Problem 85 - Physics for Scientists and Engineers with Modern Physics

Question

Almost all of naturally occurring uranium is \({ }_{92}^{238} \mathrm{U}\) with a half-life of \(4.468 \times 10^{9}\) yr. Most of the rest of natural uranium is \({ }_{92}^{235} \mathrm{U}\) with a half-life of \(7.04 \times 10^{8} \mathrm{yr} .\) Today a sample contains \(0.720 \%{ }_{92}^{235} \mathrm{U} .\) ( \(a\) ) What was this percentage 1.0 billion years ago? (b) What percentage of \({ }_{92}^{235} \mathrm{U}\) will remain 100 million years from now?

Step-by-Step Solution

Answer

(a) 1.93% (b) 0.653%

Step 1: Understanding the Decay Equation

To solve the problem, we'll use the radioactive decay law, which states that the remaining quantity of a radioactive isotope decreases exponentially over time. The formula to use is \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the amount of substance at time \( t \), \( N_0 \) is the initial amount, and \( \lambda \) is the decay constant, calculated as \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) is the half-life.

Step 2: Calculate Decay Constant for 235U

For \( _{92}^{235} \text{U} \), compute the decay constant \( \lambda \) from its half-life of \( 7.04 \times 10^8 \) years. \[ \lambda = \frac{\ln(2)}{7.04 \times 10^8 \text{ yr}} \approx 9.85 \times 10^{-10} \text{ yr}^{-1} \]

Step 3: Calculate Percentage 1 Billion Years Ago

To find the percentage of \( _{92}^{235} \text{U} \) 1.0 billion years ago, use the decay equation: \( N(t) = N_0 e^{-\lambda t} \). Set \( N_0 = 0.720\% \), \( \lambda = 9.85 \times 10^{-10} \text{ yr}^{-1} \), and \( t = -1.0 \times 10^9 \text{ yr} \). Compute: \[ N(-10^9) = 0.720 e^{(9.85 \times 10^{-10})(1.0 \times 10^9)} \] \[ N(-10^9) \approx 0.720 \times e^{0.985} \approx 0.720 \times 2.678 \approx 1.93\% \] So, the percentage was approximately 1.93% a billion years ago.

Step 4: Calculate Percentage 100 Million Years into the Future

Now calculate the percentage of \( _{92}^{235} \text{U} \) that remains 100 million years from now using the same equation with \( t = 1.0 \times 10^8 \text{ yr} \): \[ N(10^8) = 0.720 e^{-(9.85 \times 10^{-10})(1.0 \times 10^8)} \] \[ N(10^8) \approx 0.720 \times e^{-0.0985} \approx 0.720 \times 0.906 \approx 0.653\% \] Therefore, the percentage remaining in 100 million years will be approximately 0.653%.

Key Concepts

uranium-235

Uranium-235 is an isotope of uranium that is significant for its applications in nuclear energy and weapons. Unlike the more abundant Uranium-238, Uranium-235 can sustain chain reactions, making it crucial for nuclear reactors and atomic bombs.
It is a naturally occurring isotope found in small amounts in uranium ore, making up about 0.72% of the world's uranium. This isotope has a unique property called fissionability. When it absorbs a neutron, it can split into two or more smaller nuclei, releasing a significant amount of energy.

• This energy release is harnessed in nuclear power plants to produce electricity.
• Due to its scarcity, processes like enrichment are often used to increase the concentration of Uranium-235 for specific applications.

Its role in radioactive decay and nuclear reactions makes understanding its behavior over time, such as through half-life calculations, essential for safe and effective use.

half-life calculation

Half-life calculation is a crucial concept when studying radioactive decay. It refers to the time required for half of a sample of a radioactive isotope to decay.
For Uranium-235, the half-life is approximately 704 million years, which means if you start with a given amount of this isotope, half of it will have decayed into a different element after 704 million years.
Calculating the half-life involves several steps:

• Determine the initial amount of the isotope you have.
• Identify the remaining amount after a certain period.
• Use the half-life equation to find the decay constant, a fundamental parameter in these calculations.

In mathematical terms, if you wish to calculate how much of an isotope remains after a given time, you would use the equation:
\( N(t) = N_0 e^{-\lambda t} \)
where:

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

Half-life calculations are instrumental in fields ranging from archaeology, where they date ancient artifacts, to nuclear medicine, where they determine the dosage and timing of radiation therapies.

decay constant

The decay constant is a fundamental aspect of understanding radioactive decay. It is defined as the probability of decay per unit time for a radioactive isotope.
For Uranium-235, this is particularly important because it determines how quickly this isotope decays over time. The decay constant \( \lambda \) is calculated using the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
where \( t_{1/2} \) is the half-life of the isotope.
This formula illustrates the relationship between the half-life and the decay constant. A longer half-life means a smaller decay constant, indicating slower decay, which is the case for Uranium-235 with its long half-life of millions of years.
In practice, calculating the decay constant helps scientists predict how much of a substance will remain after a certain period. This prediction is crucial in fields such as geology, where it aids in dating rocks and fossils, or in nuclear waste management, where it helps in planning for long-term storage solutions.