McMillan-Mayer theory limitations

The McMillan-Mayer theory allows us to develop a formalism similar to that of a dilute interacting fluid for solute dispersed in the solvent provided that a sensible description of W can be given. At the limit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + kT In (p A Ji " ), where W(a s) is the potential of mean force for the interaction of a solute molecule with the solvent. If we define = Ll then the grand canonical partition... 

The McMillan-Mayer theory offers the most useful starting point for an elementary theory of ionic interactions, since at high dilution we can incorporate all ion-solvent interactions into a limiting chemical potential, and deviations from solution ideality can then be explicitly connected with ion-ion interactions only. Furthermore, we may assume that, at high dilution, the interaction energy between two ions (assuming only two are present in the solution) will be of the form... 

There are essentially two theories of solutions that can be considered exact the McMIllan-Mayer theory and Fluctuation Solution Theory (FST). The first Is mostly limited to solutes at low concentrations, while FST has no such issue. It is an exact theory that can be applied to any stable solution regardless of the number of components and their concentrations, and the types of molecules and their sizes. Fluctuation Theory of Solutions Applications in Chemistry, Chemical Engineering, and Biophysics outlines the general concepts and theoretical basis of FST and provides a range of applications described by experts in chemistry, chemical engineering, and biophysics. 

Many, if not most, processes of interest occnr in solutions. It is therefore somewhat unfortunate that our understanding of solutions and their properties remains rather limited. There are essentially two theories of solutions that can be considered exact. These are the McMillan-Mayer theory of solutions and Fluctuation Solution Theory (FST), or the Kirkwood-Buff (KB) theory of solutions. The former has practical issues, which limit most applications to solutes at low concentrations. The latter has no such issues. Nevertheless, the general acceptance and appreciation of FST remains limited. It is the intention of this book to outline and promote the considerable advantages of using FST/KB theory to study a wide range of solution properties. 

To conclude this section on the DH theory, we would like to point out that these last two criticisms (neglecting short range repulsive interactions and linearizing the PBE) are the only valid criticisms. In fact the McMillan-Mayer theory (MMM) showed that, provided a correct definition of the "effective interaction potential" is given, the molecular structure of the solvent needs not to be considered explicitly(1) in calculating the thermodynamic properties of ionic solutions. This conclusion has very important consequences the first one is that, as the number density of ion in a typical electrolyte solutions is of the order of 10"3 ions/A, then the solution can be considered as a dilute ionic gas as a consequence the theories available for gases can be used for ionic fluids, provided the "effective potential" (more often called potential of the mean force at infinite dilution) takes the place ot the gas-gas interaction potential. Strictly this is true only in the limit of infinite dilution, but will hold also at finite concentrations, provided the chemical potential of the solvent in the given solution is the same as in the infinitely dilute solutions. This actually... 

Chapter 6 is the extension of Chapter 5 to include mixtures of two or more liquids. The most important concepts here are ideal behavior and small deviations from it. Most of the treatment is based on the Kirkwood-Buff theory of solutions. The derivation and a sample application of this powerful theory are presented in detail. We also present the elements of the McMillan-Mayer theory, which is more limited in application. Its main result is the expansion of the osmotic pressure in power series in the solute density. The most useful part of this expansion is the first-order deviation from ideal dilute behavior, a result that may also be obtained from the Kirkwood-Buff theory. 

The above expression holds at any solute concentration. The analysis of osmotic pressure data for solutes therefore provides an indication of solute-solute affinity at the concentration of interest (Karunaweera et al. 2012). One can also show that MM theory is obtained from FST in the limiting case of an infinitely dilute solute, where both approaches then provide a series expansion in concentration (McMillan and Mayer 1945 Kirkwood and Buff 1951). A more detailed discussion is provided in the literature (Cabezas and O Connell 1993). 

In addition, because of its relative simplicity, we have used the Kirkwood-Buff theory for characterization of the various ideal solutions, as well as for the study of first-order deviations from ideal solutions. Although formal theories, such as that of McMillan and Mayer (1945), exist which provide expressions for higher-order deviations from ideality, their practical usefulness is limited to first-order terms only. Higher-order terms usually involve higher-order molecular distribution functions, about which little is known. 

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