WE HAVE NOW CONSIDERED two methods for modeling and analyzing processes: mathematical modeling based on laws and graphical analysis based on empirical measurements. In mathematical modeling we started with a general principle and substituted the specifics of the process. The general principle might be the conservation of mass, (rate of mass in) = (rate of mass out) at steady state, or a constitutive equation, such as PV = nRT. For empirical analysis we first measured data, such as the solid–liquid–vapor phase diagram of a pure substance or a vapor–liquid–composition diagram for a mixture of two components. We did not concern ourselves with why these data were the way they were. We did not dwell on why some systems are nonideal or why some systems have an azeotrope. We just used the data. We never extrapolated from the measured data. That is, if we measured vapor–liquid–composition data for mixtures of benzene-toluene at 80°C as a function of pressure, we used these data only to analyze units that separated benzene–toluene mixtures at 80° C.

What if we wanted to operate the unit at 60° C? We would have to return to the lab and remeasure the data at 60°C. We had no means of extrapolating the data to different conditions. As you will learn later, thermodynamics provides the tools needed to extrapolate equilibrium data. But lacking a knowledge of thermodynamics, is there a way to extrapolate?