Question

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?

Short answer

The fraction of the administered dose of technetium-99 that remains in a patient's body after 2.0 days is \(\frac{100}{256}\).

Step-by-step solution

Understand the half-life concept

The half-life of a radioactive substance is the time it takes for half of its atoms to decay. In this exercise, we are given that the half-life of technetium-99 is 6.0 hours.

Convert days to hours

We are given that the time passed is 2.0 days, but we need to express it in hours since the half-life is given in hours. To convert days to hours, we'll multiply the number of days by the number of hours in a day (24 hours). So, \(2.0 \mathrm{days} \times 24 \frac{\mathrm{hours}}{\mathrm{day}} = 48 \mathrm{hours}\).

Calculate the number of half-lives

Now, we need to find out how many half-lives have passed during the 48 hours. To do this, we will divide the elapsed time by the half-life of technetium-99. \(\frac{48 \mathrm{hours}}{6.0 \mathrm{hours} \, \text{per half-life}} = 8 \, \text{half-lives}\).

Calculate the remaining dose

We know that after each half-life, the quantity of technetium-99 reduces to half of what was previously present. Therefore, after 8 half-lives have passed, the remaining dose can be calculated as: Remaining dose = Initial dose × \((\frac{1}{2})^{\text{number of half-lives}}\) Remaining dose = \(100 \, \mu g \times \left(\frac{1}{2}\right)^8\) Calculating the value, we get the remaining dose = \(100 \, \mu g \times \frac{1}{256} = \frac{100}{256} \, \mu g\).

Calculate the fraction

Now, to find the fraction of the remaining dose of technetium-99, we'll divide the remaining dose by the initial dose. Fraction remaining = \(\frac{\text{Remaining dose}}{\text{Initial dose}} = \frac{\frac{100}{256} \, \mu g}{100 \, \mu g} = \frac{100}{256}\) So, the fraction of the administered dose of technetium-99 that remains in a patient's body after 2.0 days is \(\frac{100}{256}\).

Key concepts

Understanding Radioactive Half-Life

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 in Medical Imaging

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.

Calculating Technetium-99 Decay

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 in Radioactivity

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.