Free energy quadratic dependence

A simple quadratic form of Eq. (34.10) is due to an identical parabolic form of the free-energy surfaces f/, and U. Since the dependence of the activation free energy on AF is nonhnear, the symmetry factor a may be introduced by a differential relationship,... 

If the interaction between the donor and acceptor in the encounter pair (D. .. A) is weak (Scheme 4.2), the rate constant kET can be estimated by the Marcus theory. This theory predicts a quadratic dependence of the activation free energy AG versus AG° (standard free energy of the reaction). 

The quadratic rate equation [Eq. (1)] of the continuum theory arises because it implicitly assumed the parabolic dependence of the free energy profile on the solvent coordinate q. One of the consequences of this quadratic equation is the generation of a maximum in the dependence of the rate of reaction on the free energy of reaction and also in current density-overpotential dependence. 

The E° difference is a necessary but not a sufficient condition. The rate constant for either ET (in general, / et) may be described in a simple way by equation (4). The activation free energy AG is usually expressed as a quadratic function of AG°, no matter whether we deal with an outer-sphere ET or a dissociative ET. However, even if the condition (AG")c < (AG°)sj holds (hereafter, subscripts C and ST will be used to denote the parameters for the concerted and stepwise ETs, respectively), the kinetic requirements (intrinsic barriers and pre-exponential factors) of the two ETs have to be taken into account. While AGq depends only slightly on the ET mechanism, is dependent on it to a large extent. For a concerted dissociative ET, the Saveant model leads to AG j % BDE/4. Thus, (AGy )c is significantly larger than (AG )sj no matter how significant AGy, is in (AG( )gj (see, in particular. Section 4). In fact, within typical dissociative-type systems such as... 

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... 

Note that the slopes of the Br0nsted and Tafel plots, need not necesarily be constant over a large free energy range and, in fact, the Marcus—Levich theoretical treatment predicts a quadratic dependence [32c]. 

This quadratic dependence of the activation energy on the reaction free energy leads to the prediction of an inverted region in which the reaction rate constant (which depends on AG ) falls when the overall reaction free energy becomes more favourable. This is readily seen from the simple picture shown in Figure 4.14. When the intersection point of the wells leads to A G = 0 the reaction becomes free of an activation barrier, but as the products well sinks deeper the point of intersection rises again. 

Fig. 18. (a) The temperature dependence of the free energy change for the RNase transition at different pH values. The points represent AF° values calculated from the data. The solid curves are the best fit to a quadratic equation by least squares analysis. The dashed lines indicate the range of relatively high experimental accuracy, (b) The values of AH°, AS", and ACP for the RNase transition from data at pH 2.50. Reproduced from Brandts and Hunt (336). 

External stress, locally applied, can have nonlocal static effects in ferroelastics (see Fig. 4 of Ref. [7]). Dynamical evolution of strains under local external stress can show striking time-dependent patterns such as elastic photocopying of the applied deformations, in an expanding texture (see Fig.5 of Ref. [8]). Since charges and spins can couple linearly to strain, they are like internal (unit-cell) local stresses, and one might expect extended strain response in all (compatibility-linked) strain-tensor components. Quadratic coupling is like a local transition temperature. The model we consider is a (scalar) free energy density term... 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

The energy E = tfk2/ m for a free particle has a quadratic dependence on the wavevector k. 

This process is illustrated schematically in Fig. 2 [40b]. Figure 2(a) traces the alterations in the free energy of the reacting species the states O and R refer to the chemical free energies, G°B, of the oxidized and reduced forms of the redox couple, respectively, that are associated with solute-solvent interactions. Step 1 entails moving down the curve OSR the bowed shape arises from the quadratic dependence of G°s on the ionic charge that is anticipated from the Born model. State S is that formed upon completion of step 1, so that the vertical line ST refers to step 2 yielding the transition state... 

Equation [79] produces the MH quadratic energy gap law at small AEo l i il and yields a linear dependence of the activation energy on the equilibrium free energy gap at AFq - k a lai >... 

As a consequence, the rate constants for outer-sphere electron transfers depend on the free energy of the reaction as a characteristic (quadratic) function, and inner-sphere ET reactions are readily revealed by substantial deviations from the Marcus behavior described in Eq. 90. 

The free energy of stretching a real linear chain in a good solvent has a stronger dependence on size R than the quadratic dependence of the ideal chain ... 

In the classical treatment electron transfer is assumed to be adiabatic and the rate depends exponentially on the reorganization free energy (Table 1). The latter varies as the square of the displacement along the reaction coordinate. This quadratic dependence is a consequence of the assumptions that the inner-sphere reorganization energy depends on the square of the nuclear displacement from equilibrium, and that the solvent-polarization energy depends on the square of the difference between the hypotheti-... 

The WS mixing rule satisfies the low-density boundary condition that the second virial coefficient be quadratic in composition and the high-density condition that excess free energy be produced like that of currently used activity coefficient models, whereas the mixing rule itself is independent of density. This model provides a correct alternative to the earlier ad hoc density-dependent mixing rules (Copeman and... 

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