Problem 82 - Chemistry Matter and Change

Question

The half-life of plutonium-239 is \(24,000\) y. How much nuclear waste generated today will remain in 1000 years?

Step-by-Step Solution

Answer

97.18% of the plutonium-239 remains after 1000 years.

Step 1: Understand the half-life concept

The half-life is the time it takes for half of a given quantity of a substance to decay. For plutonium-239, this duration is 24,000 years.

Step 2: Use the exponential decay formula

The amount of substance remaining after time can be calculated using the formula: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]where:- \( N(t) \) is the remaining amount after time \( t \),- \( N_0 \) is the initial amount,- \( t \) is the time elapsed,- \( t_{1/2} \) is the half-life.

Step 3: Substitute the given values into the formula

Here, \( t = 1000 \) years, \( t_{1/2} = 24000 \) years. Thus, the equation becomes:\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{1000}{24000}} \]

Step 4: Simplify the exponent

Calculate the fraction and simplify the exponent:\[ \frac{1000}{24000} = \frac{1}{24} \]So the formula becomes:\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{1}{24}} \].

Step 5: Compute the remaining quantity factor

Using a calculator, compute \( \left( \frac{1}{2} \right)^{\frac{1}{24}} \) which is approximately \( 0.9718 \).

Step 6: Interpret the result

The factor \( 0.9718 \) represents the portion of the nuclear waste remaining after 1000 years given any initial quantity. So, about 97.18% of the nuclear waste will remain.

Key Concepts

exponential decay

Exponential decay is a fundamental concept in understanding how certain quantities decrease over time at a rate proportional to their current value. This is a vital tool in various scientific fields, especially when analyzing radioactive substances. In exponential decay, the decrease is not linear—instead, it happens at an exponentially decreasing rate. This means each successive half of the substance takes the same amount of time to decay as the previous half, resulting in a curve that decreases rapidly initially and then levels off. For example, in the context of radioactive decay, exponential functions can describe how an isotope like plutonium-239 diminishes over time. The formula used is:

• \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \], where:
• \( N(t) \) is the remaining amount after time \( t \),
• \( N_0 \) is the initial quantity,
• \( t \) is the time elapsed,
• \( t_{1/2} \) is the half-life of the substance.

This mathematical model helps predict how much of a radioactive sample remains after a certain period, providing insights into both environmental impacts and safety concerns.

nuclear waste

Nuclear waste refers to the radioactive residues left over after nuclear reactions, such as those that occur in nuclear reactors or during the production and testing of nuclear weapons. One of the principal challenges with nuclear waste is its long-lived radioactivity, posing potential risks to human health and the environment. Understanding the management of nuclear waste involves:

• Safe storage to prevent contamination.
• Long-term environmental responsibility given its prolonged radioactive nature.
• Ensuring the containment systems last until the waste is no longer harmful.

Radioactive isotopes such as plutonium-239, which has an exceptionally long half-life, must be handled with extreme care. Its decay happens over thousands of years, requiring robust strategies to secure and monitor these materials for extended periods.

plutonium-239

Plutonium-239 is a radioactive isotope primarily associated with nuclear weapons and nuclear power. It has fascinating properties that make it crucial, albeit dangerous. This radioactive form of plutonium is not naturally occurring but is instead manufactured in reactors. Its significance comes from:

• Its use as a fuel in nuclear reactors or in the manufacture of nuclear weapons.
• A half-life of 24,000 years, making it both potent and challenging to manage safely.
• The potential to stay hazardous in the environment for hundreds of thousands of years.

Plutonium-239, due to its long half-life, serves as a model case for studying nuclear waste management. Understanding its decay process and environmental impact is essential, considering such isotopes require careful long-term handling to mitigate associated risks.

mathematical modeling

Mathematical modeling is an essential tool that enables scientists and engineers to predict behaviors and solve problems across various fields. In the context of nuclear physics, models are particularly important for managing radioactive materials like plutonium-239. By applying mathematical models to half-life calculations, we can predict the behavior of nuclear waste over time, which helps:

• Determine how long a material remains hazardous.
• Plan safe storage durations and methods.
• Assess environmental and health risks associated with radioactive decay.

Models such as the exponential decay formula offer a structured approach to understanding the reduction of nuclear waste quantities. These models not only help in mathematical calculations but also inform policy and safety decisions. In essence, mathematical modeling allows for the practical application of theoretical concepts, providing a reliable method to forecast long-term outcomes in nuclear science.