Free energy curves

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

The environmental (i.e., solvent and/or protein) free energy curves for electron transfer reactions can be generated from histograms of the polarization energies, as in the works of Warshel and coworkers [79,80]. 

T Ichiye. Solvent free energy curves for electron transfer A non-lmear solvent response model. J Chem Phys 104 7561-7571, 1996. 

FIGURE 1-9 Free energy curve for a redox process at a potential more positive than the equilibrium value. 

Use die activated complex theory for explaining clearly how the applied potential affects the rate constant of an electron-transfer reaction. Draw free energy curves and use proper equations for your explanation. 

FIGURE 9.7 (a) A reaction that has the potential to go to completion (K> 1) is one in which the minimum of the free-energy curve (the position of equilibrium) lies close to pure products, (b) A reaction that has little tendency to form products (K < 1 lis one in which the minimum of the free-energy curve lies close to pure reactants. 

Fig. 7. The free-energy curves and the derivation of the liquidus and solidus curves.

Fig. 8. The phase diagram derived from the free-energy curves in Fig. 7.

Fig. 9. The free-energy curves calculated with the inclusion of the term ex for the solid phase.

FIGURE 35.2 Scheme of diabatic (solid line) and adiabatic (dashed line) free-energy curves for a simple electrochemical redox reaction Ox —> Red. 

While in previous ab initio smdies the reconstructed surface was mostly simulated as Au(lll), Feng et al. [2005] have recently performed periodic density functional theory (DFT) calculations on a realistic system in which they used a (5 x 1) unit cell and added an additional atom to the first surface layer. In their calculations, the electrode potential was included by charging the slab and placing a reference electrode (with the counter charge) in the middle of the vacuum region. From the surface free energy curves, which were evaluated on the basis of experimentally measured capacities, they concluded that there is no necessity for specific ion adsorption [Bohnen and Kolb, 1998] and that the positive surface charge alone would be sufficient to lift the reconstmction. 

Fig. 23 Parallel tempering of the free-energy curves in the overlapping windows as a function of the number of molten units for a single 1024-mer at a temperature of 2.967 p/A b. The y-axis is not for the absolute value of the free energy but for the relative distribution of the free energy (Hu and Frenkel, unpublished results)...

Fig. 12.2. Free energy data for electron transfer between the protein cytochrome c and the small acceptor microperoxidase-8 (MP8), from recent simulations [47]. Top Gibbs free energy derivative versus the coupling parameter A. The data correspond to solvated cytochrome c the MP8 contribution is not shown (adapted from [47]) Bottom the Marcus diabatic free energy curves. The simulation data correspond to cyt c and MP8, infinitely separated in aqueous solution. The curves intersect at 77 = 0, as they should. The reaction free energy is decomposed into a static and relaxation component, using the two steps shown by arrows a static, vertical step, then relaxation into the product state. All free energies in kcalmol-1. Adapted with permission from reference [88]...

We have seen that the free energy curves for the reactant and product states have the same curvature, so that the relaxation free energy is the same in the reactant and product states /L4plxjd = AA C. This equality reflects the fact that the dielectric susceptibility a (12.24) does not depend on the perturbing field or charge, and is the same in the reactant and product states. We then obtain... 

Fig. 13 Energy profiles for C C, Cl in DMF. Curve a potential energy in the gas phase. Curve b potential energy in the solvent (D p = 62 meV). Curve c variation of the solvation free energy. Curve d solvent reorganization energy.

Fig. 7. Alloy composition versus (a) the shift in the aluminum deposition potential, AEai and (b) the shift in the copper/zinc deposition potential Ai cu,Zn, for the deposition of fee Cu-Al and hep Zn-Al alloys, calculated by using Eq. (12) and Eq. (13) and the free energy curves from Figure 6. Reproduced from Stafford et al. [104] by permission of The Electrochemical Society.

Our problem now is to determine the functional form of this experimental free energy curve for the intrinsic rate constant ki for electron transfer. In addition to the Marcus eq 4, two other relationships are currently in use to relate the activation free energy to the free energy change in electron transfer reactions (15, JL6). 

Fig. 3.1. Variation in free energy G with reaction progress for the reaction bB + cC dD + eE. The reaction s equilibrium point is the minimum along the free energy curve.

The first microscopical computation of a free energy curve for a chemical reaction in solution was performed by the Jorgensen s group [41,52,53] ten years ago. They studied the degenerate SN2 reaction of chloride anion with methyl chloride in gas phase, in aqueous solution and in dimethylformamide (DMF) ... 

Monte Carlo simulations were carried out to determine the free energy curve for the reaction in solution. The simulations were executed for the solute surrounded by 250 water molecules (or 180 DMF molecules) in the isothermal-isobaric ensemble at 25 °C and 1 atm, including periodic boundary conditions. As a consequence, the Gibbs free energy is obtained in this case. There is sufficient solvent to adequately represent the bulk participation in the chemical reaction. 

M. J. Richardson. The Derivation of Thermodynamic Properties by DSC Free Energy Curves and Phase Stability. Thermochim. Acta 1993, 229, 1-14. 

If one now sets the potential of the working electrode more positive than that of equilibrium, the oxidation process is facilitated (as seen in Figure 5). Thus, the profile of the free energy curves becomes that illustrated in Figure 12, in which the energy barrier for the oxidation is lower than that of reduction. 

Figure 15 Relationship between the transfer coefficient a and the intersection angles of the free energy curves...

As far as the definition of a as a measure of the symmetry of the activation barrier is concerned, as shown in Figure 15, let us focus on the apical region of the intersection between the free energy curves illustrated in Figure 14. 

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