An ideal-gas mixture contains 35 % helium, 20 % methane, and...

An ideal-gas mixture contains $35 \%$ helium, $20 \%$ methane, and $45 \%$ nitrogen by volume at 2.00 atm absolute and $90^{\circ} \mathrm{C}$. Calculate (a) the partial pressure of each component, (b) the mass fraction of methane, (c) the average molecular weight of the gas, and (d) the density of the gas in $\mathrm{kg} / \mathrm{m}^{3}$.

Key Concepts

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of a gas. This principle is fundamental in the study of gases, providing a way to interconnect thermodynamic properties and calculate variables such as pressure, volume, or temperature when the others are known.

Dalton's Law of Partial Pressures

Dalton's law states that in a mixture of non-reacting gases, each gas exerts a pressure proportional to its mole fraction as if it were alone in the container. This means the total pressure of the gas mixture is the sum of the partial pressures of the individual gases, which in ideal conditions can be calculated by multiplying the total pressure by the corresponding mole fraction of each gas.

Mole Fractions and Volume Percentages

For ideal gases, the volume percentage of a gas is equivalent to its mole fraction because the volume occupied by a gas is directly proportional to the number of moles at a given temperature and pressure. This property allows one to convert volume percentages directly into mole fractions when analyzing gas mixtures.

Mass Fraction Calculation

Mass fraction is defined as the mass of a component divided by the total mass of the mixture. In gas mixtures, converting mole fractions into mass fractions requires the molecular weight of each gas. By using the mole fraction and the molecular weight, one can derive the mass contribution of each component relative to the overall mass of the gas mixture.

Average Molecular Weight of a Gas Mixture

The average molecular weight of a gas mixture is computed by taking the sum of the products of the mole fractions and the molecular weights of the individual components. This weighted average is essential for converting between molar and mass-based properties in gas calculations, such as when determining the density.

Density Calculation of a Gas

The density of a gas can be calculated using the ideal gas law rearranged in terms of mass and molecular weight. Specifically, density is given by ? = (P*M) / (RT), where P is the pressure, M is the average molecular weight of the gas, R is the gas constant, and T is the temperature in Kelvin. This formula links the macroscopic properties of the gas with its molecular characteristics.

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