Problem 47 - Calculus and its applications

Question

Cesium- 137 has a decay rate of 2.3\% per year. Suppose cesium-137 is released into the atmosphere for \(20 \mathrm{yr}\) at a rate of \(1 \mathrm{lb}\) per year. How much cesium- 137 will be present in the atmosphere after 20 yr?

Step-by-Step Solution

Answer

Approximately 16.86 lbs of cesium-137 will be present after 20 years.

Step 1: Understanding the Problem

The problem involves the continuous release of cesium-137, a radioactive substance, into the atmosphere at a rate of 1 lb per year for 20 years, with a decay rate of 2.3% annually. We need to calculate the total amount of cesium-137 present after 20 years, accounting for the decay.

Step 2: Formula for Exponential Decay

The amount of cesium-137 remaining after time can be calculated using the decay formula: \[ \text{Remaining Amount} = P_0 \times e^{-k \cdot t} \]Where \(P_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time in years. The decay constant \(k\) is calculated as \(k = 0.023\) for a 2.3% decay rate.

Step 3: Calculate Yearly Remaining Amount

For each year, 1 lb of cesium-137 is added and then time-decayed according to the exponential decay formula. Over 20 years, we calculate the remaining amount as follows: Each year's contribution decreases as it only exists for the remaining years. For example, cesium added in year 1 decays for 19 years, in year 2 for 18 years, and so on.

Step 4: Calculate Contribution from Each Year

Using the exponential decay formula for each year:- Year 1 contributes: \(1 \times e^{-0.023 \times 19}\)- Year 2 contributes: \(1 \times e^{-0.023 \times 18}\)- ...- Year 20 contributes: \(1 \times e^{-0.023 \times 0}\)Calculate these values to understand how much remains from each year’s addition.

Step 5: Total Accumulation Sum Calculation

Sum up the contributions from all the 20 years to obtain the total cesium-137 remaining present after 20 years:\[ \text{Total Cesium} = \sum_{n=0}^{19} 1 \times e^{-0.023 \cdot n} \]

Step 6: Compute the Final Total

Using a calculator or a program, sum the series from Step 5. This will provide a numerical value indicating the pounds of cesium-137 that remain in the atmosphere after 20 years.

Key Concepts

Cesium-137 Decay

Cesium-137 is a fascinating yet hazardous radioactive substance. It is commonly known for its presence in nuclear fallout. By releasing a tiny amount of radiation, the atoms of cesium-137 transform into a more stable state over time.

This process is known as radioactive decay, and for cesium-137, it has a decay rate of 2.3% per year.
It means that each year, 2.3% of the existing cesium-137 decays away and transforms into non-radioactive materials. Having such a decay rate implies that cesium-137 is relatively persistent in the environment. This persistence is why understanding the decay process is crucial when calculating how long it stays and accumulates in the environment.
The decay of cesium-137 impacts both nuclear safety and environmental monitoring, making such calculations an essential skill for scientists and engineers.

Radioactive Substances

Radioactive substances are materials that emit radiation as they decay naturally over time.
Some common features include:

• The ability to change into different elements or isotopes while releasing energy.
• The emission of alpha, beta, or gamma radiation.
• A predictable decay rate, allowing us to determine their half-life and behavior.

Cesium-137 is one of these substances, widely used in medical and industrial fields but also a safety concern when released into the environment.
Scientific tools and equations allow us to predict the behavior of such substances with ease. The formula \[ \text{Remaining Amount} = P_0 \times e^{-k \cdot t} \] captures their essence by providing a mathematical model for their decay. This ability to predict and measure the radioactive decay makes it possible to understand their long-term impact and plan necessary safety measures.

Continuous Release

Continuous release refers to the steady addition of a substance into an environment over time. For example, cesium-137 can be continuously released into the atmosphere at a rate of 1 lb per year, such as in our given exercise scenario.
Understanding continuous release helps assess how much of a substance accumulates over time, alongside its decay.
In the scenario where cesium-137 is being released yearly, each year's contribution needs to be considered separately due to its ongoing decay. The combination of continuous release and decay involves calculating how much remains from each year's addition and summing these values for an overall total.
This approach helps environmentalists and scientists understand not just how much cesium-137 enters the atmosphere but also how much remains active at any given time, allowing for informed decisions regarding environmental safety and public health.