Real Gases and Van der Waals Equation: Beyond Ideal Gas

Real gases deviate from ideal behaviour because molecules have finite size and attract each other. The van der Waals equation is the classic improved equation that includes these two corrections and can even describe condensation into liquid.

This matters because at high pressure or low temperature the ideal gas law PV = nRT can be badly wrong. Engineers compressing gases, liquefying air, or designing high-pressure vessels need a better model, and van der Waals gives physical insight with just two substance-specific constants.

Key Concepts

Excluded volume b accounts for molecular size: available volume is (V − nb), where n is moles and b is in m3/mol (about four times actual molecular volume per mole). Attraction constant a accounts for intermolecular attraction reducing wall pressure: pressure correction is a n2/V2, with a in Pa·m6/mol2 (or L2·bar/mol2). Constants a and b differ per gas (e.g. for N2 a≈0.137 Pa·m6/mol2, b≈3.87×10−5 m3/mol). Compressibility factor Z = PV/(nRT) equals 1 for ideal gas and measures deviation.

Formulas

Van der Waals equation: (P + a n2/V2)(V − n b) = n R T. For 1 mole: (P + a/Vm2)(Vm − b) = RT, Vm = molar volume.

At low density (large Vm) corrections vanish and ideal law returns. Critical constants from this equation: Vc = 3b, Pc = a/(27 b2), Tc = 8a/(27 R b) (per mole basis). Assumptions: molecules are hard spheres with weak long-range attraction; still approximate — it fails quantitatively near the critical point and for polar/associating gases, but captures liquefaction and non-ideality qualitatively.

Worked Example

Compare ideal and van der Waals pressure for 1 mole of N2 in V=0.50 L (5.0×10−4 m3) at 300 K. a=0.137, b=3.87×10−5 (SI).

Ideal: P = RT/V = 8.314×300 / 5.0×10−4 = 4.99×106 Pa ≈ 49.9 bar. Van der Waals: V−b = 5.0×10−4 − 3.87×10−5 = 4.613×10−4. RT/(V−b) = 2494.2 / 4.613×10−4 = 5.41×106. Attraction term a/V2 = 0.137 / (2.5×10−7) = 5.48×105. So P = 5.41×106 − 0.548×106 = 4.86×106 Pa ≈ 48.6 bar. The real pressure is lower, mainly because attractions pull molecules away from walls, outweighing the size effect here.

Common Misconceptions

• Misconception: Van der Waals equation is exact for real gases. Correction: It is an improved model, better than ideal but still approximate; more accurate equations (Redlich–Kwong, Peng–Robinson) are used industrially.
• Misconception: Corrections matter at all conditions. Correction: At low pressure and high temperature, Vm is large, corrections shrink and ideal law is fine.
• Misconception: Constant a means strong bonds inside molecules. Correction: a measures weak intermolecular attractions (van der Waals forces), not chemical bonds.

Real-World Applications

The equation guides liquefaction of gases (Linde process), CO2 storage and supercritical extraction, natural gas pipeline calculations, and refrigerant selection. Critical constants help design processes near phase change. It is also foundational in physical chemistry teaching and a stepping stone to modern cubic equations of state used in chemical plants.

Finite size plus attraction — two small fixes transform the ideal gas law into a model that can liquefy.