Departing from Equilibrium

If one departs from equilibrium by a potential lower than RT/F, one can easily show that the region of small net currents, called the reversible region, follows  

As the electrode potential is increased in the negative (or cathodic ) direction, it follows 

Correspondingly, if we make E depart in the positive (anodic) direction from its value at equilibrium 

Therefore, it is quite rational to say that ja is the (equal and opposite) current density in each direction at equilibrium. Thus, /, is a real finite number, for example, 10-3 A cm 2 for hydrogen evolution on platinum at 25°C. The dependence of the current on temperature would be helpful, too, but there is little of it as yet. 

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Since the interplay of theory and experiment is central to nearly all the material covered in this chapter, it is appropriate to start by defining the various concepts and laws needed for a quantitative theoretical description of the thermodynamic properties of a dilute solid solution and of the various rate processes that occur when such a solution departs from equilibrium. This is the subject matter of Section II to follow. There Section 1 deals with equilibrium thermodynamics and develops expressions for the equilibrium concentrations of various hydrogen species and hydrogen-containing complexes in terms of the chemical potential of hy-... 

Potential Difference A< Departs from Equilibrium Butler-Volmer Equation, When the interphase is not in equihbrium, a net current density i flows through the electrode (the double layer). It is given by the difference between the anodic partial current density i (a positive quantity) and the cathodic partial current density i (a negative quantity) ... 

When a system departs from equilibrium conditions, its entropy must decrease. Thus the energy of an open system not in equilibrium must always be greater than the energy of the same system when it is closed or in equilibrium, since the equilibrium state is the state of maximum entropy. Thus, broken 3-equilibrium is a broken 3-symmetry between the active vacuum and material systems, and it is a negentropic operation. 

These matters show up in terminology. For the physical electrochemist, there is the state of thermodynamic reversibility, the domain of the Nernst equation, and this state is the bedrock and the base from which he or she starts out. When a reaction departs from equilibrium in the cathodic and anodic direction, it has a degree of irreversibility in the thermodynamic sense. Thus, for overpotentials less than RT/b one refers to the linear region (i a It)I), where departure from reversibility is small enough to be measured in millivolts. If 11)1 > RT/F (about 26 mV at room temperature), the reaction is simply and straightforwardly irreversible the forward reaction has been made to become much faster than the back-reaction. 

Electrode Potential Departs from Equilibrium Overpotential... 

At equilibrium, Eq. (1) is in its dynamic condition, i.e., the rate of forward reaction equals the rate of the reverse reaction. If Eq. (1) departs from equilibrium, i.e., one of the directions of the reaction is faster than the other, then the net fiow of protons and electrons, or current, develops. The anode is defined as the electrode at which the de-electronation reaction occurs and the cathode as the electrode at which the electronation reaction occurs. The rate of the electron transfer reaction can be written in terms of current, which is defined by the movement of electrical charges carried by electrons in an electronic conductor and by ions in an ionic conductor. The more the system is away from equilibrium, the higher the current. As the... 

At first sight it might appear that the second law of thermodynamics is violated for reverse diffusion to occur. This is not so. One process may depart from equilibrium in such a sense as to consume entropy provided it is coupled to another process that produces entropy even faster. This is, of course, the basic principle of any pump, whether it moves water uphill or moves heat towards a higher temperature region. For the second law requirement <7 > 0 to hold it is allowable for to be less than zero, corresponding to reverse diffusion for 1, provided <72 and 0-3, due to species 2 and 3 diffusion, be such that the overall entropy production rate is positive (a + 0-2 + <73 > 0). 

For the purposes of this chapter, however, we do not suppose that the temperature is jumped to Fj and then kept constant instead we suppose that the temperature is raised in a slow, continuous way. The composition at the interface follows the temperature, always failing quite to attain equilibrium but departing from equilibrium only by a small amount, which we shall for the most part ignore. The evolution then resembles more the one illustrated in Figure 16.1a and, assuming as in Chapter 15 that there is a small volume-difference = F — F , the diffusion of A from the interface into the two interiors will be accompanied by a nonuniform stress effect (Tjj. (Axes are used as in Chapter 13 with x normal to the interface, cylindrical symmetry about x, all quantities uniform along y and z, and the strain rate equal to zero everywhere.)... 

The question naturally follows How might one try to determine K and experimentally Here Chapter 16 provides a start referring to Figure 16.8d, to depart from equilibrium along path (1) would permit X to be measured, while to depart from equilibrium along path (3) would yield K. Figure 16.11 is a reminder that these two paths give special or endmember behavior, dominated by one coefficient or the other without the complications of a mixed effect. 

