Problem 369
Question
True or False? If true, prove it. If false, find the true answer. The spent fuel of a nuclear reactor contains plutonium- \(239,\) which has a half-life of \(24,000\) years. If 1 barrel containing 10 \(\mathrm{kg}\) of plutonium- 239 is sealed, how many years must pass until only 10\(g\) of plutonium- 239 is left?
Step-by-Step Solution
Verified Answer
False; about 239,184 years are needed to reduce to 10 g.
1Step 1: Understanding the Concept of Half-Life
The half-life of a substance is the amount of time required for half of that substance to decay or transform. Here, the half-life of plutonium-239 is given as 24,000 years. This means every 24,000 years, only half of the initial amount of plutonium will remain.
2Step 2: Setting Up the Initial and Final Quantities
Initially, the barrel contains 10 kg of plutonium-239, which is equal to 10,000 g (1 kg = 1000 g). We want to find out how long it takes for this amount to reduce to 10 g.
3Step 3: Using the Half-Life Formula
To find out the time `t` for the substance to decay to a specific amount, we use the formula: \[N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]where \(N(t)\) is the remaining quantity of the substance at time \(t\), \(N_0\) is the initial quantity, and \(T_{1/2}\) is the half-life.
4Step 4: Applying the Formula with Values
Substitute the known values into the formula: \[10 = 10,000 \left( \frac{1}{2} \right)^{\frac{t}{24,000}} \]Solving for \(t\) involves simplifying and rearranging the equation.
5Step 5: Solving the Exponential Equation
Divide both sides by 10,000: \[\frac{1}{1000} = \left( \frac{1}{2} \right)^{\frac{t}{24,000}} \]Take the logarithm of both sides to solve for \(t\): \[\log\left(\frac{1}{1000}\right) = \frac{t}{24,000} \cdot \log\left(\frac{1}{2}\right) \]This allows us to isolate \(t\).
6Step 6: Calculating the Time
Solving for \(t\): \[\frac{t}{24,000} = \frac{\log(0.001)}{\log(0.5)} \]Calculate \(\log(0.001)\) and \(\log(0.5)\) using a calculator, then find \(t\). \[\log(0.001) \approx -3 \, \text{and} \, \log(0.5) \approx -0.301 \]Thus, \[\frac{t}{24,000} \approx \frac{-3}{-0.301} \]
7Step 7: Final Calculation of Years
Calculate the final result for \(t\): \[\frac{t}{24,000} \approx 9.966 \]Multiply both sides by 24,000 to find \(t\): \[t \approx 239,184 \, \text{years} \]
8Step 8: Conclusion
The statement is false when it suggests how many years the plutonium will need to decay to 10 g. The correct answer is that it will take approximately 239,184 years.
Key Concepts
Plutonium-239 DecayExponential Decay FormulaRadioactive Decay CalculationNuclear Physics
Plutonium-239 Decay
Plutonium-239 is a radioactive isotope that plays a significant role in nuclear reactors and weapons. It is highly radioactive and undergoes a process known as radioactive decay. Over time, Plutonium-239 naturally breaks down into lighter elements. This decay is a part of its nuclear and chaotic atomic nature. Understanding its decay is crucial for handling nuclear waste safely.
Plutonium-239 has a relatively long half-life, meaning it remains radioactive for thousands of years. This persistence poses challenges for nuclear waste management, as it must be safely stored over long periods. Its decay process releases energy, which can be harnessed for power but also requires careful control to prevent environmental contamination.
In the context of the given exercise, Plutonium-239's decay is measured by monitoring how its mass decreases over time. The rate at which it decays is described mathematically by the concept of half-life.
Plutonium-239 has a relatively long half-life, meaning it remains radioactive for thousands of years. This persistence poses challenges for nuclear waste management, as it must be safely stored over long periods. Its decay process releases energy, which can be harnessed for power but also requires careful control to prevent environmental contamination.
In the context of the given exercise, Plutonium-239's decay is measured by monitoring how its mass decreases over time. The rate at which it decays is described mathematically by the concept of half-life.
