Equilibrium states and reversibility

Thermodynamics is only concerned mth, equilibrium states, in which the state functions have constant values throughout the system. It provides us with information about the circumstances under which nonequilibrium states will move towards equilibrium, but it tells us nothing directly a bout the nonequilibrium states. 

The criteria for equilibrium are very important, and may be summarized as follows  

The mechanical properties must be uniform throughout the system and constant in time. 

The chemical composition of the system must be uniform, with no net chemical change taking place. 

The temperature of the system must be uniform and must be the same as the temperature of the surroundings. 

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The reverse process of discharging the capacitor occurs when both electrodes are electrically connected. Through this connection, the system is allowed to reach its equilibrium state, and charge flows from the negative to the positive electrode until both Fermi levels are aligned again. 

The coordinates of thermodynamics do not include time, ie, thermodynamics does not predict rates at which processes take place. It is concerned with equilibrium states and with the effects of temperature, pressure, and composition changes on such states. For example, the equilibrium yield of a chemical reaction can be calculated for given T and P, but not the time required to approach the equilibrium state. It is however true that the rate at which a system approaches equilibrium depends direcdy on its displacement from equilibrium. One can therefore imagine a limiting kind of process that occurs at an infinitesimal rate by virtue of never being displaced more than differentially from its equilibrium state. Such a process may be reversed in direction at any time by an infinitesimal change in external conditions, and is therefore said to be reversible. A system undeigoing a reversible process traverses equilibrium states characterized by the thermodynamic coordinates. 

When microscopic reversibility is present in a complex system composed of many particles, every elementary process in a forward direction is balanced by one in the reverse direction. The balance of forward and backward rates is characteristic of the equilibrium state, and detailed balance exists throughout the system. Microscopic reversibility therefore requires that the forward and backward reaction fluxes in Fig. 2.1 be equal, so that... 

Irrespective of the experiment to be done, sample preparation contains a number of necessary conditions. First, aggregation must be prevented if one wants to investigate structure and conformation of single molecules. Second, the adsorption process must be reversible, or at least, very slow in order to approach the equilibrium state and allow statistical analysis of the molecular assembly. Third, adhesion of the molecules to the substrate must be strong enough to sustain the mechanical and adhesive interactions with the tip. However, it should be relatively low to prevent the native structure from deformation. 

Bejan and Tondeur [9] make a number of other observations in their paper. One is that the relation between j and x is not necessarily linear. Another observation is that a similar analysis can show that the force x should be equipartitioned in time, which is another way of saying that the steady state is optimal. Prigogine gave an earlier proof of this principle [11]. The steady state is common in nature and often the favored state in industrial operation. It can be considered to be the "stable state" of nonequilibrium thermodynamics, comparable to the equilibrium state of reversible thermodynamics (see Figure 4.2). Of course, the latter is characterized by Sgen = 0, whereas the former is characterized by a minimum value , larger than zero. 

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."... 

Since this relation involves properties only, it must be satisfied for ch in state of any closed system of uniform T and P, without restriction conditions of mechanical and thermal reversibility assumed in its derivation/ inequality applies to every incremental change of the system between equilibrium states, and it dictates the direction of change that leads to equilibrium. The equality holds for changes between equilibrium states (reve processes). Thus Eq. (6.1) is just a special case of Eq. (13.51). 

Therefore, at equilibrium A, = 0. Equation (8.97) shows that during the time evolution, the surrogate system 2 proceeds through stable equilibrium states, and system 1 proceeds through states Xs. This condition is stated without any reference to microscopic reversibility, and applies for all values of X, which represent both the chemical equilibrium and nonequilibrium states. We can expand each of the r reactions into a Taylor series around the chemical equilibrium state at which X = 0... 

The van t Hoff-Nemst approach thus lacks a function of state associated with the chemical reaction. A second objection is that although stress is placed on the chemical reaction, consideration is in effect limited to a study of equilibrium states and of reversible changes despite the fact that quantities like the heat of reaction only have a precise and simple meaning in practice if the system considered actually undergoes a chemical reaction in a finite time. In other words a thermodynamics of chemical reactions must necessarily be a thermodynamics of irreversible phenomena. 

Because a process changes the state of a system, the process must start with the system in a particular equilibrium state and must also end with the system in a particular equilibrium state. Two such states A and B are indicated in Figure 12.1. You might wonder whether we can sketch a path on the surface of equilibrium thermodynamic states to summarize the progress of the system during a process. Only special processes of the type called reversible can be represented in this way (see discussion in the following paragraphs). 

