Problem 102 - Chemistry: An Atoms-Focused Approach

Question

A patient is administered mercury-197 to evaluate kidney function. Mercury- 197 has a half-life of 65 hours. What fraction of an initial dose of mercury-197 remains after 6 days?

Step-by-Step Solution

Answer

Answer: Approximately 18.9% of the initial dose remains after 6 days.

Step 1: Convert 6 days to hours

Since there are 24 hours in a day, to convert 6 days to hours, multiply the number of days by 24. 6 days × 24 hours/day = 144 hours

Step 2: Find the number of half-lives

In this step, we will determine how many half-lives of 65 hours have occurred within 144 hours. To do this, divide the total time that has passed (144 hours) by the half-life of mercury-197 (65 hours). Number of half-lives = 144 hours / 65 hours = 2.215

Step 3: Calculate the fraction of the remaining dose

The fraction of the initial dose that remains after each half-life can be calculated by raising 1/2 to the power of the number of half-lives. In this case, we will raise 1/2 to the power of 2.215. Fraction remaining = (1/2)^(2.215) = 0.189 So, approximately 18.9% of the initial dose of mercury-197 remains after 6 days.

Key Concepts

Half-Life Calculation

Understanding how to calculate half-life is crucial when dealing with radioactive substances. The half-life is the time it takes for half of a radioactive sample to decay. In calculations, you start with determining how many half-lives fit into a given period. This provides insight into how much of the initial substance remains.
To calculate half-life decay, follow these steps:

• Convert all time units to one consistent unit, commonly hours or days.
• Determine how many half-lives fit into the timeframe by dividing the total time by the half-life duration of the substance.
• Use the decay formula: \[ ext{Remaining Fraction} = \left( \frac{1}{2} \right)^{\text{Number of Half-Lives}} \]

In our example of mercury-197, with a half-life of 65 hours, this formula helps us find how much substance remains after 144 hours of decay, which turns out to be approximately 18.9%.

Kidney Function Evaluation

In medical practice, evaluating kidney function is essential for assessing a patient’s health. Kidneys play a crucial role in filtering and excreting waste products from the blood.
Radionuclide studies use radioactive substances like mercury-197 to measure how well the kidneys perform their tasks.
Here’s why it's significant:

• These tests can highlight any inefficiencies in kidney filtering.
• The measurement of how quickly a substance is filtered provides insights into kidney health.
• An accurate assessment can lead to early detection of kidney disorders.

The radioactive decay of a tracer like mercury-197, tracked over time, provides reliable data on kidney performance, helping doctors devise effective treatment plans.

Nuclear Medicine

Nuclear medicine is a fascinating field that combines medicine and radioactive substances to diagnose and treat diseases. This approach utilizes small amounts of radioactive materials, known as radiotracers, to examine organ function more accurately than traditional imaging methods.
A key part of nuclear medicine is understanding tracer behavior and decay, such as the process seen with mercury-197. Here's why it's important:

• Nuclear medicine procedures offer unmatched detail in visualizing organ and tissue function.
• The precise decay rates of substances like mercury-197 are crucial in ensuring accurate diagnostic information.
• This branch of medicine provides non-invasive options to understand complex medical conditions.

By harnessing the principles of radioactive decay, healthcare professionals can deliver diagnoses and treatments that improve patient outcomes significantly.