Three-Parameter Corresponding States

If we plot P versus ρ for several different fluids, we find some remarkably similar trends. As shown in Fig. 7.1 below, both methane and pentane show the saturated vapor density approaching the saturated liquid density as the temperature increases. Compare these figures to Fig. 1.4 on page 23, and note that the P versus ρ figure is qualitatively a mirror image of the P versus V figure. The isotherms are shown in terms of the reduced temperature, T_r ≡ T/T_c. Saturation densities are the values obtained by intersection of the phase envelope with horizontal lines drawn at the saturation pressures. The isothermal compressibility  is infinite, and its reciprocal is zero, at the critical point (e.g., 191 K and 4.6 MPa for methane). It is also worth noting that the critical temperature isotherm exhibits an inflection point at the critical point. This means that (∂²P/∂ρ²)_T = 0 at the critical point as well as (∂P/∂ρ)_T = 0. The principle of corresponding states asserts that all fluid properties are similar if expressed properly in reduced variables.

Figure 7.1. Comparison of the PρT behavior of methane (left) and pentane (right) demonstrating the qualitative similarity which led to corresponding states’ treatment of fluids. The lines are calculated with the Peng-Robinson equation to be discussed later. The phase envelope is an approximation sketched through the points available in the plots. The smoothed experimental data are from Brown, G.G., Sounders Jr., M., and Smith, R.L., 1932. Ind. Eng. Chem., 24:513. Although not shown, the Peng-Robinson equation is not particularly accurate for modeling liquid densities.

 The isothermal compressibility is infinite at the critical point.

Although the behaviors in Fig. 7.1 are globally similar, when researchers superposed the P-V-T behaviors based on only critical temperature, T_c and critical pressure, P_c, they found the superposition was not sufficiently accurate. For example, one way of comparing the behavior of fluids is to plot the compressibility factor Z. The compressibility factor is defined as

 The compressibility factor.

Note: The compressibility factor is not the same as the isothermal compressibility. The similarity in names can frequently result in confusion as you first learn the concepts.

The compressibility factor has a value of one when a fluid behaves as an ideal gas, but will be non-unity when the pressure increases. By plotting the data and calculations from Fig. 7.1 as a function of reduced temperature T_r = T/T_c, and reduced pressure, P_r = P/P_c, the plot of Fig. 7.2 results. Clearly, another parameter is needed to accurately correlate the data. Note that the vapor pressure for methane and pentane differs on the compressibility factor chart as indicated by the vertical lines on the subcritical isotherms. The same behavior is followed by other fluids. For example, the vapor pressures for six compounds are shown in Fig. 7.3, and although they are all nearly linear, the slopes are different. In fact, we may characterize this slope with a third empirical parameter, known as the acentric factor, ω. The acentric factor is a parameter which helps to specify the vapor pressure curve which, in turn, correlates the rest of the thermodynamic variables.²

Figure 7.2. The Peng-Robinson lines from Fig. 7.1 plotted in terms of the reduced pressure at T_r = 0.8, 0.9, 1.0, 1.1, and 1.3, demonstrating that critical temperature and pressure alone are insufficient to accurately represent the P-V-T behavior. Dashed lines are for methane, solid lines for pentane. The figure is intended to make an illustrative point. Accurate calculations should use the compressibility factor charts developed in the next section.

Figure 7.3. Reduced vapor pressures plotted as a function of reduced temperature for six fluids demonstrating that the shape of the curve is not highly dependent on structure, but that the primary difference is the slope as given by the acentric factor.

 Critical temperature and pressure are insufficient characteristic parameters by themselves. The acentric factor serves as a third important parameter.

Note: The specification of T_c, P_c, and ω provides two points on the vapor pressure curve. T_c and P_c specify the terminal point of the vapor pressure curve. ω specifies a vapor pressure at a reduced temperature of 0.7. The acentric factor was first introduced by Pitzer et al.³ Its definition is arbitrary in that, for example, another reduced temperature could have been chosen for the definition. The definition above gives values of ω ~ 0 for spherical molecules like argon, xenon, neon, krypton, and methane. Deviations from zero usually derive from deviations in spherical symmetry. Nonspherical molecules are “not centrally symmetric,” so they are “acentric.” In general, there is no direct theoretical connection between the acentric factor and the shape of the intermolecular potential. Rather, the acentric factor provides a convenient experimental vapor pressure which can be correlated with the shape of the intermolecular potential in an ad hoc manner. It is convenient in the sense that its value has been experimentally determined for a large number of compounds and that knowing its value permits a significant improvement in the accuracy of our engineering equations of state.