Clearly, the second process just described is a reversible process, while the first is irreversible. There is another important characteristic of reversible and irreversible processes. In the irreversible process just described, a single mass is placed on the piston, the stops are released, and the piston shoots up and settles in the final position. As this occurs the internal equilibrium of the gas is completely upset, convection currents are set up, and the temperature fluctuates. A finite length of time is required for the gas to equilibrate under the new set of conditions. A similar situation prevails in the irreversible compression. This behavior contrasts with the reversible expansion in which at each stage the opposing pressure differs only infinitesimally from the equilibrium pressure in the system, and the volume increases only infinitesimally. In the reversible process the internal equilibrium of the gas is disturbed only infinitesimally and in the limit not at all. Therefore, at any stage in a reversible transformation, the system does not depart from equilibrium by more than an infinitesimal amount. 

Upon cooling from above Tg to below it, which happens in most processing operations, several material state variables such as the specific volume of the material depart from equilibrium, as shown in Figure 2.11. At a fixed temperature, Tj, these variables gradually evolve back towards equilibrium, corresponding to a slow densification of the polymer [16, 17]. This phenomenon, termed structural recovery, has a profoimd impact on the performance of all glassy polymers and the amorphous domains of semi-crystalline polymers in that it causes the ... 

Regarding the initial conditions for the surface coverage, different scenarios can be envisaged depending on whether the electrochemical measurements depart from equilibrium conditions or not. Assuming the former situation and the Langmuir isotherm for the adsorption process the initial conditions of the problem are given by... 

Reaction rates at solid/solution interfaces are controlled by the area of the interface as well as by the chemical and physical conditions that occur there. Surface reactions are approximately confined to a two-dimensional region, so their rates are expressed in terms of how fast species are created per unit of surface area, and this means that the rates have imits of flux (/, mol/m sec). The flux notation (J) and terminology is used throughout this book. The environment at the solid/solution interface is a hybrid of the bulk solid and bulk solution, so models of the chemical and physical conditions controlling the reaction rates must account for this transitional character. Equilibrium thermodynamics provides a powerful starting point for constraining the surface conditions. At equilibrium the chemical potential of each component must be the same throughout the system, so the chemical potential of the components in the surface are equal to their chemical potentials in the solid and solution phases. At low temperatures, the slow rate of equilibration between the bulk solid and the surface may void this requirement for the solid but it should apply for the components in the bulk solution. Also, at equilibrium the principle of detailed balance requires that the rates of forward reactions in the interface must equal the rates of the reverse reactions. In addition, the forward and reverse reaction steps must be the same. Models of reaction rates at equilibrium are well constrained by these principles but as the system departs from equilibrium these requirements fall away and we must search for other principles to model interfacial reaction rates. 

As the relaxation processes in the glassy state and glass transition region are non-exponential and nonlinear, the theories must take account of the thermal history of glass formation and the asymmetry of the relaxations, which depend on how the system departs from equilibrium. 

The thermodynamic decomposition voltages calculated above are the equilibrium cell voltages when there is no applied current. When current passes, the system departs from equilibrium conditions to drive the reactions. The cell voltage will then he higher than the thermodynamic decomposition voltage because of the overvoltages associated with the anodic and cathodic reactions and the ohmic drops. 

In the practical situation electrodes depart from equilibrium and for the electrode process to proceed at a practical rate, a higher potential than the equilibrium value is needed to overcome the energy loss in activating ions taking part in the reaction and in transporting ions to the electrode surface. 

One of the most important principles of linear response theory relates the system s response to an externally imposed perturbation, which causes it to depart from equilibrium, to its equilibrium fluctuations. Indeed, the system response to a small perturbation should not depend on whether this perturbation is a result of some external force, or whether it is just a random thermal fluctuation. Spontaneous concentration fluctuations, for instance, occur all the time in equilibrium systems at finite temperatures. If the concentration c at some point of a liquid at time zero is (c) + 3c(r, t), where (c) is an average concentration, concentration values at time t + 8t dXt and other points in its vicinity will be affected by this. The relaxation of the spontaneous concentration fluctuation is governed by the same diffusion equation that describes the evolution of concentration in response to the external imposition of a compositional heterogeneity. The relationship between kinetic coefficients and correlations of the fluctuations is derived in the framework of linear response theory. In general, a kinetic coefficient is related to the integral of the time correlation function of some relevant microscopic quantity. 

In conditions of chemical reactivity or binding certain amount of charge transfer and the system s potential fluctuations (departing from equilibrium) are involved... 

A more complex problem is to understand the kinetics of adhesion when the system departs from equilibrium. Various mechanisms can interpose a barrier between the molecules, to cause delayed adhesion, adhesive drag and hysteresis. Contemplating these mechanisms is a great challenge for the future. 

If we depart from equilibrium electrochemistry to that of dynamic electrochemistry, consider the following electrochemical process ... 

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