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Microscopic Origin of Thermodynamics

The real-life applications of thermodynamics hinge on experimental measurements and/or theoretical predictions of thermodynamic properties for macroscopic systems of practical interest. The experimental approach is exemplified by the traditional applications of engineering and chemical thermodynamics as evidenced in the extensive use of thermodynamic tables, diagrams, and semi-empirical correlations. Conversely, the theoretical approach is based on statistical mechanics, which is of central concern in this book.

This introductory chapter presents the key hypotheses of statistical mechanics to describe the thermodynamic properties of an equilibrium system from a microscopic perspective. After a brief overview recapitulating the essential ingredients of classical thermodynamics and fundamental relations linking different thermodynamic variables, we introduce internal energy and entropy – two most fundamental quantities of thermodynamics – in terms of the properties of individual particles, i.e., the microscopic constituents of a thermodynamic system. The statistical nature of thermodynamic variables will be elucidated in the context of ensemble, ergodicity, and microstates. In addition, we will discuss flexibility in the statistical descriptions of individual particles and microstates of a thermodynamic system, as well as additivity and relativity pertaining to both internal energy and entropy when assessed from microscopic perspectives.

We assume that the readers are already familiar with the fundamentals of classical thermodynamics including their applications to chemical and phase equilibria. However, the previous exposure of statistical mechanics is not expected. The supplementary material gives a brief overview of classical and quantum mechanics for those who are unfamiliar with this subject. While an advanced understanding of quantum mechanics is not a prerequisite, basic concepts such as Hamiltonian, quantum states, and the Schrödinger equation will be used to describe particle energy and the microscopic constituents of quantum systems (e.g., photons, electrons, and phonons).

1.1 Microscopic Constituents of Thermodynamic Systems

In this section, we discuss the essential ideas of classical thermodynamics from a microscopic perspective and recapitulate the fundamental relationships among different thermodynamic variables such as temperature, pressure, entropy, and energy. In addition, we will elucidate how statistical mechanics helps to understand the macroscopic properties of thermodynamic systems based on the dynamic behavior of their constituent particles.

1.1.1 Classical Thermodynamics

Classical thermodynamics is centered around two fundamental laws of nature that are universally applicable to the collective behavior of macroscopic systems. The first law follows the conservation of the total energy, and the second law asserts that spontaneous events in nature proceed in a particular direction. These thermodynamic laws were established in the second half of the nineteenth century from repeated observations of natural phenomena underlying transformations among different forms of energy and their connections with the physical properties of matter. Extensive experience over many years renders us confidence that the thermodynamics laws are permanent, unlikely to be refuted by future scientific developments. As famously stated by Albert Einstein,1 “A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression which classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown (for the special attention of those who are skeptics on principle).”

Closely affiliated with the fundamental laws of thermodynamics are two indispensable quantities, internal energy U and entropy S. As discussed in more detail later in this chapter, both internal energy and entropy are defined by the microscopic constituents of a thermodynamic system. More precisely, internal energy refers to the total energy arising from the perpetual motions of individual particles and inter-particle interactions. The former is commonly known as kinetic energy, and the latter is potential energy. Conversely, entropy provides a measure of the uncertainty of a macroscopic system in terms of the dynamic behavior of the individual particles or, without concerning the time, in terms of possible ways that individual particles may exist (e.g., particle positions and momenta or wave functions) under a particular thermodynamic condition. Establishing the connection between thermodynamic quantities and the dynamic behavior of individual particles is an essential task of statistical thermodynamics.

Classical thermodynamics is concerned with variations in the equilibrium properties of macroscopic systems, i.e., systems consisting of many particles, typically on the order of 1023. For a macroscopic system at equilibrium, all thermodynamic quantities are fixed.2 In other words, all macroscopic properties of interest are independent of time. Here, time independence means that the duration of observation is sufficiently long compared with the time scale that characterizes the dynamics of individual particles. Thermodynamic laws cannot be applied if one is interested only in a single particle in the vacuum or even a few particles. If a system contains only a few particles, the dynamic behavior can be described with conventional equations from classical or quantum mechanics. It is the enormous number of particles in a thermodynamic system that prevents the direct use of the mechanical equations to describe particle motions at a level of certainty the same as that for a system with only a small number of particles. Uncertainty at the microscopic level is intrinsic for all thermodynamic systems.

The individual particles of a thermodynamic system embody not only kinetic and potential energies but also information concerning the precise meaning of microscopic constituents. Thermodynamics makes no assumption on the physical nature of the individual particles, i.e., the thermodynamic laws hold regardless of how the microscopic constituents are characterized or interpreted. Indeed, individual particles in a thermodynamic system are rather diverse; they may refer to atoms or molecules as commonly present in different states of matter (i.e., gases, liquids, and solids), or elementary particles such as photons and electrons, or certain aspects of elementary particles (e.g., magnetic spins), or individual units of a molecule (segments), or aggregates of molecules (e.g., colloidal particles). The flexibility in interpreting the microscopic constituents of a thermodynamic systems suggests that thermodynamic variables such as internal energy and entropy are intrinsically relative, i.e., their absolute values are dependent on the definition of individual particles. For example, internal energy and entropy may take different values when an atomic system is described in terms of electrons and atomic nuclei with quantum mechanics or as individual atoms like classical particles. In either case, the changes in thermodynamic properties predicted from a microscopic perspective should be in accordance with experimental observations.

1.1.2 The Fundamental Equation of Thermodynamics

In addition to internal energy and entropy, we use auxiliary quantities such as enthalpy H, Helmholtz energy F and Gibbs energy G to describe the macroscopic properties of a thermodynamic system. These auxiliary functions are introduced for the convenience of practical applications when thermodynamic systems are prepared under different circumstances (e.g., fixed total energy or fixed temperature, constant pressure or constant volume, etc.). All auxiliary quantities can be formally derived from internal energy and entropy along with variables that specify a thermodynamic system (e.g., temperature, pressure, and total volume). Using the auxiliary functions, we can describe heat effects for constant pressure processes simply in terms of the changes in enthalpy, and the thermodynamic limits of various isothermal processes as well as the conditions of equilibrium with different thermodynamic potentials or free energies.

For a closed system, i.e., a system free of mass transfer with its surroundings, classical thermodynamics asserts that the internal energy U and the entropy S are related through the fundamental equation

Eq. (1.1) can be obtained by applying the first and second laws to a closed system undergoing a reversible process that involves volumetric work −PdV and heat transfer TdS. Additional variables must be introduced to account for other forms of reversible work or the mass transfer of any chemical species between the system and its surroundings. According to multivariable calculus, the differential form in Eq. (1.1) suggests that the internal energy U is an analytical function of entropy S and volume V, i.e., U = U(S, V). In addition, Eq. (1.1) implies that temperature T and pressure...