Reversible limit

Figure 9. In a Motorola-type cellular phone handset, the six AA cells in series are protected by reversal-limiting diodes, and the overflow at 1.7 V is determined by the six red LEDs [38],...

The distinction between reversible and irreversible work is one of the most important in thermodynamics. We shall first illustrate this distinction by means of a specific numerical example, in which a specified system undergoes a certain change of state by three distinct paths approaching the idealized reversible limit. Later, we introduce a formal definition for reversible work that summarizes and generalizes what has been learned from the path dependence in the three cases. In each case, we shall evaluate the integrated work w 2 from the basic path integral,... 

Because wrev is the maximum available work of any type, we can say from (5.53) that AG is the maximum available non-PV work. Here, available (or free ) refers to the idealized reversible limit in which no useful work is dissipated. Practically speaking, the major non-PV work of interest to chemists is the chemical energy (as manifested, for example, in electrochemical or osmotic phenomena), associated with the chemical potential terms that will be introduced in Chapter 6. 

In Fig. 11, three such plots are represented for different values of l. Obviously, if ltV2 is sufficiently large, the curves approach the reversible limit (7rf)-1/2, as follows from eqns. (35b) and (38). It can be calculated that a significant deflection from reversible behaviour (> 10%) is observed if ltV2 2. With the condition t 100 jus raised by the experimental constraints described above, it follows that kinetics can be studied if l < 200 s-1/2, which corresponds to a rate constant of the order of 0.03— 0.3 cm s"1, depending on the values of DQ and E — E°. 

From these curves, it can be seen that the current decreases with time in all cases, and more pronounced the higher the rate constant. Moreover, for values of k° > 10-2 cm s 1, the response becomes independent of the kinetics, i.e., the reversible limit has been reached. The rate constant is related to the kinetic... 

The figure also includes the variation of the ratio k°/mo,c<, = -vA piane with log (/°) (dashed line) for comparison. Thus, it can be seen that in the reversible limit, there is a coincidence with the criterion used here, but in the irreversible one, this... 

These regions have been indicated in Fig. 5.13 for a = 0.5. Matsuda and Ayabe suggest the following ranges for classifying the electrode process [35] Kplane >15, Reversible process 10-3 < Kplane < 15, Quasi-reversible process Kplane < 10 3. Totally irreversible process. The reversible limit is similar to that proposed here, but the totally irreversible one is clearly excessive (see Fig. 5.13). In any case, this criterion has only an approximate character. 

For a fast electron transfer reaction, by introducing the condition k p 3> kc in both equations, the reversible limit given by Eqs. (6.203) and (6.206) is recovered. 

The Carnot cycle forms the basis for a thermodynamic scale of temperature. Because e = 1 — Ti/Tf, the Carnot efficiencies determine temperature ratios and thereby establish a temperature scale. The difficulty of operating real engines close to the reversible limit makes this procedure impractical. Instead, real gases at low pressures are used to define and determine temperatures (see Section 9.2). 

If all steps in the mechanism are facile, so that the exchange velocities of all steps are large compared to the net reaction rate, the concentrations of all species participating in them are always essentially at equilibrium in a local context, even though a net current flows. The result for the RDS in this nernstian (reversible) limit has already been obtained as (3.4.27), which we now rewrite in exponential form ... 

With both of these approaches one must be sure that the uncompensated resistance, / u, is sufficiently small that the resulting voltage drops (of the order of /p/ u) are negligible compared to the attributable to kinetic effects. In fact, Nicholson (14) has shown that resistive effects cannot be readily detected in the AEp-v plot, because the effect of uncompensated resistance on the AE -v behavior is almost the same as that of ip. The effect of is most important when the currents are large and when approaches the reversible limit (so that) AE differs only slightly from the reversible value). It is especially difficult not to have a few ohms of uncompensated resistance in nonaqueous solvents (such as acetonitrile or tetrahydrofuran), even with positive-feedback circuitry (Chapter 15). Many reported studies made under these conditions have suffered from this problem. 

The shapes and positions of irreversible waves can furnish only kinetic information. One may be able to determine such parameters as kf, 1, or a, but thermodynamic results, such as and free energies, are not available (28, 33, 34). As a rule of thumb, a system with > 2 X 10 cm/s appears reversible on the classical polarographic time scale of a few seconds when D is on the order of 10 cm /s. A heterogeneous charge transfer with < 3 X 10 cm/s will behave in a totally irreversible manner under the same conditions, and one can evaluate the rate parameters as described above. Systems with between these limits are quasireversible, and some kinetic information can be obtained from them through the treatment prescribed by Randles (33, 34). Naturally, the precision of the kinetic information deteriorates as the reversible limit is approached. See Section 5.5.4 for much more information about the interpretation of irreversible waves. 

Steady-state Approximation In the reverse limit of the intermediate... 

Even though a quasi-static process is driven differentially, the driving forces may still contain dissipative components. These components may arise because some properties have finite differences across boundaries or they may arise from differential effects accumulated over a finite process. If we could remove all dissipative components so the process would be driven only by conservative forces, then the change of state would be reversible. This reversible limit can be expressed as... 

Now we address the apparent contradiction between the limit in (1.3.5) and that in (1.3.2) both have A —> 0, but with different results. The resolution is that the static limit in (1.3.2) can describe a real process, while the reversible limit in (1.3.5) is an idealization. That is, a reversible "process" is not really a process at all [10], it is only a... 

To ascertain if the process actually approximates its reversible limit, it is necessary to conduct a series of experiments in which the process is carried out more and more slowly, so that the parameters can be extrapolated to an infinitely slow path and the errors estimated. 

The sequence of equiUbrium states can be approximated, as closely as desired, by the intermediate states of a real spontaneous process carried out sufficiently slowly. The reverse sequence of equiUbrium states can also be approximated, as closely as desired, by the intermediate states of another spontaneous proeess earned out suffi-eiently slowly. (This requirement prevents any spontaneous proeess with hysteresis, sueh as plastic deformation or the stretching of a metal wire beyond its elastic limit, from having a reversible limit.) During the approach to infinite slowness, very slow ehanges of the type described in item 3 on page 50 must be eliminated, i.e., prevented with hypothetical constraints. 

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