The McMillan-Mayer theory of solutions

The McMillan-Mayer (MM) theory is essentially a formal generalization of the theory of real gases. We recall that in the theory of real gases we have an expansion of the pressure in power series in the density of the form 

In the case of real gasses, the terms in the expansion (6.11.1) correspond to successive corrections to the ideal-gas behavior, due to interactions among pairs, triplets, quadruplets, etc., of particles. One of the most remarkable results of the statistical-mechanical theory of real gases is that the coefficients Bj depend on the properties of a system containing exactly j particles. For instance, B2 T) can be computed from a system of two particles in the system. 

Inherent in the MM theory is the distinction between the solute and the solvent. (Either or both may consist of one or more components. Here we treat only the case of one solute A in one solvent B.) Furthermore, the theory is useful for quite low solute densities. The most useful case is the expansion up to p, i.e., the first-order deviation from the dilute ideal behavior. Higher-order corrections to the ideal-gas equations are sometimes useful if we know the interaction energy among j particles. The situation is less satisfactory for the higher order corrections to the dilute ideal behavior. As we shall see, is expressible as an integral over the pair correlation function for two solutes in a pure solvent. B requires the knowledge of the triplet correlation function for three 

We now derive a general result which expresses the GPF of a two-component system in a form which is strikingly analogous to the expression of the GPF of a one-component system. Consider a two-component system of a solute A and a solvent 5 at a given temperature and activities Xa and respectively. The GPF of such a system is defined by 

At the limit of the dilute ideal solution, we already know the relation between Xa and Pa, i.e.. 

The SI behavior is also consistent with the condition Aab = Gaa + Gbb — 2Gab = 0. In this model, it is easy to compute each of the KB integrals. The results are 

The excess chemical potential with respect to the DI solution is obtained from (6.71) and (6.79). 

We have thus expressed the three forms of the excess chemical potentials corresponding to the three cases of ideal behaviors. 

In this case the coefficients B are called the virial coefficients of the osmotic pressure. Note that these virial coefficients depend on both the temperature and the solvent activity XB, or the solvent chemical potential 1B = exp (fipB). 

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When the adsorbent molecides are not independent, we can no longer use the relation (D.2) for the GPF of the system. In this case, we must start from the GPF of the macroscopic system from which we can derive the general form of the BI for any concentration of the adsorbent molecule. The derivation is possible through the McMillan-Mayer theory of solution, but it is long and tedious, even for first-order deviations from an ideal solution. The reason is that, in the general case, the first-order deviations would depend on many second-virial coefficients [the analogue of the quantity B2(T) in Eq. (D.9)]. For each pair of occupancy states, say i and j, there will be a pair potential [/pp(R, i,j), and the corresponding second-virial coefficient... 

The present calculations are in agreement with the conclusion of ref 59 (which employed both a lattice and the McMillan— Mayer theories of solution" ) that the solute—solute interactions in the systems investigated increase in the sequence MeOH < EtOH < 2-PrOH < 1-PrOH t-BuOH. There are, however, essential differences between the lower alcohols (MeOH and EtOH) and the higher ones. 

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. 

In the McMillan-Mayer theory of solutions, the macromolecule is considered as a small gel immersed in a solution. The gel is surrounded by the solvent, but inside the gel there are also solvent molecules (O), as shown in the diagram ... 

As a theory of solutions it is conveniently applied to the entire range of compositions. This feature makes it more useful than the McMillan-Mayer theory of solutions developed in section 6.11. 

McMillan-Mayer theory of solutions [1,2], which essentially seeks to partition the interaction potential into tln-ee parts that due to the interaction between the solvent molecules themselves, that due to die interaction between the solvent and the solute and that due to the interaction between the solute molecules dispersed within the solvent. The main difference from the dilute fluid results presented above is that the potential energy u(r.p is replaced by the potential of mean force W(rp for two particles and, for particles of solute in the solvent, by the expression... 

Explain in about 250 words the essential approach of the Mayer theory of ionic solutions and how it differs from the ionic-atmosphere view. The parent of Mayer s theory was the McMillan-Mayer theory of 1950. With what classical equation for imperfect gases might it be likened ... 

The main tenet of the McMillan-Mayer theory of nonelectrolytes is that if one is interested in the equilibrium properties of a dilute solution of solute molecules in which their mutual interaction is through short-range forces, then specific interactions between the solute and solvent can be removed by integrating over the solvent degrees of freedom. In this continuum solvent picture there is a direct correspondence between the thermodynamic equations describing this system and that of an imperfect gas. For example, the pressure of an imperfect gas translates into the osmotic pressure of the solvated system. The theory can be extended to electrolyte solutions provided the long-range interaction between ions falls off faster than the Coulomb... 

The McMillan-Mayer theory allows us to develop a fomialism 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 Ihnit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + IcT In where W(a s) is the potential of mean force for the interaction of a solute... 

The McMillan-Mayer theory offers the most usefiil starting point for an elementary theory of ionic interactions, since at high dilution we can incorporate all ion-solvent interactions into a limitmg chemical potential, and deviations from solution ideality can then be explicitly coimected with ion-ion interactions only. Furthemiore, 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 fomi... 

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... 

Special emphasis is placed upon the McMillan-Mayer theory (Sections 4 and 5) and on cluster expansions (Section 6), as these represent aspects that are both difficult and strongly established, but that are seldom given detailed exposition. Other developments that are easily accessible in the literature are treated more lightly, as are the many aspects of theory of fluids that are not yet completely developed for application to ionic solution problems. 

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... 

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. 

