Question

The noble gas with the largest atmospheric abundance is argon. Using the data in Table \(20.22\), calculate the mass of argon at \(25^{\circ} \mathrm{C}\) and \(1.0 \mathrm{~atm}\) in a room \(10.0 \mathrm{~m} \times 10.0 \mathrm{~m} \times 10.0 \mathrm{~m}\). How many Ar atoms are in this room? How many Ar atoms do you inhale in one breath (approximately \(2 \mathrm{~L}\) ) of air at \(25^{\circ} \mathrm{C}\) and \(1.0\) atm? Argon gas is inert, so it poses no serious health risks. However, if significant amounts of radon are inhaled into the lungs, lung cancer is a possible result. Explain the health risk differences between argon gas and radon gas.

Short answer

The mass of argon in a room with dimensions \(10.0 \, \text{m} \times 10.0 \, \text{m} \times 10.0 \, \text{m}\) at 25 degrees Celsius and 1.0 atm is approximately 15.05 kg, and the number of argon atoms in the room is approximately \(2.27 \times 10^{26}\) atoms. In one breath of air at the same conditions, we inhale roughly \(4.54 \times 10^{20}\) argon atoms. Although both argon and radon are noble gases, argon is inert and does not pose serious health risks. In contrast, radon is radioactive, emitting particles that can damage lung tissue and potentially lead to lung cancer when inhaled. This highlights the importance of understanding the properties and potential health risks associated with different substances.

Step-by-step solution

Calculate the volume of the room

First, calculate the volume of the room which is a cube of side 10.0m. You can compute the cube of a number by multiplying the number by itself twice. Therefore, the volume of the room is \(10.0 \times 10.0 \times 10.0 = 1000.0 \, \text{m}^3\).

Convert the volume to liters

The ideal gas law uses liters (L) as the volume unit. Convert the volume from cubic meters to liters by using the conversion factor 1 m^3 = 1000 L. Thus, the volume of the room is \(1000.0 \times 1000 = 1,000,000 \, \text{L}\).

Apply the Ideal Gas law to find the number of moles of gas in the room

According to the ideal gas law, the amount of gas (moles) can be calculated by \(\frac{PV}{RT}\), where P is pressure (1.0 atm), V is volume (1,000,000 L), R is the gas constant (0.0821 L atm/mole K), and T is absolute temperature. Convert the temperature from Celsius to Kelvin by adding 273.15 to 25, which results in 298.15 K. Thus, the number of moles of gas in the room is \( \frac{(1.0 \times 1,000,000)}{(0.0821 \times 298.15)} = 40578 \, \text{moles}\).

Find the moles and mass of argon in the room

Argon makes up about 0.93% of air by volume, so multiply the total moles of gas by this percentage to get the moles of argon. The response is \(40578 \times 0.0093 = 377 \, \text{moles of Ar}\). To find the mass, we multiply by the molar mass of argon, which is 39.95 g/mole. This gives \(377 \times 39.95 = 15054 \, \text{g} = 15.05 \, \text{Kg}\).

Compute the number of argon atoms in the room

Use Avogadro's number (\(6.022 \times 10^{23}\) atoms/mole) to convert from moles of argon to atoms. This yields \(377 \times 6.022 \times 10^{23} = 2.27 \times 10^{26} \, \text{atoms}\).

Calculate the number of argon atoms in one breath of air

In one breath, approximately 2 liters of air are inhaled. This is 0.000002 of the total volume of the room. Therefore, the number of argon atoms inhaled in one breath is \(2.27 \times 10^{26} \times 0.000002 = 4.54 \times 10^{20} \, \text{atoms}\)

Discuss the differences in health risk between inhaling argon and radon

Argon is an inert gas, meaning it does not undergo chemical reactions. Therefore, it does not pose any serious health risks when inhaled, as it doesn't interact with body tissues or cells. On the other hand, radon is a radioactive gas. When breathed in, it releases radioactive particles that can damage lung tissue, potentially leading to lung cancer. In fact, radon exposure is the second leading cause of lung cancer, after smoking. In conclusion, while both argon and radon are naturally occurring gases, they pose dramatically different health risks due to their different chemical properties. This situation demonstrates the importance of understanding the properties of the substances with which we come into contact.

Key concepts

Ideal Gas Law

The ideal gas law is a crucial equation in chemistry and physics used to relate variables such as pressure (\textbf{P}), volume (\textbf{V}), temperature (\textbf{T}), and the amount of gas in moles (\textbf{n}) for a hypothetical ideal gas. It is usually expressed as \( PV = nRT \) where \textbf{R} represents the ideal gas constant. This constant, depending on the units of the other variables, is typically 0.0821 \( L \cdot atm \cdot mol^{-1} \cdot K^{-1} \) as seen in the exercise.

Understanding the ideal gas law allows students to predict how gases will behave under various conditions. For instance, if we know the temperature, pressure, and volume of a particular sample of gas, we can calculate the amount in moles. Conversely, if we're given moles and temperature, we can deduce the volume the gas will occupy at a certain pressure.

To solve real-world problems using the ideal gas law, proper unit conversions are necessary, just like in the example where cubic meters were converted to liters before the calculation. This example intuitively demonstrates the law's practical application, which reinforces the concept in a meaningful context.

Molar Mass

Molar mass, a key concept in chemical calculations, is the mass of one mole of a substance. It is typically expressed in grams per mole (g/mol). The significance of molar mass lies in its function as a bridge between the mass of a substance and the amount in moles.

For example, in the exercise the molar mass of argon is used to convert the volume of argon in the room to its mass in kilograms. Molar mass allows us to make these conversions because by definition, one mole of an element consists of exactly the same number of atoms, ions, or molecules, known as Avogadro's number. The molar mass for every element is different and can be found on the periodic table, as indicated by the atomic weight. Therefore, knowing the molar mass of argon (39.95 g/mol) instantly informs us how many grams a single mole of argon atoms would weigh, thus enabling calculations of mass from the number of moles.

Avogadro's Number

Avogadro's number, named after scientist Amedeo Avogadro, is a constant that represents the number of particles in one mole of a substance. The actual value is approximately \( 6.022 \times 10^{23} \) particles per mole and is germane to converting moles into individual atoms, molecules, or ions for stoichiometric calculations.

In our exercise, Avogadro's number is utilized to translate the abstract concept of moles into the more tangible idea of individual argon atoms. This conversion is particularly useful when dealing with macroscopic quantities of gas and translating them into microscopic units that describe atomic-level interactions. The use of Avogadro's number exemplifies how large-scale and small-scale phenomena are interconnected in the study of chemistry.

Health Risks of Noble Gases

Noble gases, such as argon and radon, are a group of elements known for their lack of reactivity due to their full valence electron shells. However, their behavior can lead to different health implications.

Argon, being chemically inert, typically does not pose a threat to human health when inhaled because it does not react with other substances in the body. On the contrary, radon is not only a noble gas but also radioactive. As a gas, it can be easily inhaled, and once inside the lungs, it decays and emits ionizing radiation. This radiation can cause damage to lung cells, potentially leading to cancer. It is particularly dangerous because it can accumulate to high levels indoors, going undetected due to its colorless and odorless nature.

The exercise nicely contrasts these health risks, reinforcing crucial differences between nonreactive and radioactive gases—making it clear why understanding chemical properties is vital for safety and health considerations.