Question

Dry air is essentially a mixture of the following entities: \(\mathrm{N}_{2}, \mathrm{O}_{2}, \mathrm{Ar},\) and \(\mathrm{CO}_{2} .\) The composition of dry air, in mole percent, is \(78.08 \% \mathrm{N}_{2}, 20.95 \% \mathrm{O}_{2}, 0.93 \% \mathrm{Ar}\) and \(0.04 \% \mathrm{CO}_{2}\). (a) What is the mass, in grams, of a sample of air that contains exactly one mole of the entities? (b) Dry air also contains other entities in much smaller amounts. For example, the mole percent of krypton (Kr) is about \(1.14 \times 10^{-4} \% .\) Given that the density of dry air is about \(1.2 \mathrm{g} / \mathrm{L}\) at room temperature, what mass of krypton could be obtained from exactly one cubic meter of dry air?

Short answer

a) The mass of a sample of air that contains exactly one mole of entities is 28.921 g. b) The mass of krypton that could be obtained from exactly one cubic meter of air is 0.005 g.

Step-by-step solution

Determine the Molar Mass of Each Entity

Use the periodic table to find the molar mass of each entity. N_2: \(2 * 14.007 = 28.014 \, g/mol\), O_2: \(2 * 15.999 = 31.998 \, g/mol\), Ar: \(39.948 \, g/mol\), CO_2: \(12.011 + 2 * 15.999 = 43.009 \, g/mol\)

Calculate Total Molar Mass of the Air

Multiply each molar mass by its corresponding mole percent (expressed in decimal form) and sum the results: M_air = \(0.7808 * 28.014 \, g/mol + 0.2095 * 31.998 \, g/mol + 0.0093 * 39.948 \, g/mol + 0.0004 * 43.009 \, g/mol = 28.921 \, g/mol\)

Mass of One Mole of Air

The mass of one mole of air would be equal to the total molar mass of the air. Therefore, mass of one mole of air, M = 28.921 g.

Calculate Molar Mass of Krypton

Using the periodic table, we find that the molar mass of krypton (Kr) is 83.798 g/mol.

Calculate the Mass of Krypton in One Cubic Meter of Air

First calculate the number of moles in one cubic meter of air by dividing the mass of air in one cubic meter (which is Density_air * Volume_air = 1.2 g/L * 1000 L = 1200 g) with the molar mass of air (28.921 g/mol). Number of moles in one cubic meter = 1200 g / 28.921 g/mol = 41.49 mol. Then calculate the mass of Kr in the air by multiplying the number of moles of Kr (which is mole percent of Kr * total number of moles = \(1.14*10^{-6}* 41.49\)) with the molar mass of Kr (83.798 g/mol). Mass_Kr = \(1.14*10^{-6} * 41.49 mol * 83.798 g/mol = 0.005 g\).

Key concepts

Air Composition

Dry air is composed principally of nitrogen (\(\mathrm{N}_{2}\)) and oxygen (\(\mathrm{O}_{2}\)), with small quantities of argon (\(\mathrm{Ar}\)) and carbon dioxide (\(\mathrm{CO}_{2}\)). In mole percent, nitrogen dominates with 78.08%, followed by oxygen at 20.95%, argon at 0.93%, and carbon dioxide at a mere 0.04%. These percentages reflect how air is a gas mixture rather than a pure substance.
This mix of gases means that when dealing with air, we often have to think about their combined properties. Although nitrogen is the most prevalent, each gas's behavior matters. Their differences in molar mass, reactivity, and other properties all play roles in the air's collective behavior.

• Nitrogen (\(\mathrm{N}_{2}\)): Inert, comprises most of the atmosphere.
• Oxygen (\(\mathrm{O}_{2}\)): Vital for respiration, supports combustion.
• Argon (\(\mathrm{Ar}\)): Inert, makes up a small percentage.
• Carbon Dioxide (\(\mathrm{CO}_{2}\)): Essential for photosynthesis, present in trace amounts.

Understanding these components helps us appreciate what makes up the air around us and how it supports life and industrial processes.

Molar Mass Calculation

Calculating the molar mass of a gas mixture like air involves using the mole fractions of its components. To find the average molar mass of air, we consider the individual molar masses of each gas and their mole percentages.
We begin by obtaining the molar mass of each component, such as nitrogen (28.014 g/mol), oxygen (31.998 g/mol), argon (39.948 g/mol), and carbon dioxide (43.009 g/mol).
Next, we convert the mole percentages into decimals. For instance, 78.08% becomes 0.7808. We multiply each gas's molar mass by this decimal value, then sum these results to find the total molar mass of air. This calculation looks like this:\[M_{\text{air}} = 0.7808 \times 28.014 + 0.2095 \times 31.998 + 0.0093 \times 39.948 + 0.0004 \times 43.009 = 28.921 \, \text{g/mol}\]
This 28.921 g/mol represents the mass of one mole of dry air. Understanding this calculation method allows us to accurately determine the mass and behavior of mixed gases in various conditions.

Gas Mixtures

Air is a prime example of gas mixtures and their unique properties. Unlike single gases, mixtures like air involve the collective behavior of several gases combined.
Each gas retains its own properties, but together, they interact to form the characteristics of the mixture. For air, these interactions include both chemical and physical behaviors that are different from each component acting alone.
Understanding how to handle these mixtures requires a grasp of moles and mole fractions. For example, knowing the percentage of each gas allows us to determine the mixture's overall behavior, including how it reacts under changes in temperature or pressure.

The air also contains trace gases like krypton, though in minimal amounts compared to the main constituents. By accounting for these minor gases, we can refine our understanding of air’s mass and its dynamic nature. This approach allows us to predict how gases in mixtures might behave in processes such as environmental modeling, engineering applications, or scientific experiments.