Answered

Four gases were combined in a gas cylinder with these partial pressures: 3.5 atm $N_2$, 2.8 atm $O_2$, 0.2 atm Ar, and 0.15 atm He.



What is the total pressure inside the cylinder?

$P_T = P_1 + P_2 + P_3 + \ldots + P_n$



What is the mole fraction of $N_2$ in the mixture?

$\frac{P_a}{P_T} = \frac{n_a}{n_T}$

Verified Answer :

  • The total pressure \(P_T\) is calculated by summing the partial pressures: \(3.5 + 2.8 + 0.2 + 0.15 = \boxed{6.65 \text{ atm}}\).
  • The mole fraction of \(N_2\) is determined by dividing its partial pressure by the total pressure: \(\frac{3.5}{6.65} \approx \boxed{0.5263}\).

Explanation

  1. Analysis of the Problem and Data

We are given the partial pressures of four gases in a cylinder: Nitrogen (\(P_{N_2} = 3.5 \text{ atm}\)), Oxygen (\(P_{O_2} = 2.8 \text{ atm}\)), Argon (\(P_{Ar} = 0.2 \text{ atm}\)), and Helium (\(P_{He} = 0.15 \text{ atm}\)). According to Dalton's Law of Partial Pressures, the total pressure \(P_T\) in a container of non-reacting gases is the sum of the partial pressures of the individual gases. We are also asked to calculate the mole fraction of nitrogen, defined as the ratio of its partial pressure to the total pressure of the mixture: \(\chi_{N_2} = \frac{P_{N_2}}{P_T}\).

  1. Calculating Total Pressure

To find the total pressure \(P_T\), we sum the partial pressures of all the gases present in the cylinder:
\[P_T = P_{N_2} + P_{O_2} + P_{Ar} + P_{He}\]
\[P_T = 3.5 \text{ atm} + 2.8 \text{ atm} + 0.2 \text{ atm} + 0.15 \text{ atm} = 6.65 \text{ atm}\]
Thus, the total pressure inside the cylinder is \(6.65 \text{ atm}\).

  1. Calculating the Mole Fraction of Nitrogen

The mole fraction of nitrogen (\(\chi_{N_2}\)) is calculated by dividing its partial pressure by the total pressure:
\[\chi_{N_2} = \frac{P_{N_2}}{P_T}\]
\[\chi_{N_2} = \frac{3.5 \text{ atm}}{6.65 \text{ atm}} \approx 0.5263\]
The mole fraction of nitrogen in the mixture is approximately \(0.5263\).

Examples

In chemical engineering, calculating the composition of gas mixtures is essential for reactor design and safety monitoring. For example, when creating a controlled atmosphere for a laboratory experiment or a industrial storage tank, researchers need to know the exact partial pressures to ensure the safety and precision of chemical reactions. Understanding Dalton's law and mole fractions allows engineers to predict how a gas mixture will behave under different temperature and volume conditions, ensuring optimal performance in processes like gas chromatography or welding.