The Dieterici equation for a pure substance is given by

\[p=\frac{\Re T}{v-b} \mathrm{e}^{-\frac{a}{\Re T v}}\]

Determine

(a) the constants \(a\) and \(b\) in terms of the critical pressure and temperature;

(b) the compressibility factor at the critical condition;

(c) the law of corresponding states.

\[\left[a=\frac{4 \Re^{2} T_{\mathrm{c}}^{2}}{p_{\mathrm{c}} \mathrm{e}^{2}} ; \quad b=\frac{\Re T_{\mathrm{c}}}{p_{\mathrm{c}} \mathrm{e}^{2}} ; \quad z_{\mathrm{c}}=0.2707 ; \quad p_{\mathrm{R}}=\frac{T_{\mathrm{R}}}{\left(2 v_{\mathrm{R}}-1\right)} \exp \left(2\left(1-\frac{1}{v_{\mathrm{R}} T_{\mathrm{R}}}\right)\right)\right]\]