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Chapter 43: Problem 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. One such tracer is \({ }^{131} \mathrm{I}\), a \(\beta^{-}\) emitter with a half-life of \(8.0 \mathrm{~d}\). Suppose a scientist introduces a sample with an activity of \(325 \mathrm{~Bq}\) 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
The decay rate after 24 days is approximately 40.625 Bq. About 41.8 % of the tracer ended up in the thyroid gland. The isotope that remains after the I-131 decays is \({ }^{131} \mathrm{Xe}\).

Step by step solution

01

Calculate decay rate

First, let's calculate the decay rate after 24 days using the formula above. Here, \(N_{0}\) is the initial activity which is 325 Bq, \(t_{\frac{1}{2}}\) is the half-life of Isotope I-131 which is 8 days, and \(t\) is the time after which we want to calculate the decay rate which is 24 days. So, after substituting the values into the equation, we get \(N = 325 * 0.5^{\frac{24}{8}}\).
02

Compute the decay rate

After calculating N, we find that the decay rate after 24 days is about 40.625 Bq.
03

Calculate the percentage of tracer in the thyroid gland

Second part of the exercise asks what percentage of the tracer went to the thyroid gland if the decay rate in the thyroid 24 d later is measured to be 17.0 Bq. We can find this by taking the ratio of actual activity measured in the thyroid gland (17.0 Bq) to the expected activity (40.625 Bq), and then multiplying by 100 to get the percentage. That is, \(\frac{17}{40.625}*100\%.\)
04

Compute the percentage

After calculating, we find the percentage to be approximately 41.8 %.
05

Identify the isotope after decay

Lastly, we want to find out what isotope remains after the I-131 decays. I-131 is a \(\beta^{-}\) emitter, which means it undergoes beta minus decay. In beta minus decay, a neutron turns into a proton and an electron (beta particle). The proton stays in the nucleus, increasing the atomic number by 1. So, the isotope that remains after decay will be one with atomic number greater by one. Iodine (I) has atomic number 53 so the isotope after decay will be of element with atomic number 54, which is Xenon (Xe). So, the isotope will be \({ }^{131} \mathrm{Xe}\).

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Key Concepts

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

Decay Rate
The decay rate of a radioactive substance determines how quickly it loses its radioactivity over time. It is often measured in Bequerels (Bq), a unit indicating the number of decays per second. To find the decay rate at any point in time, the formula used is:
\[ N = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]This formula shows that the decay rate exponentially decreases over time as the radioactive substance loses half of its radioactivity every half-life period. For the initial activity of a sample, \( N_0 \), at a specific time, \( t \), and half-life, \( t_{1/2} \), the decay rate, \( N \), can be calculated easily.
  • Imagine a scientist monitoring a sample of iodine-131 with an activity of 325 Bq.
  • After a period of three half-lives (24 days), the decay rate can be recalculated to approximately 40.625 Bq.
Understanding decay rates is crucial for fields like nuclear medicine and radiology where precise dosage is vital.
Half-life
Half-life is an essential concept in understanding radioactive decay. It represents the time required for half of the radioactive atoms in a sample to decay. Every radioactive isotope has a unique half-life, making it possible to predict how long it takes for a substance to become less radioactive.
  • For iodine-131, the half-life is 8 days, a relatively short time compared to other isotopes.
  • This short half-life makes iodine-131 useful as a medical tracer, as it exits the body quickly and minimizes radiation exposure.
Calculating how many half-lives have passed helps in estimating the remaining activity of a substance at a given time. For instance, after 24 days, iodine-131 will have gone through approximately three half-lives, reducing its radioactivity significantly.
Beta Minus Decay
Beta minus decay is a type of radioactive decay where a neutron in an atom's nucleus converts into a proton, emitting a beta particle, which is an electron. This process leads to an increase in the atomic number by one while the mass number remains unchanged.
  • During this decay, iodine-131 emits a beta particle, transforming a neutron into a proton.
  • The atomic number of iodine (I) goes from 53 to 54, changing the element to xenon (Xe).
Beta minus decay is a common transformation in radioisotopes, leading to the formation of a new element by converting neutron-rich nuclides to more stable forms. This process is crucial in nuclear physics and medicine to understand how isotopes change within the body or environment.
Radioactive Isotopes
Radioactive isotopes are variations of elements with unstable nuclei. They release energy in the form of radiation, which is detectable and can be used in multiple applications, such as medical diagnostics and treatments.
  • Isotopes like iodine-131 are used in traces to examine the function of certain organs or detect leaks in pipes.
  • They have specific half-lives, allowing scientists to time their observations accurately.
The use of radioactive isotopes requires careful handling and understanding of decay processes, ensuring safety and effectiveness. Their role in advancing medical techniques, especially in diagnosing thyroid function or cancer treatments, highlights their importance.
Xenon
Xenon is a noble gas found naturally in the atmosphere in trace amounts. In nuclear science, it is a byproduct of beta decay processes like those involving iodine-131.
  • When iodine-131 undergoes beta minus decay, it transforms into xenon-131, adding a proton in the nucleus.
  • Xenon, atomic number 54, is chemically non-reactive, making it stable compared to other elements.
Though inert, xenon has applications in medical imaging, lighting, and anesthesia. Understanding its formation from decay processes aids scientists in tracking the life cycle of radioactive substances, contributing to advancements in nuclear and medical technologies.

