Classical equilibrium thermodynamics

In this section we limit ourselves to equilibrium. The time evolution that is absent in this section will be taken into consideration in the next two sections. We begin the equilibrium analysis with classical equilibrium thermodynamics of a one-component system. The classical Gibbs formulation is then put into the setting of contact geometry. In Section 2.2 we extend the set of state variables used in the classical theory and introduce a mesoscopic equilibrium thermodynamics. 

The variables characterizing complete states at equilibrium (we shall use hereafter the symbol y to denote them) are 

The fundamental thermodynamic relation is a function N R (e, n) i- s(e, n), where s denotes entropy per unit volume. The function s(e,n) as well as all other functions introduced below are assumed to be sufficiently regular so that the operations made with them are well defined. We can see the fundamental thermodynamic relation s = s(e,n) geometrically (as Gibbs did, see Gibbs, 1984) as a two-dimensional manifold, called a Gibbs manifold, imbedded in the three-dimensional space with coordinates (e,n,s) by the mapping (e,n) — (e, n,s(e,ri)). 

In view of the importance of Legendre transformations in equilibrium thermodynamics, we shall make, following Hermann (Hermann, 1984), an alternative formulation of the fundamental thermodynamic relation. We introduce a five-dimensional space (we shall use hereafter the symbol N to denote it) with coordinates (e,n,e ji, s) and present the fundamental thermodynamic relation as a two-dimensional manifold imbedded in the five-dimensional space N by the mapping 

We shall call this manifold a Gibbs-Legendre manifold and denote it by the symbol N. The advantage of this formulation is that the space N is naturally 

---

Particularly during its excitation discharge, the system must be an open thermodynamic system far from equilibrium in its energetic exchange with the active vacuum. In that case classical equilibrium thermodynamics does not apply, and such a system is permitted to... 

No laws of physics or thermodynamics are violated in such open dissipative systems exhibiting increased COP and energy conservation laws are rigorously obeyed. Classical equilibrium thermodynamics does not apply and is permissibly violated. Instead, the thermodynamics of open systems far from thermodynamic equilibrium with their active environment—in this case the active environment-rigorously applies [2-4]. 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

In the second half of the 20th century it is precisely the classical equilibrium thermodynamics that became a basis for the creation of numerous computing systems for analysis of irreversible processes in complex open technical and natural systems as applied to the solution of theoretical and applied problems in various fields. The methods of MP, i.e., the mathematical discipline that emerged from the Lagrange interpretation of the equilibrium state, were a main computational tool employed for the studies. 

Answer 1 The only way to find the fundamental thermodynamic relation s = s(y) inside the classical equilibrium thermodynamics is by making experimental measurements. Results of the measurements are usually presented in Thermodynamic Tables. 

Let the state variables (1) be replaced by a state variable xeM corresponding to a more detailed (more microscopic) view than the one taken in classical equilibrium thermodynamics. Several examples of x are discussed below in the examples accompanying this section. Following closely Section 2.1, we shall now formulate equilibrium thermodynamics that we shall call a mesoscopic equilibrium thermodynamics. 

Answer 2 given above invites, of course, another question Where do the fundamental thermodynamic relation h = h x) and the relation y = y x) come from An attempt to answer this question makes us to climb more and more microscopic levels. The higher we stay on the ladder the more detailed physics enters our discussion of h = h(x) and y = y(x). Moreover, we also note that the higher we are on the ladder, the more of the physics enters into y = y(x) and less into h = h x). Indeed, on the most macroscopic level, i.e., on the level of classical equilibrium thermodynamics sketched in Section 2.1, we have s = s(y) and y y. All the physics enters the fundamental thermodynamic relation s s(t/), and the relation y = y is, of course, completely universal. On the other hand, on the most microscopic level on which states are characterized by positions and velocities of all ( 1023) microscopic particles (see more in Section 2.2.3) the fundamental thermodynamic relation h = h(x) is completely universal (it is the Gibbs entropy expressed in terms of the distribution function of all the particles) and all physics (i.e., all the interactions among particles) enters the relation y = y(x). 

The time evolution that has been investigated in Section 3 is the time evolution seen experimentally in externally unforced macroscopic systems approaching, as t —> oo, equilibrium states at which their behavior is seen experimentally to be well described by classical equilibrium thermodynamics. In other words, in Section 3, we have investigated the time evolution... 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

This model of the liquid will be characterized by some macroscopic quantities, to be selected among those considered by classical equilibrium thermodynamics to define a system, such as the temperature T and the density p. This macroscopic characterization should be accompanied by a microscopic description of the collisions. As we are interested in chemical reactions, one is sorely tempted to discard the enormous number of non-reactive collisions. This temptation is strenghtened by the fact that reactive collisions often regard molecules constituting a minor component of the solution, at low-molar ratio, i.e. the solute. The perspective of such a drastic reduction of the complexity of the model is tempered by another naive consideration, namely that reactive collisions may interest several molecular partners, so that for a nominal two body reaction A + B —> products, it may be possible that other molecules, in particular solvent molecules, could play an active role in the reaction. 

In classical equilibrium thermodynamics, the simplest model of an engine that converts heat into work is the Camot cycle. The behavior of a heat engine working between two heat... 

