Classical thermodynamic state functions

The classical thermodynamic state functions [enthalpy ( ), entropy (5), free energy ( )j can be expressed in terms of the partition function as shown in the following derivations. 

In Section 2.1, we remarked that classical thermodynamics does not offer us a means of determining absolute values of thermodynamic state functions. Fortunately, first-principles (FP), or ab initio, methods based on the density-functional theory (DFT) provide a way of calculating thermodynamic properties at 0 K, where one can normally neglect zero-point vibrations. At finite temperatures, vibrational contributions must be added to the zero-kelvin DFT results. To understand how ab initio thermodynamics (not to be confused with the term computational thermochemistry used in Section 2.1) is possible, we first need to discuss the statistical mechanical interpretation of absolute internal energy, so that we can relate it to concepts from ab initio methods. 

Our aim is not to introduce the reader into the field of thermodynamic theory, its historical development and problems. The latter are numerous, starting from the general definition of entropy itself. We shall simply assume that the thermodynamic state functions are those found in tables, or computed using the classical thermodynamic relations. For example we have the relations... 

This is the same relation given as Equation 1.37 in the development of the distribution law and the introduction of temperature, which was a classical derivation of U for an ideal gas. From an expression for U, one can use the relations among thermodynamic state functions presented in Chapters 2 and 3 to find expressions for H, G, and A. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29], Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. 

The topic arises from the following sequence of aspects of entropy when entropy is introduced on a thermodynamic basis the issue is the motion of heat (Jaynes, 1988), and the assessment involves calorimetry an entropy change is evaluated. When entropy is formalized with the classical view of statistical thermodynamics, the entropy is found by evaluating a configurational integral (Bennett, 1976). But a macroscopic physical system at a particular thermodynamic state has a particular entropy, a state function, and the whole description of the physical system shouldn t involve more than a mechanical trajectory for the system in a stationary, equilibrium condition. How are these different concepts compatible ... 

In the classical formulation, the second law of thermodynamics states that there exist an absolute scale for the temperature T and an extensive function 5(p, V, called the entropy, such that for an infinitesimal process in a closed system... 

Turning to electric fields and classical Maxwell-Boltzmann statistics, soluble analytical models now exist which allow calculations of non-degenerate electron densities as a function of thermodynamic state in intense electric fields (low density high temperature). Semiclassical methods are available for switching on atomic potentials to models studied presently, though numerical results are not yet available here. 

For the fundamentals of statistical thermodynamics the reader is referred elsewhere (14) however, it is possible from a knowledge of classical thermodynamics and an acceptance of the manner in which the classical thermodynamic functions can be specified in terms of observable quantities to show how statistical theory can be usefully applied to the defect solid state. 

The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermod)mamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic fimctions are then derived from these partition fimctions, using the usual relationships of statistical thermodynamics. 

Ecosystems are open systems. Their boundaries are permeable, permitting energy and matter to cross them. Effects of environmental constraints and influences on the system play an important role in the regulation and maintenance of the system s spatio-temporal as well as trophic organization and functioning. Indeed, ecosystems operate outside the realm of classical thermodynamics. Biological, chemical, and some physical processes inside of ecosystems are nonlinear. Stationary states of ecosystems are non-equilibrium states far from thermostatic equilibrium. In the course of time, entropy does not tend to a maximum value, or entropy production to a minimum. Entropy decreases when the order of organization and structure of the ecosystem increases. Entropy production is counterbalanced by export of entropy out of the system. 

From this thermodynamic perspective, the first Hohenberg-Kohn theorem is merely an assertion that that the Legendre transform from the electronic ensemble specified by N and v(r) to that specified by p(r) exists [73]. In analogy to classical thermodynamics, the state function for the p-ensemble is ... 

Expressions for the equation of state, de Broglie thermal wavelength, and rotational and vibrational partition functions for a diatomic molecule, provided by (28-87) and (28-89), as well as multiplication by Navo, allow one to determine the chemical potential on a molar basis. The final result for p, is consistent with its definition from classical thermodynamics ... 

Classical thennodynamics deals with the interconversion of energy in all its forms including mechanical, thermal and electrical. Helmholtz [1], Gibbs [2,3] and others defined state functions such as enthalpy, heat content and entropy to handle these relationships. State functions describe closed energy states/systems in which the energy conversions occur in equilibrium, reversible paths so that energy is conserved. These notions are more fully described below. State functions were described in Appendix 2A however, statistical thermodynamics derived state functions from statistical arguments based on molecular parameters rather than from basic definitions as summarized below. 

The partition function, which is simply the sum of all Boltzmann factors for each allowable state of the system, provides the bridge between classical and statistical thermodynamics. To derive the relationships between the partition function and the classical thermodynamic properties we consider a closed system and a canonical ensemble. 

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