McMillan-Mayer model

Figure 10. Osmotic coefficient as a function of the reduced density of monomer units pp = pm.03, where prn is the number density of monomer units. Solvent primitive model (continuous lines) McMillan-Mayer model results (broken lines). From top to bottom a = 0.125,0.5, and 1.0 (each bead is charged).

From these results, the thennodynamic properties of the solutions may be obtamed within the McMillan-Mayer approximation i.e. treating the dilute solution as a quasi-ideal gas, and looking at deviations from this model solely in temis of ion-ion interactions, we have... 

The commonly used method for the determination of association constants is by conductivity measurements on symmetrical electrolytes at low salt concentrations. The evaluation may advantageously be based on the low-concentration chemical model (lcCM), which is a Hamiltonian model at the McMillan-Mayer level including short-range nonelectrostatic interactions of cations and anions [89]. It is a feature of the lcCM that the association constants do not depend on the physical... 

The study of McMillan-Mayer level models, in which the solvent coordinates have been averaged over so that only solvent-mediated ion-ion forces need be treated, is relatively well developed. However the real forces at this level are even more poorly known than the forces at the Born-Oppenheim level referred to above. It is found that McMillan-Mayer level models can be brought into good agreement with solution thermodynamic data. 

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)... 

It has been seen that reliable conductivity values are known only at low electrolyte concentrations. Under these conditions, even conductance equations for models such as the McMillan-Mayer theory (Sections 3.12 and 3.16) are known. However, the empirical extension of these equations to high concentration ranges has not been successful. One of the reasons is that conductivity measurements in nonaqueous solutions are still quite crude and literature values for a given system may vary by as much as 50% (doubtless due to purification problems). 

Many models are available for describing the thermodynamic behavior of solutions. " However, so far no one could satisfactorily simulate the solution behavior over the whole concentration range and provide the correct pressure and temperature dependencies. This generated interest in the thermodynamically rigorous theories of Kirkwood—Buff and McMillan—Mayer. In the present paper, the emphasis is on the application of the Kirkwood—Buff theory to the aqueous solutions of alcohols, because it is the only one which can describe the thermodynamic properties of a solution over the entire concentration range. The key quantities in the Kirkwood-Buff theory of solution are the so-called Kirkwood-Buff integrals (KBIs) defined as... 

In this section we consider the application of the concept of ion association to describe the properties of electrolyte solutions within the ion or McMillan-Mayer level approach. In this approach the effects of solvent molecules are taken into account by introducing the dielectric constant into Coulomb interaction law and by appropriately choosing the short-range part of ion-ion interactions. To simplify, we consider here the restrictive primitive model (RPM)... 

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. ... 

The distinguishing feature of MM-level models is that the solvent molecules do not appear explicitly in the Hamiltonian. The potential function is the potential of the forces among the ions after averaging over solvent coordinates, i.e., the forces on the ions at any fixed locations in the solvent. The rigorous foundation for the use of such models is given by the McMillan-Mayer theory described in Section 4. This theory permits all of the statistical-mechanical apparatus and approximation methods developed for the calculation of equilibrium properties of BO-level models to be applied to MM-level models. For the calculation of dynamical properties the situation is not so satisfactory. A new set of forces, not derivable from a potential, must be taken into account the fluctuating forces exerted by the solvent on the ions and the... 

While the McMillan-Mayer theory (Section 4) prescribes the iiabir) as functionals of the Hamiltonian of a BO-level model, little has been learned from this sort of direct approach. The main contributions are an analytical study of charged hard spheres in an uncharged hard-sphere solvent by Stell, " Monte Carlo and molecular dynamics studies of somewhat more realistic models, " " and a study using the mean spherical approximation (Section 7.3). ... 

Hamiltonian models are classified according to then-level of approximation. The features of Schroedinger (S), Born-Oppenheimer (BO), and McMillan-Mayer (MM) level Hamiltonian models are exemplified in Table I by a solution of NaCl in H2O. The majority of investigations on electrolyte solutions are carried out at the MM level. BO-Level calculations are a precious tool for Monte Carlo and molecular dynamics simulations as well as for integral equation approaches. However, their importance is widely limited to stractural investigations. They, as well as the S-level models, have not yet obtained importance in electrochemical engineering. S-Level quantum-mechanical calculations mainly follow the Car-Parinello ab initio molecular dynamics method. 

The primitive model of electrolytes constitutes a Arm basis for statistical-mechanical description of solutions of charged colloids. This model will be adopted throughout, and it originates from the more general McMillan-Mayer solution theory [62,63]. 

What follows will concern electrolyte solutions as well as molten salts. In fact, as we will see later, within the framework of the McMillan-Mayer theory(l), there is no difference in the mathematical treatment of a dilute aqueous solution of a given electrolyte and the corresponding molten salt. Of course, the density, temperature and potential energy will be different, but in both cases, the model to be used will be the same. It should then not be surprising that the next section starts with a discussion of the McMillan-Mayer and Debye-Hiickel theories(2) for dilute systems of charged particles. The Debye-Hiickel theory (DH) has been the most successful theory of electrolyte solutions and some of the modern approximations are simple extensions of DH theory, which are statistically consistent. 

A more fundamental approach is to attempt to model electrolyte solutions using statistical mechanical methods, of which there are two kinds of models (reviewed extensively elsewhere ° ) Born-Oppenheimer (BO) level models in which the solvent species as well as the ionic species appear explicitly in the model for the solution and McMillan-Mayer (MM) level models in which the solvent species degrees of freedom are integrated out yielding a continuum solvent approximation. Thus, for a BO level model, in addition to the interionic pair potentials one must specify the ion-solvent and solvent-solvent interactions for all of the ionic and solvent species. In this case, the interionic potentials do not contain the solvent dielectric constant in contrast to the MM-level models. Kusalik and Patey carefully discuss the distinction between these two approaches. 

In contrast to Pitzer s work, which is given in molalities (Lewis-Randall theory (LR)), the MSA naturally expresses thermodynamic quantities in terms of concentrations, in the framework of the McMillan-Mayer (MM) theory of solutions [33]. Thus, the data have to be converted from Lewis-Randall to McMillan-Mayer scale for adjusting the model to experiment. The basic ingredients of the LR-to-MM conversion have been given [34] and recently an approximate simple conversion has been tested [35]. The great advantage of this transformation is that it keeps the... 

Fig. 13. Comparison of the HNC calculations of the osmotic coefficients for the primitive-model (charged hard spheres) electrolyte with experiment. All of the results are for the McMillan-Mayer system (Rasaiah, 1970). 

Summarizing, one can say that the lattice theories need improvement and compact macromolecules need more refined treatment. We shall develop in this paper a refined and unified theory of macromolecular solutions with special emphasis on dilute solutions. We shall put our standpoint on the general theory of solutions developed by McMillan and Mayer in 1945 and Kirkwood and Buff in 1951 (9). TTiese theories do not use the lattice model and are more natural for application especially to dilute solutions. The theories extend statistical theories on gases and this is the reason why we used the name gas theories (70) in the beginning of this Introduction. 

In a general theory of solutions, McMillan and Mayer demonstrated the formal equivalence between the pressure of a gas and the osmotic pressure of a solution. Hence the ratio of the osmotic pressure O of a dilute solution to the concentration (number density) p of the solute can be expanded in a power series in p and the coefficients of the series can be expressed, as in the theory of a real gas, in terms of cluster integrals determined by intermolecular potential energy functions. The only difference is, as already mentioned, that in the solution these potentials are effective potentials of average force, which include implicitly the effects of the solvent, modelled as a continuum. 

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