The half-life of tritium, \(_{1}^{3} \mathrm{H},\) is 12.3 \(\mathrm{y} .\) How long will it take for seven-eighths of the sample to decay?

Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. Atoms of radioactive elements are unstable because of an excess of energy or mass or both. To reach stability, these atoms emit particles and energy through a process known as decay, ultimately transforming into a different element or a different isotope of the same element.

Throughout this process, a specific feature of a radioactive sample called the 'half-life' is of great interest. The half-life of a radioisotope is the time required for half the atoms in a given quantity to decay. It's a constant that's unique to each isotope. Understanding the concept of half-life is crucial for various fields, such as geology for radioactive dating, medicine for diagnostic and treatment purposes, and nuclear chemistry for the management of nuclear materials.

Tritium, symbolically represented as \( _{1}^{3}\mathrm{H} \), is a radioactive isotope of hydrogen. While most hydrogen atoms consist of just one proton and one electron, tritium also contains two neutrons, making it one of the heavy isotopes of hydrogen. Tritium is scarce and is produced naturally in the atmosphere when cosmic rays interact with nitrogen atoms.

It is also produced artificially in nuclear reactors and can be used in various applications such as in self-powered lighting, where it provides illumination without the need for batteries, and in the field of nuclear fusion as a potential fuel. Due to its radioactive nature, understanding tritium's half-life, which is 12.3 years, is essential when assessing its decay and managing its safe use and disposal.

Nuclear chemistry is the branch of chemistry that deals with the reactions and properties of atomic nuclei. It encompasses the study of both stable and unstable isotopes, the latter undergoing the process of radioactive decay. This field of chemistry is vital to our understanding of processes such as radioactivity, nuclear fission and nuclear fusion, the chemical effects resulting from the absorption of radiation, and the production and use of radioactive tracers in biochemical analysis and medical applications.

One of the key concepts in nuclear chemistry is the half-life of radioactive isotopes, as it helps chemists predict the behavior of these substances over time. Since each radioactive element has a different half-life, this concept allows us to handle each element accordingly, be it technetium in medical imaging or tritium in nuclear reactions.

Exponential decay is a mathematical description of the process by which a quantity decreases at a rate proportional to its current value. This type of decay is commonly associated with the decay of radioactive elements, where the number of atoms of the isotope decreases exponentially over time. The half-life of a radioactive substance is the most direct measure of exponential decay, indicating how long it takes for half of it to decay.

In practical terms, if you start with a certain amount of a radioactive substance, after one half-life, you will have half that amount. After two half-lives, you will have a quarter, then an eighth, and so on. This decay will continue, following an exponential curve, until the sample becomes effectively stable or non-detectable. Understanding exponential decay is not only essential for computations in nuclear chemistry but also for modeling population decline, the spread of diseases, and various financial and physical phenomena.