Question

Partial pressures of \(\mathrm{CO}_{2}\) in a mixture of \(\mathrm{CO}_{2}\) and \(\mathrm{N}_{2}\) is \(1.25\) atm while the total pressure of the gas mixture is 5 atm. Mole fraction of nitrogen in the mixture is (1) \(0.82\) (2) \(0.75\) (3) \(0.80\) (4) \(0.65\)

Short answer

The mole fraction of nitrogen in the mixture is 0.75.

Step-by-step solution

- Understanding Partial and Total Pressure

Given the partial pressure of \(\text{CO}_2\) is 1.25 atm and the total pressure is 5 atm. Partial pressure refers to the pressure exerted by an individual gas within a mixture, while the total pressure is the sum of all partial pressures in the mixture.

- Calculate Partial Pressure of \(\text{N}_2\)

To find the partial pressure of \(\text{N}_2\), subtract the partial pressure of \(\text{CO}_2\) from the total pressure: \(P_{\text{N}_2} = P_{\text{total}} - P_{\text{CO}_2}\). Substituting the given values: \ \(P_{\text{N}_2} = 5 \text{ atm} - 1.25 \text{ atm} = 3.75 \text{ atm} \).

- Determine Mole Fraction of \(\text{N}_2\)

The mole fraction of a gas in a mixture can be calculated using the formula \ \( \chi_{\text{N}_2} = \frac{P_{\text{N}_2}}{P_{\text{total}}} \). Substituting in the values: \ \( \chi_{\text{N}_2} = \frac{3.75 \text{ atm}}{5 \text{ atm}} = 0.75 \).

Key concepts

Partial Pressure

Partial pressure is the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume on its own. This concept is crucial in understanding gas mixtures. For example, in the exercise given, the partial pressure of \(\text{CO}_2\) is given as 1.25 atm. This means that the \(\text{CO}_2\) alone would exert a pressure of 1.25 atm if no other gases were present. \(\text{N}_2\) also has its own partial pressure, which can be calculated by subtracting the partial pressure of \(\text{CO}_2\) from the total pressure. This basic understanding helps us to isolate the contributions of each gas to the total pressure in the mixture.

Total Pressure

Total pressure is the sum of all the partial pressures of the gases in a mixture. In the given exercise, the total pressure of the gas mixture is 5 atm. To find the partial pressure of any individual gas, you subtract the partial pressures of the other gases from the total pressure. In this case, we subtract the partial pressure of \(\text{CO}_2\), which is 1.25 atm, from the total pressure to find the partial pressure of \(\text{N}_2\):
\[ P_{\text{N}_2} = P_{\text{total}} - P_{\text{CO}_2} = 5 \text{ atm} - 1.25 \text{ atm} = 3.75 \text{ atm} \].
Total pressure provides a comprehensive picture of the force exerted by the gas mixture on the walls of its container.

Gas Mixtures

A gas mixture is simply a combination of different gases in the same container. Each gas in the mixture behaves independently of the other gases. The pressure exerted by each gas is known as its partial pressure. The total pressure in the container is the sum of all these partial pressures.
To calculate the mole fraction of a gas within a mixture, you need to use the individual partial pressures and the total pressure. For instance, given the partial pressure of \(\text{N}_2\) as 3.75 atm and the total pressure as 5 atm, you can determine the mole fraction of \(\text{N}_2\) using the formula:
\[ \chi_{\text{N}_2} = \frac{P_{\text{N}_2}}{P_{\text{total}}} = \frac{3.75}{5} = 0.75 \].
This tells us that 75% of the gas mixture is \(\text{N}_2\), providing a clear understanding of the composition of the gas mixture.