Question

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

(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

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.

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.

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.

Key concepts

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.