This chapter combines the First and Second Laws of Thermodynamics into a single fundamental relationship that reveals the mathematical structure underlying thermodynamic systems. Using exact differentials, natural variables, Maxwell Relations, and partial derivative transformations, students will derive thermodynamic equations of state and expressions describing how thermodynamic functions change with temperature, pressure, and volume. These tools provide a powerful framework for relating difficult-to-measure thermodynamic quantities to experimentally accessible properties such as heat capacities, compressibilities, and thermal expansion coefficients.
This objective introduces the fundamental thermodynamic differential relationship that results from combining the First and Second Laws. Students will learn how this expression reveals the natural variables of the major thermodynamic functions and why those variables are important in describing thermodynamic systems.
The Maxwell Relations arise from the exact differential nature of thermodynamic state functions. Students will learn how these relationships connect seemingly unrelated partial derivatives and provide powerful tools for transforming thermodynamic expressions.
Changes in temperature, pressure, and volume alter the values of thermodynamic functions. Students will derive expressions describing these dependencies and apply them to predict how free-energy functions respond to changing conditions.
The Gibbs–Helmholtz equation provides a convenient method for estimating free-energy changes at temperatures other than those for which data are available. Students will learn how this equation is derived and how it can be used to predict the effect of temperature on reaction spontaneity.
Many thermodynamic derivatives cannot be measured directly. Students will use Maxwell Relations and derivative transformations to express these quantities in terms of measurable physical properties such as heat capacities, compressibilities, and thermal expansion coefficients.
Real systems often undergo changes in more than one state variable at a time. Students will learn how total differentials and thermodynamic equations of state can be used to combine multiple contributions and calculate overall changes in thermodynamic functions.
This objective brings together many of the concepts developed throughout Chapters 4–6, including measurable properties, Maxwell Relations, and thermodynamic equations of state. Students will derive a general expression for the difference between the constant-pressure and constant-volume heat capacities and apply it to both ideal and non-ideal systems.
\[ C_p - C_v = \frac{TV \alpha^2}{\kappa_T}\]