This chapter develops the mathematical tools needed to apply thermodynamics to real substances by expressing thermodynamic properties as total differentials and manipulating partial derivatives using exact differential relationships. Important measurable material properties, including the isothermal compressibility (\(\kappa_T\)) and isobaric thermal expansivity (\(\alpha\)), are introduced and used to describe how materials respond to changes in pressure and temperature. These concepts are then applied to phenomena such as internal pressure and the Joule-Thomson effect, allowing changes in thermodynamic quantities to be expressed in terms of experimentally measurable properties.
Thermodynamic state functions can be expressed as functions of two independent variables, allowing changes in those functions to be written as total differentials. Students should understand how exact differentials arise from state functions and use the Euler relation to verify that a differential is exact.
Reciprocal and cyclic permutation rules provide powerful tools for transforming thermodynamic partial derivatives. Students should be able to apply these relationships to express unknown derivatives in terms of measurable quantities and simplify thermodynamic expressions.
The isothermal compressibility coefficient (\(\kappa_T\)) and isobaric thermal expansivity (\(\alpha\)) describe how the volume of a substance responds to changes in pressure and temperature. Students should be able to interpret these properties physically and calculate changes in volume resulting from changes in pressure or temperature.
Real gases deviate from ideal behavior because intermolecular interactions affect how their internal energy depends on volume and pressure. Students should be able to explain the concepts of internal pressure and the Joule-Thomson coefficient, describe the experiments used to investigate them, and explain why ideal gases exhibit neither effect.
Many thermodynamic derivatives cannot be measured directly but can be expressed in terms of measurable properties such as heat capacities, \(\alpha\), and \(\kappa_T\). Students should be able to derive and apply relationships that predict how thermodynamic quantities change with temperature, pressure, and volume.