Question

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?

Short answer

It takes approximately 239,506 years.

Step-by-step solution

Understand the Exponential Decay Formula

The given formula is \( y = y_{0} e^{kt} \), where \( y \) is the final quantity, \( y_{0} \) is the initial quantity, \( k \) is the decay constant, and \( t \) is time in years. We are given the half-life and must determine how long it takes for the plutonium to decay from \( 10 \text{ kg} \) to \( 10 \text{ g} \).

Convert Units if Necessary

Convert \( 10 \text{ kg} \) to \( 10,000 \text{ grams} \) so that both beginning and end quantities are in grams: \( y_{0} = 10,000 \text{ grams} \) and \( y = 10 \text{ grams} \).

Relate Half-Life to Decay Constant

The relationship between half-life and decay constant is given by \( T_{1/2} = \frac{\ln{2}}{k} \). Given a half-life of 24,000 years, solve for \( k \): \( k = \frac{\ln{2}}{24,000} \).

Calculate the Decay Constant

Substitute the known values: \( k = \frac{0.693}{24,000} \approx 2.8875 \times 10^{-5} \) per year.

Set Up the Exponential Decay Equation

Use the equation \( 10 = 10,000 e^{kt} \). Substitute \( k \) and solve for \( t \).

Solve for Time \( t \)

First, divide each side by 10,000 to get \( 0.001 = e^{kt} \). Take the natural logarithm of both sides to solve for \( t \): \( \ln(0.001) = kt \).

Compute \( t \) Using Known Values

Use \( \ln(0.001) \approx -6.907 \) and \( k \approx 2.8875 \times 10^{-5} \) to find \( t \): \( t = \frac{-6.907}{2.8875 \times 10^{-5}} \approx 239,506 \) years.

Key concepts

Half-Life

Half-life is a term used to describe the time required for a quantity to reduce to half its initial amount. In nuclear physics, this concept is crucial as it helps in predicting how long a radioactive element will remain active. For example, plutonium-239 has a half-life of 24,000 years. This means that it takes 24,000 years for half of a given amount of plutonium-239 to decay.

Understanding half-life allows us to determine how long a radioactive substance will pose a threat, which is important for safe storage and disposal. In chemical kinetics, knowing the half-life aids in understanding reaction rates and the persistence of chemicals in nature.

To find the decay rate or decay constant, the half-life formula is used: \[ T_{1/2} = \frac{\ln 2}{k} \] This relationship between half-life and decay constant allows us to calculate one when the other is known.

Decay Constant

The decay constant is a measure of how quickly a substance undergoes exponential decay. It is denoted by the symbol \( k \) and is directly related to the substance's half-life. Given the half-life of a substance, you can calculate the decay constant using the relation:\[ k = \frac{\ln 2}{T_{1/2}} \] This equation shows that the decay constant is inversely proportional to the half-life; a shorter half-life means a larger decay constant and vice-versa.

In the exercise, we've calculated the decay constant for plutonium-239 as approximately \( 2.8875 \times 10^{-5} \) per year. This value indicates how rapidly the plutonium-239 will decay per year, which is crucial for making predictions about its future activity.

• A high \( k \) means fast decay.
• A low \( k \) indicates slow decay.

Understanding the decay constant is essential in both nuclear physics and chemical kinetics for quantifying and predicting the behavior of decaying substances.

Nuclear Physics

Nuclear physics studies the components and behavior of atomic nuclei. It involves understanding radioactive decay, including the concept of half-life and decay constant. Radioactive substances, like plutonium-239, decay over time, releasing energy and transforming into different elements or isotopes.

In nuclear reactors, understanding decay is crucial for managing spent fuel safely. Plutonium-239, a by-product of nuclear reactions, requires safe handling due to its long half-life. Knowing how long it will remain active helps in designing safe storage solutions.

Nuclear physics is not only about safety and waste management. It also encompasses the principles behind nuclear energy generation and the use of radiation in medical treatments. Mastering these principles ensures our ability to harness nuclear power efficiently and safely.

Chemical Kinetics

Chemical kinetics refers to the study of reaction rates and the factors affecting them. While traditionally applied to chemical reactions, its principles also apply to nuclear decay processes like those involving plutonium-239.

• Decay constant \( k \) in kinetics is akin to the reaction rate constant.
• Half-life is analogous to the time required for a reaction to go halfway to completion.

Understanding the kinetics of nuclear decay helps in predictions related to the behavior of nuclear materials. It also aids in designing processes and systems, like nuclear waste storage, ensuring that safety standards are met over time.

The principles of chemical kinetics enable us to better understand how various conditions, such as temperature and pressure, might affect decay rates. Though not directly applicable to nuclear decay (which is more stable against such conditions), these principles offer a broad framework for studying both chemical and nuclear transformations.