Question

An activity of 20 picocuries \(\left(20 \times 10^{-12} \mathrm{Ci}\right)\) of radon- 222 per liter of air in a house constitutes a health hazard to anyone living there. The half-life of radon-222 is \(3.82\) days. Calculate the concentration of radon in air (moles per liter) that corresponds to a 20-picocurie activity level.

Short answer

Answer: The concentration of radon-222 corresponding to a 20-picocurie activity level is approximately \(5.84\times10^{-18}\,\frac{\text{moles}}{\text{liter}}\) in the air.

Step-by-step solution

Convert half-life to decay constant

The decay constant can be calculated using the formula \(\lambda=\frac{\ln{2}}{\text{half-life}}\). The half-life of radon-222 is provided as \(3.82\) days. We first need to convert this to seconds: \(3.82 \,\text{days}\times 24\,\frac{\text{hours}}{\text{day}}\times 60 \,\frac{\text{minutes}}{\text{hour}} \times 60\,\frac{\text{seconds}}{\text{minute}}=330048 \,\text{seconds}\). Now, we can calculate the decay constant \(\lambda\). $$ \lambda=\frac{\ln{2}}{330048 \,\text{seconds}}=2.102\times 10^{-6} \,\frac{1}{\text{second}} $$

Determine the number of disintegrations from activity

In this step, we will convert the radon's activity from picocuries to atomic disintegrations per second. The conversion factor is \(1\,\text{Ci}=3.7\times10^{10}\,\frac{\text{disintegrations}}{\text{second}}\). First, convert the activity given (20 picocuries) to curies: $$ 20 \,\text{picocuries}\times\frac{1\times10^{-12}\,\text{curies}}{1\,\text{picocurie}}=2\times10^{-11}\,\text{curies} $$ Now, convert to atomic disintegrations per second: $$ 2\times10^{-11}\,\text{curies}\times\frac{3.7\times10^{10}\,\frac{\text{disintegrations}}{\text{second}}}{1\,\text{curie}}=7.4 \,\frac{\text{disintegrations}}{\text{second}} $$

Calculate the number of radon atoms

From the decay constant and the number of disintegrations, we can calculate the number of radon atoms present using the formula: $$ \text{Number of Radon Atoms}=\frac{\text{Disintegrations per second}}{\lambda}=\frac{7.4 \,\frac{\text{disintegrations}}{\text{second}}}{2.102\times 10^{-6}\,\frac{1}{\text{second}}}=3.518\times10^{6}\,\text{atoms} $$

Convert radon atoms to moles

To convert the number of radon atoms to moles, we use Avogadro's number which is approximately \(6.022\times10^{23}\,\frac{\text{atoms}}{\text{mole}}\). $$ 3.518\times10^{6}\,\text{atoms}\times\frac{1\,\text{mole}}{6.022\times10^{23}\,\text{atoms}}\approx 5.84\times10^{-18}\,\text{moles} $$

Calculate concentration in moles per liter

Since activity is given per liter of air, the concentration in moles per liter is: $$ \frac{5.84\times10^{-18}\,\text{moles}}{1\,\text{liter}}=5.84\times10^{-18}\,\frac{\text{moles}}{\text{liter}} $$ So, a 20-picocurie activity level of radon-222 corresponds to a concentration of approximately \(5.84\times10^{-18}\,\frac{\text{moles}}{\text{liter}}\) in the air.

Key concepts

Understanding Radioactive Decay

Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. This process occurs naturally in a variety of elements, such as radon-222, and is important for a range of applications, from medical treatments to geological dating.

To comprehend how atoms decay over time, it's vital to know the term 'decay product', which refers to the atom's remaining part post the decay. Different types of decay include alpha, beta, and gamma decay, each defined by the particle or radiation emitted. In the context of radon-222, it undergoes alpha decay, emitting an alpha particle and transforming into a different, more stable element.

When gauging the level of radioactive material, we consider the rate at which these disintegrations occur, which is expressed in terms of activity. This is typically measured in units called curies (Ci) or becquerels (Bq). Understanding this concept is crucial when dealing with problems of radioactive decay in various fields, including health physics and environmental science.

The Significance of Half-Life

The half-life of a radioactive element is the time required for half of the atoms in a sample to decay. It's a constant value unique to each radioactive element and is crucial in predicting how a radioactive material will behave over time. For example, radon-222 has a half-life of 3.82 days, which implies that after this period, only half of the original radon-222 atoms would remain in the sample; the rest would have decayed.

The concept of half-life is particularly important in environmental science, medicine, and nuclear industry, as it helps determine the duration a radioactive substance will remain active or hazardous. By understanding half-life, you can evaluate the potential risks associated with exposure to radioactive materials and plan accordingly for their safe handling and disposal.

Avogadro's Number and Mole Concept

Avogadro's Number

Avogadro's number, approximately equal to \(6.022 \times 10^{23}\), is the cornerstone of chemistry. It represents the number of atoms, ions, or molecules in one mole of a substance, bridging the gap between the microscopic scale of atoms and the macroscopic world we can measure.

Mole Concept

The mole is a fundamental unit in chemistry that provides a way to count particles at the atomic level, similar to how a 'dozen' counts objects at the everyday level. When calculating concentrations of substances, the mole allows for a clear connection between mass and number of particles. In the context of our radon-222 problem, we used Avogadro's number to convert the number of radon atoms to moles, enabling us to determine the concentration in the air.

Interpreting the Decay Constant

The decay constant, represented by the symbol \(\lambda\), is an important term in the formula that calculates radioactive decay. It provides the probability per unit time of a single nucleus decaying. In essence, this constant characterizes the stability of a radioactive isotope and influences its rate of decay.

Mathematically, the decay constant is inversely related to the half-life of the substance. This relationship allows us to convert between the two, thus helping in the calculation of activities, disintegration rates, and remaining quantities of a radioactive sample over time. A high decay constant indicates a more rapidly decomposing material, resulting in a shorter half-life. Understanding this relationship is essential when working on problems related to radioactive decay and subsequently determining the safety measures that need to be in place when handling such substances.