Question

A proposed system for storing nuclear wastes involves storing the radioactive material in caves or deep mine shafts. One of the most toxic nuclides that must be disposed of is plutonium- 239, which is produced in breeder reactors and has a half-life of 24,100 years. A suitable storage place must be geologically stable long enough for the activity of plutonium- 239 to decrease to \(0.1 \%\) of its original value. How long is this for plutonium- \(239 ?\)

Short answer

It takes approximately 161,890 years for the activity of plutonium-239 to decrease to 0.1% of its original value.

Step-by-step solution

Understanding the radioactive decay formula

Radioactive decay follows an exponential decay pattern, which can be represented by the formula: \[N(t) = N_0 \cdot (1/2)^{t/T}\] where: - \(N(t)\) is the remaining activity at time \(t\) - \(N_0\) is the initial activity - \(t\) is the time (in years) - \(T\) is the half-life of the nuclide (24,100 years for plutonium-239)

Set up the equation

We want to find the time it takes for the activity of plutonium-239 to decrease to 0.1% of its initial value. So, we set up the equation as follows: \[0.001 N_0 = N_0 \cdot (1/2)^{t/24100}\]

Solve for t

Now, we need to solve for \(t\). First, divide by \(N_0\) on both sides of the equation to get: \[0.001 = (1/2)^{t/24100}\] Next, we will take the natural logarithm of both sides to remove the exponential term: \[\ln(0.001) = \ln((1/2)^{t/24100})\] Using the property of logarithms, we can rewrite the equation as: \[\ln(0.001) = \frac{t}{24100} \ln(1/2)\] Now, solve for \(t\): \[t = 24100 \cdot \frac{\ln(0.001)}{\ln(1/2)}\] Evaluate the expression to find the value of \(t\): \[t \approx 161889.63\]

Conclusion

It takes approximately 161,890 years for the activity of plutonium-239 to decrease to 0.1% of its original value.

Key concepts

Radioactive Decay

Radioactive decay refers to the process by which the nucleus of an unstable atom loses energy. This transformation happens as the atom emits radiation, often in the form of alpha or beta particles, along with gamma rays. Over time, an unstable atom changes into a more stable form. This process is random and spontaneous, meaning it's hard to predict exactly when any single atom will decay. Key aspects of radioactive decay include:

• Occurrence naturally in several elements, such as uranium and thorium.
• The emission of particles resulting from the change in atomic states.
• Radioactivity can be harmful, which is why proper nuclear waste management is crucial.

The rate at which radioactive decay occurs is described using mathematical models such as the exponential decay formula. This helps in understanding how much radioactive substance remains over time.

Half-Life

The concept of half-life is crucial in understanding radioactive materials. The half-life of a substance is the time it takes for half of the radioactive atoms to decay. Each radioactive element has its own specific half-life, ranging from fractions of a second to millions of years. Some important points about half-life include:

• After each half-life, 50% of the original radioactivity remains.
• It's a constant measure for each type of atom.
• Helps in determining the duration required for a radioactive material to reach a safe level of activity.

For example, plutonium-239 has a half-life of 24,100 years. This length of time indicates how long it takes for half of any given quantity of plutonium-239 to decay. Calculating how long it takes for the material to become less hazardous is essential in planning for nuclear waste storage.

Plutonium-239

Plutonium-239 is one of the most significant byproducts of nuclear reactors, specifically bred in breeder reactors. It’s a radioactive isotope of plutonium, recognized as a key element in both nuclear weapons and sustainable energy solutions. Key characteristics of plutonium-239 include:

• Highly radioactive and toxic in nature.
• It has a half-life of 24,100 years, making it a long-term concern in waste management.
• Used as a fuel in nuclear reactors due to its ability to sustain a nuclear chain reaction.

Because of its long half-life and potential risks due to radioactivity, finding effective storage solutions for plutonium-239 is a major challenge in nuclear waste management. Implementing methods to safeguard the environment and human health is a top priority.

Exponential Decay Formula

The exponential decay formula is pivotal in modeling how radioactive substances decrease over time. This formula provides a mathematical representation of radioactive decay, highlighting the relationship between the remaining amount of radioactive material and time.The formula is written as:\[N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{t/T}\]Where:

• \(N(t)\) represents the amount of substance left at time \(t\).
• \(N_0\) indicates the initial quantity of the substance.
• \(T\) is the half-life of the substance.

This formula shows how the quantity of radioactive material diminishes by 50% with each passing half-life. In the case of plutonium-239, determining how long it takes for the activity to reduce to 0.1% of its original value requires solving the exponential decay formula, which was executed in the original problem to find that it would take approximately 161,890 years.