Quadratic driving force

Bowen, J. H. and Rimmer, P. G. Trans. Inst. Chem. Eng., 50 (1972) 168. Design of fixed bed sorbers using a quadratic driving force equation. 

This is known as the quadratic driving force equation. A plot of ln[l — (Cs/C )2] against t gives a straight line and the diffusion factor icDs/r2 may be obtained from the slope. 

Alternatively, use may be made of the quadratic driving force equation ... 

A quadratic-driving-force approximation has been developed by Vermeulen (V3). For r = 0 this fits Eq. (81) closely over the entire range of x. This can be extended to the general form ... 

For solid homogeneous diffusion (quadratic driving force), it is (6)... 

One of the most fundamental achievements of the seminal Marcus theory on electron-transfer reactions is the nowadays widely used quadratic driving force-activation free energy Marcus relationship... 

The breakdown of the linear rate approximation for nonlinear systems was noted by Vermuelen who developed modified lumped parameter approximations which represent the diffusion models more accurately than the simple linear rate expression. For solid diffusion (model 2a or 2b) a quadratic driving force approximation is recommended ... 

Possibilities for a single resistance include a linear rate expression with a lumped parameter mass transfer coefficient based either on the external fluid film or on a hypothetical solid film, depending on which film is controlling the rate of uptake of adsorbate. A quadratic driving force expression, again with a lumped parameter mass transfer coefficient, may be used instead. Alternatively, intraparticle diffusion, if the dominant form of mass transfer, may be described by the general diffusion equation (Pick s second law) with its appropriate boundary conditions, as described in Chapter 4. 

The striking correlation of the homogeneous and heterogeneous rate constants, when both processes are measured under conditions of equivalent thermodynamic driving force, is presented in Figure 8 (left) (9). Note the slope of 0.76. We now turn to the quadratic... 

In contrast with the Butler-Volmer approximation, the activation-driving force relationships [equations (1.32)] are quadratic rather than linear. This... 

We examine next the cyclic voltammetric responses expected with nonlinear activation-driving force laws, such as the quasi-quadratic law deriving from the MHL model, and address the following issues (1) under which conditions linearization can lead to an acceptable approximation, and (2) how the cyclic voltammograms can be analyzed so as to derive the activation-driving force law and to evidence its nonlinear character, with no a priori assumptions about the form of the law. 

The following expressions of the transfer coefficient (symmetry factor) result from the quadratic activation-driving force relationship in equation (3.3) and from the definition of the intrinsic barrier in equation (3.4) ... 

Convolution allows an easier and more precise derivation of the activation-driving force law and characterization of the small values of a for dissociative electron transfer. It is also a convenient means of demonstrating its quadratic character, and thus of the linear variation of a with potential, as shown in the case of the reduction of organic peroxides.7... 

Another important aspect of the Marcus theory has also been systematically investigated with organic molecules, namely the quadratic, or at least the non-linear, character of the activation-driving force relationship for outer sphere electron transfer. In other words, does the transfer coefficient (symmetry factor) vary with the driving force, i.e. with the electrode... 

RX/RX " couple but that of the RX/R + X couple. Under such conditions, the transfer coefficient should thus be smaller than 0.5 in the framework of the quadratic activation-driving force relationship predicted to hold for dissociative electron transfer by the model developed in the preceding section. 

The reduction of aryl-substituted vinyl halides by electrochemically generated aromatic anion radicals has also been investigated in DMF (Gatti et al., 1987). Counter-diffusion behaviour at low driving forces (pp. 34, 35) does not appear as clearly as in the case of aryl halides (Fig. 11). However, analysis of the log k vs E° plot according to a quadratic activation-driving force relationship gave standard potential and intrinsic barrier values that... 

Since the curvature is inversely proportional to AGq, it follows that it might be difficult to detect the expected quadratic activation/driving force relationship for reactions characterized by large AGq values. Conversely, when the latter values are not too large, the expected parabolic pattern should be detectable beyond experimental error. 

Using the values of E°roor/rovro- determined from thermochemical cycles or convolution analysis, the log fchom can be plotted as a function of the reaction free energy shown in Figs 8 and 9. The data convincingly illustrate the parabolic nature of the expected quadratic-activation driving force relationship and thus equation (7). Because of the low BDE(O-O), and thus low intrinsic barrier, the parabolic nature is much more apparent in these systems than in others that undergo dissociative ET. 

The quadratic dependence of the Gibbs energy of activation on the driving force implies that the transfer coefficient a is no longer a constant. Instead, it depends linearly on the overpotential. Figure 6.7 illustrates the free energy profile curves along the reaction coordinates for the reaction... 

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