Mayer theory of ionic solutions

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 ... 

Mayer J 1950 Theory of ionic solutions J. Chem. Phys. 18 1426... 

Among the outstanding contributors to the theory of ionic solutions after 1950 were Mayer, Friedman, Davies, Rasaiah, Blum, and Haymet. Write a few lines on the contributions due to each. 

J. C. Poirier, Thermodynamic functions from Mayer s theory of ionic solutions. II. The stoichiometric mean ionic molar activity coefficient, /. Chem. Phys. 21,972 (1953). 

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... 

Applied to ionic solutions by Mayer in 1950, this theory aimed to express the activity coefficient and the osmotic pressure of ionic solutions while neglecting the effects of solvation that had been introduced by Bjerrum and developed by Stokes and Robinson to give experimental consistency of such impressive power at high concentrations. 

Debye and Hiickel s theory of ionic atmospheres was the first to present an account of the activity of ions in solution. Mayer showed that a virial coefficient approach relating back to the treatment of the properties of real gases could be used to extend the range of the successful treatment of the excess properties of solutions from 10 to 1 mol dm". Monte Carlo and molecular dynamics are two computational techniques for calculating many properties of liquids or solutions. There is one more approach, which is likely to be the last. Thus, as shown later, if one knows the correlation functions for the species in a solution, one can calculate its properties. Now, correlation functions can be obtained in two ways that complement each other. On the one hand, neutron diffraction measurements allow their experimental determination. On the other, Monte Carlo and molecular dynamics approaches can be used to compute them. This gives a pathway purely to calculate the properties of ionic solutions. 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

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... 

The remainder of Section I is devoted to a rather brief review of earlier work in the field in order to gain a little perspective. In Sections II to IV the basic results of the cluster method are derived. In Section V a very brief account of the application of the formal equations to some systems with short-range forces is given. Section VI is devoted to a review of the application to systems with Coulomb forces between defects, where the cluster formalism is particularly advantageous for bringing the discussion to the level of modern ionic-solution theory.86 Finally, in Section VII a brief account is given of Mayer s formalism for lattice defects69 since it is in certain respects complementary to that principally discussed here. We would like to emphasize that the material in Sections V and VI is illustrative of the method. This is not meant to be an exhaustive review of results obtainable. 

For ionic defects the individual terms in the formal virial expansions diverge just as they do in ionic solution theory. The essence of the Mayer theory is a formal diagram classification followed by summation to yield new expansions in which individual terms are finite. The recent book by Friedman25 contains excellent discussions of the solution theory. We give here only an outline emphasizing the points at which defect and solution theories diverge. Fuller treatment can be found in Ref. 4. 

For simplicity we consider only the continuum limit (i.e. Mayer ionic solution theory). The last equation allows us to calculate the value of p which the association theory should predict in order to be compatible with the true value, which we assume to be given by the Mayer theory in the range considered. It is... 

The proper treatment of ionic fluids at low T by appropriate pairing theories is a long-standing concern in standard ionic solution theory which, in the light of theories for ionic criticality, has received considerable new impetus. Pairing theories combine statistical-mechanical theory with a chemical model of ion pair association. The statistical-mechanical treatment is restricted to terms of the Mayer/-functions which are linear in / , while the higher terms are taken care by the mass action law... 

Does Mayer s theory of calculating the viriai coefficients in equations such as Eq. (3.165) (which gives rise directly to the expression for the osmotic pressure of an ionic solution and less directly to those for activity coefficients) really improve on the second and third generations of the Debye-Huckel theory—those involving, respectively, an accounting for ion size and for the water removed into long-lived hydration shells ... 

Figure 3.48 shows two ways of expressing the results of Mayer s viriai coefficient approach using the osmotic pressure of an ionic solution as the test quantity. Two versions of the Mayer theory are indicated. In the one marked DHLL + B2, the authors have taken the Debye-Hiickel limiting-law theory, redone for osmotic pressure instead of activity coefficient, and then added to it the results of Mayer s calculation of the second viriai coefficient, B. In the upper curve of Fig. 3.48, the approximation within the Mayer theory used in summing integrals (the one called hypernetted chain or HNC) is indicated. The former replicates experiment better than the latter. The two approxi-... 

Since Mayer s theory originated in a theory for imperfect gases, it naturally tends to calculate the nearest analogue of gas pressure that an ionic solution exhibits—osmotic pressure. 

What is the significance of these results on dimer and trimer formation for ionic solution theory In the post-Debye and HUckel world, particularly between about 1950 and 1980 (applications of the Mayer theory), some theorists made calculations in which it was assumed that aU electrolytes were completely dissociated at least up to 3 mol dm. The present work shows that the degree of association, even for 1 1 salts, is -10% at only 0.1 mol dm . One sees that these results are higher than those of the primitive Bjerrum theory. 

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 terms in the new series are ordered differently from those in the original expansion and Mayer showed that the Debye-Huckel limiting law follows as the leading correction to the ideal behavior for ionic solutions. In principle, the theory enables systematic corrections to the limiting law to be obtained as the concentration of the electrolyte increases for any Hamiltonian which defines the short-range potential u j (r), not just the one which corresponds to the RPM. A modified (or renormalized) second virial coefficient was tabulated by Porrier (1953), while Meeron (1957) and Abe (1959) derived an expression for this in closed form. Extensions of the theory to non-pairwise additive solute potentials have been discussed by Friedman (1962). 

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