Spontaneous processes are particular examples of irreversible processes defined in Section 12.1. In stark contrast with reversible processes, they do not proceed through a sequence of equilibrium states, and their direction cannot be reversed by an infinitesimal change in the direction of some externally applied... 

This inequality applies to all incremental changes towards the equilibrium state, and the equality holds at the equilibrium state where any change is reversible. It follows immediately that when S and V are constant,... 

We have just defined reversible processes in terms of the stable equilibrium surface, but completely analogous processes are also possible on metastable equilibrium surfaces, and the definition can be extended to include these. In fact, however, most discussions of reversible processes refer to stable equilibrium states and surfaces. 

Reversible/irreversible A reversible change in a process is a change that can be reversed by an infinitesimal modification of any driving force acting on the system. The key word infinitesimal sharpens the everyday meaning of the word reversible as something that can change direction. In other words, the state of the system should always be infinitesimally close to the equilibrium state, and the process should be described as a succession of quasi-equilibrium states. 

The sol state represents principally an equilibrium state and can be reached spontaneously starting from the dry colloid and an appropriate solvent. The dissolved substance is present in true solution as sii le molecules or reversible aggregates of them. This case is characteristic of fairly all sols of Vol. II. We may discern two types ... 

The work of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy, temperature, and the second law. Clausius introduced the word entropy. The second law is a statement of existence of stable equilibrium states and distinguishes thermodynamics from mechanics and other fields of physics. The many stable equilibrium states and various other equilibrium and nonequilibrium states contemplated in thermodynamics are not contemplated in mechanics (Gyftopoulos and Beretta, 2005). The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of any material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

Processes that begin in a stable equilibrium state and proceed to another stable equilibrium state, without ever leaving the state of equilibrium more than infinitesimally, are reversible processes. 

It is necessary to consider an apparent paradox in connection with this relationship. The reversible change 2 -> 1 assumes that the system passes through a succession of equilibrium states and, in particular, that state 2 and state 1 are both equilibrium states. On the other hand, we postulate an irreversible process 1 2 without changing the external conditions. It appears as if an equilibrium... 

In order to detach the other states of an indignant chemical system evolution, let us estimate the relaxation times of intermediate and composing substances. Since the relaxation time is a time from the onset of chemical system indignation which always can be considered as being far from the equilibrium state, and putting in equation (1.49) that K 0 X KiOiXi, we will consider the reaction (1.48) as the non-reversible one. Then it can be written... 

Since elementary reactions of the chain propagation are assumed as being far from the equilibrium state and this allows to ignore their reversible directions, the stoichiometric notations of elementary (2.115) and (2.116) and also the final reactions (2.118) and (2.119) do not limit in any way the use of an algorithm of construction of the kinetic models of the routes. Furthermore, formally it can always be assumed that elementary (2.115) and (2.116) and the final reactions (2.118) and (2.119) have an anpty composite substance, the concentration of which is equal to zero for example... 

For many years, the thermodynamic description of macromolecules lagged behind other materials because of the unique tendency of pol5nneric systems to assume nonequilibrium states. Most standard sources of thermodynamic data are, thus, almost devoid of polymer information (1-7). Much of the aversion to include polymer data in standard reference sources can be traced to their nonequilibrium nature. In the meantime, polymer scientists have learned to recognize equilibrium states and utilize nonequilibrium states to explore the history of samples. For a nonequilibrium sample it is possible, for example, to thermally establish how it was transferred into the solid state (determination of the thermal and mechanical history). More recently, it was discovered with the use of temperature-modulated differential scanning calorimetry (TMDSC) that within the global, nonequilibrium structure of semicrystalline polymers, locally reversible melting and crystallization processes are possible on a nanophase level (8). 

Next let us eonsider the reversible adiabatic processes that are possible. To carry out a reversible adiabatic process, starting at an initial equilibrium state, we use an adiabatic boundary and slowly vary one or more of the work coordinates. A certain final temperature will result. It is helpful in visualizing this process to think of an A/-dimensional space in which each axis represents one of the N independent variables needed to describe an equilibrium state. A point in this space represents an equilibrium state, and the path of a reversible process can be represented as a curve in this space. 

The existenee of reversible adiabatic surfaces is the justification for defining a new state function S, the entropy. S is specified to have the same value everywhere on one of these surfaces, and a different, unique value on each different surface. In other words, the reversible adiabatic surfaces are surfaees of constant entropy in the A-dimensional space. The fact that the surfaces fill this spaee without intersecting ensures that 5 is a state function for equilibrium states, because any point in this space represents an equilibrium state and also lies on a single reversible adiabatic surface with a definite value of S. 

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