Question

Sodium- 24 in the form of \(\mathrm{NaCl}\) is given as an injection to measure the sodium electrolyte balance. If \(80 \mathrm{mg}\) of the medical radionuclide is injected, how much \(\mathrm{Na}-24\) remains after 60 hours? \(\left(t_{1 / 2}=15\right.\) hours \()\).

Short answer

5 mg of Sodium-24 remains after 60 hours.

Step-by-step solution

Understand the Half-Life Concept

The half-life of a substance is the time it takes for half of it to decay. For Sodium-24, the half-life is given as 15 hours, meaning every 15 hours, half of the remaining Sodium-24 will decay.

Determine the Number of Half-Life Periods

To find out how many half-life periods have passed in 60 hours, divide the total time by the half-life period. This is calculated as \( \frac{60}{15} = 4 \). Therefore, 4 half-life periods have passed.

Apply the Exponential Decay Formula

The amount of substance remaining after a certain number of half-lives can be calculated using the formula \( N = N_0 \times \left(\frac{1}{2}\right)^n \), where \( N_0 \) is the initial amount, and \( n \) is the number of half-lives.

Calculate the Remaining Sodium-24

Using the decay formula, where \( N_0 = 80 \,\text{mg} \) and \( n = 4 \), the calculation is:\[N = 80 \times \left( \frac{1}{2} \right)^4 = 80 \times \frac{1}{16} = 5 \,\text{mg}.\]So, 5 mg of Sodium-24 remains after 60 hours.

Key concepts

Understanding Half-Life

Half-life is a fundamental concept in radioactive decay, essential for understanding how substances like Sodium-24 break down over time. It is defined as the time required for half of the radioactive nuclei in a sample to decay. This measure provides valuable insight into the stability and duration a radioactive substance remains active. For Sodium-24, the half-life is 15 hours.

In practical terms, half-life helps us predict how much of a substance remains after a certain period. If a substance has a half-life of 15 hours, in every 15 hours, its amount will be reduced to half. For instance:

• After 1 half-life (15 hours), 50% remains.
• After 2 half-lives (30 hours), 25% remains.
• After 3 half-lives (45 hours), 12.5% remains.
• After 4 half-lives (60 hours), 6.25% remains.

By understanding half-life, we can accurately calculate how much of a radioactive material will be present after a given time period. It's important in various fields, including medicine and environmental science.

Sodium-24 and Its Applications

Sodium-24 (\(^{24}\text{Na}\)) is one of the isotopes of sodium, which is commonly used as a radioactive tracer in medicine. It has a half-life of about 15 hours, which makes it useful for short-term studies without long-term radiation exposure risks.

In medical applications, Sodium-24 is often used to assess sodium electrolyte balance, especially in cases where precise tracking of bodily sodium levels is needed. This can be particularly useful in diagnosing and managing patients with electrolyte imbalances, such as those caused by dehydration or kidney issues.Due to its radioactive properties, Sodium-24 can be tracked as it moves through the body, providing doctors with crucial data about sodium dynamics in the patients' systems. Despite these benefits, it is important to handle Sodium-24 carefully due to its radioactive nature.

Exponential Decay Formula in Radioactivity

The exponential decay formula is a key mathematical principle used to calculate the remaining amount of a radioactive substance over time. This formula reflects the nature of radioactive decay processes and relies on the concept of half-life.

The formula is given by:\[N = N_0 \times \left( \frac{1}{2} \right)^n\]Here,

• \(N\) is the final amount of the substance after decay.
• \(N_0\) is the initial amount of substance before decay.
• \(\left( \frac{1}{2} \right)\) represents the halving effect per half-life.
• \(n\) is the number of half-life periods that have passed.

For example, if you start with 80 mg of Sodium-24, and want to know the amount remaining after 60 hours, you calculate the number of half-lives first, which is 4 (since 60 divided by 15 equals 4). Applying the formula:\[N = 80 \times \left( \frac{1}{2} \right)^4 = 80 \times \frac{1}{16} = 5 \, \text{mg}\]This demonstrates how Sodium-24 decreases exponentially, and why the formula is essential for predicting the changes in radioactive substances.