Other Equations of State

The ideal gas equation fits well the experimental data only at low density regimes (at conditions of low pressure and high temperature). To fit the \(PVT\) behavior of a component at any state, many equations were proposed. The salient features of them are discussed next.

(i) van der Waals Equation

  • The van der Waals equation is given as: \[\left(P + \frac{a}{V^2}\right) (V-b) = RT \tag*{(5)}\] where

    \(P\) = Pressure of the fluid
    \(V\) = Volume of the container for containing one mole of the fluid
    \(a\) = correction term for pressure to account for the intermolecular forces of attraction
    \(b\) = correction term for volume to account for the volume of molecules
    \(R\) = Universal gas constant
    \(T\) = Absolute temperature.

    The constants of van der Waals equations, can be written in terms of the critical parameters as: \[a = \frac{27}{64}\frac{R^2T_c^2}{P_c} \qquad b = \frac{RT_c}{8P_c}\] The above equations for \(a\) and \(b\), can be obtained from applying the mathematical criteria for the saddle point, i.e., Eqn.(3) for the critical isotherm.

  • van der Waals equation can be rearranged to the form: \[P = \frac{RT}{V-b} - \frac{a}{V^2}\] The two terms on the right hand side can be considered to be \[P = P_{\text{repulsive}} + P_{\text{attractive}}\] The repulsive contribution is \(RT/(V-b)\), and it is also called as ‘hard-sphere’ term as \(b\) is related to the size of molecules; and the attractive contribution is \(-a/V^2\). The forces of attraction between the molecules themselves, lead to reduction of pressure exerted by the gas on the walls of the container.

  • The van der Waals equation of state (1873) was the first equation to predict vapor-liquid coexistence. Later, the Redlich-Kwong equation of state (1949) improved the accuracy of the van der Waals equation by proposing a temperature dependence for the attractive term.

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