Problem 2.197

Demonstrate that Eqs. \((2.6 \mathrm{a})\) and \((2.6 \mathrm{~b})\) are valid for a substance, obeying the Van der Waals equation, in critical state.

2.6(a) Relations between Van der Waals constants and the parameters of the critical state of a substance:

\[ V_{M \mathrm{c} r}=3 b, p_{\mathrm{cr}}=\frac{a}{27 b^{2}}, \quad T_{\mathrm{cr}}=\frac{8 a}{27 R b} \]

2.6(b) Relation between the critical parameters for a mole of substance:

\[ p_{\mathrm{cr}} V_{M \mathrm{cr}}=(3 / 8) R T_{\mathrm{c} r} \]

Instruction. Make use of the fact that the critical state corresponds to the point of inflection in the isothermal curve \(p(V)\).