EOSs Based on Theory

Here R is the universal gas constant, (see the end papers), and V is the volume per mol, or molar volume. This EOS can be derived from the kinetic theory of gases, in which one assumes that each gas molecule has zero volume (i.e., is a mathematical point) and that the individual molecules have no attraction for one another, but interact only by elastic collisions. Those assumptions are very close to reality for gases at low pressure and high temperatures (relative to the critical temperature of the gas), so this EOS would be expected to represent experimental PvT data very well under those conditions, and it does. 

The first theoretical improvement on the ideal gas law is due to van der Waals (vdW). He suggested that the individual molecules do have some volume, b, and that there is some attraction between one molecule and another. Using those two ideas he wrote 

The vdW EOS is not very good at representing experimental PvT data, but it has had a profound influence on thermodynamics. Fairly simple, totally empirical modifications of it by Redlich and Kwong, Soave, and Peng and Robinson are very widely used in vapor-liquid equilibrium calculations, as discussed in Chapter 10 and Appendix F. Furthermore, it led to the principle of corresponding states, discussed below, which is very useful. 

The other useful theoretical EOS, the virial EOS, begins by defining the compressibility factor 

For an ideal gas the compressibility factor is 1.00 for all T and P. Thus, the compressibility factor is a simple, dimensionless measure of the real gas behavior, compared to ideal gas behavior. With this definition, we can write the basic virial EOS 

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