Van der Waals Gas — Real Gas Isotherms and the Critical Point

Why the ideal gas law has to break down

The ideal gas law, PV = nRT, assumes molecules are point particles that never interact except in instantaneous elastic collisions. Neither assumption survives contact with a real gas: molecules have a finite size, so they cannot be squeezed into zero volume, and they attract each other weakly at short range (van der Waals forces), which pulls the pressure down below the ideal prediction. In 1873 Johannes van der Waals patched both effects into the equation of state with two extra constants, work that won him the 1910 Nobel Prize.

(P + a·n²/V²)(V − n·b) = nRT

a = attraction parameter (bigger a → stronger intermolecular attraction, pulls P down)
b = excluded-volume parameter (finite molecular size, the volume molecules cannot share)

The a·n²/V² term is added back to the measured pressure because attraction between molecules pulls them slightly together, quietly lowering the pressure a naive point-particle count would predict. The −nb term subtracts the volume the molecules themselves occupy, since only V − nb of the container is actually free space for them to move around in.

live demo · a P-V isotherm with the Maxwell construction applied● LIVE

Isotherms and the S-shaped anomaly

Plot pressure against volume at fixed temperature (an isotherm) using the van der Waals equation and, below a certain temperature, the curve stops being monotonic — it develops a wiggle where pressure appears to increase as volume increases, a physically impossible region for a stable, uniform phase. That unphysical loop is the theory's honest signal that the assumption of a single homogeneous phase has broken down: the real substance is separating into liquid and vapour coexisting side by side.

The Maxwell construction: replacing the loop with reality

James Clerk Maxwell showed how to fix the isotherm without abandoning the underlying equation. Along the unphysical wiggle, replace the S-shaped curve with a horizontal flat segment at whatever pressure makes the area enclosed above the line equal to the area enclosed below it — the equal-area rule. That flat segment is the real, experimentally observed liquid-vapour coexistence plateau: at that pressure and temperature, the substance can exist as any mixture of liquid and vapour in equilibrium, from all-liquid at one end of the plateau to all-vapour at the other, with the pressure staying constant throughout, exactly as the Clausius-Clapeyron picture of a first-order phase transition demands.

∫ (P_liquid_end → P_vapour_end) [P_vdW(V) − P_plateau] dV = 0

the flat plateau replaces the unphysical S-loop; its constant pressure
is the real vapour pressure of the liquid at that temperature

The critical point: where the two phases become identical

As temperature rises, the coexistence plateau shrinks — liquid and vapour become progressively more similar in density — until at the critical temperature T_c the plateau shrinks to a single point, called the critical point. Above T_c there is no separate liquid and vapour phase at all, only a single supercritical fluid; the distinction has literally ceased to exist. Because the van der Waals equation is a cubic in V, the critical point is exactly where the isotherm has both a zero first derivative and a zero second derivative in V — an inflection point with a horizontal tangent — which lets you solve for T_c, P_c and V_c directly from a and b:

T_c = 8a / (27Rb)         P_c = a / (27b²)         V_c = 3nb

these three, combined into P_c·V_c/(nR·T_c) = 3/8, are a universal
prediction of the van der Waals model, independent of a and b

That last ratio, 3/8, is one of the model's most striking predictions: every van der Waals gas, whatever its a and b, should have the same critical compressibility factor. Real gases cluster fairly close to it (typically 0.23-0.31) but not exactly, which is itself useful information — it tells you precisely how much real intermolecular physics the simple two-parameter model is still missing.

What the model gets right, and what it doesn't

The value of the van der Waals equation is not quantitative precision — more sophisticated equations of state fit real gases far better — it is that two extra terms, each with a clear physical meaning, are enough to qualitatively reproduce liquefaction, a critical point, and a first-order phase transition, phenomena the ideal gas law cannot describe even in principle. That is why it remains the standard first stop for teaching real-gas behaviour and phase transitions in general.

Frequently asked questions

Why does the van der Waals isotherm develop an S-shaped wiggle at low temperature?

The equation is a cubic polynomial in volume, and below the critical temperature that cubic has three real roots at a given pressure, producing a region where the curve appears to show pressure rising with volume. That region is unphysical for a uniform single phase — it signals that the substance is actually splitting into coexisting liquid and vapour.

What does the Maxwell construction actually do?

It replaces the unphysical S-shaped loop with a horizontal line at the pressure where the areas enclosed above and below the line are equal. That flat line is the real, experimentally measured constant-pressure plateau where liquid and vapour coexist at a given temperature.

What happens to the liquid-vapour distinction above the critical temperature?

It disappears entirely. Above the critical temperature there is no pressure at which the substance separates into two distinct phases — it exists only as a single supercritical fluid, and the coexistence plateau on the isotherm shrinks to zero width exactly at the critical point.

Try it live

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The Van der Waals Gas: Two Extra Terms That Predict Liquefaction