As a result of being exposed to the radiation released during the Chernobyl nuclear accident, the dose of iodine-131 in a person's body is \(7.4 \mathrm{mCi}\) \(\left(1 \mathrm{mCi}=1 \times 10^{-3} \mathrm{Ci}\right) .\) Use the relationship rate \(=\lambda N\) to calculate the number of atoms of iodine- 131 to which this radioactivity corresponds. (The halflife of \({ }^{131} \mathrm{I}\) is 8.1 d. \()\).

Nuclear chemistry deals with the processes, reactions, and properties of atomic nuclei. At the heart of this discipline is the study of radioactivity, which is the process by which unstable atomic nuclei lose energy by emitting radiation. Different isotopes, or atoms with the same number of protons but different numbers of neutrons, have varying levels of stability.

Unstable isotopes undergo radioactive decay, transforming into other elements or isotopes and releasing particles in the process. This can be alpha or beta particles, or gamma rays. For example, iodine-131, a radioactive isotope produced in nuclear fission reactions, was released during the Chernobyl accident and can be absorbed by the thyroid gland, posing health risks.

The rate of radioactive decay can be quantified using mathematical formulas involving the decay constant and the half-life of the isotope in question. These tools are essential for scientists and engineers who manage the use and cleanup of radioactive materials, and for medical professionals who use radioisotopes in treatments and diagnostics.

The decay constant, represented by the Greek letter \(\lambda\), is a probability rate at which a particular radioactive isotope will decay per unit time. Essentially, it gives us a measure of the rate at which radioactive atoms disintegrate. If you may recall from the solution example, it's computed using the natural logarithm of 2 divided by the isotope's half-life.

The formula used to calculate the decay constant is \[\lambda = \frac{\ln(2)}{t_{1/2}}\], where \(t_{1/2}\) is the half-life. Since the half-life of iodine-131 is 8.1 days, to get the decay constant \(\lambda\) in units of \(s^{-1}\), we convert that period into seconds, which gives us finer precision and suitability for more calculations. Understanding the decay constant is crucial because it both describes the inherent stability of a nucleus and serves as a way to predict how quickly a sample will reduce its radioactivity, which is important for safety and environmental considerations.

Half-life, commonly represented as \(t_{1/2}\), is one of the key concepts in the study of radioactivity. It's the time required for half of the radioactive isotopes in a sample to decay. This means that after one half-life, the radioactivity of the sample would be reduced to 50% of its original value, after two half-lives to 25%, and so on. For iodine-131, which has a half-life of 8.1 days, this means that every 8.1 days, the amount of the iodine-131 present would halve.

Knowing the half-life of a substance helps in various fields, such as determining dosages for medical treatments or estimating the time needed for radioactive contaminants to decay to safe levels. For instance, in the Chernobyl example, knowing the half-life of iodine-131 and the initial dose of \(7.4 \times 10^{-3} \mathrm{Ci}\) would allow experts to calculate the rate at which the iodine is decaying, and subsequently, when the sample will decrease to a level that's no longer harmful to human health.