Excess chemical potential method

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

Two methods may be used, in general, to obtain the thermodynamic relations that yield the values of the excess chemical potentials or the values of the derivative of one intensive variable. One method, which may be called an integral method, is based on the condition that the chemical potential of a component is the same in any phase in which the component is present. The second method, which may be called a differential method, is based on the solution of the set of Gibbs-Duhem equations applicable to the particular system under study. The results obtained by the integral method must yield... 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

A) When only one of the two components in a binary solution is volatile, the excess chemical potential of the volatile component can be determined by the methods that have been discussed. However, we require the values of the excess chemical potential of the other component or of the molar... 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

The study of a ternary system over the entire range of composition presents a formidable experimental problem. Several methods have been developed by which the excess chemical potentials of two of the components can be calculated from known values of only one component. Only two of these methods are discussed here. Throughout the discussion we assume that the values of the excess chemical potential of the first component are measured or known at constant temperature and pressure. 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

Long ago, Langmuir suggested that the rate of deposition of particles on a surface is proportional to the density of particles in the vicinity of the surface and to the available area on the surface [1], However, the calculation of the available area is still an open problem. In a first approximation, one can assume that the available area is the total area of the surface minus the area already occupied by the adsorbed particles [1]. A better approximation can be obtained if the adsorbed particles, assumed to have the shape of a disk, are in thermal equilibrium on the surface, either because of surface diffusion and/or of adsorption/desorption kinetics. In this case, one can use one of the empirical equations available for the compressibility of a 2D gas of hard disks, calculate the chemical potential in excess to that of an ideal gas [2] and then use the Widom relation between the area available to one particle and its excess chemical potential on the surface (the particle insertion method) [3], The method is accurate at low densities of adsorbed particles, where the equations of state are accurate, but, in general, poor at high concentrations. The equations of state for hard disks are based on the virial expansion and only the first few coefficients of this... 

Local concentrations can be determined accurately, if the excess chemical potential at the point is known. A good method to determine the chemical potential was developed by Widom [30] and it was later improved by Svensson et al. [31]. This technique has been used in MC simulations in order to determine PC011 and Pi0n. 

In the grand equilibrium method, a simulation of the condensed phase is done to calculate the excess chemical potentials, /x,ex, and the partial molar volumes, V,-, of all components. One may use the test-particle insertion method [59] to calculate the excess chemical potentials and the partial molar volumes as... 

In this subsection, we presented an approximate scheme to evaluate the contribution S/u. of the electron density fluctuation to the excess chemical potential. Although we saw that this contribution is minor for a QM water molecule in ambient and supercritical water, it should emphasized that 8p, can be treated quantitatively in the method of energy representation. Actually, the treatment of the electron density fluctuation is not directly possible in the PCM and RISM-SCF methods. Furthermore, the approximate 8p, is exact to second order in the solvent density and in the electron density fluctuation. Thus, when the effect of the electron density fluctuation is weak, the calculation of 5p, is expected to be accurate. 

Excess chemical potentials As early as 2002 Lynden-Bell et al. published an investigation about the chemical potential of water and organic solutes in [C,mim] [Cl] [9], The authors stated ... the chemical potential is the most important thermodynamic property of a solute in solution because it determines the solubility and chemical reactivity of a solute. Within this seminal article the authors determined the excess chemical potential (pf) by means of theoretical methods. The excess chemical potentials pA of a series of molecules dissolved in the IL [C,mim][Cl] were calculated by a sequence of transformations [9], It is defined by... 

The excess chemical potential is thus determined from the average of exp[— (r )/fcBr]-In ensembles other than the canonical ensemble the expressions for the excess chemical potential are slightly different. The ghost particle does not remain in the system and so the system is unaffected by the procedure. To achieve statistically significant results many Widom insertion moves may be required. However, practical difficulties are encountered when applying the Widom insertion method to dense fluids and/or to systems containing molecules, because the proportion of insertions that give rise to low values of i (r ) falls dramatically. This is because it is difficult to find a hole of the appropriate size and shape. 

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