Liquid solutions equilibrium energy functions

With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, Pvap and puq, whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A. 10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. 

In developing the equilibrium expression (2-10) a reaction (2-4) was described in which all of the components were in solution. From the nature of the free energy functions given in Equations 2-1 and 2-2 it follows that any gaseous component will be represented in the equilibrium expression by its partial pressure, and any pure liquid or solid by unity, since the logarithmic term is absent in Equation 2-2. Whenever the solvent appears in the chemical equation, its free energy is considered to be sufficiently close to that of the pure liquid, provided the solutions are reasonably dilute so that it too is represented in the equilibrium expression by unity based on using mole fraction as the measure of solvent water. 

Figure 1 Illustration of equilibrium solvation effects on reaction free energies as a function of the reaction coordinate s (a) for a. symmetric reaction with no variational effects and (b) for a nonsymmetric reaction in which the solvation causes the maximum of the free energy of activation profile to shift. The long-dashed curves are the gas-phase free energy of activation profile, solid cui ves are the liquid-phase free energy of activation profile, and the short-dashed curves are the solvation free energy of the solute, called AG in the text and AGsoiv in the figure. Free energies depicted in the figures are discussed in the text...

The function U fXj is called the PMF it was first introduced by Kirkwood to describe the structure of liquids [61]. It plays the role of a free energy surface for the solute. Notice that the dynamics of the solute on the free energy surface W(X) do not correspond to the true dynamics. Rather, an MD simulation on 1T(X) should be viewed as a method to sample conformational space and to obtain equilibrium, thermally averaged properties. 

The papers in the second section deal primarily with the liquid phase itself rather than with its equilibrium vapor. They cover effects of electrolytes on mixed solvents with respect to solubilities, solvation and liquid structure, distribution coefficients, chemical potentials, activity coefficients, work functions, heat capacities, heats of solution, volumes of transfer, free energies of transfer, electrical potentials, conductances, ionization constants, electrostatic theory, osmotic coefficients, acidity functions, viscosities, and related properties and behavior. 

The thermodynamic quantity 0y is a reduced standard-state chemical potential difference and is a function only of T, P, and the choice of standard state. The principal temperature dependence of the liquidus and solidus surfaces is contained in 0 j. The term is the ratio of the deviation from ideal-solution behavior in the liquid phase to that in the solid phase. This term is consistent with the notion that only the difference between the values of the Gibbs energy for the solid and liquid phases determines which equilibrium phases are present. Expressions for the limits of the quaternary phase diagram are easily obtained (e.g., for a ternary AJB C system, y = 1 and xD = 0 for a pseudobinary section, y = 1, xD = 0, and xc = 1/2 and for a binary AC system, x = y = xAC = 1 and xB = xD = 0). 

Up to now our attention was mainly devoted to calculations of the energy and other quantities referring to free, isolated molecules. The computational techniques and their applications were demonstrated to be profitable in the exploration of physico-chemical properties of free molecules and their reactivity in the gas phase (thermodynamic functions, equilibrium and rate constants). However, the gas-phase processes represent only a special minor part of chemistry. Not only processes in biological systems, but also processes in laboratory conditions proceed typically in the liquid phase - or expressed more specifically - in the solution. It is therefore not surprising that the effort for applications of ab initio calculations is also still increasing in this very important field . ... 

The activity coefficient of a component in a mixture is a function of the temperature and the concentration of that component in the mixture. When the concentration of the component proaches zero, its activity coefficient approaches the limiting activity coefficient of th component in the mixture, or the activity coefficient at infinite dilution, y . The limiting activity coefficient is useful for several reasons. It is a strictly dilute solution property and can be used dir tly in nation 1 to determine the equilibrium compositions of dilute mixtures. Thus, there is no reason to extrapolate uilibrium data at mid-range concentrations to infinite dilution, a process which may introduce enormous errors. Limiting activity coefficients can also be used to obtain parameters for excess Gibbs energy expressions and thus be used to predict phase behavior over the entire composition range. This technique has been shown to be quite accurate in prediction of vapor-liquid equilibrium of both binary and multicomponent mixtures (5). 

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