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 $^{12}_{6}$C nuclei into one $^{24}_{12}$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 fm, center-to-center? ($c$) Approximately what temperature would this require?

Key Concepts

Electrostatic Potential Energy

The potential energy associated with two charged particles at a given separation distance is calculated using Coulomb’s law. This energy consideration is important in estimating the kinetic energy that reacting nuclei must possess to overcome the repulsive forces, enabling them to approach each other to distances where nuclear forces take over.

Kinetic Energy and Temperature Relation

The relationship between kinetic energy and temperature is anchored in statistical mechanics, particularly using the Boltzmann constant. Converting a required kinetic energy per particle into an equivalent temperature provides insight into the thermal conditions needed for fusion to occur, linking microscopic energy scales to macroscopic temperature measurements.

Nuclear Binding Energy and Mass Defect

In nuclear reactions, the difference between the mass of the reactants and the products (known as the mass defect) corresponds to an energy release according to Einstein’s equation, E=mc². This concept is fundamental in calculating the energy released during fusion, as the mass loss is converted into energy, making the correct assessment of binding energies critical for understanding fusion processes.

Coulomb Barrier

The Coulomb barrier arises from the electrostatic repulsion between positively charged nuclei. Overcoming this repulsion is essential for nuclear fusion to occur, as the nuclei must come sufficiently close for the strong nuclear force to bind them together. Calculating this barrier helps determine the minimum kinetic energy necessary for the reacting nuclei to approach each other within the effective range of the nuclear force.

Transcript

00:01 For next question, we are looking at the energy that's released where we fuse two carbon nuclei into a single magnesium nuclei.

00:17 Now this is the fusion reaction equation and we want to find what is the energy released, which is the q value.

00:25 So we just take the change in the mass of the nucleides multiplied by the c square.

00:32 In this case, we'll be taking two times of the mass of the carbon truble.

00:37 Minus away the mass of our magnesium nucleate.

00:44 So this carbon charge of this magnesium.

00:51 Multiply this by c square.

00:54 C square we can use 931 .5 mavre per u.

00:59 Since we are dealing with atomic mass units, the mass of the magnesium nucleate is 23 .985 and the mass of the carbon, nucleate is 12.

01:17 From this, we should get a value of 13 .93 mvv.

01:24 So this is the energy that's released.

01:30 Now, the next question actually involves how the nuclei actually feels.

01:35 So the amount of energy that is required to bring these two nuclei together is actually dependent on the nuclear potential energy when the nuclei are at this particular distance.

01:51 So at this particular distance, the centers are given to be six firmies apart.

01:59 So what you've got to do is to find the potential difference and this potential, sorry, not potential difference, but potential energy.

02:09 So this potential energy must be equals to the total kinetic energy of the system.

02:17 When they are very far apart.

02:19 So this equals to two times of the kinetic individual kinetic energies of the nuclei.

02:25 We want to find what is the individual kinetic energy...

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