Problem 42 - Fundamentals of Physics

Question

What is the minimum energy that is required to break a nucleus of \({ }^{12} \mathrm{C}\) (of mass \(11.99671 \mathrm{u}\) ) into three nuclei of \({ }^{4} \mathrm{He}\) (of mass \(4.00151\) u each \() ?\)

Step-by-Step Solution

Answer

The minimum energy required is approximately 7.283 MeV.

Step 1: Write The Reaction Equation

Consider the nuclear reaction where one nucleus of \(^{12}C\) is transformed into three nuclei of \(^{4}He\): \[ ^{12}C \to 3 imes ^{4}He \] This is a fission process where the carbon nucleus splits into three helium nuclei.

Step 2: Calculate Mass Defect

Calculate the mass defect \( \Delta m \), which is the difference in mass before and after the reaction. Find the total mass of the products and subtract the mass of the reactants: \[ \Delta m = (3 \times 4.00151\, \text{u}) - 11.99671\, \text{u} \] \[ \Delta m = 12.00453\, \text{u} - 11.99671\, \text{u} \] \[ \Delta m = 0.00782\, \text{u} \]

Step 3: Convert Mass Defect to Energy

Use Einstein’s mass-energy equivalence \( E = \Delta m \times c^2 \) to convert the mass defect from atomic mass units to energy in MeV, recalling that 1 u = 931.5 MeV/c²: \[ E = 0.00782\, \text{u} \times 931.5\, \text{MeV/u} \] \[ E \approx 7.283 \text{ MeV} \]

Key Concepts

Mass Defect

In nuclear reactions, the concept of mass defect plays a crucial role. When a nucleus undergoes a reaction, like fission or fusion, the total mass of the resulting products usually differs from the original mass of the reactants. Simply put, mass defect is the difference in mass between the initial and final states of a reaction.
Understanding mass defect involves calculating the total mass before and after a nuclear reaction. For example, in the fission process of a \({ }^{12}C\) nucleus into three \({ }^{4}He\) nuclei, we calculate the mass of the products and subtract the mass of the \({ }^{12}C\) nucleus. This difference is the mass defect. In this exercise, the mass defect \( \Delta m \) is calculated as follows:

• Mass of three \({ }^{4}He\) nuclei: \( 3 \times 4.00151\, \text{u} = 12.00453\, \text{u} \)
• Original mass of \({ }^{12}C\) nucleus: \( 11.99671\, \text{u} \)

Thus, \( \Delta m = 12.00453\, \text{u} - 11.99671\, \text{u} = 0.00782\, \text{u} \).
This mass defect reflects the mass that has been converted into energy during the reaction, according to Einstein's mass-energy equivalence principle.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence is famously represented by the equation \( E = mc^2 \). This equation shows that mass can be converted into energy and vice versa. In nuclear physics, this principle helps explain why nuclear fission and fusion reactions release so much energy.
When you have a mass defect in a nuclear reaction, it indicates that some mass has been transformed into energy. For instance, in our example of the \({ }^{12}C\) nucleus splitting, the mass defect was calculated as \( 0.00782\, \text{u} \).
To find out how much energy corresponds to this mass defect, we apply Einstein's mass-energy equivalence using conversion factors:

• 1 atomic mass unit (u) is equal to 931.5 MeV/c².

Thus, the energy equivalent for our specific mass defect is:

• \( E = 0.00782\, \text{u} \times 931.5\, \text{MeV/u} \approx 7.283\, \text{MeV} \)

This conversion shows how a small amount of mass is equivalent to a surprisingly large amount of energy.

Nuclear Reactions

Nuclear reactions are processes in which the nuclei of atoms interact, leading to changes in their composition. These reactions can release or absorb a massive amount of energy, sometimes involving the conversion of mass into energy.
There are several types of nuclear reactions, including fission, fusion, and radioactive decay. In this exercise, we look at a nuclear fission reaction. Fission involves the splitting of a large nucleus into smaller nuclei, releasing energy due to the conversion of mass to energy. For instance, the transformation of one \( ^{12}C \) nucleus into three \( ^{4}He \) nuclei illustrates this process.
Nuclear reactions are guided by several principles:

• Conservation of energy: The total energy before and after the reaction is conserved, though it may change forms.
• Conservation of mass number: The total mass number remains constant, ensuring that the number of nucleons is preserved.

These reactions are not only foundational to understanding physics but are also the basis for significant applications, such as nuclear power and medicine.