Problem 18 - Calculus Concepts

Question

Radon-232 Isotope Radon-232 is a colorless, odorless gas that undergoes radioactive decay with a half-life of 3.824 days. It is considered a health hazard, so new homebuyers often have their property tested for the presence of radon-232. Because radon- 232 is a radioisotope, it decays (over time) at a rate that is directly proportional to the amount of the radioisotope. a. Write a differential equation for the rate at which an amount of radon- 232 decays. b. Write a general solution for this differential equation. c. If 1 gram of radon- 232 is isolated, how much of it will remain after 12 hours; after 4 days, 9 days, and 30 days?

Step-by-Step Solution

Answer

After 12 hours: 0.913g; after 4 days: 0.485g; after 9 days: 0.196g; 30 days: 0.004g.

Step 1: Understanding the differential equation needed

The decay rate of a radioactive substance is proportional to its current amount. Therefore, the differential equation can be expressed as \( \frac{dN}{dt} = -kN \), where \( N \) is the amount of radon-232 present at time \( t \), and \( k \) is the decay constant.

Step 2: Solving for the general solution of the differential equation

The differential equation \( \frac{dN}{dt} = -kN \) is a first-order linear differential equation. Its general solution is given by separating the variables and integrating: \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the initial amount of the isotope.

Step 3: Finding the decay constant using the half-life

The half-life of radon-232 is 3.824 days, which means that \( N(t) = \frac{N_0}{2} \) when \( t = 3.824 \). Substituting in the solution gives \( \frac{N_0}{2} = N_0 e^{-k(3.824)} \). Solving for \( k \) gives \( k = \frac{\ln(2)}{3.824} \approx 0.181 \).

Step 4: Computing the amount remaining after 12 hours

First convert 12 hours to days which is \( \frac{12}{24} = 0.5 \) days. Substitute into the decay formula: \( N(0.5) = 1 e^{-0.181 \times 0.5} = e^{-0.0905} \approx 0.913 \) grams.

Step 5: Computing the amount remaining after 4 days

Substitute 4 days into the decay formula: \( N(4) = 1 e^{-0.181 \times 4} = e^{-0.724} \approx 0.485 \) grams.

Step 6: Computing the amount remaining after 9 days

Substitute 9 days into the decay formula: \( N(9) = 1 e^{-0.181 \times 9} = e^{-1.629} \approx 0.196 \) grams.

Step 7: Computing the amount remaining after 30 days

Substitute 30 days into the decay formula: \( N(30) = 1 e^{-0.181 \times 30} = e^{-5.43} \approx 0.004 \) grams.

Key Concepts

Differential Equation

In the context of radioactive decay, the term 'differential equation' refers to a mathematical expression that describes how quantities change over time. For a radioactive substance like radon-232, the decay rate, or how fast the substance diminishes, is directly proportional to the amount of substance present.
This relationship can be expressed using a differential equation: \( \frac{dN}{dt} = -kN \), where:

• \( \frac{dN}{dt} \) represents the rate of change of the quantity \( N \), which is the amount of radon-232 at time \( t \).
• \( k \) is the decay constant, a positive number that indicates how quickly the substance decays.
• The negative sign indicates a decrease in the amount over time.

Understanding and solving differential equations are essential to predict how much of the isotope will remain after a given period.

Half-life

Half-life is a crucial concept in understanding radioactive decay. It is defined as the time it takes for half of the radioactive substance to decay. For radon-232, the half-life is 3.824 days. This implies that if you start with 1 gram of radon-232, only 0.5 grams will remain after 3.824 days.
The half-life is a key factor in determining the decay constant \( k \). By setting \( N(t) = \frac{N_0}{2} \) when \( t \) equals the half-life, you can solve for \( k \) using the decay equation \( \frac{N_0}{2} = N_0 e^{-k \cdot 3.824} \). This solution allows scientists and engineers to model the decay process accurately, which is vital in fields like nuclear medicine, environmental science, and safety engineering.

Decay Constant

The decay constant, represented by \( k \), is a parameter that quantifies the rate at which a radioactive substance decays. It is directly related to the half-life of the isotope. The relationship between \( k \) and the half-life \( T_{1/2} \) is given by the formula:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

For radon-232, with a half-life of 3.824 days, the decay constant is calculated as \( k = \frac{\ln(2)}{3.824} \approx 0.181 \).
The decay constant serves as a critical factor in the exponential decay equation \( N(t) = N_0 e^{-kt} \), determining how quickly the quantity diminishes over time. It helps in evaluating the safety levels of radioactive elements in environments such as homes, reducing health hazards.

Exponential Decay

Exponential decay describes a process where the quantity of a substance decreases at a rate proportional to its current value. For radon isotopes, this process is mathematically expressed as \( N(t) = N_0 e^{-kt} \), where:

• \( N(t) \) is the amount of the substance at time \( t \).
• \( N_0 \) is the initial amount at time \( t = 0 \).
• \( e \) is the mathematical constant approximately equal to 2.71828.
• \( k \) is the decay constant.

After understanding and applying this formula, one can determine how much of the substance remains after any given time period. For instance, finding out how much radon-232 remains after specific time intervals like 12 hours or 30 days helps analyze its behavior and potential environmental impact.

Radon Isotopes

Radon isotopes are naturally occurring radioactive gases that are part of the decay chain of uranium and thorium. Radon-232 is one such isotope, and like other radon variants, it is colorless and odorless, making it hard to detect without specialized equipment.
These isotopes are significant because of their health implications. When inhaled, radioactive decay products from radon can damage lung tissue, increasing the risk of lung cancer.
Radon testing in homes is especially important in areas with high natural uranium concentrations in the soil. Understanding the decay process of radon, characterized by its exponential decay and known half-life, allows us to measure and mitigate its health risks effectively. This knowledge is crucial for implementing appropriate safety measures in both residential and occupational settings.