Equilibrium thermodynamic state

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

The concepts of invertibility and reversibility must be distinguished. Invertibility is the term proposed to be used for reactions that can be made to occur in both directions, regardless of the departure from thermodynamic equilibrium that is necessary to achieve this. Reversibility of a reaction means that it occurs with a minimum departure from the thermodynamic equilibrium state. 

Creep recovery response is due to freezing in local deformation of the polymer molecules when the polymer cools rapidly. The polymer molecules are frozen into shapes that distort their Gaussian spherical equilibrium shape. If the polymer is heated or allowed to relax over a very long time there will be dimensional changes as the polymer molecules assume their thermodynamic equilibrium states (Gaussian spherical equilibrium shape). 

At first, the superconducting state was not thought to be a thermodynamic equilibrium state. But, as we know from Chapter 2, any equilibrium state must be the result of a free energy minimization. It can be shown that the superconducting and normal states will be in equilibrium when their free energies are equal, and that the free energy difference between the normal state at zero field, G H = 0), and the superconducting state at zero field, Gs H = 0), is... 

A blend s properties are largely determined by the compatibility of the component polymers since this influences morphology. When two polymers are blended polymer-polymer interactions will be exhibited if the thermodynamic equilibrium state permits. The basic thermodynamic equation... 

If we consider an experiment in which the system is set up at low flow rates, the system will settle to the lower branch (sometimes called the thermodynamic branch because it approaches the thermodynamic equilibrium state at the lowest flow rates). If the flow rate is now increased, the system moves along this branch, through the region of multistability, until it reaches the turning point. Beyond the fold, the system must jump suddenly to the... 

As first shown by J. W. Gibbs, the analytical characterization of thermodynamic equilibrium states can be expressed completely in terms of such first and second derivatives of a certain fundamental equation (as described in Section 5.1). 

With the preceding three definitions established, we can now address the important concept of state (or thermodynamic state, or thermodynamic equilibrium state ), the central concept of equilibrium thermodynamics ... 

Let us first summarize the principal features of the thermodynamic equilibrium states that are the principal focus of a thermodynamic description. According to the definitions of Section 2.10, such states are ... 

Thermodynamic equilibrium states also exhibit certain formal analogies to mechanical equilibrium states (Sidebar 2.15). The thermodynamic potentials that underlie these analogies will be discussed in Chapter 5. 

Thermodynamic equilibrium states correspond only to the stable or marginally stable states ( 0) of the mechanical analog. The first law of thermodynamics establishes the thermodynamic potential, while the second law of thermodynamics establishes the stability condition, as discussed in Chapter 5. 

These two additional properties of the H in (5.6), together with its mono-tonic decrease, has led to its identification with the entropy defined by the second law of thermodynamics. It must be realized, however, that H is a functional of a non-equilibrium probability distribution, whereas the thermodynamic entropy is a quantity defined for thermodynamic equilibrium states. The present entropy is therefore a generalization of the thermodynamic entropy the generalized entropy is... 

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

The situation becomes much less problematic if the reaction in Eq. 3.1 is considered only at equilibrium. Equilibrium states are steady states in which the net reaction fluxes to produce products or reactants, regardless of the reaction pathway, are equal and opposite. From the perspective of chemical thermodynamics, equilibrium states are unique and independent of thermodynamic path. Thus these states can be described by a unique set of chemical species irrespective of the intermediate steps of their formation. Dissolution- precipitation reactions and the chemical species that affect them at equilibrium can be described by an extension of the methodology discussed in Section 2.4 (cf. Fig. 

Assuming in (41) 8=0, we can find the ordering temperature of pure fullerite. Considering formulae (26)-(28), we receive the equation of thermodynamically equilibrium state of fullerite of any composition as follows ... 

We note that the classical equilibrium entropy (i.e., the eta-function evaluated at equilibrium states) acquires in the context of the Microcanonical Ensemble an interesting physical interpretation. The entropy becomes a logarithm of the volume of the phase space that is available to macroscopic systems having the fixed volume, fixed number of particles and fixed energy. If there is only one microscopic state that corresponds to a given macroscopic state, we can put the available phase space volume equal to one and the entropy becomes thus zero. The one-to-one relation between microscopic and macroscopic thermodynamic equilibrium states is thus realized only at zero temperature. 

