Critical constants and vdW equation

1.11.1 Critical constants and vdW equation

As illustrated in Fig. 1.4 the critical point is defined by the existence of an inflection point, i.e.

\begin{equation*} \left(\frac{\partial p}{\partial V_m} \right)_{T_c} = 0 = - \frac{R\,T_c}{(V_{m,c}-b)^2}+\frac{2a}{V_{m,c}^3} \quad \Rightarrow \quad \frac{R\,T_c}{(V_{m,c}-b)^2} = \frac{2a}{V_{m,c}^3} \label{eq:vdWcrit1} \end{equation*}

(1.16)

\begin{equation*} \left(\frac{\partial^2 p}{\partial V_m^2} \right)_{T_c} = 0 = \;\;\, \frac{2\,R\,T_c}{(V_{m,c}-b)^3}-\frac{6a}{V_{m,c}^4} \quad \Rightarrow \quad \frac{2\,R\,T_c}{(V_{m,c}-b)^3} = \frac{6a}{V_{m,c}^4} \label{eq:vdWcrit2} \end{equation*}

(1.17)

Dividing Eq. (1.16) by Eq. (1.17) we get

\begin{equation*} \frac{\frac{R\,T_c}{(V_{m,c}-b)^2}}{\frac{2\,R\,T_c}{(V_{m,c}-b)^3}} = \frac{\frac{2a}{V_{m,c}^2}}{\frac{6a}{V_{m,c}^4}} \quad \Rightarrow \quad \frac{V_{m,c}-b}{2} = \frac{V_{m,c}}{3} \Rightarrow V_{m,c} = 3\, b \label{eq:vdWcrit3} \end{equation*}

(1.18)

Including Eq. (1.18) in Eq. (1.16) we get

\begin{equation*} \frac{R\,T_C}{(3b-b)^2}=\frac{2a}{27 b^3} \Rightarrow T_c=\frac{8a}{27 Rb} \label{eq:vdWcrit4} \end{equation*}

(1.19)

Including Eq. (1.19) and Eq. (1.18) in the vdW equation we get

\begin{equation*} p_c = \frac{R\,T_c}{V_{m,c}-b}-\frac{a}{V_{m,c}^2}=\frac{a}{27 b^2} \label{eq:vdWcrit5} \end{equation*}

(1.20)

From these results e.g. the compression factor at the critical point is found:

\begin{equation*} Z_c = \frac{p_c\, V_{m,c}}{R\,T_c}=\frac{3}{8} \label{eq:vdWZc} \end{equation*}

(1.21)

In general the experimental strategy for describing real (vdW) gases means

Calculate \(a\) and \(b\) from critical constants.
Compute any relevant point in phase diagram by the vdW approach.

With frame

Maxwell construction and subcritical isotherms