Which equation correctly describes gas density ρ in terms of pressure, molar mass, and temperature?

Which equation correctly describes gas density ρ in terms of pressure, molar mass, and temperature?

• A. ρ = P / (R T)
• B. ρ = n R T / P
• C. ρ = P M / (R T)
• D. ρ = V P / T

Answer: (C)

This question hinges on connecting density to the ideal gas law and the molar mass. Start from density = mass per volume, ρ = m/V. Express the mass in terms of moles and molar mass: m = nM. From the ideal gas law, n = PV/(RT). Substituting gives ρ = (nM)/V = M(VP)/(RTV) = PM/(RT). So the density depends on pressure and temperature as well as the molar mass of the gas: heavier gases (larger M) are denser at the same P and T. This is the correct form because it correctly combines the relationships from PV = nRT with m = nM. Expressions that omit the molar mass miss the essential way density scales with the mass per mole of the gas. Using n in the numerator instead of mass per volume shifts the quantity away from density to something that doesn’t reflect mass per unit volume. And a form lacking the R in the denominator (or misplacing V) would not have the correct units or the correct dependence on P, T, and M.

This question hinges on connecting density to the ideal gas law and the molar mass. Start from density = mass per volume, ρ = m/V. Express the mass in terms of moles and molar mass: m = nM. From the ideal gas law, n = PV/(RT). Substituting gives ρ = (nM)/V = M(VP)/(RTV) = PM/(RT). So the density depends on pressure and temperature as well as the molar mass of the gas: heavier gases (larger M) are denser at the same P and T. This is the correct form because it correctly combines the relationships from PV = nRT with m = nM.

Expressions that omit the molar mass miss the essential way density scales with the mass per mole of the gas. Using n in the numerator instead of mass per volume shifts the quantity away from density to something that doesn’t reflect mass per unit volume. And a form lacking the R in the denominator (or misplacing V) would not have the correct units or the correct dependence on P, T, and M.