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Most popular questions from this chapter

Consider the nuclear reaction $$ { }_{14}^{28} \mathrm{Si}+\gamma \rightarrow{ }_{12}^{24} \mathrm{Mg}+\mathrm{X} $$where \(X\) is a nuclide. (a) What are \(Z\) and \(A\) for the nuclide \(X ?\) (b) Ignoring the effects of recoil, what minimum energy must the photon have for this reaction to occur? The mass of a \({ }_{14}^{28} \mathrm{Si}\) atom is \(27.976927 \mathrm{u}\), and the mass of a \({ }_{12}^{24} \mathrm{Mg}\) atom is \(23.985042 \mathrm{u}\).

As a health physicist, you are being consulted about a spill in a radiochemistry lab. The isotope spilled was \(400 \mu \mathrm{Ci}\) of \({ }^{131} \mathrm{Ba},\) which has a half-life of 12 days. (a) What mass of \({ }^{131}\) Ba was spilled? (b) Your recommendation is to clear the lab until the radiation level has fallen \(1.00 \mu \mathrm{Ci} .\) How long will the lab have to be closed?

The fusion reactions in the sun produce \(26.73 \mathrm{MeV}\) per alpha particle created. Each of these reactions produces two neutrinos. In total, the sun's power is about \(3.8 \times 10^{26} \mathrm{~W}\). (a) Estimate how many solar fusion reactions take place every second. (b) Estimate the number of neutrinos that leave the sun each second. (c) The distance between the earth and the sun is 150 million \(\mathrm{km},\) and the radius of the earth is \(6370 \mathrm{~km} .\) Using these figures, estimate the number of solar neutrinos that encounter the earth each second. (d) How many square centimeters does the earth present to the sun? (e) Divide your answer in part (c) by your answer in part (d) to obtain an estimate of the number of solar neutrinos that pass through every square centimeter on earth every second.

Heat flows to the surface of the earth from its interior. Twothirds of this heat is radiogenic in origin, which means it comes from the decay of radioactive elements in the earth's mantle, primarily from the isotopes \({ }^{238} \mathrm{U}\) and \({ }^{232} \mathrm{Th}\). We can estimate the earth's internal energy supply due to the decay of uranium as follows: (a) There are \(31 \mu \mathrm{g}\) of \({ }^{238} \mathrm{U}\) in each \(\mathrm{kg}\) of mantle. Estimate how many \({ }^{238} \mathrm{U}\) isotopes there are per \(\mathrm{kg}\) of mantle. (b) The half-life of \({ }^{238} \mathrm{U}\) is 4.47 billion years. Calculate the decay constant in units of \(\mathrm{s}^{-1}\). (c) Multiply the results in parts (a) and (b) to obtain the activity in each \(\mathrm{kg}\) of mantle. (d) Each decay chain \({ }^{238} \mathrm{U} \rightarrow{ }^{206} \mathrm{~Pb}\) releases \(52 \mathrm{MeV}\) of energy. Use this value to convert the activity into watts per \(\mathrm{kg}\) of mantle. (e) The mass of the earth is \(6 \times 10^{24} \mathrm{~kg},\) and twothirds of this is mantle mass. Use these values to estimate the power earth receives from uranium decays. (f) Uranium decays account for \(39 \%\) of the radiogenic supply of heat. Estimate earth's total heat power related to radioactive decays. (g) Earth also releases heat that remains from its formation, with a power comparable with that due to radioactivity. Estimate the total heat power earth receives from its interior.

How many protons and how many neutrons are there in a nucleus of the most common isotope of (a) silicon, \(\frac{28}{14} \mathrm{Si} ;\) (b) rubidium, \({ }_{37}^{85} \mathrm{Rb} ;\) (c) thallium, \({ }^{205} \mathrm{Tl} ?\)
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