Potential solvent average

The thennodynamic properties are calculated from the ion-ion pair correlation fimctions by generalizing the expressions derived earlier for one-component systems to multicomponent ionic mixtures. For ionic solutions it is also necessary to note that the interionic potentials are solvent averaged ionic potentials of average force ... 

Flere u. j(r,T,P) is the short-range potential for ions, and e is the dielectric constant of the solvent. The solvent averaged potentials are thus actually free energies that are fimctions of temperature and pressure. The... 

A chemical will be a solvent for another material if the molecules of the two materials are compatible, i.e. they can co-exist on the molecular scale and there is no tendency to separate. This statement does not indicate the speed at which solution may take place since this will depend on additional considerations such as the molecular size of the potential solvent and the temperature. Molecules of two different species will be able to co-exist if the force of attraction between different molecules is not less than the forces of attraction between two like molecules of either species. If the average force of attraction between dissimilar molecules A and B is and that between similar molecules of type B Fbb and between similar molecules of type A F a then for compatibility Fab - bb and AB - P/KA- This is shown schematically in Figure 5.5 (a). 

Note that the additional factor within the average, the n j (1 — bj), would be zero for any solvent configuration in which a solvent molecule is found in the inner shell. Thus, this expression involves a potential distribution average under the constraint that no binding in the inner shell is permitted. We can formally write the full expression for the excess chemical potential as... 

In Eq. (154), we assume indeed that only the ions (Z 0) interact with each other and that the resulting interaction is simply the Coulomb potential modified by the zero-frequency dielectric constant e of the solvent. Of course, in an exact theory, we would have to take explicitly into account the interactions with the solvent, and the dielectric constant itself should come out of the calculation. The proper way of attacking this problem is based on the theory of the potential of average forces and is carefully analyzed in H. L. Friedman s monograph.11 However, the explicit calculations always become exceedingly complicated and, in one way or another, one always has to have recourse to an approximation of the type (154). It amounts to assuming ... 

In a McMillan-Mayer level model (MM-level) for a solution, the particles are the solute particles (i.e. the ions with positive, negative, or zero charge). The ion-ion potentials can, in principle, be generated by calculations in which one averages over the solvent coordinates in a BO-level model which sees the solvent particles. (k,5,12) Pairwise additivity (we use overbars for solvent-averaged potentials)... 

Figure 2. Solvent-averaged potential for charged hard-sphere ions in a dipolar hard-sphere solvent. MC approximation by Patey and Valleau (16) and LHNC approximation by Levesque, Weis, and Patey (11). Also shown are the primitive model functions for solvent dielectric constants 9.6 and 6.

For molten salts one sets so = 1. For electrolyte solutions real fluids, eo in Eq. (11) depends on the ion density [167]. Usually, one sets so = s, where e is the dielectric constant of the solvent. A further assumption inherent in all primitive models is in = , where is the dielectric constant inside the ionic spheres. This deficit can be compensated by a cavity term that, for electrolyte solutions with e > in, is repulsive. At zero ion density this cavity term decays as r-4 [17, 168]. At... 

More refined continuum models—for example, the well-known Fumi-Tosi potential with a soft core and a term for attractive van der Waals interactions [172]—have received little attention in phase equilibrium calculations [51]. Refined potentials are, however, vital when specific ion-ion or ion-solvent interactions in electrolyte solutions affect the phase stability. One can retain the continuum picture in these cases by using modified solvent-averaged potentials—for example, the so-called Friedman-Gumey potentials [81, 168, 173]. Specific interactions are then represented by additional terms in (pap(r) that modify the ion distribution in the desired way. Finally, there are models that account for the discrete molecular nature of the solvent—for example, by modeling the solvent as dipolar hard spheres [174, 175]. 

Solvent-induced effects on phase equilibria have also been described by models based on solvent-averaged Friedman-Gumey potentials using the HNC approximation [81]. The difficulty in extracting phase transition lines from HNC calculations has been noted earlier, but only the HNC seems to be flexible enough to account for specific interactions (e.g., present in solvophobic mechanisms). 

The pmf can be decomposed into two contributions, a direct one, that is the ion-ion potential in vacuo, and an indirect one which is the solvent-averaged interaction between the solutes. The latter is obviously much more interesting, and difficult to obtain, so we are not going to discuss models for the direct ion-ion interactions. 

A multicomponent model is often used to describe spherical micelles or globular proteins in solution. In this case the ions are treated as charged hard spheres immersed in a solvent of dielectric constant uy r. In this way the micellar solution is depicted as an electrolyte where the ions grossly differ in size and charge. The solvent averaged potential in this case is given by (2aab = aa+crb) the equation,... 

However, the pair correlation function as well as the potential of average force are finite at this limit. We can think of WAA(R) in the limit of pA — 0 as the work required to bring two A s from infinite separation to the distance R in a pure solvent B at constant Tand V(or T, P depending on the ensemble we use). 

The model is a McMillan-Mayer (MM)-level Hamiltonian model. Friedman characterizes models of this type as follows With MM-models it is interesting to see whether one can get a model that economically and elegantly agrees with all of the relevant experimental data for a given system success would mean that we can understand all of the observations in terms of solvent-averaged forces between the ions. However, it must be noted that there is no reason to expect the MM potential function to be nearly pairwise additive. There is an upper Imund on the ion concentration range within which it is sensible to compare the model with data for real systems if the pairwise addition approximation is made. ... 

