Equations of State - Mathematical Formulations

The ideal gas equation of state, which relates the pressure, temperature, and specific volume, is a familiar equation  

The term p is the absolute pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the absolute temperature. The units of R have to be appropriate for the units chosen for the other variables. This equation is quite adequate when the pressure is low (such as one atmosphere). However, many chemical processes take place at very high pressure. For example, ammonia is made at pressures of 220 atmospheres or more. Under these conditions, the ideal gas equation of state may not be a valid representation of reality. 

Other equations of states have been developed, usually in conjunction with process simulators, to address chemical processes at high pressure. There are two key features (1) the equation can represent the real p-V-T behavior and (2) the parameters must be easily found, including for mixtures. This last criterion is no small requirement. There are more than 25 million chemicals, leading to an infinite number of different mixtures. Obviously, you cannot look up the properties of all those mixtures on the Web. 

The first generalization of the ideal gas law was the van der Waals equation of state  

In this equation, the b accounts for the excluded volume (a second molecule cannot use the same space already used by the first molecule), and the a accounts for the interaction force between two molecules. This extension is just a first step, however, because it will not be a good approximation at extremely high pressures. 

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