Technetium- 99 has been used as a radiographic agent in bone scans \(\left({ }_{43}^{99} \mathrm{Tc}\right.\) is absorbed by bones). If \({ }_{43}^{99} \mathrm{Tc}\) has a half-life of \(6.0\) hours, what fraction of an administered dose of \(100 . \mu \mathrm{g}\) \({ }_{43}^{99} \mathrm{Tc}\) remains in a patient's body after \(2.0\) days?

Radioactive substances undergo decay over time, spontaneously transforming into other elements or isotopes. Each radioactive isotope has a characteristic time known as its 'half-life'. The half-life is the period it takes for half the atoms of a radioactive sample to decay. This is a constant value for a given isotope, such as the 6.0-hour half-life for technetium-99. Understanding the concept of half-life is crucial, as it helps predict how long a substance remains radioactive and is a fundamental aspect of radioactive dating, medicine, and nuclear power.

When dealing with half-life problems, we work with exponential decay. For each half-life that passes, the quantity of the radioactive substance reduces by half. Therefore, after one half-life, 50% remains; after two, 25%; and so on. This exponential decrease is a key concept in understanding various natural and technological processes involving radioactivity.

Technetium-99 is widely used in the field of medical imaging as a radiographic agent, particularly in bone scans. When administrated to patients, this radioisotope localizes in areas of bone growth or repair, allowing doctors to visualize bones using a gamma camera. The choice of technetium-99 is due to its suitable half-life—which allows for effective imaging without long-term radiation exposure—and its ability to emit gamma rays that can be detected by imaging equipment.

Due to its radioactive nature, the dose administrated must be carefully calculated, so it does not pose a significant health risk. After the imaging procedure, the radioactive decay of technetium-99 commences, and the substance diminishes according to its half-life. Comprehending the decay process is vital for medical professionals to manage patient safety effectively.

To calculate the decay of technetium-99 over time, you need to apply the concepts of half-life and exponential decay. Given its half-life of 6.0 hours, you can determine how much of the initial dose remains after a period of time through a series of straightforward calculations.

First, convert the total elapsed time into the same unit as the given half-life—in this case, hours. Then, determine how many half-lives fit into this time frame. After identifying the number of elapsed half-lives, use the exponential decay formula \( R = R_0 \times (\frac{1}{2})^n \) where \( R \) is the remaining quantity, \( R_0 \) is the initial quantity, and \( n \) is the number of half-lives. This calculation reveals the remaining dose and, dividing this by the initial dose, yields the fraction of the substance that is still present.

Exponential decay is a fundamental principle describing how the quantity of a radioactive substance diminishes over time. It's not a linear relationship; instead, the rate of decay is proportional to the amount of substance present at any given moment. This results in the characteristic 'half-life' where the quantity of the radioactive material decreases by half over each elapsed half-life period.

This type of decay can be modeled by the function \( N(t) = N_0 \times e^{-\text{λt}} \), where \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, \( λ \) is the decay constant, and \( e \) is the base of the natural logarithm. In medical applications and safe handling of radioactive materials, understanding and applying the exponential decay law ensures accurate dosing and minimizes radiation exposure.