Problem 35 - Chemistry The Science in Context

Question

The half-life of radon-222, a radioactive gas found in some basements, is 3.82 days. Calculate the decay rate constant of radon- 222

Step-by-Step Solution

Answer

Answer: The decay rate constant of radon-222 is approximately 0.1812 day^{-1}.

Step 1: Identify the given information

We are given the half-life of radon-222, \(t_{\frac{1}{2}} = 3.82\) days.

Step 2: Implement the formula for decay rate constant

Use the formula for the decay rate constant with the half-life information: \(\lambda = \frac{ln(2)}{t_{\frac{1}{2}}}\)

Step 3: Substitute in the given half-life and calculate

Substitute the given half-life value, \(t_{\frac{1}{2}} = 3.82\) days, into the formula: \(\lambda = \frac{ln(2)}{3.82} = \frac{ln(2)}{3.82} \approx 0.1812 \, \mathrm{day}^{-1}\)

Step 4: Write the final answer

The decay rate constant of radon-222 is approximately \(\lambda \approx 0.1812 \, \mathrm{day}^{-1}\).

Key Concepts

Understanding Half-Life

The concept of half-life is central to the study of radioactive decay. It is defined as the time required for half of the radioactive atoms in a sample to decay. In simple terms, if you start with a certain amount of a radioactive substance, after one half-life, you'll be left with half of that original amount.

For radon-222, which is a gas that can accumulate in basements and poses health risks, the half-life is 3.82 days. This means that if you had a sample of radon-222, in just under 4 days, only half of the atoms would remain undecayed. The other half would have transformed into other elements or isotopes through the process of radioactive decay. Understanding half-life is not only important for safety and health but also in fields like archeology for dating artifacts and in medicine for administering radioactive treatments.

Radioactive Decay and Decay Constant

Radioactive decay is a stochastic (random) process where unstable atomic nuclei lose energy by emitting radiation. Each radioactive isotope, or nuclide, has a characteristic decay that can be described in terms of a decay constant. The decay constant, denoted by \(\lambda\), represents the probability of a single atom decaying per unit time.

In the case of radon-222, using the half-life, we can calculate the decay constant as \(\lambda = \frac{\ln(2)}{3.82} \) days\(^{-1}\). This value gives us an understanding of the rate at which radon-222 atoms decay. It's an essential factor in estimating the radiation exposure over time and is used in both environmental studies and radiation safety to predict the behavior of radioactive materials.

The Role of the Natural Logarithm in Decay Calculations

The natural logarithm, often represented as \(\ln\), is a mathematical function that's integral to calculating decay rates in radioactive decay processes. The natural logarithm is the inverse operation of exponentiation with the base \(e\), which is an irrational and transcendental number approximately equal to 2.71828.

The reason why the natural logarithm is used in the decay constant formula \(\lambda = \frac{\ln(2)}{t_{\frac{1}{2}}}\) is due to the exponential nature of radioactive decay. The natural logarithm of 2 is a fixed constant because the decay of a radioactive substance is based on a ratio where half of the original material decays over each half-life period. This concept clarifies the exponential decay formula and helps understand how the decay rate, expressed as a constant, relates to the observable half-life of a substance like radon-222.