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Chapter 14: Problem 17

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

Expert verified
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: The mass fractions are as follows: - Mass fraction of O₂ = 0.28 - Mass fraction of N₂ = 0.32 - Mass fraction of CO₂ = 0.4 The mole fractions are as follows: - Mole fraction of O₂ = 0.2988 - Mole fraction of N₂ = 0.3903 - Mole fraction of CO₂ = 0.3104 The average molar mass of the mixture = 32.298 g/mol.

Step by step solution

01

Identify the molar mass of each gas component

Using periodic table we gather the molar mass of each gas component: (O) Oxygen: 16 g/mol (N) Nitrogen: 14 g/mol (C) Carbon: 12 g/mol For the given gases, the molar mass are: O₂ = 2 × 16 = 32 g/mol N₂ = 2 × 14 = 28 g/mol CO₂ = 12 + (2 × 16) = 44 g/mol
02

Calculate the total mass of the mixture

Sum the mass values of the individual components. Total mass = mass of O₂ + mass of N₂ + mass of CO₂ Total mass = 7 lbm + 8 lbm + 10 lbm = 25 lbm
03

Calculate the mass fractions

(a) Determine the mass fraction of each component by dividing the mass of each component by the total mass of the mixture Mass fraction of O₂ = mass of O₂ / total mass Mass fraction of O₂ = 7 lbm / 25 lbm = 0.28 Mass fraction of N₂ = mass of N₂ / total mass Mass fraction of N₂ = 8 lbm / 25 lbm = 0.32 Mass fraction of CO₂ = mass of CO₂ / total mass Mass fraction of CO₂ = 10 lbm / 25 lbm = 0.4
04

Calculate the moles of each component

(b) Determine the mole fraction of each component by dividing the mass of each component by its respective molar mass Moles of O₂ = mass of O₂ / molar mass of O₂ Moles of O₂ = 7 lbm / 32 g/mol = 0.21875 lbm-mol Moles of N₂ = mass of N₂ / molar mass of N₂ Moles of N₂ = 8 lbm / 28 g/mol = 0.28571 lbm-mol Moles of CO₂ = mass of CO₂ / molar mass of CO₂ Moles of CO₂ = 10 lbm / 44 g/mol = 0.22727 lbm-mol
05

Calculate the total moles of the mixture

Sum the moles of each individual gas component. Total moles = moles of O₂ + moles of N₂ + moles of CO₂ Total moles = 0.21875 lbm-mol + 0.28571 lbm-mol + 0.22727 lbm-mol = 0.73173 lbm-mol
06

Calculate the mole fractions

Determine the mole fraction of each component by dividing the moles of each component by the total moles of the mixture Mole fraction of O₂ = moles of O₂ / total moles Mole fraction of O₂ = 0.21875 lbm-mol / 0.73173 lbm-mol = 0.2988 Mole fraction of N₂ = moles of N₂ / total moles Mole fraction of N₂ = 0.28571 lbm-mol / 0.73173 lbm-mol = 0.3903 Mole fraction of CO₂ = moles of CO₂ / total moles Mole fraction of CO₂ = 0.22727 lbm-mol / 0.73173 lbm-mol = 0.3104
07

Calculate the average molar mass of the mixture

(c) Determine the average molar mass of the mixture by taking the sum of the product of the mole fraction and molar mass of each component Average molar mass = Σ(mole fraction × molar mass) Average molar mass = (0.2988 × 32) + (0.3903 × 28) + (0.3104 × 44) Average molar mass = 32.298 g/mol

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