Both technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are \(6 \mathrm{h}\) and \(73 \mathrm{h}\), respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?

Radioactivity is the process by which an unstable atomic nucleus loses energy by emitting radiation. This spontaneous emission can include alpha particles, beta particles, or gamma rays, and it takes place in unstable isotopes, which are forms of elements with a different number of neutrons in the nucleus.

Why is this important? Radioactivity is a natural phenomenon that is also utilized in various applications, from medical imaging, as seen in the exercise with technetium-99 and thallium-201, to carbon dating in archaeology. It's the foundation of nuclear chemistry, which studies these processes and their results.

Nuclear chemistry delves into the reactions that take place in the nucleus of an atom, where both the atomic number and mass number can change. This field encompasses both natural processes, like radioactive decay, and artificial processes, such as nuclear fission and fusion.

Nuclear chemistry is not only about understanding these reactions but also about harnessing them. For example, radioactive isotopes are used for medical diagnostics, and knowledge of nuclear reactions is crucial for operating nuclear power plants safely.

The decay of radioactive substances is characterized by an exponential function. In the context of the given exercise, exponential decay describes how the activity of a radioactive substance decreases over time.

An exponential decay curve gets steeper over time, illustrating that the quantity of the substance decreases more rapidly at first, then slows down as less of it remains. Understanding this mathematical model is essential for scientists and technicians who work with radioactive materials.

The half-life of an isotope is the time required for half of a radioactive substance to decay. To calculate the remaining activity of an isotope after a given time, you can use the formula:\[ P = \left(\frac{1}{2}\right)^{\frac{t}{T}} \], where \( P \) is the remaining percentage, \( t \) is the time elapsed, and \( T \) is the half-life.

In the case of technetium-99 with a half-life of 6 hours, after 48 hours, which is 8 half-lives, you can calculate the remaining radioactivity. Apply the same method for thallium-201 to measure its remaining activity after 48 hours. This process is critical for safety in medical and industrial applications where managing exposure to radiation is vital.