Question

A mixture containing \(0.50 \mathrm{~mol} \mathrm{H}_{2}(g), 1.00 \mathrm{~mol} \mathrm{O}_{2}(g)\), and 3.50 \(\mathrm{mol} \mathrm{N}_{2}(g)\) is confined in a 25.0-L vessel at \(25^{\circ} \mathrm{C}\). (a) Calculate the total pressure of the mixture. (b) Calculate the partial pressure of each of the gases in the mixture.

Short answer

(a) The total pressure of the gaseous mixture is 9.86 atm. (b) The partial pressures of each gas are: H₂: 0.986 atm, O₂: 1.972 atm, and N₂: 6.902 atm.

Step-by-step solution

Convert the temperature to Kelvin

First, we need to convert the given temperature from Celsius to Kelvin. To do this, we will add 273.15 to the given value: Temperature in Kelvin = 25°C + 273.15 = 298.15 K

Use Ideal Gas Law for the total pressure

We will use the Ideal Gas Law to calculate the total pressure of the mixture: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature in Kelvin. Since we have a mixture, we will need to find the total number of moles. Thus: Total Moles (n) = Moles of H₂ + Moles of O₂ + Moles of N₂ = 0.50 + 1.00 + 3.50 = 5.00 moles Now that we know the total moles, we can solve for the total pressure P: P = nRT / V

Calculate the total pressure

Substitute the values for n, R, T, and V into the equation to find the total pressure: P = (5.00 moles) x (0.0821 L atm/mol K) x (298.15 K) / (25.0 L) = 9.86 atm Thus, the total pressure of the mixture is 9.86 atm.

Calculate the mole fractions of each gas

To find the partial pressure of each gas, we need to find the mole fraction of each gas in the mixture. The mole fraction is defined as the number of moles of a specific component divided by the total number of moles in the mixture: Mole fraction for H₂ (x_H2) = Moles of H₂ / Total moles = 0.50 / 5.00 = 0.10 Mole fraction for O₂ (x_O2) = Moles of O₂ / Total moles = 1.00 / 5.00 = 0.20 Mole fraction for N₂ (x_N2) = Moles of N₂ / Total moles = 3.50 / 5.00 = 0.70

Calculate the partial pressures of each gas

Now we can find the partial pressure of each gas by multiplying the mole fraction of each gas by the total pressure: Partial Pressure of H₂ (P_H2) = x_H2 x P = 0.10 × 9.86 atm = 0.986 atm Partial Pressure of O₂ (P_O2) = x_O2 x P = 0.20 × 9.86 atm = 1.972 atm Partial Pressure of N₂ (P_N2) = x_N2 x P = 0.70 × 9.86 atm = 6.902 atm

Summary of the Results

In conclusion, (a) The total pressure of the gaseous mixture is: 9.86 atm (b) The partial pressures of each gas are: - H₂: 0.986 atm - O₂: 1.972 atm - N₂: 6.902 atm

Key concepts

partial pressure

In any gas mixture, each gas exerts pressure as if it occupied the entire volume of the container by itself. This specific pressure is known as the partial pressure of that gas. Using Dalton's Law of Partial Pressures, we find that the total pressure of the mixture is the sum of all the individual partial pressures.
The formula to calculate partial pressure is:

• Partial Pressure of a gas = Mole Fraction of the gas x Total Pressure

For example, in the given gas mixture, each gas contributes to the total pressure based on its mole fraction. The total pressure is 9.86 atm, as calculated using the Ideal Gas Law. From this, by multiplying the mole fraction of each gas (found earlier) with this total pressure, we can find the partial pressures.
By understanding and calculating partial pressures, we can determine how gases interact in a mixture and their individual contribution to the total pressure.

mole fraction

Mole fraction is a crucial concept in understanding gas mixtures. It represents a ratio that describes the number of moles of one component to the total number of moles in the mixture. This fraction indicates how much of the total composition of the gas mixture is made up of a particular gas.
The formula for mole fraction is:

• Mole Fraction = Number of Moles of the Component / Total Moles of the Mixture

In the problem, we saw how to calculate the mole fraction for each gas: hydrogen ( ext{H}_2), oxygen ( ext{O}_2), and nitrogen ( ext{N}_2). These were 0.10, 0.20, and 0.70, respectively.
The mole fraction simplifies the calculation of properties like partial pressure and helps understand the concentration of each gas in the mixture. It is important to note that the sum of all mole fractions in a mixture always equals 1.

gas mixture

A gas mixture consists of different gases occupying the same volume but behaves as a single entity. Understanding gas mixture behavior is facilitated through principles such as the Ideal Gas Law and Dalton's Law.
Each gas in the mixture obeys the ideal gas laws independently, and the total pressure is the cumulative effect of each gas's partial pressure. This is why we consider both total and individual gas quantities in a gas mixture.
Using the given example, the gas mixture in the problem contains hydrogen, oxygen, and nitrogen gases in a specific volume at a given temperature. These gases, although different, contribute uniformly to the total pressure of the system, which is 9.86 atm. Such calculations are pivotal for chemical processes involving gas mixtures and allow for precise control and prediction of reactions and behaviors.