vdW equation and Leiden form

1.11.2 vdW equation and Leiden form

According to Eq. (1.10) the Leiden form of the virial coefficients is

\begin{equation*} p = \frac{RT}{V_m} \left(1 + \frac{B}{V_m} + \frac{C}{V_m^2} + \cdots \right) \label{eq:p_Leiden} \end{equation*}

(1.22)

According to Eq. (1.12) the pressure can be written as

\begin{equation*} p=\frac{RT}{V_m-b}-\frac{a}{V_m^2} \approx \frac{RT}{V_m} \left(1 + \frac{1}{V_m} \left( b - \frac{a}{RT}\right) +\frac{b^2}{V_m^2} \right) \label{eq:p_vdW} \end{equation*}

(1.23)

Here we have used the property of the geometrical series

\begin{equation*} \frac{1}{1-x}=1 + x + x^2 + \cdots \label{eq:geom_series} \end{equation*}

(1.24)

Comparing the coefficients of Eq. (1.22) and Eq. (1.23) we get

\begin{equation*} B = b - \frac{a}{RT} \quad \mbox{and} \quad C = b^2 \label{eq:vdW_vs_Leiden} \end{equation*}

(1.25)

Since \(B\) contains two terms with opposite sign we find the expected behavior:

At high \(T\) the repulsive forces are dominant, i.e. \(B \gt 0\).
At low \(T\) the attractive forces are dominant, i.e. \(B \lt 0\).
\(B\) increases with \(T\) (consistent with experimental data), i.e.

\begin{equation*} \frac{dB}{dT}= \frac{a}{R\,T^2} \gt 0 \label{eq:dBdT} \end{equation*} (1.26)
The Boyle temperature \(T_B\) is found for \(B=0\), i.e.

\begin{equation*} T_B=\frac{a}{b\,R} \label{eq:TB_vdW} \end{equation*} (1.27)

With frame

Maxwell construction and subcritical isotherms