Problem 369 - Calculus Volume 1

Question

For the following exercises, use \(y=y_{0} e^{k t}\). 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

Answer

It takes about 239,940 years for 10 kg of plutonium-239 to decay to 10 g.

Step 1: Write Down the Exponential Decay Formula

The exponential decay formula provided is \( y = y_0 e^{kt} \), where \( y \) is the amount of plutonium left, \( y_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is time. We need to find \( t \) when \( y = 10 \text{ g} \).

Step 2: Convert Units

Since the initial amount is given in kilograms (10 kg), and the final amount is given in grams (10 g), convert one unit to match the other. Thus, \( 10 \) kg = \( 10000 \) g. This means \( y_0 = 10000 \text{ g} \) and \( y = 10 \text{ g} \).

Step 3: Determine the Decay Constant

The half-life of plutonium-239 is 24,000 years. The decay constant \( k \) can be determined using the formula for half-life: \[ \frac{1}{2} = e^{k \cdot 24000} \]. This gives \( k \) as \( k = \frac{\ln(0.5)}{24000} \).

Step 4: Set Up the Equation to Solve for Time

Use the exponential decay formula and plug in the known values: \( 10 = 10000 e^{k \cdot t} \).

Step 5: Solve for \( t \)

Isolate \( e^{kt} \) by dividing both sides by 10000: \( \frac{10}{10000} = e^{kt} \), which simplifies to \( 0.001 = e^{kt} \). Take the natural logarithm: \( \ln(0.001) = kt \). Substitute \( k = \frac{\ln(0.5)}{24000} \) and solve for \( t \): \[ t = \frac{\ln(0.001)}{\frac{\ln(0.5)}{24000}} \].

Step 6: Calculate the Time \( t \)

Perform the calculation: \( t \approx \frac{-6.907755}{-0.00002885} \approx 239,940.2 \text{ years} \). Round as needed based on the precision required.

Key Concepts

Half-life Calculation

The concept of half-life is crucial in understanding exponential decay, especially in nuclear chemistry. A half-life is the time it takes for half of a substance to decay or transform into another substance through radioactive decay. For instance, the half-life of plutonium-239 is 24,000 years. This means that every 24,000 years, half of any given amount of plutonium-239 will decay. To calculate how long it takes for the amount of a radioactive substance to reduce to a certain level, we use the formula:\( \frac{1}{2} = e^{k \cdot T} \)where:

• \( T \) is the half-life period (24,000 years in this example)
• \( k \) is the decay constant

Understanding half-life helps us determine how nuclear materials will behave over time. By applying half-life calculations, one can predict when radioactive materials will reduce to safer levels.

Decay Constant

The decay constant \( k \) is a pivotal parameter in exponential decay equations. It gives the rate at which a particular substance decays. The decay constant depends on the half-life of the material.To find the decay constant for a substance like plutonium-239, we use the formula:\[ k = \frac{\ln(0.5)}{T} \]where \( T \) is the half-life of the material.In the case of plutonium-239 with a half-life of 24,000 years, the decay constant \( k \) can be calculated as:\[ k = \frac{\ln(0.5)}{24000} \]Understanding the decay constant is essential for predicting how quickly a radioactive substance will decrease in quantity, allowing us to calculate future contamination levels or potential remaining material in practical applications.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function used extensively in exponential decay problems. The natural logarithm is the inverse function of the exponential function with base \( e \approx 2.71828 \). It is particularly useful in converting exponential equations into linear forms that are easier to manipulate.In our problem, we utilized the natural logarithm to solve for time \( t \) by transforming the equation:\[ \ln(0.001) = kt \]Using known values and solving for \( t \) through the expression:\[ t = \frac{\ln(0.001)}{\frac{\ln(0.5)}{24000}} \]This method simplifies the complex exponential relations into manageable figures for computational purposes.

Nuclear Chemistry

Nuclear chemistry is a branch of chemistry focused on the radioactive processes, properties, and transformations within atomic nuclei. A key topic in nuclear chemistry is radioactive decay, which includes understanding how substances like plutonium-239 transform over time. The application of nuclear chemistry concepts is vital in predicting material stability and safety. For instance, when dealing with nuclear reactor spent fuel, it's crucial to know how long radioactive elements last and how quickly they decay. Using exponential decay formulas, scientists can calculate the future concentrations of hazardous materials. This understanding helps in managing nuclear waste, ensuring effective safety measures, and supporting the design of nuclear facilities. By understanding the chemical properties and behaviors of radioactive elements, we can better protect the environment and human health.