Van der Waals Equation: Real Gases & Phase Transitions

Where the ideal gas law breaks down

PV = nRT assumes molecules are point particles that exert no force on each other except during collisions. That works well at low pressure and high temperature, where molecules are far apart and moving fast. At higher pressure or lower temperature, two real effects intrude: molecular volume becomes a non-negligible fraction of the container, and intermolecular attraction pulls molecules together, reducing the pressure they exert on the walls. Johannes Diderik van der Waals won the 1910 Nobel Prize in Physics for a two-parameter fix that captures both — and, remarkably, predicts condensation itself, a phenomenon completely absent from the ideal model.

Two corrections, one equation

The pressure correction +a/V² accounts for the net inward pull molecules near the wall feel from their neighbours, which reduces the momentum actually delivered to the wall — so the ideal pressure must be corrected upward, by an amount that scales with the square of number density. The volume correction −b accounts for the finite hard-core size of real molecules: the space available for motion is the container volume minus the volume the molecules themselves occupy.

( P + a/V² ) · ( V − b ) = RT          (per mole)

a — strength of intermolecular attraction (Pa·m⁶/mol²)
b — excluded molecular volume (m³/mol), ≈ 4× the actual molecular volume

He: a=0.034, b=0.0237   CO₂: a=3.59, b=0.0427   H₂O: a=5.54, b=0.0305  (L²·atm/mol², L/mol)

The critical point: where liquid and gas merge

The critical point is the inflection point of the P-V isotherm, where both (∂P/∂V)_T and (∂²P/∂V²)_T vanish simultaneously. Solving those two conditions with the van der Waals equation gives closed-form critical constants — and predicts a universal compressibility factor Z_c = 3/8 = 0.375 at that point (real gases land closer to 0.23-0.29, one of the equation's known limitations). Above T_c, no pressure can force condensation; the substance becomes a supercritical fluid instead.

T_c = 8a / (27Rb)      P_c = a / (27b²)      V_c = 3b
Z_c = P_c·V_c / (R·T_c) = 3/8 = 0.375  (ideal vdW prediction)

Maxwell's fix for the unphysical loop

Below T_c, the raw van der Waals equation predicts an S-shaped "loop" where pressure would rise as volume increases — a mechanically unstable, unphysical result. James Clerk Maxwell resolved this in 1875 with the equal-area construction: draw a horizontal line through the loop such that the areas enclosed above and below are equal, and that line gives the true liquid-gas coexistence pressure. This is equivalent to requiring equal chemical potentials for the two phases, and it correctly separates the S-loop into a stable liquid branch, a stable gas branch, and a flat two-phase region between them where the lever rule sets the relative amounts of each.

Frequently asked questions

Try it live

Everything above runs in your browser — open Van der Waals Gas and drag temperature and volume to watch the P-V isotherm, the Maxwell equal-area construction, and the critical point respond live. Nothing is installed, nothing is uploaded.