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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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