Thermodynamic potential more useful

Equilibrium potentials can be calculated thermodynamically (for more details, see Chapter 3) when the corresponding electrode reaction is known precisely, even when they cannot be reached experimentally (i.e., when the electrode potential is nonequilibrium despite the fact that the current is practically zero). The open-circuit voltage of any galvanic cell where at least one of the two electrodes has an nonequilibrium open-circuit potential will also be nonequilibrium. Particularly in thermodynamic calculations, the term EMF is often used for measured or calculated equilibrium OCV values. 

In contrast to the weights formalism, the partition function approach directly employs the ideal flat-histogram expression in (3.36). Its goal is not to determine q but Q(N. V, T) directly, or more precisely in this case, the N dependence of Q. Due to numerical reasons, we usually work instead with the associated thermodynamic potential which is the logarithm of the partition function of interest in this case it is In Q = — 7/1 =. A, where we have used script i as an abbreviation. Thus our sampling scheme becomes... 

The reactions that are more favored thermodynamically tend to be also favored kineti-cally. Semiconductor electrodes can be stabilized by using this effect. For this purpose, redox couples in the electrolyte are established with the redox potential more negative than the oxidative decomposition potential, or more positive than reductive decomposition potential in such a manner that the electrolyte redox reaction occurs preferentially compared to the electrode decomposition reaction. 

We shall treat more compUcated cases, such as systems with a larger number of identical or different sites, and also cases of more than one type of ligand. But the general rules of constructing the canonical PF, and hence the GPF, are the same. The partition functions, either Q or have two important properties that make the tool of statistical thermodynamics so useful. One is that, for macroscopic systems, each of the partition functions is related to a thermodynamic potential. For the particular PFs mentioned above, these are... 

Defect thermodynamics is more complicated when applied to binary (or multi-component) compound crystals. For binary systems, there is one more independent thermodynamic variable to control. In the case of extended binary solid solutions, one would normally choose a composition variable for this purpose. For compounds with very narrow ranges of homogeneity (i.e., point defect concentrations), however, the composition is obviously not a convenient variable. The more natural choice is the chemical potential of one of the two components of the compound crystal. In practice one will often use the vapor pressure ( activity) of this component. 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

The beauty of the fundamental equation for U (equation 2.2-8) is that it combines all of this information in one equation. Note that the Ns + 2 extensive variables S, V, and n, are independent, and the Ns + 2 intensive variables T, P, and /uj obtained by taking partial derivatives of U are dependent. This is wonderful, but equations of state 2.2-10 to 2.2-12 are not very useful because S is not a convenient independent variable. Fortunately, more useful equations of state will be obtained from other thermodynamic potentials introduced in Section 2.5. 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

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