Reversible process mathematics applied

Equation (6.87) is a condensed mathematical statement of the second law the inequality applies to any real process, which is necessarily irreversible, and the equality applies to the limiting case of the reversible process. 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

The continuous succession of equilibrium states represented by the line in Figure 3.7b is one example of a reversible process. The explanation of reversible processes takes up considerable space in most texts, and usually it seems to have some connection with entropy changes. However, the importance of reversible processes is much more fundamental than furnishing an explanation for the entropy. It is a direct result of our desire to apply mathematics to physical properties. 

Thus we have completed the proof of that part of the mathematical statement of the second law that applies to reversible processes. We have shown that the differential dQ for a reversible process becomes a perfect differential when divided by T ... 

In this chapter we have established a mathematical model that fully describes a cychc voltammetry experiment of a one-electron reversible process at a planar macrodisc electrode where the diffusion coefficients of both chemical species are equal. The model consists of a one-dimensional partial differential equation that describes the evolution of the concentration of some chemical species in both time and space starting from some initial conditions at time t = 0, and boimded by some finite spatial region 0 < X < Xmax- At X = 0 is the electrode boundary which alters the concentration in a manner that depends on the potential applied to it. At X = Xmax the concentration is imaffected by the processes occurring at the electrode and so is equal to that of the bulk solution. The potential at the electrode is varied and the resultant current is recorded and plotted as a voltammogram. 

Why in the world would we be interested in such a strange kind of impossible process It s simple, really. The reason the reversible process (defined as a continuous succession of equilibrium states) is important in the thermodynamic model is that it is the only kind of process that our mathematical tools of differentiation and integration can be applied to - they only work on continuous functions. Once our crystal of diamond leaves its state of equilibrium at 25 °C, practically anything could happen to it, but as long as it settles back to equilibrium at 50 °C, all of its state variables have changed by fixed amounts from their values at 25 °C. We have equations to calculate these energy differences, but they refer to lines and surfaces in our model, and that means that they must refer to continuous equilibrium between the two states. 

In a review regarding piezoelectricity, Ballato (1996) revealed that Coulomb was the first to suspect that electricity generation could be attained through applying pressure to materials. Katzir (2006) pointed out that Jacques Curie and Pierre Curie were the first to observe piezoelectricity in 1880. It is interesting to note that in 1881, it was not the Curie brothers but Lippmann (1881) who announced the existence of a converse piezoelectric effect. Basically, this converse effect is deformation of a piezoelectric material due to influence of an applied electrical field. Lippmann (1881) postulated the existence of this effect through mathematical prediction by applying basic thermodynamic principles to reversible processes. Curie and Curie (1881) verified and estab-hshed the converse piezoelectric effect experimentally soon after. 

An equation of state is a mathematical expression relating the amount, volume, temperature, and pressure of a substance (usually applied to gases). Equilibrium refers to a condition where (1) the forward and reverse processes proceed at equal rates (2) for reactions at constant T and constant P, the Gibbs energy of reaction, Afi, is equal to zero. For a system that has reached equilibrium, no further net change occurs. For example, amounts of reactants and products in a reversible reaction remain constant over time. 

Although more complex models have been proposed to describe the process [57, 85, 86], involving the Ps bubble state and its shrinking upon reaction, the equations based on a reversible reaction with a forward and reverse reaction rate constants as in scheme (X) enables the fitting of the data perfectly, as shown by the solid line in Figure 4.9. The kinetic equations corresponding to such a scheme are tedious to derive, particularly as concerns the intensities (still more when a magnetic field is applied). However, they do not present insuperable mathematical difficulties and should be used instead of the approximate expressions that have appeared casually (e.g., "steady state" treatment of the reversible reaction). From scheme (X), it is not expected that the variation of X3 with C be linear, but the departure from linearity may be rather small, so that the shape of the X3 vs C plots may not be taken as a criterion to ascribe the nature of the reaction. 

Overall, liophilic ions (usually small ions capable for dispersive interactions) provide a useful means for selective alteration of the retention of basic analytes. Influence of these ions on the column properties is fully reversible, and equilibration requires minimal time (usually less than an hour, or about 10 to 20 column volumes). On the other hand, the mechanism of their effect is very complex and is dependent on the type of organic modifier used and on the concentration applied. Theoretical description and mathematical modeling of this process is a subject for further studies. 

The disadvantage of the rigorous approach to this type of problem is that the mathematics are very difficult for any case except the most simple e.g., this first order reaction). Instead, it is common to consider the somewhat vague concept of the reaction layer . This is an approach which gives a physical idea of the processes involved as well as allowing rate coefficients to be derived for more complicated kinetic mechanisms. The reaction layer is a hypothetical layer surrounding the electrode within which all the HA molecules produced by reaction (9) reach the electrode and are reduced. Its thickness p depends on the reverse rate coefficient, ky, (p = Suppose the applied potential is only sufficient to discharge... 

The general phenomenon of polymer adsorption/retention is discussed in some detail in Chapter 5. In that chapter, the various mechanisms of polymer retention in porous media were reviewed, including surface adsorption, retention/trapping mechanisms and hydrodynamic retention. This section is more concerned with the inclusion of the appropriate mathematical terms in the transport equation and their effects on dynamic displacement effluent profiles, rather than the details of the basic adsorption/retention mechanisms. However, important considerations such as whether the retention is reversible or irreversible, whether the adsorption isotherm is linear or non-linear and whether the process is taken to be at equilibrium or not are of more concern here. These considerations dictate how the transport equations are solved (either analytically or numerically) and how they should be applied to given experimental effluent profile data. 

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