Low-overpotential Approximation

More complex reductions of CO2 by enzyme cascades have also been achieved. A combination of an electron mediator and two enzymes, formate dehydrogenase and methanol dehydrogenase, was used to reduce CO2 to methanol. This system operates with current efficiencies as high as 90% and low overpotentials (approximately —0.8 V vs. SCE at pH 7) [125]. The high selectivity and energy efficiency of this system indicate the potential of enzyme cascades. There are also drawbacks to these systems. In general, enzymes are... 

Using (16) and linearizing the Butler-Volmer for low overpotential approximation, one can obtain the following expression ... 

In the high-overpotential case (cf. Section 7.2.3b.2), the first exponential term can be neglected for n 0, i.e., for net electronation, and the second exponential term for T) 0, i.e., for net deelectronation. In the low-field approximation, where both exponential terms in the Butler-Volmer equation can be linearized, Eq. (7.136) becomes... 

What is the width of the linear region represented by Eq. 6E. There is no unique answer to this question, since it depends on the level of accuracy desired. We can readily calculate the range of overpotential for which the deviation from linearity of the i/r plot does not exceed, say, 5%. This turns out to be about + 28 mV for P = 0.5. For other values of P the curve is no longer symmetrical and the linear region is shorter, as can be seen in Fig. 2E, Thus, while p does not appear in the equation for the low overpotential linear approximation, its numerical value does influence the region over which this linear approximation applies. It is interesting to note that in Section 11.1 we estimated the linear region to be approximately + 5 mV, whereas here we find it to be + 28 mV for P = 0.5. This discrepancy arises... 

This system of differential equations has been solved for the potentiostatic case [2.24-2.27]. Many authors have solved the problem using different additional assumptions and approximations [2.30, 2.31]. A comprehensive review can be found in ref. [2.27]. The more informative galvanostatic case has not yet been solved. However, a quite instructive equivalent electric circuit may be used for simulation of galvanostatic transients at low overpotentials as described below. Three points should be emphasized ... 

Two limiting cases may be considered (1) at low overpotentials, when the tanh expression may be linearized, /, varies linearly with Iq (2) at higher values of overpotential, when the tanh expression approximates to unity, 7 changes linearly with Vig. 

Note that, in contrast to the low-field approximation, the Tafel equation is sometimes referred to as the high-field approximation since it is only valid for large values of overpotential. 

At values of ZoAl ratio lower than 1, the complete diffusion cOTitrol of the electrodeposition process arises at aU overpotentials. The lower limit of the region of the complete diffusion control can be determined as follows it is obvious that the convex shape of the polarization curve characterizes the diffusion control of deposition process and the concave one the activation control of deposition process. The Z/Zl ratio as function of tj is shown in Fig. 1.3 and the Z as a function of in Fig. 1.4. In both cases, the convex shape of curves changes in the concave one at approximately Zo/ l 01> meaning that the diffusion control changes in the activation one at the beginning of the polarization curve at low rj and At larger overpotentials, the diffusion control occurs. Hence, the diffusion ccaitrol at aU overpotentials appears at 0.1 activation control appears at Zq/Zl< 0.1 at low overpotentials. 

At low overpotentials, where approximation (i) is rather poor, the size of the nucleus becomes large and the sum in the denominator in Eq. (52), being composed of a large number of terms, can be replaced by an integral. 

The polarization overpotential (Fpoiarizatioii) is the sum of the polarization overpotentials at the anode and cathode and given by the Butler-Volmer equation. At low overpotentials, the polarization overpotential can be approximated by (24.6), where R is the gas constant, T is the temperature in Kelvin, F is the Faraday s constant, i is the current density, and j o is the exchange current density. 

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