Mayer theory

By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

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... 

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... 

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... 

Most interestingly, the lifetimes for 3,3-isotope effect on the lifetime of the vibrational energy redistribution requirements, not the sorts of factors associated with standard Bigeleisen and Mayer theory. °... 

Justice, M.C. and Justice, J.C. Ionic interactions in solutions I association concepts and the McMUlan-Mayer theory. J. Solution Chem. 1976,5, 543-561. 

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-... 

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. 

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... 

In a parallel series of developments, starting with the Mayer theory and continuing with the so-called mean spherical approximation, the effects of hydration and ion association were arbitrarily removed from consideration, in spite of their undeniable presence in nature. 

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). 

Fig. 17. Experimental molar cycUzation e iUbrium con nts (in mol dm ) for cyclics Mj in the melt at 525 K (o) are plotted as logJSTx against log x. They are compared with theoretical values calculated by the Jacobson and Stock-mayer theory which assumes that the corresponding open chain molecules obey Gaussian statistics (these values are denoted )...

Theoretical discussion of the osmotic pressure of polyelectrolytes has been made by two methods, one using the Donnan equilibrium and the other the McMillan and Mayer theory. Both methods are equivalent but in order to obatin explicitly the osmotic pressure we should know in the former case the activities of component systems and in the latter case the potential of average force between the solute molecules. 

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. 

Chain Branching. A branched molecule is more compact, having greater density and lower [iq] than its linear counterpart. The Zimm-Stock-mayer theory defines the g factor for a polymer as the ratio of [iq] for the branched polymer to [iq] of the linear polymer, at the same molecular weight, with s being the shape factor (—0.75). 

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