Equations of State & Properties of Real Systems Notes

Equations of State & Properties of Real Systems

The ideal-gas law and its limits, the compressibility factor, van der Waals, virial, Redlich–Kwong and Peng–Robinson equations, critical constants and reduced properties, the principle of corresponding states, gas mixtures (Dalton, Amagat), vapour pressure and phase equilibrium (Clausius–Clapeyron, Raoult's and Henry's laws) and fugacity — with worked examples.

Last reviewed 4 Nov 2026 · 3 min read

1. Ideal gas

FormulaIdeal-gas law

P V = n R T

R = 8.314 J/mol·K = 0.08206 L·atm/mol·K. At STP (0 °C, 1 atm), 1 mol occupies 22.414 L.

Assumes molecules have no volume and no intermolecular forces.
Good at low pressure and high temperature; fails near condensation and at high pressure.

2. Compressibility factor

FormulaCompressibility factor

Z = \frac{P V}{n R T}

Z = 1 for an ideal gas. Z < 1 when attractive forces dominate (moderate pressures); Z > 1 when molecular volume dominates (very high pressures).

3. Cubic equations of state

van der Waals (1873)

Formulavan der Waals equation

(P + \frac{a n ^{2}}{V ^{2}}) (V - nb) = n R T

a corrects for intermolecular attraction; b for the volume of molecules.
From critical constants: $a = \frac{27 R ^{2} T _{c}^{2}}{64 P _{c}}$ , $b = \frac{R T _{c}}{8 P _{c}}$ .
Predicted critical compressibility $Z_{c} = 3/8 = 0.375$ (real gases are about 0.23–0.31, so vdW is approximate).

Equation	Feature
Redlich–Kwong (1949)	Temperature-dependent attraction term ( $a / T$ ) — better for gases
Soave–Redlich–Kwong	Adds the acentric factor ω
Peng–Robinson (1976)	Better liquid densities; widely used for hydrocarbons and in process simulators

Virial equation

Z = 1 + \frac{B}{V} + \frac{C}{V ^{2}} + \dots or Z = 1 + B^{'} P + C^{'} P^{2} + \dots

B, C — second and third virial coefficients, functions of temperature only; rooted in statistical mechanics; truncated after B at moderate pressures.

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1. Ideal gas

2. Compressibility factor

3. Cubic equations of state

van der Waals (1873)

Virial equation

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