Question

The atmosphere contains \(9.0 \times 10^{-6} \%\) Xe by volume at \(1.0 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\). a. Calculate the mass of Xe in a room \(7.26 \mathrm{~m}\) by \(8.80 \mathrm{~m}\) by \(5.67 \mathrm{~m} .\) b. A typical person takes in about \(2 \mathrm{~L}\) of air during a breath. How many Xe atoms are inhaled in each breath?

Short answer

a. The mass of Xe in the room is approximately 0.849 g. b. A person inhales about \(1.16 × 10^{14}\) Xe atoms in each breath.

Step-by-step solution

Calculate the volume of the room

The volume of the room can be calculated using the formula for the volume of a rectangular prism, which is the product of its length, width, and height. In this case: Volume = Length × Width × Height So, we need to multiply the given dimensions of the room: Volume = \(7.26 m × 8.80 m × 5.67 m\)

Calculate the number of moles of Xe in the room

First, we need to find the moles of air in the room. The Ideal Gas Law helps us to find the number of moles (n) of a gas, using the formula: n = \(PV/RT\) Where: - P is the pressure (1.0 atm) - V is the volume (calculated in Step 1) - R is the gas constant (0.0821 L*atm/(K*mol)) - T is the temperature (25°C = 298 K) Next, we need to determine the number of moles of Xe in the room. As the atmosphere contains \(9.0 × 10^{-6} \%\) Xe by volume, we can use this proportion to calculate the number of moles of Xe from the total moles of air in the room.

Calculate the mass of Xe in the room

To find the mass of Xe in the room, use the formula: Mass = moles of Xe × molar mass of Xe (131 g/mol) Now we need to multiply the moles of Xe calculated in the previous step by the molar mass of Xe.

Calculate the number of Xe atoms inhaled per breath

We are given that a person takes in about 2 L of air during a breath. To find out how many Xe atoms are inhaled in each breath, we need first to calculate the moles of Xe in 2 L of air. Using the Ideal Gas Law formula n = \(PV/RT\), calculate the moles of air in 2 L at 1 atm and 25°C: - P= 1 atm - V = 2 L - R = 0.0821 L*atm/(K*mol) - T = 298 K Now, find the moles of Xe in 2 L of air by using the proportion of Xe in the air (as we did in Step 2). Finally, to find the number of Xe atoms, we must convert the moles of Xe inhaled to the number of atoms using Avogadro's number (6.022 × 10^23 atoms/mol). Multiply the moles of Xe by Avogadro's number to get the number of Xe atoms in each breath.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. This law combines the three gaseous laws into one equation and is expressed as \[ PV = nRT \]where:

• P is the pressure of the gas, usually in atmospheres (atm).
• V is the volume occupied by the gas, usually in liters (L).
• n is the number of moles of gas present.
• R is the gas constant, which is 0.0821 L·atm/(K·mol).
• T is the temperature in Kelvin (K).

The Ideal Gas Law is used to predict how a gas will behave under different conditions. To solve for the number of moles of a gas, as in this exercise, you rearrange the equation to: \[ n = \frac{PV}{RT} \]This formula allows you to find out how many moles of a gas are present in a given volume at a specific temperature and pressure. Understanding this law is key to solving many problems in chemistry, such as calculating the amount of a specific gas in a room or determining how a gas will expand when heated.

Molar Mass

The molar mass is a crucial concept when dealing with gases and their calculations. It is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol). Molar Mass of xenon (Xe) is approximately 131 g/mol. To calculate the mass of xenon present in a given volume or number of moles, you use the formula: \[ \text{Mass} = n \times \text{Molar Mass} \]where n is the number of moles calculated with the Ideal Gas Law. Knowing molar mass is useful for converting between moles and grams, allowing us to measure the physical amount of substance present. For instance, if we have determined the number of moles of xenon in a room, we can easily find the total mass of xenon by multiplying those moles by the molar mass of xenon. This method is often used in chemistry to relate quantities in calculations.

Avogadro's Number

Avogadro's Number is a cornerstone of chemistry. It tells us the number of constituent particles (usually atoms or molecules) contained in one mole of a substance. Avogadro's Number is approximately \( 6.022 \times 10^{23} \) particles/mol.This means that one mole of any element or compound, no matter what it is, contains this many atoms or molecules. It is particularly useful when converting between the number of moles of a substance and the number of atoms or molecules in that substance. For example, in the context of the exercise, to find out how many xenon atoms are inhaled in a single breath, you first calculate the moles of xenon in 2 liters of air. Then, multiply this by Avogadro's Number to find out the total number of atoms. This concept helps bridge the gap between atom-level (microscopic) and bulk (macroscopic) quantities.

Volume of a Rectangular Prism

Volume measurements are essential in chemistry, especially when calculating the amount of gas in a given space. A rectangular room, for example, has a volume calculated by multiplying its length, width, and height: \[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]In this exercise, the room's dimensions are 7.26 m by 8.80 m by 5.67 m. So, the volume is calculated by: \[ 7.26 \times 8.80 \times 5.67 \text{ cubic meters} \]Knowing how to compute the volume of a rectangular prism is important for understanding how much space a gas will occupy, as gases expand to fill the container they are in. It lays the groundwork for using the Ideal Gas Law, where the volume V plays a critical role in calculating the number of moles of a gas present. It's a useful formula not only in chemistry but also in everyday practical scenarios.