Question

\(\textbf{Radioactive Tracers}\). 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. One such tracer is \(^1$$^3$$^1\)I, a \(\beta$$^-\) emitter with a half-life of 8.0 d. Suppose a scientist introduces a sample with an activity of 325 Bq and watches it spread to the organs. (a) Assuming that all of the sample 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 measured to be 17.0 Bq, what percentage of the tracer went to that gland? (c) What isotope remains after the I-131 decays?

Short answer

(a) 40.63 Bq; (b) 41.84%; (c) The remaining isotope is \(^{131}\text{Xe}\).

Step-by-step solution

Calculate Number of Half-Lives Passed

To determine the number of half-lives that have passed after 24 days, use the formula:\[n = \frac{t}{T_{1/2}}\]where \( n \) is the number of half-lives, \( t = 24 \) days is the time elapsed, and \( T_{1/2} = 8 \) days is the half-life of \( ^{131}\text{I} \).Calculate:\[ n = \frac{24}{8} = 3\] Thus, 3 half-lives have passed.

Determine Remaining Activity After 3 Half-Lives

Use the formula for radioactive decay:\[A = A_0 \times \left(\frac{1}{2}\right)^n\]where \( A_0 = 325 \) Bq is the initial activity, and \( n = 3 \) is the number of half-lives.Calculate:\[A = 325 \times \left(\frac{1}{2}\right)^3 = 325 \times \frac{1}{8} = 40.625 \text{ Bq}\]So, the decay rate in the thyroid gland 24 days later would be approximately 40.63 Bq.

Calculate Percentage of Tracer in Thyroid

If the measured activity is 17.0 Bq, we need to determine what percentage of the initial sample this represents. Use the formula:\[\text{Percentage} = \left(\frac{A_{measured}}{A_{calculated}}\right) \times 100\%\]where \( A_{measured} = 17.0 \) Bq and \( A_{calculated} = 40.63 \) Bq.Calculate:\[\text{Percentage} = \left(\frac{17.0}{40.63}\right) \times 100\% \approx 41.84\%\]Approximately 41.84% of the tracer went to the thyroid gland.

Identify the Remaining Isotope

After \( ^{131}\text{I} \) undergoes beta decay, it transforms into another element. The decay changes the element from iodine (I) to xenon (Xe) as a beta particle is emitted and a neutron becomes a proton. Thus, the remaining isotope is \( ^{131}\text{Xe} \), which is xenon.

Key concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. When atoms undergo radioactive decay, they transmute into different atoms. This process is central to applications like radioactive tracers.
The decay is random, and each atom of the radioactive substance has a specific probability of decaying at a given time. This probability is described by the half-life, a crucial concept in understanding the time taken for half of the radioactive atoms to decay.
In decay processes like beta decay, a neutron turns into a proton. This emission changes the elemental identity, exemplifying isotope transformation.

Half-Life Calculation

A half-life is the time required for half of the radioactive atoms in a sample to decay. Calculating the half-life is essential in predicting how long a radioactive substance remains active.
The formula to calculate half-lives passed is: \[n = \frac{t}{T_{1/2}}\]where \(n\) is the number of half-lives, \(t\) is the elapsed time, and \(T_{1/2}\) is the half-life.
For instance, the half-life for iodine-131 (^{131} ext{I}) is 8 days. If 24 days have passed, it means three half-lives have elapsed. This can significantly reduce the activity, affecting how tracers work in medical diagnostics.

Isotope Transformation

Isotope transformation occurs when a radioactive element changes into another element during decay. This transformation is a key part of nuclear reactions and applications of radioactive tracers.
For iodine-131 ( ^{131} ext{I}), a beta decay process occurs. A beta particle, which is a fast-moving electron, is emitted, converting a neutron in the iodine nucleus into a proton. This process changes the element into xenon ( ^{131} ext{Xe}).
Such transformations play a critical role in understanding the movement and eventual endpoint of radioactive materials in scientific and medical settings.

Activity Measurement

Activity measurement refers to the number of decay events occurring per unit time in a radioactive sample. It is conventionally measured in becquerels (Bq), where one becquerel corresponds to one decay per second.
To calculate remaining activity, use the formula:\[A = A_0 \times \left(\frac{1}{2}\right)^n\]where \(A_0\) is the initial activity and \(n\) is the number of half-lives passed.
In practical scenarios, such as using tracers in the thyroid, monitoring changes in activity provides insights into how much tracer has localized in specific tissues. Comparing expected versus measured activity helps determine the distribution of radioactive tracers, serving as a diagnostic tool.