Problem 10 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A mixture of nitrogen and carbon dioxide contains \(38.4 \% \mathrm{N}_{2}\) by mass. What is the mole fraction of nitrogen in the mixture? If the total pressure is 1.2 atm, what is the partial pressure of each gas in Pa? (Section 8.3)

Step-by-Step Solution

Answer

The mole fraction of nitrogen is 0.494. Its partial pressure is 60065 Pa, and CO₂'s partial pressure is 61525 Pa.

Step 1: Determine Moles of Each Component

Assume a 100 g sample of the mixture for simplicity. Thus, it contains 38.4 g of nitrogen (N₂) and 61.6 g of carbon dioxide (CO₂). Calculate the moles of N₂: \[\text{Molar mass of } \mathrm{N}_2 = 28.02 \, \text{g/mol} \\text{Moles of } \mathrm{N}_2 = \frac{38.4 \, \text{g}}{28.02 \, \text{g/mol}} = 1.37 \, \text{mol}\] Next, calculate the moles of CO₂: \[\text{Molar mass of } \mathrm{CO}_2 = 44.01 \, \text{g/mol} \\text{Moles of } \mathrm{CO}_2 = \frac{61.6 \, \text{g}}{44.01 \, \text{g/mol}} = 1.40 \, \text{mol}\]

Step 2: Calculate the Mole Fraction of Nitrogen

The mole fraction of a component in a mixture is the ratio of the moles of that component to the total number of moles of all components in the mixture. Calculate the mole fraction of N₂:\[\text{Total moles} = 1.37 \, \text{mol (N₂)} + 1.40 \, \text{mol (CO₂)} = 2.77 \, \text{mol}\]\[\text{Mole fraction of } \mathrm{N}_2 = \frac{1.37 \, \text{mol}}{2.77 \, \text{mol}} = 0.494\]

Step 3: Calculate Partial Pressures using Total Pressure

The partial pressure of each gas is calculated by multiplying its mole fraction by the total pressure. Convert the total pressure from atm to Pa (1 atm = 101325 Pa).\[\text{Total Pressure in Pa} = 1.2 \, \text{atm} \times 101325 \, \text{Pa/atm} = 121590 \, \text{Pa}\]Calculate the partial pressure of N₂:\[\text{Partial pressure of } \mathrm{N}_2 = 0.494 \times 121590 \, \text{Pa} = 60065 \, \text{Pa}\]Calculate the partial pressure of CO₂:\[\text{Partial pressure of } \mathrm{CO}_2 = (1 - 0.494) \times 121590 \, \text{Pa} = 61525 \, \text{Pa}\]

Step 4: Summarize the Results

The mole fraction of nitrogen in the mixture is calculated as 0.494. For partial pressures, nitrogen has a partial pressure of 60065 Pa, and carbon dioxide has a partial pressure of 61525 Pa.

Key Concepts

Partial Pressure

Partial pressure is a fundamental concept in chemistry, particularly when dealing with gas mixtures. It refers to the pressure exerted by a single component of a gas mixture as if it occupied the entire volume by itself. For each gas in a mixture, partial pressure is calculated using its mole fraction and the total pressure of the gas mixture.

To find the partial pressure of nitrogen in our example, we multiply its mole fraction (0.494) by the total pressure (in Pascals). The formula used is \( P_{\text{partial}} = x_i \times P_{\text{total}} \), where \( x_i \) is the mole fraction and \( P_{\text{total}} \) is the total pressure of the system. Thus, nitrogen's partial pressure is 60065 Pa, which reflects the influence of nitrogen alone in the mixture.

Partial pressures are important because they allow chemists to understand and predict the behavior of each component in a mixture under various conditions, such as in chemical reactions or when gases are dissolved in liquids.

Moles Calculation

Calculating the moles of a substance in a mixture serves as the first critical step in determining other properties like the mole fraction or partial pressure. Moles are calculated by dividing the mass of the substance by its molar mass, which is the weight of one mole of that substance in grams.

In the given exercise, we calculated the moles of nitrogen (N₂) and carbon dioxide (CO₂) present in a 100 g sample. With nitrogen, for instance, we used its molar mass of 28.02 g/mol and calculated there were 1.37 moles of N₂. Similarly, with the molar mass of carbon dioxide being 44.01 g/mol, we found there were 1.40 moles of CO₂.

These calculations help in simplifying complex problems and are the foundation for further steps like calculating mole fractions or partial pressures. Knowing the amount in moles of each gas component is crucial, especially when applying the Gas Laws to predict their behaviors under different conditions.

Molecular Weight

Molecular weight, or molar mass, is the sum of the atomic weights of all atoms present in a molecule. It is an essential factor in converting between grams and moles, a process needed in various chemical calculations.

In our problem, the molecular weight of nitrogen (N₂) is 28.02 g/mol, and for carbon dioxide (CO₂), it's 44.01 g/mol. These values are derived by adding up the atomic weights of their constituent atoms found on the periodic table. For N₂, it consists of two nitrogen atoms, making the molar mass \( 2 \times 14.01 \), while CO₂, made of one carbon and two oxygen atoms, computes to \( 12.01 + 2 \times 16.00 \).

These molecular weights allow chemists to relate the measurable mass of materials to the amount of substance in moles, providing a bridge for further quantitative analysis and application of gas laws.

Gas Laws

The Gas Laws establish relationships between various properties of gases such as pressure, volume, temperature, and the number of moles. These relationships are essential for predicting how gases will behave in different situations.

In the context of our exercise, the ideal gas law, among other gas laws, plays a role in understanding gas behaviors like when calculating pressures. While we did not directly use PV = nRT in this exercise, knowing pressure conversion from atm to Pascals was vital for calculating partial pressures. The relation \( 1 \, \text{atm} = 101325 \, \text{Pa} \) was crucial in ensuring accurate pressure conversion to solve the problem.

These laws help in various chemical processes by predicting how conditions like temperature or volume changes will affect the gas's behavior, thus enabling better control over reactions and processes involving gases.