Given that the energy released in the fusion of two deuterium atoms to a 3 He and a neutron is \(3.3 \mathrm{MeV},\) and in the fusion to tritium and a proton it is 4.0 \(\mathrm{MeV}\) . Calculate the energy change for the process \(^{3} \mathrm{He}+1 \mathrm{n} \longrightarrow^{3} \mathrm{H}+1 \mathrm{p} .\) Suggest an explanation for why this process occurs at much lower temperatures than either of the first two.

Nuclear fusion is a reaction in which two lighter atomic nuclei combine to form a heavier nucleus while releasing energy. This process powers our sun and other stars, as the extreme pressure and temperatures in their cores force nuclei to collide with sufficient force to overcome the repulsive electric forces between them.
In the fusion of two deuterium atoms, one example involves the production of a helium-3 nucleus and a neutron. Deuterium, or heavy hydrogen, possesses an additional neutron compared to the common hydrogen nucleus. When two deuterium nuclei merge, they form a new isotopes and release a specific amount of energy measured in million electron volts (MeV).
Understanding the principles of nuclear fusion is crucial as it promises a future of practically limitless and clean energy, should we learn how to control and sustain the reaction efficiently on Earth. Simplifying, nuclear fusion is the process of 'sticking together' light atoms under extreme conditions to release an immense amount of energy, which is several times greater than that of nuclear fission, the principle behind today's nuclear reactors.

During nuclear reactions, energy is either absorbed or released, chiefly guided by the principle of mass-energy equivalence, encapsulated by Einstein's famous equation, E=mc^2. In the context of fusion, the mass of the resulting nucleus is slightly less than the sum of the masses of the original nuclei. The 'missing' mass is what's been converted into energy.
In our example problem, the fusion of deuterium atoms can lead to different products, each with a distinct energy signature. By measuring the energy released in fusion to either helium-3 or tritium, we can deduce the energy change for subsequent reactions involving these nuclei. The fact that the energy released when forming tritium is higher than when forming helium-3 (4.0 MeV vs 3.3 MeV) results in an energy change, when helium-3 goes on to react with a neutron to form tritium and a proton.
Computing this energy change is foundational to understanding nuclear reactions as it reflects the nature and feasibility of such processes and their potential applications in energy generation, like fusion power plants.

To achieve nuclear fusion, the temperature must be incredibly high, often reaching millions of degrees. This is because nuclei are positively charged, and like charges repel. Only at immense temperatures, where particles have sufficiently high velocities, can these repulsive forces be overcome, allowing the nuclei to come close enough for the strong nuclear force to take over and bind them together.
However, not all fusion reactions require the same temperature conditions. The reaction between helium-3 and a neutron to produce tritium and a proton can take place at much lower temperatures than the fusion of deuterium nuclei. The primary reason for this diminished thermal requirement is that reaction products, such as helium-3 nuclei and neutrons, do not have to overcome the same substantial electrostatic barriers that deuterium nuclei do. This is because neutrons are neutral and do not repel the nucleus they interact with, facilitating the fusion at a lower kinetic energy, which is directly related to temperature.
Appreciating these temperature dynamics is essential for developing practical fusion energy solutions. It informs the condition designs of reactors such as tokamaks or inertial confinement systems, indicating a path forward in harnessing fusion as a viable source of power.