At \(8 : 00\) A.M., a patient receives a 58 -mg dose of I- 131 to obtain an image of her thyroid. If the nuclide has a half-life of 8 days, what mass of the nuclide remains in the patient at \(5 : 00\) p.M.the next day? (Assume the nuclide is not excreted from the body.)

Radioactive decay is a spontaneous process wherein an unstable atomic nucleus loses energy by emitting radiation. This transformation results in the nucleus changing to a different element or a lower-energy state of the same element. Imagine a class of balloons slowly losing air; similarly, an unstable atomic nucleus will 'deflate' by emitting particles such as alpha particles, beta particles, or gamma rays. The decay occurs according to a predictable rate characterized by the half-life, which is specific to each isotope.

For instance, in our textbook problem, the isotope Iodine-131 (I-131) is used in medical imaging for its ability to highlight thyroid issues. It decays by beta emission, transforming into a stable xenon isotope over time. This decay process is crucial for both diagnostic procedures in medicine and in understanding the behaviors of materials in nuclear chemistry.

Nuclear chemistry delves into reactions that change the composition of atomic nuclei. This entails the study of radioactive decay, nuclear fission, and nuclear fusion. It's the core science behind various technologies, from medical imaging to power generation. In the heart of nuclear chemistry is the balance between protons and neutrons in an atom; if this balance is disturbed, the nucleus may become unstable and subject to decay, just like a tower of blocks may topple if not stacked properly.

In clinical applications, such as the example with I-131, nuclear chemistry allows us to use the principles of radioactive decay to help diagnose and treat illnesses. The meticulous calculations and understanding of half-life periods underpin the dosages used in treatments to ensure safety and efficacy.

Exponential decay refers to the decrease in an amount at a rate proportional to its current value. This process follows the mathematical expression \( M = M_0 \times \left(\frac{1}{2}\right)^n \), where \( M_0 \) is the original quantity, \( n \) is the number of half-lives, and \( M \) is the quantity at time \( t \). It's like watching a savings account dwindle by a fixed percentage each period; the greater your balance, the more you'd lose each time.

For radioactive substances, the exponential decay model perfectly captures the predictable pattern of decay over time. This model is integral to solving problems such as our textbook scenario, where we need to know how much of a substance remains after a certain period. Knowledge of exponential decay is vital not just in nuclear chemistry, but in understanding a wide array of natural and financial processes.