Problem 32 - Essential Calculus

Question

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{239} \mathrm{Pu} \quad 24,100 \quad \quad \quad \quad \quad 0.4 \mathrm{~g} $$

Step-by-Step Solution

Answer

After 1,000 years, 0.386g of Pu-239 will remain and after 10,000 years, 0.272g will still be present.

Step 1: Determine the number of half-lives

First, we need to understand that half-life is the time it takes for the amount of a radioactive substance to decrease by half. To calculate the number of half-lives that have passed, we divide the number of years that have passed by the half-life of the isotope. Thus, after 1000 years, the number of half-lives for the isotope Pu-239 is \( \frac{1000}{24100} = 0.0415\) and after 10,000 years it is \( \frac{10000}{24100} = 0.4147\).

Step 2: Calculate the amount remaining

The amount of the radioactive isotope remaining can be calculated using the formula \( A = A_0(0.5)^n \), where \( A \) is the remaining amount, \( A_0 \) is the initial quantity, and \( n \) is the number of half-lives. So, after 1000 years, the amount of Pu-239 remaining is \(0.4 * (0.5)^{0.0415}\) and after 10,000 years it is \(0.4 * (0.5)^{0.4147}\).

Step 3: Result

Substituting the values into the formula gives us approximately: \(0.4 * (0.5)^{0.0415} = 0.3856g\) after 1000 years and \(0.4 * (0.5)^{0.4147} = 0.2719g\) after 10,000 years. Thus, after 1000 years, approximately 0.386g of Pu-239 will remain and after 10,000 years, about 0.272g will still be present.

Key Concepts

Radioactive Isotope Decay

Radioactive isotopes, or radionuclides, are unstable atoms that emit radiation as they transform into more stable forms. This process, known as radioactive decay, occurs spontaneously and continuously until a stable isotope is formed. Each radioactive isotope has a characteristic rate of decay, which is unaffected by external factors such as temperature or pressure.

During decay, an isotope may release alpha particles, beta particles, or gamma rays. This release alters the number of protons and neutrons in the atom's nucleus, resulting in a different element or a different isotope of the same element. For example, plutonium-239 (( ^{239}Pu )), used in the exercise, decays over time by emitting alpha particles and transforms into uranium-235, another isotope.

Understanding the decay of radioactive isotopes is crucial in various fields, such as nuclear medicine for cancer treatments, archeological dating with carbon-14, and nuclear power generation. Knowing how much of an isotope remains after a certain period can help us calculate the original amount or predict how long it will remain hazardous.

Radioactive Half-Life

The radioactive half-life of an isotope is the time required for half of the substance to decay. It's a critical measure that helps us understand the longevity and decay pattern of radioactive material. Each isotope has a specific half-life, which can range from fractions of a second to millions of years. The half-life of an isotope remains constant, serving as a clock that helps us track the decay process over time.

Using the isotope ( ^{239}Pu ) with a half-life of 24,100 years as mentioned in the exercise, we can predict that after 24,100 years, only half of the original amount will remain. After another 24,100 years, half of that amount will decay, leaving us with a quarter of the original quantity. This exponential decay continues indefinitely.

The concept of half-life is not only crucial for scientists dealing with nuclear materials but also for safety planning and waste management, especially in determining the safe storage period for radioactive waste before it becomes harmless.

Exponential Decay Formula

The mathematical description of radioactive decay is expressed through the exponential decay formula. This formula reflects the fact that the decay process decreases the amount of a substance at a rate proportional to its current value. Mathematically, the remaining quantity of a radioactive isotope is calculated using the formula ( A = A_0(0.5)^n ), where:

• ( A ) represents the remaining amount after time ( t ),
• ( A_0 ) is the initial quantity of the isotope,
• ( n ) is the number of half-lives that have elapsed.

The number of half-lives ( n ) is found by dividing the time elapsed by the half-life of the isotope. In our exercise, applying the formula to calculate the remaining amount of ( ^{239}Pu ) after 1,000 and 10,000 years demonstrates how we can predict the deterioration of a radioactive substance over time. This formula is pivotal in fields like environmental science, nuclear physics, and healthcare, offering a practical way to model the decay of radioactive elements.