Compressors increase the pressure and density of gases, whereas pumps increase the pressure of a liquid at constant density. To understand the compression process in gases a thermal equation of state is needed to give the relationships between the intensive thermodynamic variables (p, T and v) and a caloric equation of state for internal energy and enthalpy (u and h). The simplest equation of state is that for a perfect gas. Centrifugal compressors are used with such a wide range of fluids that complex real gas models are often needed to characterise the behaviour of the fluid. The law of corresponding states and the van der Waals equation of state are introduced. Generalised cubic equations of state are described, and an example of this type, the Aungier–Redlich–Kwong equation of state, is given in more detail. The real gas equations not only change the relationship between the intensive properties but also require special care in determination of the integration of a polytropic process to determine the aerodynamic work. Different methods of carrying out this integration (from Schulz, Mallen and Saville and from Huntington) are discussed.