Which equation correctly relates gas density ρ to pressure P, molar mass M, the gas constant R, and temperature T?

Which equation correctly relates gas density ρ to pressure P, molar mass M, the gas constant R, and temperature T?

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

Answer: (A)

Density of a gas comes from how much mass sits in a given volume. Start with the ideal gas law PV = nRT and relate mass to moles by m = nM, where M is the molar mass. Since density is ρ = m/V, substitute m = nM to get ρ = (nM)/V. From PV = nRT, n/V = P/(RT). Combine these to get ρ = M · (n/V) = M · P/(RT) = PM/(RT). This shows why heavier gases are denser at the same pressure and temperature, and why density increases with pressure but decreases with temperature. If you tried forms that put M in the denominator or RT in the numerator, you’d predict the opposite behavior—density would be lower for heavier gases or increase with temperature than with pressure, which contradicts the relationship derived from the ideal gas law.

Density of a gas comes from how much mass sits in a given volume. Start with the ideal gas law PV = nRT and relate mass to moles by m = nM, where M is the molar mass. Since density is ρ = m/V, substitute m = nM to get ρ = (nM)/V. From PV = nRT, n/V = P/(RT). Combine these to get ρ = M · (n/V) = M · P/(RT) = PM/(RT).

This shows why heavier gases are denser at the same pressure and temperature, and why density increases with pressure but decreases with temperature. If you tried forms that put M in the denominator or RT in the numerator, you’d predict the opposite behavior—density would be lower for heavier gases or increase with temperature than with pressure, which contradicts the relationship derived from the ideal gas law.