The half-life of \(\mathrm{Tc}^{99}\) is \(6.0 \mathrm{hr}\). The total residual activity in a patient \(30 \mathrm{hr}\) after receiving an injection containing \(\mathrm{Tc}^{99}\) must be more than \(0.01 \mu \mathrm{C}_{i}\). What is the maximum activity (in \(\mu \mathrm{C}_{i}\) ) that the sample injected an have? (a) \(0.16\) (b) \(0.32\) (c) \(0.64\) (d) \(0.08\)

Half-life is a crucial concept in nuclear chemistry, particularly when dealing with radioactive isotopes. Simply put, a half-life is the amount of time it takes for half of a radioactive sample to decay. This concept is central to understanding how radioactive materials degrade over time and is used to calculate the remaining quantity of a substance after a certain period.

To perform half-life calculations, you need to follow a set of steps. First, determine the total number of half-lives that have passed, which is found by dividing the total time elapsed by the duration of one half-life, as seen in the provided problem. After knowing how many half-lives have passed, you can use this information to compute the residual activity of the substance.

In the case of the exercise question, the half-life of \( \mathrm{Tc}^{99} \) is given as 6 hours. After 30 hours, we calculate that 5 half-lives have passed. Knowing this helps us determine the initial activity that the \( \mathrm{Tc}^{99} \) sample could have had.

Radioactive decay is the process through which an unstable atomic nucleus loses energy by emitting radiation. There are several types of radioactive decay, including alpha decay, beta decay, and gamma decay, among others. In this process, the nucleus of a radioactive atom disintegrates and transforms into a different nucleus or into a lower energy state.

This phenomenon is random but can be characterized statistically through decay rates and half-lives. Understanding the decay process is essential for various applications, such as in medicine for diagnostic imaging and treatment, as well as in archaeology for dating artifacts. The decay is inherently exponential, as each decay event is independent and its probability does not change over time, leading to the exponential decay formula utilized in calculations of this nature.

The exponential decay formula mathematically describes how a quantity diminishes over time at a rate proportional to its current value. For radioactive decay, the formula is typically expressed as \( N(t) = N_0 \cdot (1/2)^{t/T} \), where \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, and \( T \) is the half-life of the substance.

By applying the exponential decay formula, we can compute the quantity of a radioactive substance that remains after a given period. The formula underlines the exponential nature of decay, where each successive half-life reduces the quantity by half, no matter the starting amount. In the context of the exercise example, we utilize this formula to extrapolate the initial amount of \( \mathrm{Tc}^{99} \) that a patient could have been given, based on the required residual activity after 30 hours and knowing that this time span covers 5 half-lives.