Derivation of critical constants from van der Waals constant

The critical constants $T_c$, $P_c$ and $V_c$ are not independent, freely-chosen numbers -- they can be derived directly from the van der Waals constants a and b of a gas, since the critical point is a special feature of the van der Waals equation itself.

Start from the van der Waals equation for one mole, $\left(P+\dfrac{a}{V^2}\right)(V-b)=RT$. Multiplying out and clearing the $V^2$ denominator turns this into a cubic equation in V:

A cubic in general has three roots for V at fixed P and T -- and indeed, below the critical temperature, the van der Waals equation genuinely predicts three physically meaningful volumes for a given pressure (visible as the wiggle a van der Waals isotherm develops in the two-phase region). But at the critical point itself, all three roots coincide at a single value, the critical volume $V_c$. That means the cubic above, evaluated at $T=T_c$ and $P=P_c$, must be identical to the algebraic expression $(V-V_c)^3=0$, which expands to

Since these two cubics describe the same equation, their coefficients must match term by term:

Dividing (iii) by (ii) cancels $a/P_c$ and leaves $b=\dfrac{V_c^3}{3V_c^2}=\dfrac{V_c}{3}$, i.e.

Substituting $b=V_c/3$ back into (ii): $\dfrac{a}{P_c}=3V_c^2=3(3b)^2=27b^2$, so

$P_c=\frac{a}{27b^2}$ …