Question

A mixture of gases contains \(0.75 \mathrm{~mol} \mathrm{~N}_{2}, 0.30 \mathrm{~mol} \mathrm{O}_{2}\) and \(0.15 \mathrm{~mol} \mathrm{CO}_{2}\). If the total pressure of the mixture is \(1.56 \mathrm{~atm}, \mathrm{what}\) is the partial pressure of each component?

Short answer

The partial pressures of each gas component in the mixture are approximately: N₂: \(0.975 \mathrm{~atm}\), O₂: \(0.390 \mathrm{~atm}\), and CO₂: \(0.195 \mathrm{~atm}\).

Step-by-step solution

Calculate the sum of moles of all components

Add the moles of each gas component: Nitrogen, Oxygen, and Carbon Dioxide, to get the total moles in the mixture. Total moles = moles of N₂ + moles of O₂ + moles of CO₂ Total moles = 0.75 + 0.30 + 0.15

Calculate mole fraction of each component

The mole fraction 'x_i' of a particular gas component 'i' in the mixture is calculated by dividing the moles of the component i by the total moles in the mixture. Mole fraction of N₂ = Moles of N₂ / Total moles Mole fraction of O₂ = Moles of O₂ / Total moles Mole fraction of CO₂ = Moles of CO₂ / Total moles

Calculate partial pressure of each component

The partial pressure of each gas in the mixture can be calculated by multiplying the mole fraction of each component by the total pressure of the mixture. This can be calculated using the following formula: Partial pressure of a component i (P_i) = Mole fraction of i (x_i) * Total pressure (P_total) Partial pressure of N₂ = Mole fraction of N₂ * Total pressure Partial pressure of O₂ = Mole fraction of O₂ * Total pressure Partial pressure of CO₂ = Mole fraction of CO₂ * Total pressure Now we have all the information we need to find the partial pressures of each gas component in the mixture.

Calculate the partial pressures and provide the final answer

Total moles = 0.75 + 0.30 + 0.15 = 1.20 Mole fraction of N₂ = Moles of N₂ / Total moles = 0.75/1.20 Mole fraction of O₂ = Moles of O₂ / Total moles = 0.30/1.20 Mole fraction of CO₂ = Moles of CO₂ / Total moles = 0.15/1.20 Partial pressure of N₂ = Mole fraction of N₂ * Total pressure = (0.75/1.20) * 1.56 atm Partial pressure of O₂ = Mole fraction of O₂ * Total pressure = (0.30/1.20) * 1.56 atm Partial pressure of CO₂ = Mole fraction of CO₂ * Total pressure = (0.15/1.20) * 1.56 atm Partial pressure of N₂ ≈ 0.975 atm Partial pressure of O₂ ≈ 0.390 atm Partial pressure of CO₂ ≈ 0.195 atm Therefore, the partial pressures of each gas component in the mixture are approximately: N₂: 0.975 atm, O₂: 0.390 atm, and CO₂: 0.195 atm.

Key concepts

Mole Fraction

The mole fraction is an essential concept when dealing with gas mixtures. It introduces a way to express the concentration of a particular component within a mixture of gases. To calculate the mole fraction, you take the number of moles of the gas of interest and divide it by the total number of moles of all gases in the mixture.

Here is the formula used:

• Mole fraction of gas i, \( x_i = \frac{n_i}{n_{total}} \)

where \( n_i \) is the number of moles of gas i and \( n_{total} \) is the sum of moles of all the gases in the mixture.

The mole fraction has a few interesting properties:

• Mole fractions are always less than one, as they represent a part of the whole.
• They are dimensionless, meaning they have no units.
• The sum of all mole fractions in the mixture equals one.

Using the mole fraction allows you to determine the proportionate effect that each gas has on the overall behavior of the gas mixture.

Gas Mixtures

Gas mixtures consist of more than one type of gas, each contributing to the properties of the mixture as a whole. Understanding gas mixtures requires considering how the different gases interact and share space. A fundamental property of gas mixtures is that all components distribute evenly among one another, given sufficient time and space for the molecules to mix thoroughly. However, each gas retains its own chemical identity while mixed.

In a mixture, each gas exerts a pressure as if it were the only gas present. This pressure is called partial pressure. The sum of the partial pressures of all gases in the mixture will give you the total pressure of the gas mixture. Therefore, knowing the individual contributions of each gas is crucial when analyzing or predicting the behavior of gas mixtures.

Other than pressure, mole fraction is widely used in describing gas mixtures. It provides information about the relative amount of each gas present in the mixture.

Ideal Gas Law

The Ideal Gas Law is a cornerstone concept in chemistry and physics, describing the relationship between the four critical properties of gases: pressure, volume, temperature, and moles. It is usually written as:\[ PV = nRT \]Where:

• \( P \) stands for pressure
• \( V \) is volume
• \( n \) denotes the number of moles
• \( R \) is the ideal gas constant
• \( T \) is temperature in Kelvin

This formula allows you to calculate any one of these variables if the other three are known. While it assumes gases behave ideally, meaning the gas particles do not interact and occupy no volume, this model works well for many gases under a wide range of conditions.

The Ideal Gas Law can be particularly handy when dealing with gas mixtures. Given that partial pressure is linked to the number of moles and the conditions in the container, you can use the law to explore how changes in one variable can affect the others. It's a very powerful tool for predicting the behavior of gases and their interactions in mixtures.