The internal energy of the electromagnetic field is distinct from that of the matter exposed to the field. The choice of field variable is determined by physical considerations concerning the electric, displacement, magnetic and induction fields. Legendre transform are worked out in order to define electric and magnetic enthalpies and free enthalpies. Spatial derivatives of the enthalpies yield the force densities that dielectrics and magnets experience in inhomogeneous electric and magnetic induction fields. Either internal energy or electric enthalpy must be used to analyse the force on a dielectric inserted in a capacitor, depending on the constraint (constant charge or constant field). Likewise, internal energy or magnetic enthalpy must be considered to analyse the force on a paramagnetic material inserted in a coil or in between the poles of a magnet. A complete analysis of adiabatic demagnetisation offers an example of application of Mayer’s relation to specific heat at constant induction field or constant magnetisation. The effect is predicted for the case of a paramagnetic material with a magnetisation that obeys Curie’s law.

By applying the first and second laws to systems consisting of two subsystems separated by a wall, it is possible to determine the equilibrium conditions relevant for that wall's properties, e.g. a fixed diatermal wall, a mobile diathermal wall, a fixed permeable wall. The second law imposes a condition on the entropy production rate which implies relations between heat and temperature difference, matter transfer rate and chemical potential difference, volume rate of change and pressure difference. Thus, transport equations are introducted, akin to Fourier law, Fick law and Poiseuille law. These processes are examples of dissipative processes. A worked solution shows that when two subsystems are subjected to a mechanical action, non-symmetric heat flow may occur.

The principle of a Carnot cycle is discussed and the operation of a Carnot engine with an ideal gas is calcuated. Engine efficiency, heating and cooling coefficient of performance are defined and their values calculated for a Carnot engine operating on an ideal gas. In order to bring out the importance of irreversibility during heat transfers in a heat engine, the endoreversible cycle is analysed; the heat transfers at the hot and cold reservoirs are the only source of dissipation. The Stirling engine is described. Work and heat exchange are caculated for an ideal gas undergoing the processes that are involved in Diesel, Otto, Lenoir, Atkinson, Brayton and Rankine engines. The Rankine cycle of a biphasic fluid is analysed also.

Continuity equations are derived first for extensive scalar quantities, then extended to vectorial quantities. Thus, material derivatives, source densities and current densities are introduced. The formalism is applied to express the conservation of mass and charge. The continuity equation for linear momentum corresponds to Newton’s second law with a stress tensor included to account for deformations in continuous media. Evolution equations for energy are obtained. This approch yields a local Gibbs relation, the relationship betwen intenal energy current density, conductive and convective heat currents and an expression for the entropy source density in terms of generalised currents and forces. This presentation of the thermodynamics of continuous media ends with volume integrations formalises the conceptual link between the local description of continuous media with the description of simple systems subjected to thermal and mechanical processes.

The stability of a system is discussed in terms of the curvature of entropy as a function of internal energy and volume, then in terms of internal energy as a function of entropy and volume. Global and local conditions are given. The most difficult mathematical developments are differed to worked solutions and exercices. This analysis introduces the notion that phase diagrams may contain regions where distinct phases coexist. The slope of phase coexistence lines are deduced from thermodynamic principles and give the Clausius-Clapeyron formula. Equilibrium between coexisting phases is shown to imply the Gibbs phase rule which gives the number of degrees of freedom of a system in terms of the number of substances and phases present in the system. The van der Waals equation of state is discussed. In the worked solutions, a model is presented for a concrete case of phase coexistence, and observations from every day life are analysed, such as the melting temperature of salt water or the gas pressure of a bottle containing liquid in which gas is dissolved.