Question

Each statement that follows refers to a comparison between two radioisotopes, \(A\) and \(X .\) Indicate whether each of the following statements is true or false. (a) If the half-life for \(\mathrm{A}\) is shorter than the half-life for \(\mathrm{X}, \mathrm{A}\) 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 yr, and \(X\) has a half-life of \(10,000 \mathrm{yr}\), A would be a more suitable radioisotope to measure processes occurring on the 40 -yr time scale.

Short answer

(a) True - A shorter half-life implies a larger decay rate constant. (b) False - A non-radioactive isotope has an infinite half-life, not zero. (c) True - Isotope A is more suitable for measuring processes on a 40-year time scale.

Step-by-step solution

(a: Half-life vs. Decay rate constant relationship)

Since the relationship between half-life and the decay constant is \(T=\frac{ln(2)}{\lambda}\), we can deduce that the half-life and the decay rate constant are inversely related. Therefore, if the half-life of isotope A is shorter than the half-life of isotope X, the decay rate constant for A would be larger. So, the statement is true.

(b: Half-life of non-radioactive isotope)

If isotope X is "not radioactive," that means it does not decay or disintegrate by emitting radiation, and it has a stable nucleus; hence, its half-life is not essentially zero. Instead, it would be infinite since it would not undergo decay. The statement is false.

(c: Suitability for measuring 40-year time scale processes)

We need to determine whether A or X would be more suitable for measuring processes occurring on a 40-year time scale. Given that isotope A has a half-life of 10 years, and isotope X has a half-life of 10,000 years, isotope A with a half-life closer to the given time scale (40 years) would be better suited for measuring the processes of interest, as that would mean more significant changes in its decay in a 40-year period compared to isotope X. The statement is true.

Key concepts

Half-life

The concept of "half-life" is central in understanding how radioisotopes decay. A half-life is the time required for half of the radioactive nuclei in a sample to undergo decay. It's a measure of how quickly or slowly a radioisotope "loses" its radioactivity. For example, isotope A with a short half-life of 10 years will decay more rapidly than isotope X with a longer half-life of 10,000 years.
Knowing the half-life helps scientists understand the stability and longevity of a radioisotope. A shorter half-life implies rapid decay, making such isotopes useful for short-term applications like medical diagnostics. On the other hand, isotopes with long half-lives are better suited for long-term processes such as geological dating. This is because they release energy slowly over extended periods, minimizing radioactive exposure. In calculations, the half-life can be linked to the decay constant via the formula: \[ T = \frac{\ln(2)}{\lambda} \] where \( T \) represents the half-life and \( \lambda \) the decay rate constant.

Decay rate constant

The decay rate constant, denoted as \( \lambda \), describes how rapidly a radioisotope decays. This constant helps quantify the likelihood of a decay event occurring per unit time. A larger decay rate constant indicates a higher probability of decay each second, leading to a shorter half-life for the isotope.This inverse relation between half-life and decay constant, established by the equation \[ T = \frac{\ln(2)}{\lambda} \], is fundamental in radioactivity studies:

• High decay rate constant: faster decay, shorter half-life.
• Low decay rate constant: slower decay, longer half-life.

When comparing isotopes A and X, if A has a shorter half-life than X, it necessarily means isotope A has a larger decay rate constant. This means isotope A is actively decaying more rapidly than isotope X.

Radioactivity measurement

Measuring radioactivity is crucial for various applications like dating historical artifacts, diagnosing medical conditions, and monitoring environmental changes. The choice of a suitable radioisotope for measurement depends heavily on its half-life relative to the timeframe of the process being studied. For instance, if we have a process that unfolds over 40 years, an isotope like A (with a 10-year half-life) would be preferable over one like X (with a 10,000-year half-life). The reason is isotope A would undergo significant decay over the 40-year period, providing clear data. In contrast, isotope X would show little change, making it difficult to accurately track or measure changes within that timeframe. In addition to the half-life, effective radioactivity measurement requires:

• Understanding the decay rate: Faster decaying isotopes can give more immediate results.
• Proper instrumentation: Tools like Geiger counters or scintillation detectors are used to measure emitted radiation.
• Safe handling: Given the potential hazards, it's crucial to manage exposure to radioactive materials carefully.