In this section we consider two ions a and b, of arbitrary species but assumed for simplicity to be spherical, at an arbitrary but fixed center-to-center distance r in a large mass of solvent. The potential Uahir) of the force between the ions is called a solvent-averaged potential. Its relation to the intermolecular forces in a model at the BO level is given in Section 4 but it has some features that can be discussed here in a less formal way. 

The solvent-averaged pair potential as given by Eqs. (6) and (4) can be expected to be quite realistic for large r or for small r. In the range... 

Although these studies may provide the basis for important future developments, in most of them the solvent-averaged potentials Uab for anion-cation pairs have such deep minima that in a physical model the ions would be largely associated into ion pairs or larger clusters at concentrations above 10 M. One should therefore be cautious when comparing the results of these studies directly to those real ionic systems in which the solute is largely ionized, even above 1M electrolyte concentration. 

The McMillan-Mayer theory shows that the osmotic pressure of a solution, the thermodynamic functions that may be derived from the osmotic pressure as a function of composition, and the solute-solute correlation functions can all be expressed as functionals of the solvent-averaged potentials... 

The solvent-averaged potentials Un are strong functions of the temperature and of the activity of the solvent, and hence of the pressure Pom in Fig. 1. This must be borne in mind when differentiating expressions such as Eq. (42) with respect to temperature or Pom, for truly realistic models. It is well known that temperature-dependent potentials in statistical mechanics, whether at the BO or MM level, correspond closely to free energies in thermodynamics. 

Perhaps at this time the best procedure is to use MM models in which the solvent-averaged potentials are pairwise additive but to limit their application to solutions that are dilute enough so that configurations in which three or more ions are mutually close together are quite improbable. 

Hess B, van der Vegt NFA (2007) Solvent-averaged potentials for alkali-, earth alkali-, and alkylammonium halide aqueous solutions. J Chem Phys 127 234508 Collins KD (1997) Charge density-dependent strength of hydration and biological structure. Biophys J 72 65-76... 

Solvent potential. The averaged solvent electrostatic field, , is important for inhomogeneous media, such as enzymes, membranes, miscelles and crystalline environments systems. Due to the existence of strong correlations, such a field does not cancel out. This factor becomes an important contribution to solvent effects at a microscopic level. In a study of non-rigid molecules in solution, Sese et al. [25] constructed a by using the solute-solvent atom-atom radial distribution function. Electrostatic interactions in three-dimensional solids were treated by Angyan and Silvi [26] in their self-consistent Madelung potential approach such a procedure can be traced back to a calculation of . An earlier application of the ISCRF theory to the study of proton mechanisms in crystals of hydronium perchlorate both [Pg.441]

Parametrization of the thermodynamic properties of pure electrolytes has been obtained [18] with use of density-dependent average diameter and dielectric parameter. Both are ways of including effects originating from the solvent, which do not exist in the primitive model. Obviously, they are not equivalent and they can be extracted from basic statistical mechanics arguments it has been shown [19] that, for a given repulsive potential, the equivalent hard core diameters are functions of the density and temperature Adelman has formally shown [20] (Friedman extended his work subsequently [21]) that deviations from pairwise additivity in the potential of average force between ions result in a dielectric parameter that is ion concentration dependent. Lastly, there is experimental evidence [22] for being a function of concentration. There are two important thermodynamic quantities that are commonly used to assess departures from ideality of solutions the osmotic coefficient and activity coefficients. The first coefficient refers to the thermodynamic properties of the solvent while the second one refers to the solute, provided that the reference state is the infinitely dilute solution. These quantities are classic and the reader is referred to other books for their definition [1, 4],... 

We now define the A -particle potential of average force in a pure solvent B by... 

This formal analogy assures us that if we expand the osmotic pressure in power series in the solute density Pa, the analogy between the virial expansion of the pressure and the virial expansion of the osmotic pressure will be maintained. For instance. Bo is given by an integral over the pair potential C/(Xi, X2) likewise, Bt will be given by the same form of an integral, but one in which the pair potential U Xx, X2) is replaced by the potential of average force )T0i (Xi, X2 z. = 0) for two solute particles in a pure solvent. 

In the virial expansion of the pressure, each coefficient Bj depends on the properties of a system of exactly j molecules. Likewise, in the virial expansion of the osmotic pressure, each coefficient Bf depends on the properties of j solutes in a pure solvent. It is true that in the expressions Bf, the solvent does not feature explicitly. This apparent simplification is quite deceiving, however. It should be realized that the potential of average force is dependent on the solvent properties. Therefore, in any actual calculation one must evaluate the relevant potential of average force, taking into account the properties of the solute as well as of the solvent. 

The first-order term involves the pair potential of average force. This may be sometimes approximated by viewing the solvent as a continuum. An example of such an approximation is the Debye-Hiickel theory of ionic solutions. 

The potential of average force is defined as the work (in the appropriate ensemble) required to bring particle i from infinity to a location R relative to a at the origin of our coordinate system. If the distance R is short, then Wia(R) includes contributions due to the electrostatic interactions as well as indirect interactions due to the solvent. The latter are present even when the particles i and a are uncharged (see section 6.4). In making the approximation (6.12.36), we have to consider the following three points ... 

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