Problem 104 - Chemistry The Science in Context

Question

A gas mixture contains \(7.0 \mathrm{g}\) of \(\mathrm{N}_{2}, 2.0 \mathrm{g}\) of \(\mathrm{H}_{2},\) and \(16.0 \mathrm{g}\) of \(\mathrm{CH}_{4}\). What is the mole fraction of \(\mathrm{H}_{2}\) in the mixture? Calculate the pressure of the gas mixture and the partial pressure of each constituent gas if the mixture is in a \(1.00 \mathrm{L}\) vessel at \(0^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

Answer: The mole fraction of H₂ in the mixture is 0.442, the total pressure of the gas mixture is 50.13 atm, and the partial pressures of N₂, H₂, and CH₄ are 5.61 atm, 22.15 atm, and 22.37 atm, respectively.

Step 1: Calculate moles of each gas

To find the amount of each gas in moles, we need to divide the given mass of each gas by its respective molecular weight. For nitrogen, hydrogen, and methane we have: Molecular Weight of N₂ = 28.02 g/mol Molecular Weight of H₂ = 2.02 g/mol Molecular Weight of CH₄ = 16.04 g/mol Moles of N₂ = (7.0 g) / (28.02 g/mol) = 0.250 moles Moles of H₂ = (2.0 g) / (2.02 g/mol) = 0.990 moles Moles of CH₄ = (16.0 g) / (16.04 g/mol) = 0.998 moles

Step 2: Calculate mole fractions of each gas

Now, we need to find the mole fraction of each gas by dividing the moles of each gas by the total moles in the mixture: Total moles = Moles of N₂ + Moles of H₂ + Moles of CH₄ = 0.250 + 0.990 + 0.998 = 2.238 moles Mole fraction of N₂ = (0.250 moles) / (2.238 moles) = 0.112 Mole fraction of H₂ = (0.990 moles) / (2.238 moles) = 0.442 Mole fraction of CH₄ = (0.998 moles) / (2.238 moles) = 0.446

Step 3: Calculate total pressure of the mixture

To find the total pressure of the gas mixture, we will use the Ideal Gas Law: PV = nRT Where: P = pressure (atm, to be calculated) V = volume of the vessel = 1.00 L n = total moles of the mixture = 2.238 moles R = gas constant = 0.0821 L*atm/mol*K T = temperature in Kelvin = 0°C + 273.15 K = 273.15 K Rearranging the Ideal Gas Law to solve for P: P = nRT / V P = (2.238 moles * 0.0821 L*atm/mol*K * 273.15 K) / (1.00 L) = 50.13 atm

Step 4: Calculate partial pressures of each gas

Using the mole fractions and the total pressure, we can now compute the partial pressure of each gas: Partial Pressure of N₂ = Mole fraction of N₂ * Total pressure = 0.112 * 50.13 atm = 5.61 atm Partial Pressure of H₂ = Mole fraction of H₂ * Total pressure = 0.442 * 50.13 atm = 22.15 atm Partial Pressure of CH₄ = Mole fraction of CH₄ * Total pressure = 0.446 * 50.13 atm = 22.37 atm So, the mole fraction of H₂ in the mixture is 0.442, the total pressure of the gas mixture is 50.13 atm, and the partial pressures of N₂, H₂, and CH₄ are 5.61 atm, 22.15 atm, and 22.37 atm, respectively.

Key Concepts

Partial Pressure

Understanding partial pressure is crucial for students working with gas mixtures, as it helps explain how each gas within a blend contributes to the overall pressure. When we talk about partial pressure, we are referring to the pressure that a single gas within a mixture would exert if it were alone occupying the entire volume. It can be thought of as that gas's share of the total pressure.

To calculate partial pressure, one needs to know the mole fraction of the gas in the mixture and the total pressure of the mixture. The mole fraction is simply the ratio of the moles of one particular gas to the total moles of all gases present. Multiplying the mole fraction by the total pressure yields the partial pressure of each constituent gas. This is beautifully illustrated in the step-by-step solution using the gas mixture of nitrogen, hydrogen, and methane. The individual pressures calculated are essential for understanding the behavior of each gas under the specified conditions.

Ideal Gas Law

The Ideal Gas Law is a foundational piece in chemistry that describes the behavior of an ideal gas under various conditions. Expressed as PV = nRT, the law relates pressure (P), volume (V), total number of moles (n), temperature (T), and the universal gas constant (R). To find the total pressure of a gas mixture, as shown in the problem, one rearranges the Ideal Gas Law formula to solve for P, given the constants and conditions of the system.

In a classroom setting or when providing educational content online, it's important to ensure that the real-world relevance of such equations is clear. For instance, explaining that the Ideal Gas Law has applications ranging from predicting the behavior of gases in our atmosphere to the engineering of efficient combustion engines creates context that resonates with students.

Molar Mass

The concept of molar mass is integral when it comes to understanding molecular compositions and reactions. It represents the mass of one mole of a substance, usually expressed in grams per mole (g/mol). To determine the molar mass, one sums the atomic masses of all atoms in a molecule. As seen in the solution, calculating moles of each gas requires the molar mass of each gas, allowing us to convert grams to moles.

The molar mass is also connected to the mole fraction calculations. When students understand the connection between the mass of a substance and the number of particles contained in a specific sample, they better grasp stoichiometry and the quantitative analysis of substances in chemical reactions. It's the bridge between the microscopic world of molecules and the macroscopic world we can measure and observe.