Problem 49 - Calculus

Question

Different isotopes (versions) of the same element can have very different half-lives. With \(t\) in years, the decay of plutonium- 240 is described by the formula $$Q=Q_{0} e^{-0.00011 t}$$ whereas the decay of plutonium-242 is described by $$Q=Q_{0} e^{-0.0000018 t}$$ Find the half-lives of plutonium- 240 and plutonium-242.

Step-by-Step Solution

Answer

Plutonium-240 has a half-life of approximately 6301.34 years, and plutonium-242 has a half-life of about 385081.67 years.

Step 1: Understand Half-Life Concept

The half-life of a radioactive substance is the time required for the amount of substance to reduce to half its initial quantity. Mathematically, this means finding the time \( t \) when \( Q = \frac{1}{2}Q_0 \).

Step 2: Setup Equation for Plutonium-240

For plutonium-240, set the equation \( Q = Q_0 e^{-0.00011t} \) to \( \frac{1}{2}Q_0 \). This means solving \( \frac{1}{2}Q_0 = Q_0 e^{-0.00011t} \).

Step 3: Solve for Half-Life of Plutonium-240

Cancel \( Q_0 \) from both sides to get \( \frac{1}{2} = e^{-0.00011t} \). Take the natural logarithm (ln) of both sides to solve for \( t \): \( \ln\left(\frac{1}{2}\right) = -0.00011t \) \( t = \frac{\ln\left(\frac{1}{2}\right)}{-0.00011} \).

Step 4: Calculate Half-Life of Plutonium-240

Compute \( t \): \( t = \frac{-0.693147}{-0.00011} \approx 6301.34 \) years. So, the half-life of plutonium-240 is approximately 6301.34 years.

Step 5: Setup Equation for Plutonium-242

For plutonium-242, set the equation \( Q = Q_0 e^{-0.0000018t} \) to \( \frac{1}{2}Q_0 \). This means solving \( \frac{1}{2}Q_0 = Q_0 e^{-0.0000018t} \).

Step 6: Solve for Half-Life of Plutonium-242

Cancel \( Q_0 \) from both sides to get \( \frac{1}{2} = e^{-0.0000018t} \). Take the natural logarithm of both sides to solve for \( t \): \( \ln\left(\frac{1}{2}\right) = -0.0000018t \) \( t = \frac{\ln\left(\frac{1}{2}\right)}{-0.0000018} \).

Step 7: Calculate Half-Life of Plutonium-242

Compute \( t \): \( t = \frac{-0.693147}{-0.0000018} \approx 385081.67 \) years. So, the half-life of plutonium-242 is approximately 385081.67 years.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where the nucleus of an unstable atom loses energy by emitting radiation. This process results in the transformation of an element into another element or isotope over time. The rate at which radioactive decay occurs is described using the concept of half-life. Understanding radioactive decay is crucial for fields like nuclear physics, medicine, and environmental science.

• Atoms decay through various modes such as alpha decay, beta decay, and gamma decay.
• The time taken for half of a given amount of radioactive substance to decay is known as its half-life.

The decay follows an exponential pattern, which can be mathematically described using exponential functions. These functions make it possible to predict how long it will take for a certain percentage of a substance to decay. This also helps in understanding the long-term behavior of nuclear waste and its environmental impact.

Plutonium Isotopes

Plutonium isotopes are different forms of the element plutonium which have varying numbers of neutrons. Isotopes like plutonium-240 and plutonium-242 are particularly important due to their differing half-lives and properties.

• Plutonium-240: This isotope features a relatively shorter half-life of about 6301.34 years. Its medium half-life contributes to less long-term storage concerns compared to some other isotopes like plutonium-242.
• Plutonium-242: Known for its very long half-life of about 385,081.67 years. This characteristic makes it significant for considerations surrounding nuclear waste management, as it remains hazardous for a much longer time.

The differing half-lives and decay rates of these isotopes are crucial in applications such as nuclear energy production and nuclear weapons development. Understanding these isotopes helps in designing better safety protocols and storage techniques for radioactive materials.

Exponential Functions

Exponential functions play a crucial role in modeling radioactive decay processes. These mathematical functions describe the decaying amounts of substances over time using an exponential decay formula.

• An exponential function can be generally expressed as: \(Q = Q_0 e^{-kt}\)
• Here, \(Q_0\) is the initial quantity, \(k\) is the decay constant that determines how quickly decay occurs, and \(t\) represents time.
• The base of the natural exponential function, denoted by \(e\), is approximately equal to 2.71828.

In the context of half-life calculations:

• The formula is rearranged to find the specific time when half of the substance remains: \[t_{1/2} = \frac{\ln(0.5)}{-k}\]
• In cases like plutonium isotopes, precise knowledge of the decay constant allows scientists to accurately calculate half-life and understand how long the material remains active.

Exponential functions simplify the complex processes of radioactive decay, allowing us to make accurate predictions and informed decisions about radioactive material management.