Question

A gas mixture consists of \(7 \mathrm{lbm}\) of \(\mathrm{O}_{2}, 8 \mathrm{lbm}\) of \(\mathrm{N}_{2}\), and \(10 \mathrm{lbm}\) of \(\mathrm{CO}_{2}\). Determine \((a)\) the mass fraction of each component, \((b)\) the mole fraction of each component, and \((c)\) the average molar mass of the mixture.

Short answer

Question: Calculate the mass fractions, mole fractions, and average molar mass of a gas mixture containing 7 lbm of O2, 8 lbm of N2, and 10 lbm of CO2. Answer: (a) Mass fractions: O2 = 0.28, N2 = 0.32, CO2 = 0.40 (b) Mole fractions: O2 = 0.30, N2 = 0.39, CO2 = 0.31 (c) Average molar mass of the mixture: 34.2 g/mol

Step-by-step solution

Calculate the total mass of the mixture

To calculate the total mass, add the masses of all components: Total mass = Mass of O2 + Mass of N2 + Mass of CO2 Total mass = 7 lbm + 8 lbm + 10 lbm Total mass = 25 lbm

Calculate the mass fraction of each component

Mass fraction = mass of component / total mass Mass fraction of O2 = (7 lbm) / (25 lbm) = 0.28 Mass fraction of N2 = (8 lbm) / (25 lbm) = 0.32 Mass fraction of CO2 = (10 lbm) / (25 lbm) = 0.40

Calculate the number of moles for each component

Number of moles = mass / molar mass Molar mass of O2 = 32 g/mol Molar mass of N2 = 28 g/mol Molar mass of CO2 = 44 g/mol (Note: Convert lbm to grams: 1 lbm = 453.59 g) Number of moles of O2 = (7 lbm * 453.59 g/lbm) / 32 g/mol = 99.66 mol Number of moles of N2 = (8 lbm * 453.59 g/lbm) / 28 g/mol = 129.71 mol Number of moles of CO2 = (10 lbm * 453.59 g/lbm) / 44 g/mol = 102.66 mol

Calculate the mole fraction of each component

Mole fraction = number of moles of component / total number of moles Total number of moles = 99.66 mol + 129.71 mol + 102.66 mol = 332.03 mol Mole fraction of O2 = 99.66 mol / 332.03 mol = 0.30 Mole fraction of N2 = 129.71 mol / 332.03 mol = 0.39 Mole fraction of CO2 = 102.66 mol / 332.03 mol = 0.31

Calculate the average molar mass of the mixture

Average molar mass = sum(molar mass * mole fraction) for each component Average molar mass = (32 g/mol * 0.30) + (28 g/mol * 0.39) + (44 g/mol * 0.31) = 9.6 g/mol + 10.92 g/mol + 13.64 g/mol = 34.2 g/mol Final results: (a) Mass fractions: O2 = 0.28, N2 = 0.32, CO2 = 0.40 (b) Mole fractions: O2 = 0.30, N2 = 0.39, CO2 = 0.31 (c) Average molar mass of the mixture: 34.2 g/mol