Problem 14 - An Introduction to Thermal Physics

Question

Calculate the mass of a mole of dry air, which is a mixture of \(\mathrm{N}_{2}\) \((78 \% \text { by volume }), \mathrm{O}_{2}(21 \%),\) and argon \((1 \%)\).

Step-by-Step Solution

Answer

The mass of a mole of dry air is approximately 28.98 g/mol.

Step 1: Determine Molar Masses

To calculate the mass of a mole of dry air, we start by finding the molar masses of each gas in the mixture. - Nitrogen ( N_2): 28.02 g/mol - Oxygen ( O_2): 32.00 g/mol - Argon (Ar): 39.95 g/mol

Step 2: Convert Volume Percentages to Mole Fractions

Since the gases are mixed by volume, we can use the volume percentages as mole percentages for our calculation. - Mole fraction of N_2: 0.78 - Mole fraction of O_2: 0.21 - Mole fraction of Ar: 0.01

Step 3: Calculate Molar Mass of Dry Air

The molar mass of dry air is the sum of the products of the molar mass and mole fraction of each component:\[\text{Molar Mass of Dry Air} = (28.02 \, \text{g/mol} \times 0.78) + (32.00 \, \text{g/mol} \times 0.21) + (39.95 \, \text{g/mol} \times 0.01)\]Calculate each term:- N_2: 21.86 g/mol- O_2: 6.72 g/mol- Ar: 0.40 g/molAdd them together to get the total molar mass of dry air.

Step 4: Summarize the Calculation

Add up the contributions from each gas to find the total molar mass:\[\text{Molar Mass of Dry Air} = 21.86 + 6.72 + 0.40 = 28.98 \, \text{g/mol}\]Thus, the mass of a mole of dry air is approximately 28.98 g/mol.

Key Concepts

Mole Fraction

The concept of "mole fraction" is frequently used in chemistry to describe the composition of mixtures. It represents the ratio of the number of moles of a particular component to the total number of moles in the mixture. In simple terms, it tells us how much of one substance is present compared to the whole mixture.

To find the mole fraction, we use the following formula:

• Mole Fraction (\(x_i\)) = Number of Moles of Component (\(n_i\)) / Total Number of Moles (\(n_{total}\))

For gases, the mole fraction is often equivalent to the volume percentage, since gases behave ideally under certain conditions. In the dry air example, nitrogen (\(\text{N}_2\)) has a mole fraction of 0.78, which means 78% of the air consists of nitrogen molecules. Similarly, oxygen (\(\text{O}_2\)) has a mole fraction of 0.21, and argon (\(\text{Ar}\)) has 0.01. The mole fractions are essential for calculations, especially when determining the properties of a mixture like molar mass.

Volume Percentage

Volume percentage is a way to express the concentration of a component in a gas mixture. It tells us how much of the volume of the mixture is taken up by a specific gas. This measure is particularly handy for gas mixtures because their components often combine volumetrically.

Since gases at the same temperature and pressure occupy equal volumes, volume percentages can directly translate into mole percentages. Therefore, in dry air, if nitrogen is present at 78% by volume, this means that 78% of the total volume of air is made up of nitrogen gas. The same idea applies to oxygen, at 21% by volume, and argon, at 1% by volume. This approach simplifies calculations by allowing us to use these percentages as mole fractions in further calculations.

Mixture Composition

Mixture composition in chemistry refers to how the different substances in a mixture are combined. For gases, the composition is often specified as parts of the whole, either by volume or moles, due to their ideal behavior under a range of conditions.

The mixture composition affects physical properties like density and molar mass. With dry air, composed of nitrogen, oxygen, and argon, we determine the molar mass by giving each component a weight according to its proportion, either in terms of volume percentages or mole fractions. Thus, knowing the composition allows chemists to predict how the mixture will behave and interact, which is vital in applications ranging from environmental science to industrial chemistry.

Gas Components

Understanding gas components means recognizing each individual element or compound present in a gaseous mixture. Each gas has its own properties, like molar mass, that contribute to the overall characteristics and behavior of the mixture. Nitrogen, oxygen, and argon in dry air are three primary components with distinct properties.

• Nitrogen (\(\text{N}_2\)): The most abundant, known for its inertness and lightness.
• Oxygen (\(\text{O}_2\)): Essential for combustion and respiration, contributing to a higher molar mass.
• Argon (\(\text{Ar}\)): A noble gas that is chemically inactive, but adds to the molar mass due to its heavier atomic weight.

Each gas component affects the mixture’s properties, like the molar mass of dry air which we calculated to be 28.98 g/mol. This value helps in various scientific and practical calculations, such as determining the air’s density or its buoyancy in weather balloons.