## Mathematical prelude

Of all the thermodynamic properties that have been introduced in the preceding chapters, only a few can be directly measured by laboratory experiments. Pressure, temperature and volume are obviously among the measurable properties. However, there is no instrument to measure entropy or any of the properties related to energy (u, h, f, and g). Except for entropy, which according to the 3rd law, is zero at 0 K, these quantities cannot be assigned absolute values; only changes in them as a result of a process have quantitative meaning. All of the thermodynamic information concerning simple substances (i.e., one-component systems) is based upon measurements of the equation of state v(p,T), the temperature-dependent heat capacity CP(T) at a particular pressure, and the enthalpy changes associated with phase transitions.

One of the significant achievements of classical thermodynamics is to connect various properties, so that only a few measurements are needed for a complete description of a substance. For example, the difference CP - CV and the effect of pressure on the enthalpy can be obtained from the p-v-T equation of state of the substance. For the ideal gas and the approximate model of condensed phases presented in Chap. 2, there was no need for a formalized approach to these relations.

For a more accurate description, particularly for nonideal gases and condensed phases under extreme conditions, relationships between thermodynamic properties need to be developed. Establishing these connections is the purpose of the present chapter. First, certain mathematical fundamentals need to be reviewed. These are based on the fact that all thermodynamic properties of a simple substance are smoothly-varying functions of any two.