## What Is An Equation Of State

There are two types of equations of state, or EOS. In general, they are thermodynamic relationships between three properties of a pure {one-component) substance. In discussing the fundamental meaning of equations of state, it is useful to eliminate the quantity of the substance in order to deal only with intensive properties <see Section 1.6.2). Extensive properties can be converted to intensive properties by dividing the formed by the quantity of the substance, (e.g., on a per-mole basis v = V/n, u = U/n).

The volumetric EOS refers to the relationship of the p-v-T properties of a gas. Solids and liquids are also described by p-v-T equations of state, although the quantitative forms are very different from those applicable to gases.

In functional form, the volumetric EOS can be written as v =fip,T), or v(p,T) for short. This form indicates that the specific volume (or molar volume) is expressed as a function of pressure and temperature. However, the EOS can be equally well written as pOOD or T(p,v). Because pressure and temperature are usually specified in an experiment or in a process, the form v(p,T) is most commonly employed.

The EOS relating v, p, and T provides no information about the other thermodynamic properties, in particular about the internal energy u and the entropy s. However, according to the phase rule (Section 1.12), specifying any two properties of a pure substance fixes all properties. Nonetheless, the EOS in the form p(T,V), for example, gives no hint about the functions w(7>), s(T,v\ or of the remaining auxilliary properties h,f, and g (see Section 1.6.1). Knowledge of (7» requires information about the substance beyond that contained in its p-v-T relationship. Functional relationships such as w(7>) are sometimes called thermal equations of state. Two gases, helium and nitrogen for example, may follow the ideal -gas law with reasonable accuracy, but show significantly different u(T,v) behavior. 