Question

When two protons fuse in a star, the product is \({ }^{2} \mathrm{H}\) plus a positron (Equation 21.26 ). Why do you think the more obvious product of the reaction, \({ }^{2} \mathrm{He},\) is unstable?

Short answer

The more obvious product of proton fusion, \({ }^{2}\mathrm{He}\), is unstable due to its N/Z ratio of 0, which indicates that the repulsive force between the two protons in the nucleus goes unbalanced and leads to instability. In contrast, the actual product \({ }^{2}\mathrm{H}\) has a stable N/Z ratio of 1, with the neutron helping to balance the repulsive force, making it the preferred and more stable product of proton fusion in a star.

Step-by-step solution

Understand Proton Fusion

Proton fusion is a nuclear reaction that takes place in the core of stars. During this process, two protons (hydrogen nuclei) come together under extreme temperatures and pressures to form a new nucleus. In the specific case we're investigating, the fusion reaction produces deuterium (\({ }^{2}\mathrm{H}\)), which is a stable isotope of hydrogen consisting of one proton, one neutron, and one electron, and a positron (a positively charged electron). The potential product we are questioning, \({ }^{2}\mathrm{He}\), is an unstable isotope of helium. It consists of two protons and two electrons, but no neutrons.

Understand Nuclear Stability

Nuclear stability is determined by the balance between the attractive strong nuclear force, which holds protons and neutrons together in the nucleus, and the repulsive electromagnetic force, which causes protons to repel each other. The stability of a nucleus can be predicted by examining the neutron-to-proton ratio (N/Z). If the number of neutrons relative to protons falls within a certain range, usually close to 1 for lighter elements, the nucleus is more likely to be stable. As elements become heavier, the stable N/Z ratio tends to increase slightly.

Examine the N/Z Ratio for \({ }^{2}\mathrm{He}\) and \({ }^{2}\mathrm{H}\)

For the potential product \({ }^{2}\mathrm{He}\), the nucleus contains two protons and no neutrons, which gives us an N/Z ratio of 0/2 or 0. This ratio is far from 1, indicating that the nucleus is unstable due to the overwhelming repulsive force among the protons. On the other hand, for the actual product \({ }^{2}\mathrm{H}\), the nucleus contains one proton and one neutron, which gives us an N/Z ratio of 1/1 or 1. This ratio is close to 1, indicating that the nucleus is stable with a balanced attractive and repulsive forces in the nucleus.

Explain the Stability of \({ }^{2}\mathrm{H}\)

The reason why \({ }^{2}\mathrm{H}\) is more stable than \({ }^{2}\mathrm{He}\) can be explained by the strong nuclear force and the neutron-to-proton ratio. The deuterium nucleus (\({ }^{2}\mathrm{H}\)) has one proton and one neutron, giving it a stable N/Z ratio of 1. This means that the neutron helps to balance the repulsive force between the protons, making the nucleus stable.

Conclude the Reason for \({ }^{2}\mathrm{He}\) Instability

The more obvious product of proton fusion, \({ }^{2}\mathrm{He}\), is unstable because its nucleus has an N/Z ratio of 0, far away from the ideal N/Z ratio for stability. This means that the repulsive force between the two protons in the \({ }^{2}\mathrm{He}\) nucleus goes unbalanced, leading the nucleus to be unstable. On the other hand, the actual product \({ }^{2}\mathrm{H}\) has a stable N/Z ratio, making it the preferred and more stable product of proton fusion in a star.

Key concepts

Nuclear Stability

Nuclear stability plays a crucial role in understanding why some atomic nuclei are stable while others are not. At the heart of nuclear stability is a delicate balance between two primary forces:

• The **strong nuclear force**: This attractive force holds protons and neutrons (nucleons) together in the nucleus. It is very strong but acts over a very short range.
• The **electromagnetic force**: This repulsive force exists between protons due to their positive charges. It is weaker than the strong nuclear force but acts over a longer range.

When these forces are balanced, the nucleus remains stable. If the repulsive force overtakes the attractive force, the nucleus becomes unstable, leading to potential decay or transformation. Understanding nuclear stability helps explain why certain isotopes are common in nature while others are rare or nonexistent. Stable nuclei tend to have a balance where these forces naturally reach equilibrium, providing the foundation for chemical elements.

Neutron-to-Proton Ratio

The neutron-to-proton ratio (\( \frac{N}{Z} \)) is a significant factor in determining nuclear stability. Here's why this ratio is essential:

• For lighter elements, a 1:1 ratio (one neutron for each proton) often indicates stability. This balance helps manage the repulsive forces between protons, as neutrons act as a buffer.
• For heavier elements, more neutrons are usually needed to maintain stability. The ratio increases slightly, as the additional neutrons help counteract the stronger electromagnetic forces arising from more protons.
• If the neutron-to-proton ratio falls outside the stable range, the nucleus is likely unstable and may undergo radioactive decay to reach a more stable state.

In our example, **deuterium** has a 1:1 ratio with one neutron and one proton, resulting in a stable nucleus. However, \( ^2 \mathrm{He} \), composed only of two protons, lacks neutrons to balance the repulsive electromagnetic force, making it unstable.

Deuterium

Deuterium is an isotope of hydrogen characterized by its unique composition. Unlike regular hydrogen, deuterium's nucleus contains one neutron in addition to one proton. This difference makes deuterium significant in various scientific applications.

• **Stability**: The presence of a neutron allows deuterium to have a balanced 1:1 neutron-to-proton ratio, contributing to its stability. This stability makes it a more common outcome in nuclear reactions such as those occurring in stars.
• **Relevance in stars**: Deuterium plays an essential role in nuclear fusion processes within stars. When stars form deuterium from proton fusion, they also produce a positron (a positively charged electron), illustrating the centrality of deuterium in these high-energy environments.
• **Applications**: Deuterium is used in various fields, including nuclear reactors, as a tracer in biochemistry, and in nuclear magnetic resonance (NMR) spectroscopy, demonstrating its versatility and importance in science and industry.

By understanding deuterium's properties and applications, we gain insight into both natural processes in the stars and practical uses in technology and research.