Problem 6 - Applied Physics

Question

Find the percent of a sample of \({ }_{92}^{-38} \mathrm{U}\) that will decay in the next 975 years. Its half-life is \(4.47 \times 10^{9}\) years.

Step-by-Step Solution

Answer

The percent of the sample that will decay in 975 years is approximately 0.015%.

Step 1: Identify Initial and Half-Life Conditions

We know that the isotope \({ }_{92}^{-38} \mathrm{U}\) has a half-life of \(4.47 \times 10^9\) years. We want to find the percentage that will decay over 975 years.

Step 2: Determine the Decay Constant

The decay constant \(\lambda\) is related to the half-life by the formula \( \lambda = \frac{\ln(2)}{T_{1/2}} \). Substituting the known half-life, we get \[ \lambda = \frac{\ln(2)}{4.47 \times 10^9} \] years^{-1}.

Step 3: Use the Exponential Decay Formula

The fraction remaining after a time \t\ is given by \( N(t) = N_0 e^{-\lambda t} \), where \( N_0 \) is the initial amount. We need the fraction that decays, given by \( 1 - \frac{N(t)}{N_0} \).

Step 4: Calculate Remaining Fraction After 975 Years

Substitute \lambda\ and \t = 975\ years into the formula. Compute \[..., N(t) = e^{-\lambda \cdot 975} \].

Step 5: Plug Values and Solve for Decay Percentage

Substitute the values to compute \[..., decay\ percentage = (1 - e^{-\lambda \cdot 975}) \times 100\] percent. Exact calculation gives \( decay\ percentage \approx 0.015\% \).

Key Concepts

Half-life calculation

The half-life of a radioactive substance is the time it takes for half of the substance to decay. This is a fundamental concept in nuclear physics and is crucial for understanding how radioactive decay works. In the case of the isotope \( ext{ }_{92}^{-38} ext{U}\), its half-life is given as \(4.47 \times 10^9\) years. To solve problems involving half-life, like finding how much of a sample will decay in a given time, follow these steps:

• Identify the half-life of the substance. Here, it's \(4.47 \times 10^9\) years.
• Determine the total time period for which you are calculating the decay. In this exercise, it's 975 years.
• Use the half-life in decay equations to find the decay constant and the fraction of the remaining sample.

By understanding how to calculate half-life, you can predict how long it takes for radioactive substances to reduce by specific amounts over time, which is central to many real-world applications.

Exponential decay

Exponential decay describes the process by which the quantity of a radioactive material decreases over time. This means the material reduces at a rate proportional to its current value. The governing formula for exponential decay is:\[ N(t) = N_0 e^{-\lambda t} \]where:

• \(N(t)\) is the remaining quantity after time \( t \).
• \(N_0\) is the initial quantity.
• \(\lambda\) is the decay constant.
• \(t\) is the elapsed time.

To find how much will decay, calculate \(1 - \frac{N(t)}{N_0}\), which gives the fraction that has decayed over time. This fraction is then converted to a percentage to express the decay in terms of a percentage of the original sample. Understanding exponential decay allows scientists and engineers to model natural processes and predict future conditions, which is vital in fields like nuclear medicine and environmental science.

Decay constant

The decay constant (\(\lambda\)) is a pivotal factor in the equation of radioactive decay, as it quantifies the rate at which a substance undergoes decay. It is specifically calculated using the formula:\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]where:

• \(\ln(2)\) is the natural logarithm of 2, approximately equal to 0.693.
• \(T_{1/2}\) is the half-life of the substance.

This value defines how quickly or slowly a sample will decay. A larger decay constant indicates a faster decay rate. In our exercise, substituting the half-life for \( ext{ }_{92}^{-38} ext{U}\) provides the decay constant, giving insights into how the sample decreases over time. In practical applications, knowing the decay constant helps in managing radioactive materials in medical or industrial contexts, ensuring safety and efficacy.

Nuclear physics concepts

Nuclear physics encompasses the study of the constituents and behavior of atomic nuclei. Understanding radioactive decay is a crucial part of this field. Here are some essential concepts:

• Radioactive decay: The process where an unstable atomic nucleus loses energy by emitting radiation.
• Isotopes: Different forms of elements with the same number of protons but different numbers of neutrons, influencing stability and decay rates.
• Decay modes: Includes alpha, beta, and gamma decay, each with unique particles and energy emissions.

Studying these concepts helps explain phenomena such as nuclear fission and fusion, which power both reactors and stars. With this knowledge, scientists can harness energy from nuclear reactions, develop medical imaging techniques, and understand the fundamental principles governing matter in the universe. This field is vast, touching on everything from the composition of stars to the safety protocols of radioactive waste management.