Problem 40 - Calculus and its applications

Question

Cesium- 137 has a decay rate of \(2.3 \%\) per year. Suppose a nuclear accident causes cesium- 137 to be released into the atmosphere perpetually at the rate of 1 lb per year. What is the limiting value of the radioactive buildup?

Step-by-Step Solution

Answer

The limiting value of the radioactive buildup is approximately 43.48 pounds.

Step 1: Understanding the Decay Process

Cesium-137 decays at a rate of 2.3% per year, which means that if you have an amount of Cesium-137, 2.3% of it will transform into another element each year. The remaining 97.7% will persist.

Step 2: Formulate the Differential Equation

Let the amount of Cesium-137 at time \( t \) be \( C(t) \). The differential equation modeling the rate of change of Cesium is given by \( \frac{dC}{dt} = 1 - 0.023C(t) \), where 1 lb is added per year.

Step 3: Find the Limiting Value

To find the steady-state or equilibrium (limiting value), set \( \frac{dC}{dt} = 0 \). This gives us the equation \( 1 - 0.023C(t) = 0 \). Solving this, we get \( C(t) = \frac{1}{0.023} \).

Step 4: Calculate the Limiting Value

Calculate \( C(t) = \frac{1}{0.023} \). Therefore, \( C(t) \approx 43.478 \) pounds. This is the limiting or equilibrium amount of Cesium-137 buildup in the atmosphere.

Key Concepts

Differential Equations

Understanding differential equations is crucial in modeling many natural processes, including radioactive decay. In the context of Cesium-137, a differential equation describes how the amount of cesium changes over time, considering both its decay and the continuous addition from a nuclear accident. A differential equation might seem intimidating at first, but it's essentially a mathematical expression defining how one quantity changes with respect to another. In our case, it denotes how the amount of cesium changes with respect to time.

The differential equation given for Cesium-137 is \(\frac{dC}{dt} = 1 - 0.023C(t)\). Here's the breakdown:

• \(\frac{dC}{dt}\) is the rate at which the cesium amount is changing over time.
• The term \(1\) represents the 1 lb of cesium added to the environment annually.
• The term \(-0.023C(t)\) represents the natural decay of cesium, where 2.3% of the cesium decays each year.

Solving this equation helps to understand how the cesium levels vary over time and is fundamental to predicting its long-term behavior.

Steady State Analysis

Steady state analysis is a concept used to determine when a system reaches equilibrium. In terms of cesium-137 in the environment, this is when the rate of cesium addition equals the rate of its decomposition, leading to consistent levels over time.

To find this state, we use the principle that at a steady state, there's no net change in the system, thus \(\frac{dC}{dt} = 0\). Applying this to our earlier equation \(\frac{dC}{dt} = 1 - 0.023C(t)\), setting it to zero gives us the equation \(1 - 0.023C(t) = 0\).

Solving for \(C(t)\) provides the level at which the amount of cesium stays constant, assuming no other variables change.

Limiting Value Calculation

The limiting value is the long-term value of a quantity as time progresses towards infinity. For Cesium-137 leakage, this value is where the buildup levels off due to continuous addition and decay. It's calculated by solving our steady state equation.

Starting from \(1 - 0.023C(t) = 0\), rearrange it to find:

• "\(C(t) = \frac{1}{0.023}\)" which simplifies to approximately 43.478 lbs. This means the environment will stabilize around 43.478 pounds of cesium-137 in the atmosphere.

This is the equilibrium point where inputs and outputs in the system balance each other out.

Understanding the limiting value helps in anticipating the long-term environmental impacts and preparing appropriate mitigation strategies to manage radioactive substances effectively.