forces

Nuclear forces refer to the interactions that hold the nucleus together, overcoming the repulsive Coulomb forces between positively charged protons. These forces are predominantly the strong nuclear force, which is one of the fundamental forces of nature.

The strong force acts between all nucleons, irrespective of their charge, and it is much stronger than the Coulomb force but operates over a much shorter range. It is this strong interaction that allows protons and neutrons to bind together despite the electromagnetic repulsion between protons. The nuclear forces working within the tritium ({ }_{1}^{3}H) nucleus easily overcome the minimal Coulomb repulsion due to the presence of just one proton, resulting in a higher binding energy. The helium-3 ({ }_{2}^{3}He) nucleus, however, must contend with stronger Coulomb repulsion between its two protons, necessitating additional strong force interaction to maintain stability, and thus exhibits a lower binding energy.