Now we want to leave our discussion of what might be called the ancient and early modem periods of solution theory history and concentrate on the modem period, characterized by the theories of Mayer and McMillan (McMillan and Mayer 1945) and of Kirkwood and Buff (Kirkwood and Buff 1951). The McMillan-Mayer theory was the earlier of the two, by some 6 years, and had already captured the attention of the experimental community by the time the Kirkwood-Buff theory appeared. 

The theory of McMillan and Mayer is exact, but only useful in dilute solutions. It delivers thermodynamic functions as a power series in the solute concentrations and it is quite difficult to compute, or even to interpret the coefficients higher than the second virial coefficient, Bj. About 6 years after the McMillan-Mayer theory was developed a new solution theory appeared, not subject to this difficulty, that of Kirkwood and Buff (Kirkwood and Buff 1951), of course this new theory had computational problems of its own. KB (Kirkwood-Buff) theory is also known as fiuctuation theory for reasons that will become obvious below. It is the basis for the rest of this volume and therefore will occupy the remainder of this chapter. 

A mean field theory of solvent structure has been employed by Marcelja(146) to describe the effect of solvent correlation on solute-solute interactions of both hydrophobic and hydrophilic solutes. The interactions between hydrophilic solutes in water has also been considered in a group of papers(141,147-150) where the heats of dilution and of the mixing at constant molality for various non electrolytes (alcohols, amides, sugars, urea, aminoacids and peptides) are interpreted in the framework of the McMillan-Mayer theory(151) and the enthalpy effects arising from interactions between each functional group on one molecule and every functional group on the other molecule are evaluated. 

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. 

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. 

There are generally two ways to proceed in determining the equilibrium properties of a solution. The configurational averages can be carried out simultaneously over the solute and solvent species, or they may be carried out successively over the solvent, at fixed positions of the solute, and then over the coordinates of the solute species. The first method is due to Kirkwood and Buff (1951) and has been applied most successfully, until recently, to simple nonelectrolyte solutions and mixtures. The second is the McMillan-Mayer theory which has been more widely used in the study of electrolytes, polymers, and proteins. Our discussion of electrolytes will be in terms of the McMillan-Mayer (MM) formalism. For recent applications of the Kirkwood Buff theory see Perry et al. (1988). 

By using solvent-averaged potentials the emphasis in the McMillan-Mayer theory is on the solute molecules. Van t Hoff suggested in 1907 that the osmotic coefficient <(), in the McMillan-Mayer system, measures the deviation of the osmotic pressure of the solution from the ideal gas behavior and is defined by... 

The McMillan-Mayer theory provides the theoretical foundation for van t Hoff s supposition of ideal gas behavior for the osmotic pressure of the solute it also shows how deviations from this behavior, which are observed at higher solute concentrations, can be studied with the same theory. The assumption of pair-wise additivity of the solvent-averaged solute potentials, which we have assumed to simplify our discussion, is not completely justified, but even this can be taken into account in... 

Another approach to the thermodynamic properties of solutions is to calculate them from the solute-solute distribution functions rather than from the virial coefficients. Approximations to these functions, which correspond to the summation of a certain class of terms in the virial series to all orders in the solute concentration (or density), have already been worked out for simple fluids, and the McMillan-Mayer theory states that the same approximations may be applied to the solute particles in solution provided the solvent-averaged potentials are used to determine the solute distribution functions. Examples of these approximations are the Percus-Yevick (PY) (1958), Hypernetted-Chain (HNC), mean-spherical (MS), and Born-Green-Yvon (BGY) theories. Before discussing them we will review some of the properties of distribution functions and their relationship to the observed thermodynamic variables. 

Our general approach is a proper adaptation and generalization of the gas-type theories of McMillan and Mayer and of Kirkwood and Buff. These were originally developed for simple (monomer) solutions. We use the cluster development of McMillan and Mayer, which itself is an adaptation of the original (Ursell)-Mayer cluster development. We... 

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. 

We shall now derive the virial expansion of the osmotic pressure following McMillan and Mayer and Hill (P). but simplifying the derivation. The virial expansion plays an imjwrtant role in the theory of solutions. For our purpose we introduce the grand partition function of... 

The application of McMillan-Mayer theory to high polymer solutions was first made by B. H. ZiMM. J. Chem. Phys. 14, 104 (1946). 

In this chapter some aspects of the present state of the concept of ion association in the theory of electrolyte solutions will be reviewed. For simplification our consideration will be restricted to a symmetrical electrolyte. It will be demonstrated that the concept of ion association is useful not only to describe such properties as osmotic and activity coefficients, electroconductivity and dielectric constant of nonaqueous electrolyte solutions, which traditionally are explained using the ion association ideas, but also for the treatment of electrolyte contributions to the intramolecular electron transfer in weakly polar solvents [21, 22] and for the interpretation of specific anomalous properties of electrical double layer in low temperature region [23, 24], The majority of these properties can be described within the McMillan-Mayer or ion approach when the solvent is considered as a dielectric continuum and only ions are treated explicitly. However, the description of dielectric properties also requires the solvent molecules being explicitly taken into account which can be done at the Born-Oppenheimer or ion-molecular approach. This approach also leads to the correct description of different solvation effects. We should also note that effects of ion association require a different treatment of the thermodynamic and electrical properties. For the thermodynamic properties such as the osmotic and activity coefficients or the adsorption coefficient of electrical double layer, the ion pairs give a direct contribution and these properties are described correctly in the framework of AMSA theory. Since the ion pairs have no free electric charges, they give polarization effects only for such electrical properties as electroconductivity, dielectric constant or capacitance of electrical double layer. Hence, to describe the electrical properties, it is more convenient to modify MSA-MAL approach by including the ion pairs as new polar entities. 

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