Van der Waals equation: critical constants from a and b, Zc = 3/8, and a test with CO₂

Starting from the van der Waals equation, derive the critical constants $V_c$, $T_c$ and $P_c$ in terms of $a$ and $b$, and show that the critical compressibility factor $Z_c = P_cV_c/RT_c = 3/8$ for every van der Waals gas. For CO₂, $a = 3.640$ L² bar mol⁻² and $b = 0.04267$ L mol⁻¹: calculate $T_c$, $P_c$ and $V_c$ and compare them with the experimental values 304.1 K, 73.8 bar and 94 cm³ mol⁻¹. Comment on the agreement.
($R = 0.08314$ L bar K⁻¹ mol⁻¹)