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 would take approximately 180,609 years for the activity of plutonium-239 to decrease to 0.1% of its original value. Therefore, a geologically stable storage place must be capable of securing the nuclear waste for at least 180,609 years.

Step-by-step solution

Understand the concept of half-life

Half-life is the time required for a quantity to reduce to half of its initial value. In the context of nuclear decay, it means that half of the radioactive nuclei will have decayed after one half-life period. In this problem, we are given the half-life of plutonium-239 as 24,100 years.

Set up the decay equation

The general equation to describe radioactive decay is given by: \[A(t) = A_0 \cdot (1/2)^{\frac{t}{T}}\] Where: - \(A(t)\) is the activity of the radioactive substance at time \(t\) - \(A_0\) is the initial activity of the radioactive substance - \(T\) is the half-life of the radioactive substance - \(t\) is the time elapsed In our problem, we want to find the time \(t\) when the activity of plutonium-239 decreases to 0.1% (0.001) of its initial activity \(A_0\). Hence, \[0.001 A_0 = A_0 \cdot (1/2)^{\frac{t}{24100}}\]

Solve for the time \(t\)

To find the time \(t\) when the activity reaches 0.1% of its initial value, we divide both sides of the equation by \(A_0\) and solve for \(t\). We have: \[0.001 = (1/2)^{\frac{t}{24100}}\] Now, we can take the natural logarithm of both sides of the equation: \[\ln(0.001) = \ln((1/2)^{\frac{t}{24100}})\] Using the logarithm property \(\ln(a^b) = b \ln(a)\), we get: \[\ln(0.001) = \frac{t}{24100} \ln(1/2)\] Finally, we can solve for \(t\): \[t = \frac{24100 \cdot \ln(0.001)}{\ln(1/2)}\] An approximation gives: \[t \approx 180609 \text{ years}\]

Interpret the result

It would take approximately 180,609 years for the activity of plutonium-239 to decrease to 0.1% of its original value. Therefore, a geologically stable storage place must be capable of securing the nuclear waste for at least 180,609 years.

Key concepts

Half-Life

Half-life is an essential concept in understanding nuclear waste storage. It represents the time required for a radioactive material to decay to half of its initial quantity. In simple terms, if you start with a certain amount of radioactive substance, after one half-life, only half of it will remain.
The remaining amount continues to halve with each subsequent half-life period, and this continues over time.
Understanding half-life is crucial because it helps us predict how long a radioactive material will remain hazardous. In the case of plutonium-239, which is a common byproduct in nuclear reactors, its half-life is an astonishing 24,100 years. This means that it remains harmful for a very long period, necessitating careful consideration in its disposal.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This natural process leads to the transformation of elements, such as uranium, into other elements like thorium or plutonium, over time.
Different radioactive elements have varying rates of decay, which is characterized by their half-lives.

• Decay occurs randomly but predictably, providing a measurable rate over specified times.
• For plutonium-239, its decay process is slow, making it a significant concern in nuclear waste storage.
• The decay equation: \[A(t) = A_0 \cdot (1/2)^{\frac{t}{T}}\] helps us understand how the activity of a radioactive material decreases over time.

Plutonium-239

Plutonium-239 is a powerful radioactive element produced in nuclear reactors, particularly in breeder reactors. It poses a major challenge in nuclear waste management due to its long half-life of 24,100 years. This means that it remains radioactive and potentially harmful for an extended period, which complicates its safe disposal.
Plutonium-239 is used for generating energy in nuclear reactors, but it must be stored safely after its functional life. This ensures that it does not contaminate the environment or pose risks to human health.

• The need for geological stability in storage sites is essential to contain plutonium-239 safely over millennia.

Geological Stability

Geological stability is a critical factor when selecting a location for nuclear waste storage, such as plutonium-239. It refers to the ability of a geological area to remain structurally sound and unchanged over extensive periods, usually thousands or millions of years.
For a nuclear storage site, geological stability ensures that the site is resistant to natural disasters.

• It should not be prone to earthquakes, volcanic activity, or erosion.
• The integrity of the containment structures must be preserved for the entire duration of the radioactive decay, in the case of plutonium-239, at least 180,609 years.
• Natural barriers such as stable rock formations help in minimizing any potential leakages.

In summary, ensuring geological stability is vital to contain the hazards of long-lived nuclides securely.