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43.24. 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 \(B^{-}\) 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 \(\mathrm{d}\) (about 2\(\frac{1}{2}\) weeks) later? (b) If the decay rate in the thyroid 24 d later is actually measured to be 17.0 \(\mathrm{Bq}\) , what percentage of the tracer went to that gland? (c) What isotope remains after the \(\mathrm{I}-131\) decays?

Step by step solution

01

Understand the Problem

We are given a radioactive tracer, iodine-131, which has an initial activity of 375 Bq and a half-life of 8 days. We need to find the decay rate in the thyroid gland after 24 days (part a), determine the percentage of tracer that went to the gland if the measured activity is 17.0 Bq after 24 days (part b), and identify the isotope remaining after iodine-131 decays (part c).

02

Calculate the Decay Constant

The decay constant \(\lambda\) is calculated using the formula \(\lambda = \frac{0.693}{T_{1/2}}\) where \(T_{1/2}\) is the half-life. For iodine-131, \(T_{1/2} = 8\) days. So, \(\lambda = \frac{0.693}{8}\approx 0.0866 \ \text{d}^{-1}.\)

03

Determine the Remaining Activity After 24 Days

The remaining activity \(A\) after time \(t\) can be calculated using \(A = A_0 e^{-\lambda t}\), where \(A_0 = 375 \ \text{Bq}\) and \(t = 24\) days. Substituting, we get \(A = 375 e^{-0.0866 \times 24} \approx 47.9 \ \text{Bq}.\)

04

Calculate the Percentage of Tracer in Thyroid

If the actual measured activity is 17.0 Bq, the percentage of tracer that went to the thyroid is given by \( \frac{17.0}{47.9} \times 100\% \approx 35.5\%.\)

05

Identify the Remaining Isotope

When iodine-131 decays, it undergoes beta decay, converting to xenon-131. Therefore, the remaining isotope is xenon-131 after the decay process.

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Key Concepts

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

Radioactive Isotopes

Radioactive isotopes, or radionuclides, are atoms with an unstable nucleus, which are best known for their ability to emit radiation. These isotopes can be used in various fields, particularly in medicine, to track biological processes.
Iodine-131 is a common radioactive isotope used in medical diagnostics. When introduced into the body, it helps scientists and doctors monitor the function and structure of organs, like the thyroid gland.
Key aspects of radioactive isotopes include:

• They have a specific number of protons and neutrons.
• Their nuclei are unstable, causing them to release energy (radiation) until they reach a stable state.
• They are naturally occurring or artificially produced in labs.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiating particles or electromagnetic waves. This process continues until the nucleus reaches a stable state, transforming into a different element in the process.
There are different types of decay processes including alpha, beta, and gamma decay, each emitting different particles or energy forms. For iodine-131, it undergoes beta decay, where the nucleus releases beta particles to become more stable.
Important points about radioactive decay:

• It's a random process but can be measured statistically.
• The decay rate is constant for a given isotope and is a key factor in the isotope's half-life.
• This process is utilized in many industries, from medicine to archaeology for dating artifacts.

Half-Life

The half-life of a radioactive substance is the time it takes for half of the sample to decay. It's a fundamental concept that helps us understand how long a radioactive isotope will remain active in a given environment.
For iodine-131, the half-life is 8 days. This implies that every 8 days, half of the remaining iodine-131 will decay, transforming into another element, such as xenon-131 after a series of transformations.
Key insights on half-life include:

• It is unique to each radioactive isotope and remains constant.
• Knowing the half-life allows calculation of the remaining amount of an isotope at any given time.
• It is crucial for determining dosage and timing in medical treatments using tracers.

Beta Decay

Beta decay is a type of radioactive decay where a beta particle, either electron (\( \beta^- \)) or positron (\( \beta^+ \)), is emitted from the nucleus. This occurs in isotopes with an excess of neutrons or protons, helping them achieve a stable neutron to proton ratio.
In the case of iodine-131, it goes through beta-minus (\( \beta^- \)) decay, releasing electrons and transforming into xenon-131.
Key features of beta decay:

• It changes the original element into a new one without changing the mass number.
• Beta-minus decay turns a neutron into a proton, while beta-plus decay turns a proton into a neutron.
• It plays a crucial role in radiotracers used for medical imaging and treatments.