Exponential Decay Formula
Exponential decay is a mathematical concept used to describe the process by which a quantity decreases at a rate proportional to its current value. This principle is applied in various fields, such as physics, chemistry, finance, and more. In the case of radioactive substances like Plutonium-239, exponential decay is fundamental.
The specific formula for calculating decay in the context of the exercise is: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]Here, \(N(t)\) represents the remaining quantity of the substance after time \(t\), \(N_0\) is the initial quantity, and \(T_{1/2}\) is the half-life of the substance. Understanding the use of this formula is essential to accurately predict how much of a substance remains after a given period, which is crucial in nuclear physics and safety assessments.
This formula helps you solve problems like the one in the exercise, where you determine the time needed for a certain amount of Plutonium-239 to decay to a lower mass.
The specific formula for calculating decay in the context of the exercise is: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]Here, \(N(t)\) represents the remaining quantity of the substance after time \(t\), \(N_0\) is the initial quantity, and \(T_{1/2}\) is the half-life of the substance. Understanding the use of this formula is essential to accurately predict how much of a substance remains after a given period, which is crucial in nuclear physics and safety assessments.
This formula helps you solve problems like the one in the exercise, where you determine the time needed for a certain amount of Plutonium-239 to decay to a lower mass.
Radioactive Decay Calculation
When performing radioactive decay calculations, it's important to apply mathematical techniques to determine how much of a radioactive substance remains over time. The decay follows an exponential pattern, requiring knowledge of logarithms and the decay formula.
In the exercise, you started with 10 kg, wanting to find how long it would take to decay to 10 grams. After setting up the decay formula \[ 10 = 10,000 \left( \frac{1}{2} \right)^{\frac{t}{24,000}} \], solving involves isolating the time variable \(t\).
This involves dividing both sides by the initial mass and then applying the logarithm, allowing you to handle the exponent. By solving for \(t\), you determine how long the decay will take. Such calculations are indispensable for safely managing nuclear materials and predicting their behavior over long periods.
These processes validate how abstract mathematical concepts have concrete applications in fields like nuclear physics.
In the exercise, you started with 10 kg, wanting to find how long it would take to decay to 10 grams. After setting up the decay formula \[ 10 = 10,000 \left( \frac{1}{2} \right)^{\frac{t}{24,000}} \], solving involves isolating the time variable \(t\).
This involves dividing both sides by the initial mass and then applying the logarithm, allowing you to handle the exponent. By solving for \(t\), you determine how long the decay will take. Such calculations are indispensable for safely managing nuclear materials and predicting their behavior over long periods.
These processes validate how abstract mathematical concepts have concrete applications in fields like nuclear physics.
Nuclear Physics
Nuclear physics explores the fundamental principles governing atomic nuclei and their interactions. It's a diverse field, with applications in energy production, medical technology, and more. Understanding nuclear physics is crucial for addressing complex problems involving radioactive materials, like Plutonium-239.
This field explains how unstable nuclei, such as those found in Plutonium-239, decay into more stable forms by emitting energy. Such processes are the foundation upon which nuclear reactors operate, providing power but also creating waste that needs careful handling. Nuclear physics provides the tools and understanding necessary for developing and maintaining safe nuclear technologies.
In the exercise, Plutonium-239's decay is an example of nuclear reactions occurring over extended timescales. By applying principles from nuclear physics, such as half-life and decay calculations, you gain insights into managing and estimating the lifecycle of radioactive materials.
This field explains how unstable nuclei, such as those found in Plutonium-239, decay into more stable forms by emitting energy. Such processes are the foundation upon which nuclear reactors operate, providing power but also creating waste that needs careful handling. Nuclear physics provides the tools and understanding necessary for developing and maintaining safe nuclear technologies.
In the exercise, Plutonium-239's decay is an example of nuclear reactions occurring over extended timescales. By applying principles from nuclear physics, such as half-life and decay calculations, you gain insights into managing and estimating the lifecycle of radioactive materials.
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