Free-energy dependence

Sekiguchi S, Kobori Y, Akiyama K and Tero-Kubota S 1998 Marcus free energy dependence of the sign of exchange interactions in radical ion pairs generated by photoinduced electron transfer reactions J. Am. Chem. Soc. 120 1325-6... 

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

The parameters, Aj and Bj, can be obtained via thermodynamic integration using titration simulations of the model compound at different 9 values [58]. Finally, UpH (0j) models the free energy dependence on the external pH by... 

Fig. 4 Free energy dependence of the rate constants for charge separation and charge recombination for hairpins in which two A T base pairs separate the linker acceptor from the nucleobase donor. The dashed line is a fit of the charge separation data to the Marcus-Levitch-Jortner equation...

Fig. 24 Bell-shaped free-energy dependence of the formation constants Kec of the encounter complex [ArH, Q ] in dichloromethane. Reproduced with permission from Ref. 260a.

Relative activation enthalpies (Aif) in Table 2 were converted to o% kx k ) at 298 K, and were plotted against Hammett a constants. Here, we used enthalpies, because the size of the entropy and hence the free energy depend much on low frequencies, which are less reliable than higher frequencies, especially for compounds with weak interactions such as TS (8). The use of free energy (AG ) gave similar correlations with more scattered points. As for the Hammett o constant, we used dual-parameter o constants in the form of the Yukawa-Tsuno equation (LArSR equation) (9) as defined in eq 3. Here, the apparent a constant (aapp) has a variable resonance contribution parameter (r), which varies depending on the nature of the reaction examined for t-cumyl... 

T.W. Hamann, F. Gstrein, B.S. Brunschwig, N.S. Lewis, Measurement of the free-energy dependence of interfacial charge-transfer rate constants using Zn0/H2 0 semiconductor/liquid contacts, J. Am. Chem. Soc. 127 (2005) 7815-7824. 

Contrasts in the Predicted Free Energy Dependence of Reaction Rates... 

However, the identicalness of protein molecules possessing the same macroconformation is not absolute. Within each structurally determined conformational macrostructure, there exists a microdisordering which is similar to that observed in amorphous solids and glasses.(U,14) It is associated with the presence of multiple relative minima of the free energy depending on small shifts and variations in orientation of certain groups within the limits of available space. 

According to eqs 7-10, the 5 values of peptides separated with a particular RPC system can be derived from the experimental plots of In k t versus if at specified T values by regression analysis. Moreover, the 5 value of a peptide in the presence of a RPC sorbent can be related1 7 31 32 34 via implicit linear free energy dependencies on extra-thermodynamic parameters, such as the accessible molecular surface area, AAmol, through the expression... 

When two phases of a substance are in equilibrium, their molar free energies are equal. Each molar free energy depends on the pressure and the temperature, so the condition for equilibrium of two phases a and (3 can be written... 

A catalyst speeds up both the forward and reverse reactions by the same amount. Therefore, the dynamic equilibrium is unaffected. The thermodynamic justification of this observation is based on the fact that the equilibrium constant depends only on the temperature and the value of AGr°. A standard reaction free energy depends only on the identities of the reactants and products, and is independent of its rate or the presence of any substances that do not appear in the overall chemical equation for the reaction. 

As explained in the previous section, truncatable here means that the excess free energy / depends only on K moment densities p,. Note that, in the first (ideal) term of (6), we have included a dimensional factor R(a) inside the logarithm. This is equivalent to subtracting T dap a) In R(a) from the free energy. Since this term is linear in densities, it has no effect on the exact thermodynamics it contributes harmless additive constants to the chemical potentials p a). However, in the projection route to a moment free energy, it will play a central role. 

In summary, we have shown that the projection and combinatorial methods for obtaining moment free energies give equivalent results. The only difference between the two approaches is that within the projection approach, one need not necessarily retain the zeroth moment, which is the overall density p0 = p, as one of the moment densities on which the moment free energy depends. If p0 does not appear in the excess free energy, this reduces the minimum number of independent variables of the moment free energy by one (see Section V for an example). 

As expected from the general discussion in Section III. A, the criterion (57) can also be derived from the exact free energy an alternative form involving the spinodal determinant Y is given in Appendix D. Equation (57) shows that the location of critical points depend only on the moment densities p[t py, and pijk [11, 46]. For a system with an excess free energy depending only on power-law moments up to order K - 1, the critical point condition thus involves power-law moments of the parent only up to order 3 (K — 1). 

As pointed out in Section III.A, the definition of the moment free energy depends on a prior R(p) and represents the properties of systems with density distributions p(a) in the corresponding maximum entropy family (7). Instead of identifying R(o) = p (cr), we now allow a general prior R(ff). Concep-... 

We now illustrate how the moment method is applied and demonstrate its usefulness for several examples. The first two (Flory-Huggins theory for length-polydisperse homopolymers and dense chemically polydisperse copolymers, respectively) contain only a single moment density in the excess free energy and are therefore particularly simple to analyze and visualize. In the third example (chemically polydisperse copolymers in a polymeric solvent), the excess free energy depends on two moment densities, and this will give us the opportunity to discuss the appearance of more complex phenomena such as tricritical points. 

Its free energy depends on both its enthalpy and its entropy ... 

Viscosity and density of the component phases can be measured with confidence by conventional methods, as can the interfacial tension between a pure liquid and a gas. The interfacial tension of a system involving a solution or micellar dispersion becomes less satisfactory, because the interfacial free energy depends on the concentration of solute at the interface. Dynamic methods and even some of the so-called static methods involve the creation of new surfaces. Since the establishment of equilibrium between this surface and the solute in the body of the solution requires a finite amount of time, the value measured will be in error if the measurement is made more rapidly than the solute can diffuse to the fresh surface. Eckenfelder and Barnhart (Am. Inst. Chem. Engrs., 42d national meeting, Repr. 30, Atlanta, 1960) found that measurements of the surface tension of sodium lauryl sulfate solutions by maximum bubble pressure were higher than those by DuNuoy tensiometer by 40 to 90 percent, the larger factor corresponding to a concentration of about 100 ppm, and the smaller to a concentration of 2500 ppm of sulfate. 

Figure 4 Free energy dependence of electron transfer reaction rate for case A where —X is approximately equal to AGS (or AGp) and case B where —X is closer in energy to AGq.

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