Suppose you are designing a proton decay experiment and you can detect 50 percent of the proton decays in a tank of water. (a) How many kilograms of water would you need to see one decay per month, assuming a lifetime of \(10^{31} \mathrm{y} ?\) (b) How many cubic meters of water is this? (c) If the actual lifetime is \(10^{33} \mathrm{y}\), how long would you have to wait on an average to see a single proton decay?

Particle physics is a branch of physics that studies the nature of particles that are the constituents of what is usually referred to as matter or radiation. In the realm of particle physics, the proton is a subatomic particle with a positive electric charge, found within the nucleus of an atom. Experiments investigating proton decay seek to understand the stability of matter by determining if protons decay into more fundamental particles over a very long period.

These experiments are significant because they can provide insights into the Grand Unified Theories (GUTs), which aim to describe how the three fundamental forces of the Standard Model (electromagnetism, the weak nuclear force, and the strong nuclear force) are actually manifestations of a single, more fundamental force at extremely high energies. Proton decay, if it occurs, would be a rare event, taking place over timescales vastly exceeding the age of the universe, and thus requires specialized detection methods and a large number of protons to observe even a single decay event.

Avogadro's number, approximately \(6.022 \times 10^{23}\), is a key constant in chemistry and physics named after the Italian scientist Amedeo Avogadro. It defines the number of constituent particles, often atoms or molecules, that are contained in one mole of a substance. The mole is one of the seven base SI units and is used to bridge the microscopic world of atoms and molecules with the macroscopic world we interact with every day.

In the context of the proton decay experiment, Avogadro's number is used to calculate the total number of protons in a mass of water. Since each water molecule (H2O) consists of 10 protons, by determining the number of water molecules in one kilogram of water through Avogadro's number, we can then easily find the total number of protons. This large number is critical in understanding the sheer volume of protons necessary in such experiments, given that proton decay is hypothesized to be an exceedingly rare process.

Half-life is a concept widely used in nuclear physics and chemistry to describe the time required for one-half of the unstable nuclei (or in this case, hypothetical unstable protons) in a sample to undergo radioactive decay. In typical radioactive substances, this parameter is crucial in determining how quickly a material will decay, presenting important information on its stability and potential risks.

When dealing with proton decay, however, the term 'lifetime' is used rather than half-life, since we are not observing a rapid series of decay events but are instead waiting for an exceedingly rare event. The provided exercise requires estimating the number of kilograms of water needed to detect one event per month, assuming a hypothetical proton lifetime. This is essentially an inverse problem compared to typical half-life calculations, in which the volume and mass necessary to have a reasonable chance of observing a single decay are computed based on an assumed rate of decay. In this advanced context, half-life estimation is instrumental for researchers to design experiments with sufficient sensitivity to detect phenomena predicted to occur over vast timescales.