Problem 24 - Calculus Concepts

Question

Uranium- 238 Decay The rate at which 100 milligrams of \({ }^{238} \mathrm{U}\) is decaying can be modeled as \(r(t)=-1.55\left(0.9999999845^{t}\right) 10^{-6}\) milligrams per year where \(t\) is the number of years since the 100 milligrams began decaying. a. How much of the \({ }^{238} \mathrm{U}\) will decay during the first 100 years? During the first 1000 years? b. How much of the \({ }^{238} \mathrm{U}\) will eventually decay?

Step-by-Step Solution

Answer

Over 100 years, about 0.000155 mg decays; over 1000 years, about 0.00155 mg decays; eventually, 0.1 mg will decay.

Step 1: Understand the Decay Model

The decay rate function given is \( r(t) = -1.55 \times 10^{-6} \cdot (0.9999999845)^t \), where \( t \) is the time in years. This function describes how quickly uranium-238 is decaying over time.

Step 2: Setup the Integral for Total Decay

To find the total amount of decay, we integrate the rate function from 0 to \( t \). The integral \( \int r(t) \, dt \) from 0 to \( t \) gives us the amount of uranium that has decayed over that time period.

Step 3: Integrate to Find Decay Over 100 Years

Let's integrate the decay rate function from 0 to 100: \[ \int_{0}^{100} -1.55 \times 10^{-6} (0.9999999845)^t \, dt \]Use substitution \( u = 0.9999999845^t \), leading to \[ \text{Decay}_{100} = \left[ -\frac{1.55 \times 10^{-6}}{\ln(0.9999999845)}(0.9999999845)^t \right]_{0}^{100} \] Calculate this to find the decay amount in 100 years.

Step 4: Calculate Result for 100 Years

Calculate the definite integral result for 100 years using the expression obtained: \[ \text{Decay}_{100} = -\frac{1.55 \times 10^{-6}}{\ln(0.9999999845)}(0.9999999845^{100} - 1) \]This gives approximately 0.000155 milligrams decayed over 100 years.

Step 5: Integrate to Find Decay Over 1000 Years

Integrate the decay rate from 0 to 1000:\[ \int_{0}^{1000} -1.55 \times 10^{-6} (0.9999999845)^t \, dt \]Use the same substitutions and method as in Step 3 to calculate the decay over 1000 years.

Step 6: Calculate Result for 1000 Years

Calculate the definite integral for 1000 years using:\[ \text{Decay}_{1000} = -\frac{1.55 \times 10^{-6}}{\ln(0.9999999845)}(0.9999999845^{1000} - 1) \]This results in approximately 0.00155 milligrams decayed over 1000 years.

Step 7: Evaluate the Limit for Total Decay

To determine how much \(^{238}U\) will eventually decay, evaluate:\[ \lim_{t \to \infty} \int_{0}^{t} -1.55 \times 10^{-6} (0.9999999845)^t \, dt \]Since \(0 < 0.9999999845 < 1\), the term \((0.9999999845)^t \) approaches 0 as \(t\) approaches infinity, leading to a converging geometric series for the total decay.

Step 8: Calculate Total Decay Eventually

Using the properties of converging geometric series:\[ \text{Total Decay} = \frac{1.55 \times 10^{-6}}{\ln(0.9999999845)}(1) \]The uranium-238 will eventually decay around 0.1 milligrams in total.

Key Concepts

Uranium-238

Uranium-238 is a naturally occurring radioactive isotope of uranium. It accounts for about 99.3% of the uranium found on Earth, making it the most abundant uranium isotope. Uranium-238 has a half-life of approximately 4.5 billion years, which means it decays very slowly. This slow decay makes it useful in geological dating, as well as in nuclear reactors where it can be used to produce plutonium-239, a fissile material. Due to its long half-life, uranium-238 emits low levels of radiation compared to other isotopes like uranium-235, which is more commonly used in nuclear power and weapons. Understanding its decay is crucial in multiple scientific fields, including geology, archeology, and physics.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a consistent rate over time. This type of decay is mathematically characterized by an exponential function, common in natural processes such as radioactive decay. For uranium-238, the decay is modeled by an exponential function indicating how the quantity of the substance diminishes over time. This function is represented as:\[ r(t) = a imes b^t \]where \( a \) is the initial amount, \( b \) is the decay factor, and \( t \) is time.The decay factor \( b \) is always between 0 and 1 for decay processes, ensuring the quantity decreases over time. As time moves forward, the base \( b^t \) results in smaller values, reflecting the reduction in substance.

Integration

Integration is a fundamental concept in calculus used to find the total accumulation of a quantity. In the context of radioactive decay, integration helps determine the total amount of uranium-238 that has decayed over a specific time period.We perform integration on the decay rate function, like:\[ \int r(t) \, dt \]This integral calculates the total decay from time 0 to \( t \). In our equation, integrating gives the total amount of uranium lost over the given time frame, whether 100 years, 1000 years, or for the entire decay process.By analyzing these integrals, we can understand how much uranium-238 turns into other elements as it decays over years, providing insights into its long-term behavior.

Geometric Series

A geometric series involves the sum of terms, each multiplied by a constant ratio from the previous term. This type of series is significant when evaluating the total decay of radioactive substances like uranium-238 over an indefinite period.In the context of radioactive decay, the process can be likened to a geometric series where the common ratio is a number less than 1. This ratio corresponds to the decay factor used in exponential decay functions. Therefore, as the function continues over time, the cumulative decay approaches a finite limit despite the continuous process.Mathematically, a convergent geometric series for uranium decay can be expressed as:\[ \text{Total Decay} = \frac{a}{1 - b} \]In this equation, \( a \) represents the initial term, and \( b \) is the decay ratio, ensuring that the series converges to a finite amount, showing how much uranium will eventually decay.