McMillan-Mayer theory

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  

The gel acts like a semipermeable membrane. Only the solvent molecule can pass through the gel. The temperature remains the same inside and outside. The outside system has a specified value of the chemical potential pj, while the inside system has specified values of Pi for the solvent and p2 for the solute. If the solution is in osmotic equilibrium with pure solvent, the inside (solution) would have pressure p + n to balance the outside (pure solvent) pressure p. The osmotic pressure inside the system is the excess pressure created to give p, the same value (which tends to be lowered in the presence of the solute) inside as outside. 

The grand partition function (See Appendix C) for the inside solution (according to Hill, 1960) is 

With a messy mathematical manipulation on the equation that is related to Q and Pj, one can reach the theoretical expressions for n and Aa  

The value of Aa depends on w. If the solute molecule is considered as a hard sphere of diameter a, then 

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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... 

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... 

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... 

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 McMillan-Mayer theory is an alternative to the Debye-Htickel theory. It is called the virial coefficient approach and its equations bear some conceptual resemblance to the virial equation of state for gases. The key contribution in... 

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). 

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

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. 

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 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). ... 

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. 

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. 

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 starting point of McMillan-Mayer theory is a relationship between distribution functions at different activity sets. The derivation of this relationship is the difficult part of the theory. But once obtained, the relation leads to an expression for the osmotic pressure of a solution, since the components permeable to the osmotic membrane have the same chemical potential on both sides of the membrane while those impermeable have differing chemical potentials. A lengthy computation then leads to an expansion for the osmotic pressure, completely analogous to the activity expansion of the pressure in the theory of imperfect gases. Indeed, for the purpose of comparing gas theory with solution theory, it helps to regard the gas as a solute in a very special and very simple solvent— vacuum. The X expansion is. 

The results of McMillan-Mayer theory have been used primarily in the area of solutions of macromolecules in low molecular weight solvents. The osmotic second virial coefficient, which can be measured either by osmometry or light scattering, gives information on the size of the solute molecules. We shall see why in more detail later when we discuss fluctuation theory. 

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. 

Second, one might ask, since McMillan-Mayer and Kirkwood-Buff theories are both exact, what is the relation between them McMillan-Mayer theory is formulated in terms of potentials of mean force at infinite dilution, albeit of increasing numbers of particles. Kirkwood-Buff theory is formulated in terms of the potential of mean force between pairs only, but at the actual concentration of the solution. The answer to this question is given by Equation KB23, written down without derivation. A future publication with a derivation is promised but, as far as I know, now 60 years later, none has appeared. This is an unsatisfactory state of affairs. 

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... 

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