Some stars, in a later stage of evolution, may begin to fuse two (12)/(6) C nuclei into one (24)

step 1 In this question we are given that at some stage of stars, they will start to fuse carbon nucleu

Some stars, in a later stage of evolution, may begin to fuse...

Some stars, in a later stage of evolution, may begin to fuse two $\frac{12}{6} \mathrm{C}$ nuclei into one $\frac{24}{12} \mathrm{Mg}$ nucleus (a) How much energy would be released in such a reaction? (b) What kinetic energy must two carbon nuclei each have when far apart, if they can then approach each other to within $6.0 \mathrm{fm},$ center-to-center? (c) Approximately what temperature would this require?

Some stars, in a later stage of evolution, may begin to fuse two $\frac{12}{6} \mathrm{C}$ nuclei into one $\frac{24}{12} \mathrm{Mg}$ nucleus (a) How much energy would be released in such a reaction? (b) What kinetic energy must two carbon nuclei each have when far apart, if they can
then approach each other to within $6.0 \mathrm{fm},$ center-to-center?
(c) Approximately what temperature would this require?

Key Concepts

Thermal Energy and Temperature Estimation

Temperature in a stellar context is related to the average kinetic energy of particles. Estimating the temperature corresponding to a given kinetic energy involves using the relation E ~ kT, where k is Boltzmann’s constant. This estimate helps determine the thermal conditions necessary for overcoming the Coulomb barrier in fusion reactions.

Kinetic Energy Considerations

The kinetic energy required for the nuclei is determined by the need to overcome the Coulomb barrier. This involves calculating the electrostatic potential energy at the distance where the nuclei must approach each other, which sets a lower limit on the kinetic energy of each nucleus when they are far apart.

Mass Defect and Energy Release

The mass defect refers to the difference between the total mass of the reacting nuclei and the mass of the resulting nucleus. This difference, when multiplied by c² (the speed of light squared), yields the energy released during the reaction. This principle is fundamental in explaining energy production in stellar environments.

Nuclear Fusion

Nuclear fusion is the process by which two light atomic nuclei combine to form a heavier nucleus. This process releases energy because the mass of the resulting nucleus is less than the sum of the masses of the individual nuclei, with the missing mass being converted into energy according to the mass-energy equivalence principle.

Coulomb Barrier

The Coulomb barrier is the energy barrier due to electrostatic repulsion between two positively charged nuclei, such as carbon nuclei. For fusion to occur, the nuclei must have sufficient kinetic energy to overcome this repulsive force and come into close proximity so that the strong nuclear force can take over and bind them together.

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