A more realistic cycle than the Carnot cycle is a modified cycle taking into account the processes time of heat transfer between the system and its surroundings, in which the working temperatures are different of those its reservoirs [1], obtaining the efficiency t]CAN =1 Jtc / TH, first found in references [2] and [3], and known as Curzon-Ahlbom-Novikov-Chambadal efficiency. At present, the duration of heat transfer processes is important. Based on this model, at the end of the last century, a theory was developed as an extension of classical equilibrium thermodynamics, the finite time thermodynamics, in which the duration of the exchange processes heat becomes important. 

In the classical equilibrium thermodynamics, Stirling and Ericsson cycles have an efficiency that goes to the Carnot efficiency, as it is shown in some textbooks. These three cycles have the common characteristics, including two isothermal processes. The objection to the classical point of view is that reservoirs coupled to the engine modeled by any of these cycles do not have the same temperature as the working fluid because this working fluid is not in direct thermal contact with the reservoir. Thus, an alternative study of these cycles is using finite... 

The existence of a finite heat transfer in the isothermal processes is affected with the assumption of a non-endoreversible cycle with ideal gas as working substance. Power output and ecological function have also an issue that shows direct dependence on the temperature of the working substance. Expressions obtained with the changes of variables have the virtue of leading directly to the shape of the efficiency through Z, function. Thus, in classical equilibrium thermodynamics, the Stirling cycle has its efficiency like the Carnot cycle efficiency in finite time thermodynamics, this cycle has an efficiency in their limit cases as the Curzon-Ahlborn cycle efficiency. 

We begin with a synopsis of some familiar notions, pertinent to both classical equilibrium thermodynamics [39] and to its nonequilibrium extension due to Onsager [1,3,4]. 

The information provided in this Chapter can be divided into four parts 1. Introduction, 2. Thermodynamic theories of polymer blends, 3. Experimental methods, and 4. The characteristic thermodynamic parameters for polymer blends. Introduction presents the basic principles of the classical, equilibrium thermodynamics, describes behavior of the single component materials, then focuses on the two-component systems solutions and polymer blends. 

In this chapter we discuss uniform systems, the properties of which change only in time. Similarly as in [1-8] our main aim here is to demonstrate the method of rational thermodynamics and application of its principles in simple material models. In other words, the main aim is pedagogical—to begin with simple issues, demonstrations, and examples. Nevertheless, even this chapter contains practical results which can be applied on many simple real systems. Among others the principal results of classical equilibrium thermodynamics will be obtained and this will be shown also for reacting mixtures and heterogeneous (multiphase) uniform systems. 

Model A is in fact the main material model studied in the classical equilibrium thermodynamics. Each state in this model A may be an equilibrium state from postulate S4 in Sect. 1.2 by fixing not only V, T, but also V = 0, t = 0 then not only power but, by (2.1), (2.6)i, also heating should be zero 2 = 0. Its persistence is assured (cf. Rems. 6 in this chapter, 11 in Chap. 1) by stability conditions. ... 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

In this Sect. 4.8 we discuss also the results concerning chemical potentials and activities, studied mainly by classical equilibrium thermodynamics of mixtures [129, 138, 141, 152]. These are also valid in our models, among others in non-equilibrium (e.g. in transports or/and chemical reactions), because of the validity of local equilibrium, cf. Sect.4.6. 

Thus, from the point of view of modem phenomenological thermodynamics, the current outputs of classical equilibrium thermodynamics (e.g. the description of thermochemistry of mixtures) and the tasks of irreversible thermodynamics, like the description of linear transport phenomena and nonlinear chemical kinetics, are valid much more generally, e.g. even when all these processes mn simultaneously. As we noted above, these properties are not expected to be valid in any material models in some models the local equilibrium may not be valid, reaction rates may depend not only on concentrations and temperature, etc. 

The infortnation provided in this chapter can be divided into four parts 1. introduction, 2. thermodynamic theories of polymer blends, 3. characteristic thermodynamic parameters for polymer blends, and 4. experimental methods. The introduction presents the basic principles of the classical equilibrium thermodynamics, describes behavior of the single-component materials, and then focuses on the two-component systems solutions and polymer blends. The main focus of the second part is on the theories (and experimental parameters related to them) for the thermodynamic behavior of polymer blends. Several theoretical approaches are presented, starting with the classical Flory-Huggins lattice theory and, those evolving from it, solubility parameter and analog calorimetry approaches. Also, equation of state (EoS) types of theories were summarized. Finally, descriptions based on the atomistic considerations, in particular the polymer reference interaction site model (PRISM), were briefly outlined. 

Contribution to the fundamental science will influence development of macrscopic kinetics, classical equilibrium thermodynamics, and joint application of these disciplines to study the macroworld. The capabilities of kinetic analysis will be surely expanded considerably, if traditional kinetic methods that are reduced to the analysis of trajectory equations are sup>plemented by novel numerical methods. The latter are to be based on consideration of continuous sequences of stationary processes in infinitesimal time intervals. The problems of searching for the trajectories, being included into the subject of equilibrium thermodynamics, would make deserved the definition of this discipline as a closed theory that allows the study of any macroscopic systems and processes on the basis of equilibrium principles. Like the equilibrium analytical mechanics of Lagrange the thermodynamics may be called the unified theory of statics and dynamics. Joint application of kinetic and thermodynamic models further increases the noted potential advantages of the discussed directions of studies. 

---