Problem 28 - Chemistry The Molecular Science

Question

Sodium- 24 is a diagnostic radioisotope used to measure blood circulation time. Calculate the mass (mg) of a \(20-\mathrm{mg}\) sample of sodium- 24 that remains after 1 day and 6 hours if sodium- 24 has \(t_{1 / 2}=15\) hours.

Step-by-Step Solution

Answer

After 1 day and 6 hours, 5 mg of sodium-24 remains.

Step 1: Understanding Half-life

The half-life (\( t_{1/2} \)) of a radioactive substance is the time it takes for half of the sample to decay. For sodium-24, \( t_{1/2} = 15 \) hours.

Step 2: Total Elapsed Time Calculation

Determine the total time elapsed from the initial measurement to the given point in time. \(1 \) day is \( 24 \) hours and the problem mentions an additional \( 6 \) hours, totaling \( 24 + 6 = 30 \) hours.

Step 3: Calculate Number of Half-lives

Calculate the number of half-lives (\( n \)) that has elapsed using:\[n = \frac{\text{Total Time Elapsed}}{t_{1/2}} = \frac{30}{15} = 2\]

Step 4: Calculate Remaining Mass

Use the formula for radioactive decay to find the remaining mass:\[\text{Remaining Mass} = \text{Initial Mass} \times \left(\frac{1}{2}\right)^n\]Substituting the given values:\[20 \times \left(\frac{1}{2}\right)^2 = 20 \times \frac{1}{4} = 5 \, \text{mg}\]

Key Concepts

Half-life

Half-life refers to the duration required for half of a radioactive substance to decay. This concept is significant in understanding how long a sample remains active. For sodium-24, the half-life is 15 hours. This means every 15 hours, half of the sodium-24's mass decreases due to radioactive decay. The importance of half-life lies in various fields, such as nuclear medicine and radiochemistry, where it helps predict the behavior of radioactive substances over time. The shorter the half-life, the faster the decay process. Thus, understanding half-life is crucial for determining how long a radioactive material will be useful in practical applications like medical diagnostics.

Sodium-24

Sodium-24 is a radioactive isotope of sodium with a half-life of 15 hours. It plays a vital role in medical diagnostics due to its relatively short half-life and harmless decay process. Sodium-24 is particularly used in measuring blood circulation time, as it can quickly provide insights into body functions without lingering in the body for too long. The rapid decay ensures that the radioactive exposure is minimized, making sodium-24 a safe choice for diagnostic procedures. Sodium-24 decays by beta emission, which leads to the formation of magnesium-24, a stable isotope, indicating that it does not form long-lasting hazardous materials.

Blood circulation measurement

Measuring blood circulation with sodium-24 involves using isotopes to track how quickly blood flows through the body. Since sodium-24 emits radiation detectable by medical instruments, it allows doctors to understand blood flow dynamics effectively. The procedure starts with the injection of sodium-24 into the bloodstream. As it circulates, detectors measure the time taken for the isotope to pass through various parts of the circulatory system. Understanding blood circulation is crucial for diagnosing certain conditions and ensuring the proper function of bodily organs. This non-invasive method offers valuable insights without the need for surgical intervention, making it a popular choice in medical diagnostics.

Radiochemistry

Radiochemistry is the study of radioactive substances and their applications. It combines principles of chemistry and nuclear science to explore the behavior of radioactive materials such as sodium-24. This field is critical in producing and analyzing medical isotopes. For example, radiochemists design protocols for creating isotopes like sodium-24 for medical tests, ensuring they are safe and effective for patient use. Radiochemistry also investigates ways to manage radioactive waste and optimize the use of isotopes in various fields. Its applications extend beyond medicine, contributing to fields like environmental science and nuclear power. By understanding the properties of radioactive substances, radiochemistry helps develop innovative solutions in science and industry.