Question

The molar analysis of a gas mixture at \(290 \mathrm{~K}\) and \(250 \mathrm{kPa}\) is 65 percent \(\mathrm{N}_{2}, 20\) percent \(\mathrm{O}_{2}\), and 15 percent \(\mathrm{CO}_{2}\). Determine the mass fraction and partial pressure of each gas.

Short answer

Question: Calculate the mass fractions and partial pressures of N2, O2, and CO2 in a gas mixture with a total pressure of 250 kPa, given the following molar analysis: N2 - 65%, O2 - 20%, and CO2 - 15%. Answer: The mass fractions of N2, O2, and CO2 in the gas mixture are 0.60, 0.25, and 0.15, respectively. The partial pressures of N2, O2, and CO2 in the mixture are 162.50 kPa, 50 kPa, and 37.50 kPa, respectively.

Step-by-step solution

Calculate the molar fractions of each gas

We are given the molar analysis of the gas mixture, so we can directly use the given percentages to find the molar fractions. The molar fraction of a gas is the ratio of its molar ratio to the total molar ratio. Molar fractions: - Nitrogen (N2): 65% - Oxygen (O2): 20% - Carbon Dioxide (CO2): 15%

Calculate the mass fraction

The mass fraction of a gas is the ratio of its mass to the total mass of the mixture. To find the mass fraction, we first need the molar mass of each gas: - Molar mass of N2: \(28 \mathrm{g/mol}\) - Molar mass of O2: \(32 \mathrm{g/mol}\) - Molar mass of CO2: \(44 \mathrm{g/mol}\) Next, we'll multiply the molar fraction of each gas by its respective molar mass, and then divide the result by the sum of the weighted molar masses of all the gases: Mass fraction of N2: \(\frac{0.65 \times 28}{(0.65\times 28) + (0.2\times 32) + (0.15\times44)}\) Mass fraction of O2: \(\frac{0.2 \times 32}{(0.65\times 28) + (0.2\times 32) + (0.15\times44)}\) Mass fraction of CO2: \(\frac{0.15 \times 44}{(0.65\times 28) + (0.2\times 32) + (0.15\times44)}\)

Calculate the partial pressures

The partial pressure of each gas can be determined by multiplying the total pressure of the mixture by its respective molar fraction: Partial pressure of N2: \(250\ \mathrm{kPa} \times 0.65\) Partial pressure of O2: \(250\ \mathrm{kPa} \times 0.20\) Partial pressure of CO2: \(250\ \mathrm{kPa} \times 0.15\) Now we can calculate the mass fractions and partial pressures for each gas. Mass fraction of N2: \(\frac{0.65 \times 28}{(0.65\times 28) + (0.2\times 32) + (0.15\times44)} = 0.60\) Mass fraction of O2: \(\frac{0.2 \times 32}{(0.65\times 28) + (0.2\times 32) + (0.15\times44)} = 0.25\) Mass fraction of CO2: \(\frac{0.15 \times 44}{(0.65\times 28) + (0.2\times 32) + (0.15\times44)} = 0.15\) Partial pressure of N2: \(250\ \mathrm{kPa} \times 0.65 = 162.50\ \mathrm{kPa}\) Partial pressure of O2: \(250\ \mathrm{kPa} \times 0.20 = 50\ \mathrm{kPa}\) Partial pressure of CO2: \(250\ \mathrm{kPa} \times 0.15 = 37.50\ \mathrm{kPa}\) Therefore, the mass fractions of N2, O2, and CO2 are 0.60, 0.25, and 0.15, respectively, and their partial pressures are 162.50 kPa, 50 kPa, and 37.50 kPa, respectively.

Key concepts

Gas Mixture Composition

Understanding the composition of a gas mixture is key to solving a variety of chemical and physical problems. It involves determining the percentage of each gas present in a mixture. The molar analysis, often provided as percentages, indicates how much of each component there is relative to the total amount.

For instance, when a molar analysis says a mixture contains 65% \textbf{N}\(_2\), 20% \textbf{O}\(_2\), and 15% \textbf{CO}\(_2\), we interpret these percentages as molar fractions. This means that out of every 100 parts by mole of the gas mixture, 65 parts are nitrogen, 20 are oxygen, and 15 are carbon dioxide. Having this understanding of molar fractions sets the stage for calculating important derived properties such as mass fractions and partial pressures, which can then be applied in engineering calculations, environmental monitoring, and various scientific research.

Molar Mass

When dealing with gas mixtures, the molar mass is a vital concept. The molar mass tells us the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is crucial for converting between moles and grams because it serves as a conversion factor.

Molar Mass in Calculations

As an example, nitrogen (\textbf{N}\(_2\)) has a molar mass of 28 g/mol, oxygen (\textbf{O}\(_2\)) has a molar mass of 32 g/mol, and carbon dioxide (\textbf{CO}\(_2\)) has a molar mass of 44 g/mol. These values are used to translate the molar fractions from our gas mixture composition into mass fractions. Knowing the molar masses allows us to how much actual mass each component contributes relative to the entire mixture, which is especially useful for chemical reactions and physical processes within various fields such as respiratory sciences, meteorology, and materials engineering.

Mass Fraction

Another fundamental concept in understanding gas mixtures is the mass fraction, which describes the proportion of a component's mass in relation to the total mass of the mixture. Unlike the molar fraction, which is based on the number of moles, the mass fraction accounts for the differing molar masses of gases in the mixture.

To calculate mass fractions, we multiply the molar fraction by the molar mass of each gas and then divide by the total mass of all gases, weighted by their molar fractions and molar masses. To simplify, the mass fraction is essentially the part of the total mass contributed by that specific gas. It's an important measure when considering the physical properties of the gas mixture, such as its density or how it will behave when subjected to different temperatures and pressures in thermodynamic processes.