Problem 24 - University Physics with Modern Physics

Question

BIO Radioactive Tracers. Radioactive isotopes are often introduced into the body through the bloodstream. Their spread through the body can then be monitored by detecting the appearance of radiation in different organs. \(^{131} \mathrm{I},\) a \(\beta^{-}\) emitter with a half-life of 8.0 \(\mathrm{d}\) , is one such tracer. Suppose a scientist introduces a sample with an activity of 375 \(\mathrm{Bq}\) and watches it spread to the organs. (a) Assuming that the sample all went to the thyroid gland, what will be the decay rate in that gland 24 d (about 3\(\frac{1}{2}\) weeks) later? (b) If the decay rate in the thyroid 24 d later is actually measured to be 17.0 Bq, what percentage of the tracer went to that gland? (c) What isotope remains after the I-131 decays?

Step-by-Step Solution

Answer

(a) 46.875 Bq; (b) 36.26%; (c) Xenon-131.

Step 1: Understanding the Concept of Half-Life

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. For Iodine-131 (I-131), the half-life is given as 8 days.

Step 2: Calculate Remaining Activity After 24 Days

To find the remaining activity after 24 days, we use the formula: \[ A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where \( A \) is the remaining activity, \( A_0 = 375 \) Bq is the initial activity, \( t = 24 \) days, and \( T_{1/2} = 8 \) days. Substitute these values. \[ A = 375 \times \left(\frac{1}{2}\right)^{\frac{24}{8}} = 375 \times \left(\frac{1}{2}\right)^3 = 375 \times \frac{1}{8} = 46.875 \text{ Bq} \]

Step 3: Calculate Observed Percentage of the Tracer in Thyroid

We know the measured decay rate in the thyroid after 24 days is 17.0 Bq. To find the percentage of the original tracer that ended up in the thyroid gland, we use the formula: \[ \text{Percentage} = \left(\frac{\text{Observed activity}}{\text{Calculated remaining activity}}\right) \times 100 \]Substitute the values:\[ \text{Percentage} = \left(\frac{17.0}{46.875}\right) \times 100 \approx 36.26\% \]

Step 4: Identify the Decay Product of Iodine-131

Iodine-131 ( \(^{131}\mathrm{I}\)) decays by beta decay into Xenon-131 ( \(^{131}\mathrm{Xe}\)). Therefore, after Iodine-131 decays, Xenon-131 remains as a stable by-product.

Key Concepts

Isotope Half-life Calculation

The concept of half-life is essential for understanding how radioactive isotopes behave over time. In nuclear physics, a half-life is the time required for half of the radioactive atoms in a sample to decay. This means that after one half-life, only half of the original radioactive material remains, and this fraction gets halved with each subsequent half-life.

For example, in the case of the isotope Iodine-131 (\(^{131} \text{I}\)), which has a half-life of 8 days, if you start with 375 Becquerels (Bq) of radioactivity, you'll have half of that, approximately 187.5 Bq, after 8 days. After another 8 days (which totals 16 days), it will have decreased to about 93.75 Bq, and so on.

Understanding this decay pattern is crucial for predicting how long an isotope will remain active in a biological system or how long it can be effectively used in applications such as medical diagnostics or treatments.

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. It's a fundamental concept in nuclear physics that explains how elements transform over time. In radioactive decay, isotopes may emit particles such as alpha particles, beta particles, or gamma rays.

In the case of Iodine-131, it undergoes beta decay. This means the isotope emits a beta particle, resulting in the transformation of a neutron into a proton, which changes the element from iodine to xenon (\(^{131} \text{I} \rightarrow ^{131} \text{Xe}\)). This process decreases the atomic number by one, but the mass number remains the same, as it's a balance maintained by a neutron-proton swap.

Radioactive decay is used in various fields to determine ages of archaeological findings, track biological processes, and as seen in medicine, to diagnose and treat diseases with radioactive tracers.

Essential Diagnostics in Medicine

Radioactive tracers are pivotal in medical diagnostics. They offer a non-invasive way to diagnose and monitor various conditions. Doctors introduce a small amount of a radioactive isotope into the body, often through the bloodstream, which then spreads to specific areas like organs.

Tools like gamma cameras detect the radiation emitted by these isotopes, creating images that help physicians see how organs and tissues function in real-time. For instance, Iodine-131 is often used to study and treat thyroid disorders. Once introduced, it accumulates in the thyroid gland, enabling doctors to observe thyroid activity through imaging derived from emitted radiation.

These techniques allow for an effective diagnosis with minimal discomfort to the patient, ensuring radioactive substances are used carefully, following strict safety standards to ensure that the benefits outweigh any potential risks.

Nuclear Physics Concepts

Nuclear physics covers a broad range of concepts essential for understanding how radioactive isotopes work in real-life applications. At its core, nuclear physics investigates the components and structure of atomic nuclei and the forces that bind them.

Fundamental topics within nuclear physics include the behavior of isotopes, radioactive decay mechanisms, nuclear reactions, and the energy released during these processes. These concepts are critical in both theoretical and applied physics, influencing areas such as medicine, energy production, and environmental science.

Understanding these principles empowers scientists to harness nuclear reactions for various uses, such as generating electricity in nuclear power plants or creating medical isotopes for diagnostics and treatments. In everyday life, these principles underpin a host of technologies that benefit society.