Problem 39 - Thomas Calculus in SI Units

Question

Physicists using the radioactivity equation \(y=y_{0} e^{-k t}\) call the number \(1 / k\) the mean life of a radioactive nucleus. The mean life of a radon nucleus is about \(1 / 0.18=5.6\) days. The mean life of a carbon- 14 nucleus is more than 8000 years. Show that \(95 \%\) of the radioactive nuclei originally present in a sample will disintegrate within three mean lifetimes, i.e., by time \(t=3 / k .\) Thus, the mean life of a nucleus gives a quick way to estimate how long the radioactivity of a sample will last.

Step-by-Step Solution

Answer

After three mean lifetimes, 95% of the sample will disintegrate as \( e^{-3} \approx 0.05 \).

Step 1: Understand the Equation

The equation given is \( y = y_0 e^{-kt} \), where \( y \) is the remaining quantity of radioactive nuclei at time \( t \), \( y_0 \) is the initial quantity, and \( k \) is the decay constant.

Step 2: Calculate for Three Mean Lifetimes

We need to show that by time \( t = \frac{3}{k} \), 95% of the original sample has disintegrated. Substitute \( t = \frac{3}{k} \) into the equation: \( y = y_0 e^{-k(\frac{3}{k})} = y_0 e^{-3} \).

Step 3: Determine the Remaining Quantity

Calculate \( e^{-3} \) to find the fraction of the original sample that remains: \( e^{-3} \approx 0.05 \). This means that 5% of the nuclei remains after three mean lifetimes.

Step 4: Find the Percentage Disintegrated

Since 5% of the nuclei remain, it means that 95% have disintegrated: \(100\% - 5\% = 95\% \).

Step 5: Conclusion

After three mean lifetimes \( t = \frac{3}{k} \), 95% of the original radioactive nuclei would have disintegrated, as \( e^{-3} \approx 0.05 \). Thus, the mean life provides a quick estimate for the duration of radioactivity.

Key Concepts

Decay Constant

The decay constant, often represented by the symbol \( k \), is a fundamental parameter in the study of radioactivity. It describes the rate at which a radioactive substance decays over time. In essence, it is a measure of how quickly the nuclei in a radioactive sample disintegrate. This constant is crucial because it helps us understand the stability of different isotopes and predict their behavior.

Here's why it matters:

• A larger decay constant means that the substance decays more quickly. It indicates that the radioactive nuclei have a higher probability of disintegrating in a given timeframe.
• A smaller decay constant suggests a slower decay rate, meaning the substance is more stable over time.

Understanding the decay constant allows scientists to determine various characteristics, such as the mean life and half-life of a substance, which are pivotal in Nuclear Physics.

Mean Life

Mean life is a term used to describe the average lifetime of a radioactive nucleus before it decays. It is mathematically expressed as the reciprocal of the decay constant, \( \frac{1}{k} \). For example, if a sample has a decay constant \( k = 0.18 \), its mean life is \( \frac{1}{0.18} = 5.6 \) days. This term is central in predicting how long a radioactive sample will remain active.

Key points about Mean Life:

• It provides a general estimate of the time it takes for a large number of identical nuclei to undergo decay.
• Unlike the half-life, which indicates when half of the nuclei have decayed, mean life focuses on the entire process over a prolonged period.

Mean life is particularly useful in applications such as radiocarbon dating, which relies on the decay of carbon-14 to estimate the age of organic materials.

Exponential Decay

Exponential decay is a process in which the quantity of a substance decreases at a rate proportional to its current amount. It is described by the mathematical formula \( y = y_0 e^{-kt} \), where \( y \) is the remaining amount at time \( t \), and \( y_0 \) is the initial quantity. This concept is at the core of understanding radioactive decay, as it accurately models how substances disintegrate over time.

Why it matters:

• Exponential decay allows for predictions about how quickly a sample will reduce in quantity, based on known parameters like the decay constant.
• This model applies not only to radioactivity but also to other phenomena, such as population decline and cooling processes.

The beauty of exponential decay is that it provides a seamless way to predict future quantities without complex calculations, making it invaluable in nuclear physics and beyond.

Nuclear Physics

Nuclear physics is the field of science concerned with the constituents and interactions of atomic nuclei. It delves into the understanding of radioactive substances, nuclear decay processes, and the forces that hold nuclei together. In the context of the exercise, nuclear physics provides the theoretical framework for analyzing mean life, decay constants, and exponential decay.

In nuclear physics, one studies:

• The forces and particles that make up atomic nuclei.
• How unstable isotopes transform and release energy.
• The applications of radioactive decay, such as in energy production, medical treatments, and carbon dating.

This field is crucial for both theoretical research and practical applications, providing insights into the fundamental nature of matter and energy.