Solvation chemical potential

The RISM integral equations in the KH approximation lead to closed analytical expressions for the free energy and its derivatives [29-31]. Likewise, the KHM approximation (7) possesses an exact differential of the free energy. Note that the solvation chemical potential for the MSA or PY closures is not available in a closed form and depends on a path of the thermodynamic integration. With the analytical expressions for the chemical potential and the pressure, the phase coexistence envelope of molecular fluid can be localized directly by solving the mechanical and chemical equilibrium conditions. 

For the KH approximation (4.13), the solvation chemical potential keeps an analytical form too ... 

Solvation chemical potential of an ionic cluster in electrolyte solution... 

The closed analytical expressions (4.14) or (4.15) for the excess chemical potential of solvation are no more valid in the SC-3D-RISM approach, since the orientational averaging (4.56) breaks the symmetry of the 3D-RISM integral equation with respect to the solvent indices. Nevertheless, the solvation chemical potential obtained from the SC-3D-RISM/HNC equation (4.59) does take the HNC form (4.14) within the additive approximation (4.64), (see Appendix). The use of the 3D-KH closure leads with account of (4.64) to the solvation chemical potential (4.15). 

Applying Eq. (4.68) to the solvation chemical potential (4.A. 17) yields the solvation energy in the form... 

With the ion-dipole asymptotics (4.74) and (4.75), calculation of the solvation chemical potential from expression (4.A.17) requires analytical treatment of the long-range contributions. They appear in the terms (/r (r)) and h r)c (r) but cancel out in c (r) owing to the electroneutrality of the solvent molecule. On separating out the electrostatic terms, the integration can be performed merely over the supercell volume,... 

Derivation of this expression in the case of the solvation chemical potential specified in the 3D-KH approximation is given in Appendix. Notice that the mean field potential (4.101) follows essentially from the use of the solvation chemical potential in either form (4.14) or (4.15). 

Similarly to the expressions found by Singer and Chandler [80] for the RISM/HNC equations, the KH approximation (4.f3) allows one to obtain the free energy functions in a closed analytical form avoiding the necessity of numerical coupHng parameter integration. The derivation is analogous for both RISM and 3D-RISM/KH equations [28], and is shown here in the context of the 3D approach. The excess part of the solvation chemical potential, in excess over the ideal translational term, can be related to the 3D site correlation functions by the Kirkwood s charging formula... 

Transfer chemical potentials solvation metal ions, 2,298 Transferrins, 2, 772 6, 669... 

The normal pressure Pn In the fiuld confined between the walls varies with wall separation and Is not. In general, equal to the bulk pressure P3 of fiuld at the same chemical potential. The difference Pn - Pb Is the solvation force per unit area, fg, and can be calculated from the equilibrium density profiles by... 

The ions in solution are subject to two types of forces those of interaction with the solvent (solvation) and those of electrostatic interaction with other ions. The interionic forces decrease as the solution is made more dilute and the mean distance between the ions increases in highly dilute solutions their contribution is small. However, solvation occurs even in highly dilute solutions, since each ion is always surrounded by solvent molecules. This implies that the solvation energy, which to a first approximation is independent of concentration, is included in the standard chemical potential and has no influence on the activity. 

In Equation (1) we assume particles are spherical with radius r. The chemical potentials are and for the particle and the solvated atoms or molecules, respectively, n is the number of moles per unit volume and a is the surface energy (or tension). Since the particle has formed, we can take the bulk term as negative with Ap = p — Ps<0 hence favorable, but formation of the surface costs energy so is positive and unfavorable. These two functionalities yield a maximum in AG. Differentiation of Equation (1) finds this maximum to be at a critical size Vc given by... 

This dependence is fundamental for electrochemistry, but its key role for liquid-liquid interfaces was first recognized by Koryta [1-5,35]. The standard transfer energy of an ion from the aqueous phase to the nonaqueous phase, AGf J, denoted in abbreviated form by the symbol A"G is the difference of standard chemical potential of standard chemical potentials of the ions, i.e., of the standard Gibbs energies of solvation in both phases. 

Equation (31) is true only when standard chemical potentials, i.e., chemical solvation energies, of cations and anions are identical in both phases. Indeed, this occurs when two solutions in the same solvent are separated by a membrane. Hence, the Donnan equilibrium expressed in the form of Eq. (32) can be considered as a particular case of the Nernst distribution equilibrium. The distribution coefficients or distribution constants of the ions, 5 (M+) and B X ), are related to the extraction constant the... 

The elementary step of ion transfer is considered to take place between positions x and X2, and therefore the electrical potential drop affecting this transfer is Ao02- The ion transfer involves the renewal of the solvation shell. The change in standard chemical potential Ao f associated with this process takes place over very short distances in the interfacial region [51] and can be assumed to occur between positions X2 and x - Thus, the BV equation for the flux density /, of an ionic species i is [52]... 

We shall look more closely at this equation. On one hand, the standard chemical potentials of Ox and Red depend on their standard Gibbs solvation energies, AG ox and AG°Red, and, on the other hand, on the standard Gibbs energy of ionization of Red in the gas phase, AGlon Red. This quantity is connected with the ionization potential of Red, /Rcd, which is, however, a sort of enthalpy so that it must be supplemented by the entropy term, -TA5 on Red. Thus, Eq. (3.1.17) is converted to the form... 

In Equation 50 the chemical potential of non-electrolyte A depends on the usual choice of standard-state conventions described above, and the chemical potentials of both H2(g) and H+(sod are taken to be zero (this defines e.s.s., the electrolyte standard state). By setting the standard-state free energy of the solvated proton equal to zero, this standard-state convention... 

Note, in using Equations 50 and 53 above, that tabulations of thermodynamic data for electrolytes tend to employ a 1 molar ess concentration for all species in solution. For situations defined to have a standard-state pH value different from 0 (which corresponds to a 1 molar concentration of solvated protons), the standard-state chemical potentials for anions and cations are determined as... 

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