Chapter 5: Fundamental Thermodynamic Equations

The combined First and Second Laws, based on Internal Energy (dependent on Entropy and Volume), forms the initial foundation. The first auxiliary function defined is Enthalpy, which is the sum of the Internal Energy and the product of Pressure and Volume. Its natural independent variables are Entropy and Pressure, and significantly, the change in Enthalpy at constant pressure represents the thermal energy (heat) exchanged with the system. Next, the Helmholtz Free Energy is introduced, defined as Internal Energy minus the product of Temperature and Entropy, and serves as the characteristic potential for systems held at constant temperature and constant volume. For these constant temperature and volume systems, equilibrium is achieved when the Helmholtz Free Energy reaches its minimum value, and its decrease quantifies the maximum non-volume work obtainable during a reversible, isothermal process (leading to its designation as the "work function"). The most useful potential for practical materials engineering is the Gibbs Free Energy, defined as Enthalpy minus the product of Temperature and Entropy, which has the easily measurable intensive variables, Temperature and Pressure, as its natural independent variables. The critical criterion for equilibrium in systems maintained at constant Temperature and constant Pressure is the minimization of the Gibbs Free Energy. To analyze systems that change composition, such as those undergoing chemical reactions, the text incorporates the Chemical Potential for species i. This intensive variable is defined as the partial derivative (rate of change) of the Gibbs Free Energy (or other potentials) with respect to the number of moles of species i, holding the appropriate conjugate variables constant. The relationships between these potentials allow for the derivation of Maxwell Relations, which are equalities between mixed second partial derivatives, linking properties that are difficult to measure (like the change in Entropy with Pressure) to those that are readily measurable (like the change in Volume with Temperature). Furthermore, the chapter establishes three TdS equations, which express the product of Temperature and the change in Entropy in terms of experimentally measurable quantities such as heat capacity at constant pressure or volume, the coefficient of thermal expansion, and isothermal compressibility. Practical thermodynamic predictions are enhanced by the Gibbs–Helmholtz equations, which relate the temperature dependence of the Gibbs Free Energy (at constant pressure) to the Enthalpy, and similarly relate the Helmholtz Free Energy to the Internal Energy. The chapter also explores complex systems by incorporating non-volume work terms, such as that introduced by an applied magnetic field, defining specialized caloric effects, including the magnetocaloric effect (the temperature change of a material when a magnetic field is applied adiabatically) and pyromagnetism (the dependence of magnetization on temperature).