Problem 83 - Chemistry Matter and Change

Question

Red blood cells A medical researcher is using a chromium-51 source to study red blood cells. The gamma-emission intensity at a distance of 1.0 \(\mathrm{m}\) is 0.75 \(\mathrm{mrem} / \mathrm{s} * \mathrm{m}^{2}\) . At what distance would the intensity drop to 0.15 mrem \(/\) \(\mathrm{s} \cdot \mathrm{m}^{2}\) ?

Step-by-Step Solution

Answer

The distance at which the intensity drops to 0.15 mrem/s·m² is approximately 2.24 m.

Step 1: Understand Intensity Decay

The intensity of radiation diminishes with the square of the distance from the source. This is described by the inverse square law: \( I_2 = \frac{I_1 \cdot r_1^2}{r_2^2} \), where \( I_1 \) and \( I_2 \) are the initial and final intensities, and \( r_1 \) and \( r_2 \) are the initial and final distances from the source. In this problem, \( I_1 = 0.75 \ \text{mrem/s·m}^2 \), and \( I_2 = 0.15 \ \text{mrem/s·m}^2 \), with \( r_1 = 1.0 \ \text{m} \). We need to find \( r_2 \).

Step 2: Solve for Final Distance

Re-arrange the inverse square law equation to solve for \( r_2 \): \( r_2 = r_1 \sqrt{\frac{I_1}{I_2}} \). Substitute the known values: \( r_2 = 1.0 \ \text{m} \cdot \sqrt{\frac{0.75}{0.15}} \).

Step 3: Perform Calculation

Calculate the ratio \( \frac{0.75}{0.15} = 5 \), and then find the square root: \( \sqrt{5} \approx 2.236 \). Therefore, the final distance \( r_2 = 1.0 \ \text{m} \cdot 2.236 = 2.236 \ \text{m} \).

Key Concepts

Radiation Intensity

Radiation intensity refers to the power per unit area carried by a wave, in this case, gamma rays emitted from a chromium-51 source. As radiation moves away from its source, its intensity, or how strong it feels at a point, decreases with distance. This change follows the inverse square law, which is known in physics to describe how a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity.

The inverse square law formula is given by \( I_2 = \frac{I_1 \cdot r_1^2}{r_2^2} \). Here, \( I_1 \) is the initial intensity at a distance \( r_1 \), and \( I_2 \) is the intensity at a new distance \( r_2 \). If you know the initial intensity and distance, you can calculate how the intensity changes, which is particularly useful in studies involving radiation like chromium-51. Understanding this concept ensures that medical researchers can precisely measure and control radiation exposure levels, safeguarding both lab workers and patients.

Chromium-51

Chromium-51 is a radioactive isotope of chromium, often used in medical research. Its primary application is the labeling of red blood cells for various studies, including those related to the lifespan of these cells, their volume, and the diagnosis of disorders such as anemia. Because chromium-51 emits gamma rays, it enables researchers to track the behavior and life cycle of red blood cells within the human body.

Using isotopes like chromium-51 requires awareness of radiation safety due to its gamma-emitting nature. The understanding of radiation intensity as per the inverse square law becomes crucial here to avoid excessive exposure, which could be harmful. Different isotopes are chosen for specific long-term or short-term studies, depending heavily on their half-lives and emission properties.

Red Blood Cells

Red blood cells, or erythrocytes, are the most common type of blood cell and serve to carry oxygen from the lungs to the body's tissues and transport carbon dioxide back to the lungs to be expelled. The function and lifespan of red blood cells can be areas of interest for medical research, especially when analyzing blood disorders.

To study these cells, medical researchers often employ chromium-51 because it helps in tracking the red blood cells via gamma radiation. The behavior and movement of these cells can then provide valuable information regarding the efficiency of oxygen transport and the patient's overall health. When using radioactive markers, accurate measurement of radiation intensity with distance is paramount, as it affects both the reliability of the study and the safety of those involved. Thus, principles such as the inverse square law become crucial tools in the methodology of such research.