*Forming an intermediate state between liquids, in which we assume no external pressure, and gases, in which we omit molecular forces, we have the state in which both terms occur. As a matter of fact, we shall see further on, that this is the only state which occurs in nature.*

*van der Waals (1873, ch2)*

In Unit I we focused predominantly on a relatively small number of pure fluids. But the number of chemical compounds encountered when considering all possible applications is vast, and new compounds are being invented and applied every day. Imagine how many charts and tables would be necessary to have properties available for all those compounds. Then imagine how many charts would be necessary to represent the properties of all the conceivable mixtures of those compounds. Clearly, we cannot address all problems by exactly the same techniques as applied in Unit I. We must still use the energy and entropy balance, but we need to be able to represent the physical properties of pure compounds and mixtures in some condensed form, and we desire to predict physical properties based on very limited data.

As one might expect, an excellent shorthand is offered by the language of mathematics. When we sought values in the steam tables, we noticed that specification of any two variables was sufficient to determine the variable of interest (e.g, *S* or *H*). This leads to an excellent application of the calculus of two variables. Changes in each value of interest may be expressed in terms of changes in whatever other two variables are most convenient. It turns out that the most convenient variables mathematically are temperature and density, and that the most convenient variables experimentally are temperature and pressure.

There is a limit to how condensed our mathematical analysis can be. That limit is dictated by how much physical insight is required to represent the properties of interest to the desired accuracy. With no physical insight, we can simply measure the desired values, but that is impractical. With maximum physical insight, we can represent all the properties purely in terms of their fundamental electronic structure as given by the periodic table and their known molecular structure. The current state-of-the-art lies between these limits, but somewhat closer to the fundamental side. By developing a sophisticated analysis of the interactions on the molecular scale, we can show that three carefully selected parameters characterizing physical properties are generally sufficient to characterize properties to the accuracy necessary in most engineering applications. This analysis leads to an equation of state, which is then combined with the necessary mathematics to provide methods for computing and predicting physical properties of interest. The development of van der Waals’ equation of state provides an excellent case study in the development of engineering models based on insightful physics and moderately clever extrapolation. Note that before van der Waals the standard conception was that the vapor phase was represented by what we now refer to as an ideal gas, and the liquid was considered to be an entirely different species. Van der Waals’ analysis led to a unification of these two conceptions of fluids that also set the stage for the analysis of interfaces and other inhomogeneous fluids. Van der Waals’ approach lives on in modern research on inhomogeneous fluids.

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