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Chapter 21: Problem 31

Each statement that follows refers to a comparison between two radioisotopes, \(\mathrm{A}\) and \(\mathrm{X}\). Indicate whether each of the following statements is true or false, and why. (a) If the half-life for \(A\) is shorter than the half-life for \(X, A\) has a larger decay rate constant. (b) If \(X\) is "not radioactive," its half-life is essentially zero. (c) If A has a half-life of 10 years, and \(X\) has a half-life of 10,000 years, A would be a more suitable radioisotope to measure processes occurring on the 40 -year time scale.

Short Answer

Expert verified
(a) True. A shorter half-life implies a larger decay rate constant because \( t_{1/2} = \frac{ln(2)}{k}\). (b) False. A non-radioactive element has an infinite half-life, not essentially zero. (c) True. A, with a half-life of 10 years, is more suitable for measuring processes on a 40-year time scale than X with a half-life of 10,000 years.

Step by step solution

01

Statement (a) - Half-life and decay rate constant relationship.

We are given that the half-life of A is shorter than the half-life of X. We need to determine if A has a larger decay rate constant. The decay rate constant, \(k\), is related to the half-life, \(t_{1/2}\), by the equation: \[ t_{1/2} = \frac{ln(2)}{k}\] A shorter half-life means smaller \(t_{1/2}\), and since the numerator \(ln(2)\) is constant, the decay rate constant, \(k\), should be larger to maintain the relationship. Therefore, A has a larger decay rate constant than X. Hence, the statement (a) is true.
02

Statement (b) - Non-radioactive isotope's half-life.

We are given that X is not radioactive. We need to determine if its half-life is essentially zero. A non-radioactive isotope does not decay, meaning that its decay rate constant, \(k\), is zero. Using the previous equation related to the half-life and decay rate constant: \[ t_{1/2} = \frac{ln(2)}{k}\] When \(k = 0\), the equation becomes undefined as we are dividing by zero. A non-radioactive element has an infinite half-life and not essentially zero. Therefore, the statement (b) is false.
03

Statement (c) - Suitability as a measure of processes.

We are given that A has a half-life of 10 years, and X has a half-life of 10,000 years. We need to determine which radioisotope is more suitable for measuring processes on a 40-year time scale. A radioisotope with a half-life that is closer to the time scale of the process will be a more suitable radioisotope for measuring because the activity will be more significant and easier to measure during that time frame. In this case, A has a half-life of 10 years, which is closer to the 40-year time scale than X's half-life of 10,000 years. Therefore, A would be a more suitable radioisotope to measure processes on the 40-year time scale. Hence, the statement (c) is true.

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Key Concepts

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

Half-life
When we talk about radioisotopes, one essential concept is the half-life. Half-life is the time it takes for half of the radioactive nuclei in a sample to decay.
This concept is crucial because it helps us understand how quickly a radioisotope will lose its radioactivity.
  • A shorter half-life means the radioisotope decays more quickly.
  • A longer half-life implies a slower decay process.
The half-life is an important measure because it can tell us how long a radioisotope will remain active in a given state. The relationship between half-life and the decay rate constant, \(k\), is given by the formula:
\[ t_{1/2} = \frac{\ln(2)}{k} \]This formula tells us that if the half-life is shorter, \(k\) has to be larger for the equation to hold true. For instance, in statement (a) from the exercise, radioisotope A, having a shorter half-life than X, results in A having a larger decay rate constant, making the statement "true".
Decay Rate Constant
The decay rate constant, denoted as \(k\), is a parameter that describes the rate at which a radioactive isotope decays. This constant directly relates to the radioisotope's stability and how quickly it undergoes radioactive decay.
  • A high decay rate constant means the isotope decays rapidly.
  • A low decay rate constant indicates slower decay.
Understanding \(k\) helps in determining the characteristics of a radioisotope. It is strongly linked to the half-life through the formula \( t_{1/2} = \frac{\ln(2)}{k} \), showing how faster decay (higher \(k\)) results in a shorter half-life.
When an isotope is termed "not radioactive," like in statement (b) for isotope X, it technically has a decay rate constant of zero. This means it does not decay, resulting in an infinite half-life, contrary to the idea of having a zero half-life.
Radioactive Measurement Suitability
When we want to use radioisotopes for measuring processes, we must consider their half-lives relative to the time scale of the process we are studying.
Suitability depends significantly on this relationship between half-life and time scale. An ideally suitable isotope will have a half-life close to the time period we are interested in measuring.
  • If the half-life is much shorter than the process duration, the isotope may become inactive too soon.
  • If the half-life is much longer, changes may not be detectable within the time frame.
In the given exercise, statement (c) compares radioisotopes A and X for measuring processes over 40 years. With half-lives of 10 years for A and 10,000 years for X, isotope A is more suitable because its half-life is closer to the 40-year measurement window.
This makes measurements more precise and manageable, and therefore, in this statement, A is indeed the most fitting choice.

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Most popular questions from this chapter

One nuclide in each of these pairs is radioactive. Predict which is radioactive and which is stable: (a) \({ }_{20}^{40} \mathrm{Ca}\) and \({ }_{20}^{45} \mathrm{Ca},(\mathbf{b}){ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C},(\mathrm{c})\) lead- 206 and thorium- \(230 .\) Explain your choice in each case.

Iodine- 131 is a convenient radioisotope to monitor thyroid activity in humans. It is a beta emitter with a half-life of 8.02 days. The thyroid is the only gland in the body that uses iodine. A person undergoing a test of thyroid activity drinks a solution of NaI, in which only a small fraction of the iodide is radioactive. (a) Why is NaI a good choice for the source of iodine? (b) If a Geiger counter is placed near the person's thyroid (which is near the neck) right after the sodium iodide solution is taken, what will the data look like as a function of time? (c) A normal thyroid will take up about \(12 \%\) of the ingested iodide in a few hours. How long will it take for the radioactive iodide taken up and held by the thyroid to decay to \(0.01 \%\) of the original amount?

It takes 5.2 min for a 1.000 -g sample of \({ }^{210} \mathrm{Fr}\) to decay to \(0.250 \mathrm{~g}\). What is the half-life of \({ }^{210} \mathrm{Fr}\) ?

A \(65-\mathrm{kg}\) person is accidentally exposed for \(240 \mathrm{~s}\) to a \(15-\mathrm{mCi}\) source of beta radiation coming from a sample of \({ }^{90}\) Sr. (a) What is the activity of the radiation source in disintegrations per second? In becquerels? (b) Each beta particle has an energy of \(8.75 \times 10^{-14} \mathrm{~J},\) and \(7.5 \%\) of the radiation is absorbed by the person. Assuming that the absorbed radiation is spread over the person's entire body, calculate the absorbed dose in rads and in grays. (c) If the \(\mathrm{RBE}\) of the beta particles is 1.0, what is the effective dose in mrem and in sieverts? (d) How does the magnitude of this dose of radiation compare with that of a mammogram ( \(300 \mathrm{mrem}\) )?

Complete and balance the nuclear equations for the following fission or fusion reactions: (a) \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{2}^{3} \mathrm{He}+\) (b) \({ }_{92}^{239} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{98} \mathrm{Nb}+{ }_{-0}^{1} \mathrm{n}\)
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