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\(\textbf{Radioactive Tracers}\). 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

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

Step by step solution

01

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.
02

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.
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

\(\textbf{We Are Stardust.}\) In 1952 spectral lines of the element technetium-99 (\(^9$$^9\)Tc) were discovered in a red giant star. Red giants are very old stars, often around 10 billion years old, and near the end of their lives. Technetium has no stable isotopes, and the half-life of \(^9$$^9\)Tc is 200,000 years. (a) For how many halflives has the \(^9$$^9\)Tc been in the red giant star if its age is 10 billion years? (b) What fraction of the original \(^9$$^9\)Tc would be left at the end of that time? This discovery was extremely important because it provided convincing evidence for the theory (now essentially known to be true) that most of the atoms heavier than hydrogen and helium were made inside of stars by thermonuclear fusion and other nuclear processes. If the \(^9$$^9\)Tc had been part of the star since it was born, the amount remaining after 10 billion years would have been so minute that it would not have been detectable. This knowledge is what led the late astronomer Carl Sagan to proclaim that "we are stardust".

The United States uses about 1.4 \(\times\) 10\(^1$$^9\) J of electrical energy per year. If all this energy came from the fission of \(^2$$^3$$^5\)U, which releases 200 MeV per fission event, (a) how many kilograms of \(^2$$^3$$^5\)U would be used per year, and (b) how many kilograms of uranium would have to be mined per year to provide that much \(^2$$^3$$^5\)U? (Recall that only 0.70% of naturally occurring uranium is \(^2$$^3$$^5\)U.)

\(\textbf{Radioactive Fallout.}\) One of the problems of in-air testing of nuclear weapons (or, even worse, the \(use\) of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 (\(^9$$^0\)Sr), which breaks down by \(\beta$$^-\) decay with a half- life of 28 years. It is chemically similar to calcium and therefore can be incorporated into bones and teeth, where, due to its rather long half-life, it remains for years as an internal source of radiation. (a) What is the daughter nucleus of the \(^9$$^0\)Sr decay? (b) What percentage of the original level of \(^9$$^0\)Sr is left after 56 years? (c) How long would you have to wait for the original level to be reduced to 6.25% of its original value?

Consider the fusion reaction \(^{2}_{1}H\) + \(^{2}_{1}H \rightarrow ^{3}_{2}He + ^{1}_{0}n\). (a) Estimate the barrier energy by calculating the repulsive electrostatic potential energy of the two \(^{2}_{1}H\) nuclei when they touch. (b) Compute the energy liberated in this reaction in MeV and in joules. (c) Compute the energy liberated \(per\) \(mole\) of deuterium, remembering that the gas is diatomic, and compare with the heat of combustion of hydrogen, about \(2.9 \times 10^{5} J/mol\).

Consider the nuclear reaction \(^{4}_{2}He\) + \(^{7}_{3}Li\) \(\rightarrow\) X + \(^{1}_{0}n\) where X is a nuclide. (a) What are Z and A for the nuclide X? (b) Is energy absorbed or liberated? How much?

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