Given that the energy released in the fusion of two deuterons to a ³He and a neutron is 3.3 MeV, and in the fusion to tritium and a proton it is 4.0 MeV, calculate the energy change for the process ³He + ¹n → ³H + ¹p. Suggest an explanation for why this process occurs at much lower temperatures than either of the first two.

Verified step by step guidance

1. Identify the given reactions and their energy changes: \(2 \text{D} \rightarrow \text{^3He} + \text{n} + 3.3 \text{ MeV}\) and \(2 \text{D} \rightarrow \text{^3H} + \text{p} + 4.0 \text{ MeV}\).
2. Write the target reaction: \(\text{^3He} + \text{n} \rightarrow \text{^3H} + \text{p}\).
3. Use the principle of conservation of energy to relate the given reactions to the target reaction. Consider the energy changes involved in the formation of \(\text{^3He}\) and \(\text{^3H}\).
4. Calculate the energy change for the target reaction by considering the difference in energy released between the two given reactions.
5. Discuss why the target reaction might occur at lower temperatures, considering factors such as the kinetic energy of particles, potential energy barriers, and the nature of the particles involved.

Key Concepts

Nuclear Fusion

Nuclear fusion is the process where two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This energy release occurs because the mass of the resulting nucleus is less than the sum of the masses of the original nuclei, with the mass difference converted into energy according to Einstein's equation, E=mc². Understanding fusion is crucial for analyzing energy changes in nuclear reactions.

Energy Release in Fusion Reactions

The energy released during fusion reactions varies depending on the specific isotopes involved and the reaction pathway. In the given question, the fusion of deuterons produces different products with distinct energy outputs, which can be quantified in MeV (mega-electronvolts). This energy difference is essential for calculating the energy change in subsequent reactions, such as the one involving 3He and a neutron.

Temperature and Reaction Rates

Temperature plays a critical role in nuclear fusion, as it affects the kinetic energy of the particles involved. Higher temperatures provide the necessary energy to overcome the electrostatic repulsion between positively charged nuclei. The process involving 3He and a neutron occurs at lower temperatures because it has a lower Coulomb barrier compared to the fusion of deuterons, allowing the reaction to proceed more readily under less extreme conditions.