Problem 29 - Fundamentals of Physics

Question

A radioactive isotope of mercury, \({ }^{197} \mathrm{Hg}\), decays to gold, \({ }^{197} \mathrm{Au},\) with a disintegration constant of \(0.0108 \mathrm{~h}^{-1} .\) (a) Calculate the half-life of the \({ }^{197} \mathrm{Hg}\). What fraction of a sample will remain at the end of (b) three half-lives and (c) 10.0 days?

Step-by-Step Solution

Answer

(a) 64.17 hours; (b) 1/8 (12.5%) remains; (c) 7.4% remains.

Step 1: Calculate the Half-life

The half-life of a radioactive isotope is calculated using the formula: \[ t_{1/2} = \frac{\ln(2)}{k} \]where \( t_{1/2} \) is the half-life and \( k \) is the disintegration constant. Here, \( k = 0.0108 \, \text{h}^{-1} \). First, compute the natural logarithm of 2: \( \ln(2) \approx 0.693 \). Substitute the values into the formula:\[ t_{1/2} = \frac{0.693}{0.0108} \approx 64.17 \, \text{hours} \]

Step 2: Determine Remaining Fraction After Three Half-lives

After each half-life, the remaining fraction of the substance is halved. After one half-life, \( \frac{1}{2} \) of the substance remains; after two half-lives, \( \left(\frac{1}{2}\right)^2 \); and after three half-lives, \( \left(\frac{1}{2}\right)^3 \). Calculate this:\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]Thus, \( \frac{1}{8} \) or 12.5% of the original sample remains after three half-lives.

Step 3: Calculate Fraction Remaining After 10 Days

First, convert 10 days to hours: \( 10 \, \text{days} \times 24 \, \frac{\text{hours}}{\text{day}} = 240 \, \text{hours} \). The decay formula is:\[ N(t) = N_0 e^{-kt} \]where \( N(t) \) is the amount remaining after time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant. Substitute the values:\[ N(t) = N_0 e^{-0.0108 \times 240} \]\[ N(t) = N_0 e^{-2.592} \]Compute the exponential value: \( e^{-2.592} \approx 0.074 \). Therefore, 7.4% of the original sample remains after 10 days.

Key Concepts

Half-life calculation

The concept of half-life is fundamental to understanding radioactive decay. It is the amount of time required for half of the radioactive nuclei in a sample to decay. This doesn't mean that after one half-life, the radioactive material completely disappears; rather, only half of it transforms, leaving the other half still radioactive. In the calculation of half-life, we use the formula:\[ t_{1/2} = \frac{\ln(2)}{k} \]where:

• \( t_{1/2} \) is the half-life, typically measured in units of time.
• \( \ln(2) \) is the natural logarithm of 2, approximately 0.693.
• \( k \) is the disintegration constant, which indicates the rate of decay.

For example, the disintegration constant for a radioactive mercury isotope is given as \( 0.0108 \, \text{h}^{-1} \). By substituting this into the formula, the half-life of the isotope can be calculated as:\[ t_{1/2} = \frac{0.693}{0.0108} \approx 64.17 \, \text{hours} \]This result tells us that approximately every 64.17 hours, half of the isotope will have decayed.

Disintegration constant

The disintegration constant, noted as \( k \), is a crucial component in understanding how quickly a radioactive substance decays. It quantifies the probability of decay of a single nucleus per unit time and is usually expressed with units that match the time of decay, such as per hour or per second.Intuitively, a larger disintegration constant means the substance decays more quickly, while a smaller value indicates a slower rate of decay. It is directly used in calculating both the half-life and in the exponential decay formula that predicts how much of a substance remains after a certain period.Revisiting the example of radioactive mercury, the given disintegration constant \( k = 0.0108 \, \text{h}^{-1} \) conveys how rapidly this isotope decays each hour. This constant is fundamental for computing how much of a substance remains as it decays over time, using various calculations.

Exponential decay formula

Exponential decay describes how the quantity of a radioactive isotope decreases over time. This decay process follows a specific mathematical pattern, known as the exponential decay formula:\[ N(t) = N_0 e^{-kt} \]where:

• \( N(t) \) is the quantity remaining after time \( t \).
• \( N_0 \) is the initial quantity of the substance.
• \( e \) is the base of the natural logarithms, approximately equal to 2.718.
• \( k \) is the disintegration constant.
• \( t \) is the time elapsed.

In practical terms, by knowing the disintegration constant and the time elapsed, you can use this formula to determine how much of the original substance remains. For instance, if you want to know how much of a mercury isotope remains after 10 days, convert the time to hours (as the disintegration constant is per hour) and substitute into the formula:Given that 10 days is equivalent to 240 hours, and using \( k = 0.0108 \, \text{h}^{-1} \), the formula becomes:\[ N(t) = N_0 e^{-0.0108 \times 240} \approx N_0 e^{-2.592} \approx N_0 \times 0.074 \]Thus, about 7.4% of the initial quantity remains after 10 days.