Before starting to discuss (116), we make an observation. The fast time evolution (116) is also observed in driven systems that cannot be described on the level Ath- For example, let us consider the Rayleigh-Benard system (i.e., a horizontal layer of a fluid heated from below). It is well established experimentally that this externally driven system does not reach thermodynamic equilibrium states but its behavior is well described on the level of fluid mechanics (by Boussinesq equations). This means that if we describe it on a more microscopic level, say the level of kinetic theory, then we shall observe the approach to the level of fluid mechanics. Consequently, the comments that we shall make below about (116) apply also to driven systems and to other types of systems that are prevented from reaching thermodynamical equilibrium states (as, e.g., glasses where internal constraints prevent the approach to Ath)-... 

A system reaches the thermodynamic equilibrium state when it is left for a long time with no external disturbances. At equilibrium the internal properties are fully determined by the external properties. This makes it easy to describe such systems for example, if the temperature is not uniform within the system, heat is exchanged with the immediate surrounding until the system reaches a thermal equilibrium, at which the total internal eneigy U and entropy S are completely specified by the temperature, volume, and number of moles. Therefore, at equilibrium there are no thermodynamic forces operating within the system (Figure 2.1). Equilibrium systems are stable. For small deviations, the system can spontaneously return to the state of equilibrium. Equilibrium correlations result from short-range intermolecular interactions. Existence of the extremum principles is a characteristic property of equilibrium thermodynamics. 

In the natural environment, however, there are components of states differing in their composition or thermal parameters from thermodynamic equilibrium state. These components can undergo thermal and chemical processes. Therefore, they are natural resources with positive exergy. Only for commonly appearing components can a zero value of exergy be accepted. A correct definition of the reference level is essential for the calculation of external exergy losses. The most probable chemical interaction between the waste products and the environment occurs with the common components of the environment. 

A system is said to he in Thermodynamic Equilibrium State when its state properties have defined values, which do not tend to change with time (Khangaonkar 1967). 

When the CO disproportionation is catalyzed by cobalt, some ordered metastable structures are detected inside the active metal nanoparticles after the reaction. These structures are regular thin (approximately 5 atoms in thickness) alternating cobalt layers of different crystallographic modifications (Figure 4.17). Note that the appearance of such structures at thermodynamically equilibrium states of the catalyst substance is contrary to the Gibbs phase rule for the phase equilibria in solids. Thus, the metastable layered structures may be considered an analogue of spatial dissipative structures. 

Our world is fuU of a variety of substances and materials that are used in human practice for very different purposes. Note that terms material and substance are not identical. A substance (chemical compound) is an ensem ble of a great number of atoms, molecules, ions, and/or radicals that define properties of the substance as an object for investigations, whereas materi als are the phase separated forms of substances (chemical individuals) or their ensembles (for example, specialty mixtures and composites) that feature a set of properties necessary for practical apphcations of the material. To create the material that possesses the required properties for particular practical applications, a specialty substance must be prepared and stabilized. Of importance is that the required substance modifications are often in a metastable state that differs from the thermodynamically equilibrium state and is, nevertheless, appropriate for the material storage and operation. 

This section briefly reviews some elementary aspects of the thermodynamics of chemical reactions, (e.g. Atkins (1978)) which are used to analyze a-Si H. The thermodynamic equilibrium state of a system is described by a minimum of the Gibbs free energy function... 

A metal CMP process involves an electrochemical alteration of the metal surface and a mechanical removal of the modified film. More specifically, an oxidizer reacts with the metal surface to raise the oxidation state of the material, which may result in either the dissolution of the metal or the formation of a surface film that is more porous and can be removed more easily by the mechanical component of the process. The oxidizer, therefore, is one of the most important components of the CMP slurry. Electrochemical properties of the oxidizer and the metal involved can offer insights in terms of reaction tendency and products. For example, relative redox potentials and chemical composition of the modified surface film under thermodynamically equilibrium condition can be illustrated by a relevant Pourbaix diagram [1]. Because a CMP process rarely reaches a thermodynamically equilibrium state, many kinetic factors control the relative rates of the surface film formation and its removal. It is important to find the right balance between the formation of a modified film with the right property and the removal of such a film at the appropriate rate. 

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

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