Carbon detonations are powerful nuclear reactions that...

Carbon detonations are powerful nuclear reactions that temporarily tear apart the cores inside massive stars late in their lives. These blasts are produced by carbon fusion, which requires a temperature of approximately $6 \times 10^{8} \mathrm{K}$ to overcome the strong Coulomb repulsion between carbon nuclei. (a) Estimate the repulsive energy barrier to fusion, using the temperature required for carbon fusion. (In other words, what is the average kinetic energy of a carbon nucleus at $6 \times 10^{8} \mathrm{K}^{2}$ ) (b) Calculate the energy (in MeV) released in each of these "carbon-burning" reactions: $$ \begin{array}{l}{^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow^{20} \mathrm{Ne}+^{4} \mathrm{He}} \\ {^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow^{24} \mathrm{Mg}+\gamma}\end{array} $$ (c) Calculate the energy in kilowatt-hours given off when 2.00 $\mathrm{kg}$ of carbon completely fuses according to the first reaction.

Carbon detonations are powerful nuclear reactions that temporarily tear apart the cores inside massive stars late in their lives. These blasts are produced by carbon fusion, which requires a temperature of approximately $6 \times 10^{8} \mathrm{K}$ to overcome the strong Coulomb repulsion between carbon nuclei. (a) Estimate the repulsive energy barrier to fusion, using the temperature required for carbon fusion. (In other words, what is the average kinetic energy of a carbon nucleus at $6 \times 10^{8} \mathrm{K}^{2}$ ) (b) Calculate the energy (in MeV) released in each of these "carbon-burning" reactions:
$$
\begin{array}{l}{^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow^{20} \mathrm{Ne}+^{4} \mathrm{He}} \\ {^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow^{24} \mathrm{Mg}+\gamma}\end{array}
$$
(c) Calculate the energy in kilowatt-hours given off when 2.00 $\mathrm{kg}$ of carbon completely fuses according to the first reaction.

Key Concepts

Energy Unit Conversions

In the context of nuclear physics and astrophysics, energy is often expressed in various units such as electron volts (eV), mega-electron volts (MeV), joules, and kilowatt-hours. Mastery of energy unit conversions is important for comparing and contextualizing nuclear reaction energies with more familiar units of energy, facilitating communication between theoretical predictions and practical implications.

Nuclear Fusion Reaction Energetics

This concept involves the calculation of the energy released in nuclear fusion reactions, often determined by the Q-value, which is the difference in binding energy between the reactants and the products. Understanding nuclear reaction energetics is essential for evaluating how much energy is liberated during fusion processes, which is key to both astrophysical phenomena and potential energy applications.

Coulomb Barrier

The Coulomb barrier is the energy barrier due to electrostatic repulsion that charged nuclei must overcome in order to get close enough for the strong nuclear force to act and cause fusion. It is a crucial concept in the study of fusion reactions because it determines the minimum kinetic energy required for the nuclei to fuse, directly linked to the needed temperature for ignition in stellar environments.

Thermal Kinetic Energy

This concept relates to the average kinetic energy of particles in a system at a given temperature. In nuclear physics, it is used to estimate the energy available for particles to overcome barriers, such as the Coulomb repulsion between positively charged nuclei. The average thermal kinetic energy is often represented as kT (where k is Boltzmann’s constant and T is the temperature), which provides a rough measure of the energy scale for nuclear reactions.

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