Potential-dependent free energy

Later in this chapter, we ll see that cell potentials, like free-energy changes, depend on the composition of the reaction mixture. The standard cell potential E° is the cell potential when both reactants and products are in their standard states—solutes at 1 M concentrations, gases at a partial pressure of 1 atm, solids and liquids in pure form, with all at a specified temperature, usually 25°C. For example, E° for the reaction... 

Cell potentials, like free-energy changes (Section 17.10), depend on temperature and on the composition of the reaction mixture—that is, on the concentrations of solutes and the partial pressures of gases. This dependence can be derived from the equation... 

It is easy to see that Eqs. (17-59), (17-60), and (17-61) are equivalent to Eq. (17-58). It must be noted that Eq. (17-60) expresses the solvation free energy of a molecule with a pairwise additive potential, hence the theory of energy representation described in Section 17.3.4 can be applied without any further approximations. An appropriate choice of E and h(r) will make the contribution E + Ap major in the total excess chemical potential. The free energy change expressed by Eq. (17-61) directly depends on the choice of the standard energy E and involves many-body effects since the solute-solvent interaction is described by Eqm/mm (n, X) at the final state... 

The chemical potential or free energy of a species, the latter being a component in a solid or in a solution, depends on the chemical environment. In the case of a charged species, such as an ion or an electron, we have to consider in addition the electrical energy required for bringing a charge to the site of the species. Accordingly, an electrochemical potential Mi s defined instead of the chemical potential. Both are related by... 

Eor clarity we only derive Eq. 1 for the simplest case. However, Eq. 1 is also valid for arbitrary many-body interactions [7] as well as time-dependent external potentials V r i). The latter simply leads to a time-dependent free energy functional F[p f] which is of the same form as the functional in Eq. 2. 

From the computation of different reaction pathways, potential or free energies for various mechanisms are compared with each other and the most likely mechanism is identified based on energetic considerations. Comparison with experimental data is then used to substantiate or invalidate the proposed mechanism, as it has to agree with and/or explain experimental findings such as the importance of a certain residue, the dependence of a metal cofactor, the preference for certain substrates, and so on. Once a likely mechanism is determined, the structural and energetic contributions to catalysis are determined using a variety of methods. 

Fredrickson and Binder [9] further improved this theory to describe the kinetics of the ordering process. Their concentration dependent free energy potential shows two side minima, which have the same depth as the middle one at the microphase separation transition temperature Tmst- Therefore, they presume a coexistence of the disordered and the lamellar phase at Tmot- As the temperature is further lowered, these side minima become dominant and the transition comes to completion. For a supercooled material, they expect after a completion time an Avrami-type ordering transformation with an exponent of 4 equivalent to spherically growing droplets of ordered material. This characteristic time corresponds to the time to form stable droplets of ordered material plus the time needed for the structures to grow to a size that they can be detected by the used technique. 

Ruckenstein (Ruckenstein and Chi, 1975 Friberg and Venable, 1983 Flanagan and Singh, 2006) explored the stability and size of microemulsion domains based on a thermodynamic approach where interactions (Van der Waals forces, electric double layer potential, and free energy and entropy of formation) were considered. He studied the dependence of the free... 

Voltammetry is an important tool for evaluating electrochemical and electro-catalytic processes. In a voltammetric experiment, the potential of a working electrode is varied with time relative to a reference electrode. The current of the working electrode is measured and reported as a function of potential. If the potential is swept linearly with time, peaks or waves are observed, which can be attributed to the various electrochemical processes possible in the system. For comparison with experiment, DFT calculated energetics can be used to predict voltammetry results in much the same way microkinetic models are used to predict catalytic kinetics. In the sections above, we have discussed DFT methods to calculate elementary reaction or adsorption free energies as a function of electrode potential. These free energy differences can be used to calculate potential dependent equilibrium constants. Section 3.2.5 will present a method to calculate potential dependent activation barriers. With these values for all possible elementary reaction steps, we could use microkinetic modeling to simulate voltammetry and compare with experiment. 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

A potential advantage of methods based on a series expansion of the free energy is that the convergence of the series is determined by the A dependence of the potential energy function meaning that the efficiency of the approach could be enhanced by a judicious choice of coupling scheme. 

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