How much energy would be released if six hydrogen atoms and six neutrons were combined to form \(^{12}_{6} \mathrm{C} ?\)

Imagine the famous equation from Albert Einstein, E=mc^2, which allows us to connect the realms of mass and energy, a concept known as mass-energy equivalence. Essentially, it tells us that mass can be converted into energy and vice versa. That 'c' in the equation stands for the speed of light, which is a rather large number (\(3 \times 10^8 \text{ m/s}\)), making 'c^2' extremely large. Therefore, even a small amount of mass (\(m\)) can be converted into a huge amount of energy (\(E\)). This isn't just a theoretical concept — it's the core principle behind nuclear power and weapons, and it is also why the sun and other stars shine, through the process of nuclear fusion. The energy calculated in our textbook exercise is derived from this fundamental principle.

In practical terms, for the exercise we discussed, this means that when several small particles like hydrogen atoms come together to form a heavier nucleus, like that of carbon, they lose some mass. That lost mass is not gone — instead, it is transformed into energy, which follows from the mass-energy equivalence concept.

Mass defect is a term we use to describe the phenomenon where the mass of a nucleus is less than the sum of its individual nucleons (protons and neutrons). This 'missing' mass is key to understanding nuclear fusion and fission. The steps of the textbook problem guide us through calculating the mass defect by subtracting the mass of the final nucleus from the total mass of the individual protons and neutrons before they combined. This difference represents the binding energy, the energy needed to hold the nucleus together, or conversely, the energy released when the nucleus is formed from these protons and neutrons.

Using the example from our exercise, the combined initial mass of six hydrogen atoms and six neutrons is greater than the mass of the resulting carbon-12 nucleus. The mass defect quantifies that difference and, when plugged into Einstein's equation, tells us exactly how much energy we can expect to be released during the process.

The atomic mass unit (amu) is a standard unit of mass that quantifies mass on an atomic or molecular scale. It's defined as one twelfth of the mass of an unbound neutral atom of carbon-12 at rest and in its ground state. Because atoms and subatomic particles are so minuscule, standard units of mass like grams or kilograms aren't practical — they would involve inconveniently small decimal numbers. This is where amus come in handy, helping scientists and students work with more manageable figures.

In our example, we used the fact that the mass of a carbon-12 nucleus is defined as 12 amu. By comparing this with the initial mass of the hydrogen atoms and neutrons in amus, we were able to calculate the mass defect in the same unit, making it simpler to then convert that mass into energy (using Einstein's equation again), a crucial step for solving the problem at hand.

Nuclear fusion is a powerful process that occurs when two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. It's the exact opposite of nuclear fission, where a heavy nucleus splits into two lighter nuclei. Fusion powers the sun and stars, as hydrogen atoms fuse to form helium and other elements in stellar cores under extremely high pressure and temperature conditions.

In the educational exercise we're considering, the fusion of six hydrogen atoms and six neutrons to form a carbon-12 nucleus is a simplified model of what happens in stars, albeit under different conditions. Understanding fusion is not only fundamental to astrophysics but also has practical implications. Researchers are working tirelessly to harness nuclear fusion as a clean energy source here on Earth. This technology has the potential to provide a near-limitless supply of energy with relatively benign waste products compared to current nuclear fission reactors.