$word_{count}$: 200

Critical constants from van der Waals: $T_c$ = 8$\frac{a}{27Rb}$, $P_c$ = $\frac{a}{27b^2}$, $V_c$ = 3b. $Z_c$ = $\frac{P_cV_c}{RT_c}$ = 3/8. Above $T_c$: gas cannot be liquefied regardless of pressure (supercritical fluid). Below $T_c$: isotherms show liquid-gas phase transition (flat region where P is constant during phase change). Boyle temperature: $T_B$ = $\frac{a}{Rb}$ = 27$T_c$/8. Inversion temperature: $T_i$ = 2$\frac{a}{Rb}$ = 2$T_B$. Joule-Thomson effect: expansion below $T_i$ causes cooling. Above $T_i$: expansion causes heating. Liquefaction methods: Linde (Joule-Thomson cooling with countercurrent heat exchange) and Claude (adiabatic expansion against piston). For H2 ($T_i$ = 202 K) and He ($T_i$ = 40 K): must pre-cool below inversion temperature first. From critical constants, a and b can be calculated: a = 27$R^{2T}_c$^$\frac{2}{64P_c}$, b = $\frac{RT_c}{8P_c}$.