Linear systems, irreversible processes

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

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 fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

We consider a system made of a solid phase (denoted by s) containing a liquid phase (denoted by L). The latter is composed by water (denoted by e) and by two kinds of ions (denoted by + and —). An electric field is applied. The methods of the linear thermodynamics of irreversible processes permits the description of transport phenomena by linear relations. For the liquid phase [9] (in this paper, the indices or exponents k and m refer to cartesian coordinates) ... 

Irreversible processes may promote disorder at near equilibrium, and promote order at far from equilibrium known as the nonlinear region. For systems at far from global equilibrium, flows are no longer linear functions of the forces, and there are no general extremum principles to predict the final state. Chemical reactions may reach the nonlinear region easily, since the affinities of such systems are in the range of 10-100 kJ/mol. However, transport processes mainly take place in the linear region of the thermodynamic branch. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

We saw in Chapter 2 that in the range of validity of linear nonequilibrium thermodynamics (i.e., in the scope of linear Onsager relations), a system approaching its stationary state is characterized by a monotonous decrease in the rate of entropy production (energy dissipation rate) resulting from the existence of internal irreversible processes dP < 0 and < 0. 

Current/Voltage Relationships for Irreversible Reactions Many voltammetric electrode processes, particularly those associated with organic systems, are partially or totally irreversible, which leads to drawn-out and less well defined waves. The quantitative description of such waves requires an additional term (involving the activation energy of the reaction) in Equation 23-11 to account for the kinetics of the electrode process. Although half-wave potentials for irreversible reactions ordinarily show some dependence on concentration, diffusion currents are usually still linearly related to concentration many irreversible processes can, therefore, be adapted to quantitative analysis. 

Onsager relations - An important set of equations in the thermodynamics of irreversible processes. They express the symmetry between the transport coefficients describing reciprocal processes in systems with a linear dependence of flux on driving forces. 

When there are several different fluxes and forces brought into play in a system, then the thermodynamics of linear irreversible processes postulates that there is a matrix relationship between the different fluxes and forces (with a symmetrical matrix ONSAGER relations). The non-zero non-diagonal terms in the matrix signify that a coupling of the phenomena is taking place. 

In most electrochemical systems, except for poor conducting mediaone can assume that electroneutrality applies on a macroscopic scale in the zone located away from the interfaces. The mathematical description of the system therefore combines the electroneutrality and the various volume mass balances of each species with the equations for the molar flux densities in the framework of the thermodynamics of linear irreversible processes. The result is the following system with (n+1) differential equations for [n+1) unknowns (n concentrations and the potential) ... 

In the earlier chapters, transport phenomena involving a barrier have been discussed from the angle of (i) basic understanding of the physico-chemical phenomena and (ii) test of the linear thermodynamics of irreversible processes. Similar phenomena in continuous systems such as thermal diffusion (Soret effect)/Dufour effect are of equal... 

In view of the complexity in biological systems, different theoretical techniques have to be applied for different non-equilibrium regions. The range of applicability of linear thermodynamics of irreversible processes is quite limited on account of very thin... 

Thus, in the formalism of extended irreversible thermodynamics by inclusion of gradients in the basic formalism of linear thermodynamics of irreversible processes, a small correction in the local entropy due to flow appears. With this modification, the new formalism can be applied to the systems such as shock waves, where there are large gradients. 

The extensive properties of the overall system that is not in equilibrium, such as volume or energy, are simply the sums of the (almost) equilibrium properties of the subsystems. This simple division of a sample into its subsystems is the type of treatment needed for the description of irreversible processes, as are discussed in Sect. 2.4. Furthermore, there is a natural limit to the subdivision of a system. It is reached when the subsystems are so small that the inhomogeneity caused by the molecular structure becomes of concern. Naturally, for such small subsystems any macroscopic description breaks down, and one must turn to a microscopic description as is used, for example, in the molecular dynamics simulations. For macromolecules, particularly of the flexible class, one frequently finds that a single macromolecule may be part of more than one subsystem. Partially crystalhzed, linear macromolecules often traverse several crystals and surrounding liquid regions, causing difficulties in the description of the macromolecular properties, as is discussed in Sect. 2.5 when nanophases are described. The phases become interdependent in this case, and care must be taken so that a thermodynamic description based on separate subsystems is still valid. 

The change of total entropy is dS = dgS + diS. The term deS is the entropy exchange through the boundary, which can be positive, zero, or negative, while the term diS is the rate of entropy production, which is always positive for irreversible processes and zero for reversible ones. The rate of entropy production is diS/dt = JkXk. A near-equilibrium system is stable to fluctuations if the change of entropy production is negative, i.e. Ai5 < 0. For isolated systems, dS/dt > 0 shows the tendency toward disorder as d S/dt = 0 and dS = diS > 0. For nonisolated systems, diS/dt > 0 shows irreversible processes, such as chemical reactions, heat conduction, diffusion, or viscous dissipation. For states near global equilibrium, d S is a bilinear form of flows and forces that are related in linear form. 

Before introducing the notion of nonequilibrium thermodynamics we shall first summarize briefly the linear and nonlinear laws between thermodynamic fluxes and forces. A key concept when describing an irreversible process is the macroscopic state parameter of an adiabatically isolated system These parameters are denoted by. At equilibrium the state parameters have values A , while an arbitrary state which is near or far from the equilibrium may be specified by the deviations from the equilibrium state ... 

Hence, for a reversible system, the well-known linear relation is obtained between the potential E and log (/iim -///). Other equations have been derived for those reversible systems that involve semiquinone formation, dimerization, or the formation of complex compounds with mercury. Logarithmic analysis of the polarographic wave is often the only proof of reversibility which is considered but recently several authors, in particular Zuman and Delahay, " have pointed out that it is inadequate to assume that an electrode process is reversible on this evidence alone. For a reversible reaction, plots of E vs. In (/lim - ///) give the electron number z from the slope of the plot, RT/zF, A clearer indication of irreversibility is the evaluation of slopes of log i-E curves for higher concentrations (for i < /lim). Irreversible processes will give Tafel behavior. 

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