Problem 37 - Chemistry Structure and Dynamics

Question

The threat to people's health from radon in the air trapped in their houses has received attention in recent years. If the average level of radon in a house is approximately 1 picocurie (pCi) per liter of air, how many radon atoms are there per liter? (Assume that \({ }^{222} \mathrm{Rn}\) is the principal source of this activity and that the half-life for the decay of this nuclide is 3.823 days.)

Step-by-Step Solution

Answer

Approximately \(2.22 \times 10^{10}\) radon atoms per liter.

Step 1: Understand the Problem

We need to calculate the number of radon atoms in 1 liter of air given a radon activity level of 1 picocurie (pCi). We'll use the information about radon ( 222Rn) half-life and radioactive decay to find our answer.

Step 2: Convert Curie to Decays per Second

1 picocurie (pCi) equals to \(10^{-12}\) curies. We know that 1 curie is \(3.7 \times 10^{10}\) decays per second. Therefore, \(1\) pCi equals \(3.7 \times 10^{-2}\) decays per second.

Step 3: Calculate the Decay Constant

Decay constant \(\lambda\) can be calculated using the formula \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2} = 3.823\) days. Convert half-life to seconds: \(t_{1/2} = 3.823 \times 24 \times 3600\) seconds. Calculate \(\lambda\).

Step 4: Calculate the Number of Atoms

Use the formula for activity \(A = \lambda N\), where \(A = 3.7 \times 10^{-2}\) decays per second, and solve for \(N\), the number of radon atoms per liter. Substitute \(\lambda\) from the previous step into this equation and solve for \(N\).

Step 5: Express the Result

After solving \(N = \frac{A}{\lambda}\), substitute the calculated values to find the number of radon atoms per liter. Verify the final calculation.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei release energy in the form of radiation until they transform into a more stable form. During this process, a parent isotope (for example, radon-222) loses particles or radiation over time, changing into a different element or isotope.

There are several types of radioactive decay, such as alpha, beta, and gamma decay. Radon-222, the primary isotope responsible for radon exposure, undergoes alpha decay.

In simple terms, during alpha decay, the radon nucleus emits an alpha particle, consisting of 2 protons and 2 neutrons, which greatly reduces its size and changes its identity, turning into a different element- polonium.

This step-by-step change can continue until a stable form is reached, which involves several decay processes. When studying properties like activity level (measured in curies or becquerels), we focus on the rate at which these transformations happen.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms in a sample to decay. This property is crucial because it helps determine the decay rate of the substance and is specific to each isotope.

For radon-222, the half-life is known to be 3.823 days. This means that every 3.823 days, half of the radon atoms in the air will have decayed into another form.

To use the half-life in calculations, we often rely on the decay constant, \( \lambda \), which is used to describe the speed of the decay process.

The decay constant can be calculated using the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) is the half-life time in seconds. Knowing the decay constant allows us to compute the number of radioactive atoms in a given sample by using the formula \( A = \lambda N \), where \( A \) is the measured activity.

Radon Health Effects

Radon is a naturally occurring radioactive gas found in the environment, and is particularly concerning because it is odorless, colorless, and tasteless, making it imperceptible without special equipment. The health risks mainly arise when radon decays into other radioactive elements, which can attach to dust particles and be inhaled. Once inside the lungs, these particles can continue to decay, releasing radiation that can damage lung tissue.

Long-term exposure to high levels of radon is associated with an increased risk of lung cancer. This risk is higher for smokers as their lungs are already exposed to harmful substances. It's estimated that radon exposure is the second leading cause of lung cancer after smoking. Given these potential health effects, it is important to test homes for radon levels and implement measures such as improved ventilation if necessary to reduce exposure.