Research: Science & Education
Predicting Saturation Curve of a Pure Substance Using Maxwell’s Rule Jaime Wisniak* and Moshe Golden Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel The vapor–liquid saturation curve of a pure substance is important from both the practical and the thermodynamic viewpoints because it provides basic information on the behavior of the fluid as well as on its properties. Prediction of the phenomenon using equations of state (EOS) developed for a gas provides an interesting didactic application of the first and second laws of thermodynamics. It will be shown here how the laws can be used independently to develop the pertinent thermodynamic operator, which will then be applied to the van der Waals EOS and three other powerful EOS (Redlich–Kwong, Soave, and Peng–Robinson) to test for their ability to predict the vapor–liquid dome. Theory When plotted on the P-v plane the isotherms of most EOS have the typical behavior illustrated in Figure 1. Inspection of this figure indicates the following interesting features. 1. Above the critical temperature (Tc) the curves are continuous and single-valued (one root) for a given pressure. 2. Below the critical temperature the curves are S-shaped showing a maximum value, two inflection points, and a minimum value, implying the continuity of the liquid and vapor phases. 3. The minimum and maximum volumes become identical at the critical point and the first inflection points occur at a volume equal to the critical volume, since the critical point and the inflection points fulfill the same mathematical requirements. 4. For temperatures sufficiently low, negative pressures are predicted for the liquid phase. 5. For every temperature below the critical one the curve has three real roots.
The S-shape is characteristic of all cubic equations and some which are not cubic, such as the Benedict–Webb– Rubin equation; but this is not so of the virial equation. Let us now analyze the physical implications of an Sshaped curve like ABCDE. Liquid and vapor states that coexist in equilibrium must have the same pressure and temperature. Furthermore, since the two states may be present in any proportion, it is possible to bring about a change of state at constant pressure and at constant temperature (path ACE). On the other hand, the EOS indicates that an isothermal path (like ABCDE) alone can bring the saturated liquid to the saturated vapor state. Portion AB represents an overexpanded liquid, which may exist but is metastable. Similarly, DE represents a supersaturated vapor phase, which is also metastable and disappears spontaneously if condensation nuclei are introduced. Between B and D we have
∂P ∂v
