Problem 86 - Chemistry Matter and Change

Question

What is the half-life of radon-222 if a sample initially contains 150 \(\mathrm{mg}\) and only 18.7 \(\mathrm{mg}\) after 11.4 days?

Step-by-Step Solution

Answer

The half-life of radon-222 is approximately 3.8 days.

Step 1: Understanding the Problem

We need to find the half-life of radon-222 given that a sample initially contains 150 mg and decays to 18.7 mg in 11.4 days.

Step 2: Applying the Formula for Exponential Decay

The formula for exponential decay in the context of half-life is given by: \[ A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \]where \( A \) is the final amount of the substance, \( A_0 \) is the initial amount, \( t \) is the time elapsed, and \( T \) is the half-life. In this case, \( A = 18.7 \), \( A_0 = 150 \), and \( t = 11.4 \). We need to solve for \( T \).

Step 3: Calculating the Ratio of Final to Initial Amount

Divide the final amount by the initial amount:\[ \frac{A}{A_0} = \frac{18.7}{150} \approx 0.1247 \]

Step 4: Setting Up the Equation

Using the exponential decay formula, set the equation:\[ 0.1247 = \left(\frac{1}{2}\right)^{\frac{11.4}{T}} \]

Step 5: Taking the Logarithm to Solve for T

To solve for \( T \), take the logarithm of both sides of the equation:\[ \log \left( 0.1247 \right) = \frac{11.4}{T} \log \left( \frac{1}{2} \right) \]

Step 6: Calculating the Half-Life

Rearrange the equation to solve for \( T \):\[ T = \frac{11.4}{\log \left( 0.1247 \right) / \log \left( \frac{1}{2} \right)} \]Calculate numerically to find \( T \approx 3.8 \) days.

Key Concepts

Exponential Decay Formula

Exponential decay is a fundamental concept when dealing with substances or quantities that decrease at a rate proportional to their current value. In the context of half-life calculations, this principle is used to determine how long it takes for a radioactive substance to reduce to half its initial quantity. The formula to address this is:

• \( A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \)

Here, \( A \) represents the remaining amount of the substance, \( A_0 \) is the initial amount, \( t \) refers to the elapsed time, and \( T \) is the half-life.
By applying this formula, we can model the decay of substances like radon-222 over time. Understanding this equation is essential in solving problems related to radioactive decay, such as determining the half-life from given amounts and timeframes.

Logarithmic Calculations

Logarithms are extremely useful in solving exponential decay problems, especially when it comes to isolating the variable related to time or the half-life. In the context of radon-222 decay, logarithms allow us to solve for the unknown half-life by transforming the exponential decay equation into a linear form.
For instance, when determining the half-life \( T \), one can take the logarithm of both sides of the equation:

• \( \log \left( 0.1247 \right) = \frac{11.4}{T} \log \left( \frac{1}{2} \right) \)

This transformation makes it much easier to isolate and solve for \( T \). Understanding how to work with logarithms in such scenarios is vital, as they simplify complex exponential relationships into more manageable linear ones.

Radon-222 Decay

Radon-222 is a radioactive isotope of radon and is commonly studied regarding its decay properties. It serves as a classic example of exponential decay due to its relatively short half-life and natural occurrence in the environment.
This decay process involves the emission of radiation, which reduces the amount of radon over time. As with all radioactive materials, knowing how to calculate its half-life is crucial for both scientific applications and safety assessments. The change from 150 mg to 18.7 mg within 11.4 days highlights the rapid reduction typical of radioactive isotopes.
Such information provides insight not only into the behavior of radon-222 but also underscores broader principles applicable to other radioactive elements.

Radioactive Decay Principles

Radioactive decay is governed by a set of principles explaining how unstable nuclei lose energy through radiation. It's a random process at the level of a single atom, but it reflects predictable patterns when considering large numbers of atoms, characterized by their half-life.

• Half-life is the time it takes for half of a radioactive sample to decay.
• Each decay event reduces the amount of the initial substance, occurring at a consistent percentage rate.
• The process continues in successive half-lives, always halving the remaining amount.

The principles underpinning radioactive decay are essential in nuclear physics, impacting fields such as nuclear energy, medical imaging, and archaeology. By understanding these principles, you can predict how a radioactive substance will behave over time, making these concepts integral to applied sciences.