The van der Waals equation for carbon dioxide, a = 3.640 L² bar mol⁻² and b = 0.04267 L mol⁻¹…

Step 1. The idea. The equation of van der Waals gives a loop below $T_c$, which is not real. The true isotherm replaces the loop with a horizontal line at the vapour pressure. The line is placed so that the area of the loop above the line equals the area below it.

Step 2. The search. For each pressure between $41.5$ and $57.2\ \text{bar}$, solve the cubic $PV^3 - (Pb + RT)V^2 + aV - ab = 0$. Take the lowest and the highest root. Add up the area between the curve and the line. The area is zero at $P = 52.8\ \text{bar}$.

Step 3. The volumes. At $52.8\ \text{bar}$ the lowest root is $0.0809\ \text{L/mol}$, the liquid, and the highest is $0.2655\ \text{L/mol}$, the gas. The middle root is not stable.