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The half-life of a substance is a measure of how quickly it undergoes radioactive decay. It tells us the time it takes for half of a sample of a radioactive element to disintegrate. This concept is crucial in understanding the stability of the substance over time. The half-life formula is given by:

• \( t_{1/2} = \frac{0.693}{k} \)

where \( t_{1/2} \) is the half-life and \( k \) is the decay constant. In the given exercise, the decay constant, \( k \), is provided as \( 0.0108 \, \mathrm{h}^{-1} \). By substituting this value into the half-life formula, we can calculate the half-life of the radioactive isotope mercury, \( ^{197} \mathrm{Hg} \), as follows:

• \( t_{1/2} = \frac{0.693}{0.0108} \approx 64.17 \, \mathrm{hours} \)

This means it takes approximately 64.17 hours for half of a given sample of \( ^{197} \mathrm{Hg} \) to decay into another element.

The decay constant is a fundamental parameter in radioactive decay calculations. It is denoted by \( k \) and represents the probability per unit time that a single nucleus will decay. The decay constant is unique for each radioactive isotope and provides essential information on how rapidly the isotope will decay.

Given that the decay constant \( k \) for \( ^{197} \mathrm{Hg} \) is \( 0.0108 \, \mathrm{h}^{-1} \), we can deduce that:

• The higher the decay constant, the faster the substance will decay.
• A lower decay constant indicates a slower decay process.

The relationship between the decay constant and half-life is inversely proportional. As seen in the half-life formula,

• \( t_{1/2} = \frac{0.693}{k} \)

As \( k \) increases, \( t_{1/2} \) decreases, indicating quicker decay. Understanding the decay constant allows us to predict how a radioactive isotope behaves over time. This principle is applied throughout fields like geology, archeology, and medicine.

Exponential decay is a mathematical model describing how quantities decrease rapidly at first, then slowly, with time. This model is highly relevant in understanding radioactive decay processes. The main formula for exponential decay in the context of radioactivity is:

• \( N_t = N_0 \cdot e^{-kt} \)

where:

• \( N_t \) = the quantity remaining at time \( t \)
• \( N_0 \) = the initial quantity
• \( k \) = decay constant
• \( t \) = time elapsed

In the given exercise, exponential decay helps us determine how much of \( ^{197} \mathrm{Hg} \) remains after a certain period. For instance, after 240 hours (10 days), the remaining fraction of \( ^{197} \mathrm{Hg} \) can be calculated by substituting \( k = 0.0108 \, \mathrm{h}^{-1} \) and \( t = 240 \, \mathrm{h} \) into the formula:

• \( N_t / N_0 = e^{-0.0108 \times 240} \approx e^{-2.592} \approx 0.0746 \)

This calculation shows that only about 7.46% of the original sample remains after 10 days, illustrating the principle of exponential decay.