Problem 97 - Chemistry A Molecular Approach

Question

Given that the energy released in the fusion of two deuterons 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.

Step-by-Step Solution

Answer

The energy change for the process \({}^{3}He + {}^{1}n \longrightarrow {}^{3}H + {}^{1}P\) is \(7.3 MeV\). This process occurs at lower temperatures because it is exothermic, meaning it releases energy, making it energetically favorable.

Step 1: Identifying the Energy Released in Each Reaction

First, identify the energy released when two deuterons fuse into \({}^{3}He\) and a neutron, which is \(3.3 MeV\) and when they fuse into tritium and a proton, which is \(4.0 MeV\). These values are given in the problem.

Step 2: Writing the Energy Changes for the Individual Processes

Express the individual reactions and their energy releases: \(\text{Deuteron fusion to } {}^{3}He + \text{n}: 3.3 MeV\) \(\text{Deuteron fusion to } {}^{3}H + \text{p}: 4.0 MeV\). The question asks to calculate the energy change for \(\text{}^{3}He + \text{}^{1}H \longrightarrow \text{}^{3}H + \text{}^{1}P\), which can be thought of as the reverse of the deuteron fusion processes.

Step 3: Calculating the Energy Change of the Given Process

The energy change of the process \(^{3}He + {}^{1}n \longrightarrow {}^{3}H + {}^{1}P\) will be equal to the difference between the energy released in forming \(^{3}H + {}^{1}P\) (4.0 MeV) and the energy required to break up \(^{3}He + {}^{1}n\) (-3.3 MeV, since we are reversing the process). The net energy change is therefore \(4.0 MeV - (-3.3 MeV) = 4.0 MeV + 3.3 MeV = 7.3 MeV\).

Step 4: Providing an Explanation for the Temperature Requirement

As the resulting energy change of the process is positive (\(7.3 MeV\)), it indicates that the process is exothermic and thus favorable in terms of energetics. Reactions occurring at lower temperatures are typically those that are energetically favorable, meaning they release energy. Therefore, as this process is exothermic, it is likely to occur at lower temperatures compared to the initial deuteron fusion processes.

Key Concepts

Nuclear Fusion

Nuclear fusion is a compelling topic in the realm of physics, involving the merger of two light atomic nuclei to form a heavier nucleus. This process is not only at the heart of powering our Sun but it is also a promising energy source for the future. During fusion, a significant amount of energy is released, which is derived from the strong nuclear forces that bind protons and neutrons together in the nucleus.

Understanding nuclear fusion requires a grasp of how these forces overcome the natural repulsion between positively charged protons at very high temperatures and pressures. In man-made fusion reactors, achieving the conditions necessary for fusion is a complex challenge. The energy released from nuclear fusion comes from a small fraction of mass that is converted into energy, described by Einstein's famous equation, \( E=mc^2 \), where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light.

When deuterons, which are isotopes of hydrogen with one proton and one neutron, undergo fusion, they can create different products such as helium-3 (\(^{3}He\)) and a neutron (\(n\)), or tritium (\(^{3}H\)) and a proton (\(p\)), releasing substantial amounts of energy in the process. These are just two possible outcomes in the vast landscape of fusion reactions; each with its own characteristic energy yield.

Energy Change in Reactions

The concept of energy change in reactions is a cornerstone of chemical and nuclear physics, reflecting the conservation of energy principle. Every reaction involves a change in energy due to the breaking and forming of bonds or, in the case of nuclear reactions, changes in the nucleus. Such energy changes can either absorb energy (endothermic reaction) or release energy (exothermic reaction).

In nuclear fusion, for example, when lighter nuclei combine to form a heavier nucleus, the energy difference between the initial and final states is emitted as kinetic energy, often in the form of gamma rays or as kinetic energy of the products. This is due to the binding energy per nucleon increasing as you go from light to medium mass nuclei, which makes these reactions exothermic. The concept of binding energy is crucial here, as it is a measure of how tightly bound an electron, proton, or neutron is within an atom or a nucleus.

The calculation of energy change for the helium-3 and neutron fusion to tritium and proton shows a significant positive energy yield of \(7.3 MeV\). The positive figure signifies that this particular nuclear conversion releases energy, hence it is exothermic. This is a prime example of how understanding the energy profile of a reaction can explain the conditions, such as temperature, under which the reaction is likely to occur.

Neutron and Proton Interactions

Neutrons and protons, collectively known as nucleons, are the building blocks of atomic nuclei, and their interactions are fundamental to the stability of atoms. The strong nuclear force, which acts at very short ranges, is responsible for the attractive interactions that hold the nucleons together inside the nucleus despite the repulsion between protons due to their positive charge.

When we delve into the realm of nuclear reactions, the behavior of neutrons and protons becomes even more intriguing. Neutrons, having no charge, are particularly effective at penetrating the nucleus without being repelled, making them essential players in initiating and sustaining nuclear reactions. This makes reactions involving neutrons, such as the one that converts helium-3 into tritium with a subsequent proton release, more likely to occur at lower energies compared to reactions involving solely charged particles like protons.

The interaction energy between neutrons and protons, which in part determines the binding energy of the nucleus, is responsible for significant yield in fusion reactions. Keeping track of neutron and proton interactions is not only crucial for predicting the energy outcome, as seen in the provided exercise, but also for understanding the broader conditions, such as temperature and pressure, that facilitate these atomic dance partners to join in a nuclear fusion.