TNK LAB

Van der Waals: The Maxwell Construction

The van der Waals equation of state predicts an S-shaped isotherm below the critical temperature — including a stretch where pressure would rise as volume increases, which is unphysical. Real substances skip that stretch entirely, transitioning from gas to liquid at constant pressure instead. The Maxwell construction finds that pressure: it's the one horizontal line that carves out equal areas above and below itself in the loop — exactly equivalent to requiring liquid and gas have the same Gibbs free energy.

Floored at 0.3, not 0 — below this, Vgas/Vc exceeds 10,000× and keeps exploding (millions, then more), so there's no useful scale left to plot it on even on this log axis.

Maxwell Construction

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Inspected Point

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Reduced van der Waals Equation

$$\left(p + \frac{3}{v^2}\right)(3v - 1) = 8t$$
where p = P/Pc, v = V/Vc, t = T/Tc — the same reduced form for every van der Waals substance (the law of corresponding states). Below t=1, solving for p(v) at fixed t gives the S-shaped loop on the right.

Gibbs Free Energy Along the Isotherm

Since dG = V dP at fixed T, integrating v dp along the isotherm (closed-form here, not numerical) gives g(v) up to an additive constant that doesn't affect where curves cross — that's the self-intersecting curve on the left. The crossing point is exactly the Maxwell pressure: it's the one point where gas and liquid share both P and G.
Why is the loop's middle section unstable?
Between the two spinodal points (3 and 5), ∂P/∂V > 0: squeezing the fluid would make its pressure drop, which would let it get squeezed further — a runaway collapse, not equilibrium. No real substance sits there (except fleetingly, as a supersaturated/superheated metastable state slightly past 2 or 6).
Points 2 and 6 are the only two states on this isotherm where liquid and gas genuinely coexist — note they land at exactly the same point in the left-hand G–P plot, since equal G at equal P is precisely what makes them coexist.

Pressure vs. Volume (Isotherm)

Stable branch
Unstable (∂P/∂V>0)
Maxwell tie line
Equal areas
Faint background = isotherm family & coexistence dome

Gibbs Free Energy vs. Pressure

Stable branch
Unstable branch
Coexistence (2,6 coincide)

Pressure vs. Temperature (Phase Diagram)

Coexistence curve Psat(T)
Current T
Open circle = critical point, where liquid and gas stop being distinct