Question

The half-life of radon-222, a radioactive gas of concern as a health hazard in some homes, is \(3.82\) days. What is the decay constant of \({ }^{222} \mathrm{Rn}\) ?

Short answer

The decay constant of radon-222 is approximately \(0.1814\, \text{days}^{-1}\).

Step-by-step solution

Understanding Half-Life

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms in a sample to decay. In this problem, the half-life of radon-222 is given as \(3.82\) days.

Decay Constant Formula Introduction

The decay constant (\(\lambda\)) is a probability rate at which a radioactive substance decays, which can be found using the formula \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2}\) is the half-life of the substance.

Substitute the Given Values

Substitute the given half-life value into the decay constant formula: \[ \lambda = \frac{\ln(2)}{3.82} \].

Calculate the Decay Constant

Calculate the value of the decay constant by evaluating the expression: \(\ln(2)\) is approximately \(0.693\). Therefore, calculate \(\lambda = \frac{0.693}{3.82}\approx 0.1814\, \text{days}^{-1}\).

Unit Verification

Verify that the decay constant \(\lambda\) is in the correct units of \(\text{days}^{-1}\) because we divided by time in days.

Key concepts

radon-222 half-life

The half-life of a substance is a fundamental concept in radioactive decay. It represents the time required for half of a given amount of a radioactive substance to transform into another element or isotope. Radon-222, an inert radioactive gas, has a specific half-life of 3.82 days. This means that if you start with a certain amount of radon-222, after 3.82 days, precisely half of that initial amount will have decayed.

This property makes the half-life crucial in determining how long a radioactive substance remains hazardous. In the case of radon-222, its relatively short half-life implies that it decays quickly, which is good as it limits prolonged exposure, but it can also mean higher initial activity levels. Understanding half-life is vital in fields like radiology and environmental safety, especially when considering radon's potential health risks in enclosed spaces like homes.

decay constant calculation

The decay constant, often represented by the Greek letter \(\lambda\), is a key parameter in the study of radioactive decay. It quantifies the likelihood of a particle decaying per unit time and is inversely related to the half-life of a substance. Simply put, the higher the decay constant, the faster the rate of decay.

To calculate the decay constant of radon-222, we use the known half-life and the formula:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]where \(t_{1/2}\) is the half-life. Substituting the value for radon-222, we get:\[ \lambda = \frac{0.693}{3.82} \] With \(\ln(2)\) approximately equal to 0.693, the calculation gives a decay constant of 0.1814 \text{days}^{-1}\. This calculation is essential because it gives us insight into the rapidity of decay, which is directly connected to both safety considerations and practical applications involving radioactive materials.

half-life formula

The half-life formula is a crucial tool in understanding and predicting the behavior of radioactive substances. It is: \[ \lambda = \frac{\ln(2)}{t_{1/2}} \]This formula allows us to calculate the decay constant \(\lambda\), where \(\ln(2)\) is the natural logarithm of 2, a constant value approximately equal to 0.693. The term \(t_{1/2}\) represents the half-life of the substance in question.

To use this formula effectively, follow these simple steps:

• Identify the half-life of the radioactive element you are working with, like the 3.82 days for radon-222.
• Substitute this value into the formula to calculate the decay constant.
• Evaluate with \(\ln(2) \approx 0.693\), for straightforward computation.

The half-life formula is not only fundamental in physics but also practically useful in areas such as medical diagnosis and treatment, radiocarbon dating, and nuclear energy, where understanding decay processes is critical.