This chapter follows a logic of exposition initiated by Gibbs in 1902. On the one hand, some theoretical results in statistical mechanics have been derived in Chapter 3, while, on another hand, some theoretical/experimental results are expressed within thermodynamics, and parallels are drawn between the two approaches. To this end, the theory of thermodynamics and its laws are presented. The chapter takes an approach where each stated law is attached to a readable source material and a person’s writing. The exposition of the second law follows the axiomatics of Carathéodory, for example. This has the advantage of decoupling the physics from the mathematics. The structure of thermodynamic theory with the scaling behaviour of thermodynamic variables, Massieu potentials and Legendre transformations is also developed. Finally, correspondence relations are postulated between thermodynamics and statistical mechanics, allowing one to interpret thermodynamic variables as observational states associated to certain probability laws. Applications are given, including the Gibbs paradox. The equivalence between the canonical and the microcanonical ensembles is analysed in detail.

We define the laws of black hole thermodynamics by first reviewing the laws of regular thermodynamics, and seeing what the analog of the zeroth, first, second, and third laws are. After stating them, we show some partial proofs. As part of this, we show a simple proof and a general argument for the Hawking radiation and the Hawking temperature of a black hole, and the corresponding Bekenstein–Hawking entropy of the black hole. We finish by defining the gravitational thermodynamic potential.

The use of energy has defined our civilization and governs our lives. Throughout the day and night modern humans consume enormous quantities of energy resources for: food preparation; transportation; lighting, heat, ventilation and air-conditioning of buildings; entertainment; and a myriad other applications that define modern life. A gigantic global energy industry transports and inconspicuously transforms the energy resources to convenient forms (gasoline, diesel, electricity) that are vital to the functioning of the modern society. This introductory chapter surveys the types of the global primary energy sources, how they are transformed to useful energy, and how they are used. The chapter introduces the two laws of thermodynamics that govern the conversion of energy from one form to another; explains the methodology of thermodynamics, which is essential for the understanding of energy conversion processes; and delineates the limitations on energy conversion. The thermodynamic cycles for the generation of power and refrigeration are reviewed and the thermodynamic efficiencies of the cycles and energy conversion equipment (turbines, compressors, solar cells, etc.) are defined.

The canonical ensemble describes systems which can exchange energy with their surroundings, which may be modelled as a heat bath.The statistical mechanical quantity that characterizes systems in the canonical ensemble is the partition function, which is shown to be related to the Helmholtz free energy.The connections between statistical mechanics and the laws of thermodynamics are discussed.The application of the canonical ensemble is illustrated through a variety of examples: two-level systems, the quantum and classical simple harmonic oscillator, rigid rotors and a particle in a box.The differences in the statistical properties of distinguishable and indistinguishable particles are considered and used to derive the thermodynamic properties of ideal and non-ideal gases, including the ideal gas equation, the Sackur--Tetrode equation and the Van der Waals equation.The chapter concludes with a discussion of the equipartition theorem and its application to the Dulong--Petit Law.