The fusion reaction \({ }_1^2 \mathrm{H}+{ }_1^3 \mathrm{H} \rightarrow{ }_2^4 \mathrm{He}+\mathrm{n}\) releases about \(17.5 \mathrm{MeV}\). The available energy is \(14.1 \mathrm{MeV}\) given to the neutron. The world's energy consumption in 2013 has been estimated as \(4 \times 10^{20} \mathrm{~J}\). About one out of every 6420 hydrogen atoms in the earth's oceans is deuterium. The number of kilograms of water that would be enough to supply the world's annual energy needs is most nearly (A) \(10^8 \mathrm{~kg}\) (B) \(10^9 \mathrm{~kg}\) (C) \(10^{10} \mathrm{~kg}\) (D) \(10^{11} \mathrm{~kg}\) (E) \(10^{12} \mathrm{~kg}\)

Nuclear fusion is a process where two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This is the same type of reaction that powers the sun and other stars, where hydrogen atoms fuse to form helium. In the textbook exercise, a fusion reaction combines deuterium (hydrogen with one neutron) and tritium (hydrogen with two neutrons) to create a helium nucleus and a free neutron.

Each fusion reaction releases a considerable amount of energy, characteristically measured in millions of electron volts (MeV). In our day-to-day experience, energy is more commonly referred to in joules (J). The conversion from MeV to joules is crucial to understand the large-scale energy yield of fusion reactions. Converting these immense energies into more tangible units allows one to appreciate the potential of nuclear fusion as an energy source for our planet.

Energy conversion calculations are essential to compare different forms and scales of energy. As mentioned, physicists often use units like MeV to quantify the energy released in nuclear reactions. However, to relate this to practical energy requirements, such as the world's annual energy consumption, we convert MeV to joules.

To convert the energy released per fusion reaction to joules, we use the relationship 1 MeV = 1.60218 x 10^{-13} J. This step transforms the abstract concept of nuclear energy into a form that can be used to calculate how much water would be needed to fulfil global energy needs through nuclear fusion. It highlights the incredible efficiency of nuclear fusion, where a small amount of matter can release vast amounts of energy.

Avogadro's number, approximately 6.022 x 10^23, is a fundamental constant in chemistry that represents the number of particles (usually atoms or molecules) in one mole of a substance. It serves as a bridge between the macroscopic and atomic scales. When we calculate the number of fusion reactions that can occur per kilogram of water, Avogadro's number is used to determine the number of deuterium atoms available to fuse.

In the calculation from the exercise, deuterium atoms within water are counted because only they are involved in the specific fusion reaction considered. By dividing the moles of deuterium in a kilogram of water by Avogadro's number, we can estimate the number of fusion reactions, and thus the energy, we can harness from a given amount of water. This not only solidifies the linkage between atomic-scale phenomena and real-world applications but also illustrates the potential power tucked away in the water we encounter every day.