The Dieterici equation of state is p a RT y b b exp a a

Chapter 11, Problem 11.17

(choose chapter or problem)

The Dieterici equation of state is

\(p=\left(\frac{R T}{v-b}\right) \exp \left(\frac{-a}{R T v}\right)\)

(a) Using Eqs. 11.3, show that

\(a=\frac{4 R^{2} T_{\mathrm{c}}^{2}}{p_{\mathrm{c}} e^{2}}, \quad b=\frac{R T_{\mathrm{c}}}{p_{\mathrm{c}} e^{2}}\)

(b) Show that the equation of state can be expressed in terms of compressibility chart variables as

\(Z=\left(\frac{v_{\mathrm{R}}^{\prime}}{v_{\mathrm{R}}^{\prime}-1 / e^{2}}\right) \exp \left(\frac{-4}{T_{\mathrm{R}} v_{\mathrm{R}}^{\prime} e^{2}}\right)\)

(c) Convert the result of part (b) to a virial series in \(v_{\mathrm{R}}^{\prime}\). (Hint: Expand the \(\left(v_{\mathrm{R}}^{\prime}-1 / e^{2}\right)^{-1}\) term in a series. Also expand the exponential term in a series.)