Excess entropy of solution

Other ordering systems show striking discrepancies with the predictions of the quasi-chemical theories. Cu-Pt,67 Co-Pt,38 and Pb-Tl36 are binaries the solid solutions of which exhibit a positive partial excess free energy for one of their components, as well as positive excess entropies of solution. Co-Pt goes even further in deviating from theory in that it has a positive enthalpy of solution,... 

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

Four of the solid solutions of Table III have excess entropies of solution which include contributions from magnetic disordering in both the alloy and in one or both of the pure components. These contributions can be quite large, and since there is no assurance... 

Bismuth, excess entropy of solution of noble metals in liquid bismuth, 133 Block polymers, 181 Bond energies in the halogens, 61 Boron fluoride as initiator in polymerization, 156... 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

Gas chromatography is primarily an analytical separation technique. However, since the basic process is an equilibration of a solute between two immiscible phases, the chromatographic technique may be used to measure such physical properties as activity coefficients, second virial coefficients of gas mixtures, partition coefficients, adsorption and partition isotherms, and complex formation constants. Other properties which can be measured with less accuracy, from secondary measurements or from temperature variation studies, include surface areas, heats of adsorption, and excess enthalpies and excess entropies of solution. A number of reviews and discussions on these measurements have appeared in the literature. The present work is restricted to a review of activity-coefficient measurements. 

A zero excess chemical potential does not imply that both the enthalpy of solution and the excess entropy of solution are zero as in the case of true ideal solutions where AH = 0 and = 0. Rather, a theta state means only... 

In general exothermic heats and negative partial molar excess entropies of solution indicate that the solution process involves strong interactions between Y and the ions in the melt. ... 

Table III presents integral excess entropies of formation for some solid and liquid solutions obtained by means of equilibrium techniques. Except for the alloys marked by a letter b, the excess entropy can be taken as a measure of the effect of the change of the vibrational spectrum in the formation of the solution. The entropy change associated with the electrons, although a real effect as shown by Rayne s54 measurements of the electronic specific heat of a-brasses, is too small to be of importance in these numbers. Attention is directed to the very appreciable magnitude of the vibrational entropy contribution in many of these alloys, and to the fact that whether the alloy is solid or liquid is not of primary importance. It is difficult to relate even the sign of the excess entropy to the properties of the individual constituents.

TABLE III. Integral Excess Entropies of Formation of Concentrated Solutions a... 

The ideal solution approximation is well suited for systems where the A and B atoms are of similar size and in general have similar properties. In such systems a given atom has nearly the same interaction with its neighbours, whether in a mixture or in the pure state. If the size and/or chemical nature of the atoms or molecules deviate sufficiently from each other, the deviation from the ideal model may be considerable and other models are needed which allow excess enthalpies and possibly excess entropies of mixing. 

At 42°C the enthalpy of mixing of 1 mole of water and 1 mole of ethanol is — 343.1 J. The vapor pressure of water above the solution is 0.821 p and that of ethanol is 0.509 P2, in which p is the vapor pressure of the corresponding pure liquid. Assume that the vapors behave as ideal gases. Compute the excess entropy of mixing. 

The regular solution approximation is introduced by assuming definition) that the excess entropy of mixing is zero. This requires that the excess free energy equal the excess enthalpy of mixing. For binary mixtures the excess enthalpy of mixing is ordinarily represented by a function of the form... 

This conclusion implies that the excess entropy of mixing is non-zero and that the mixed micelles presumably acquire more internal order than they would by random mixing. An examination of the magnitude of the deviations from the regular solution approximation shows that there must be a large TS contribution to the excess free energy of mixing. 

Haselton H. T, Hovis G. L., Hemingway B. S., and Robie R. A. (1983). Calorimetric investigation of the excess entropy of mixing in analbite-sanidine solid solutions Lack of evidence for Na, K short range order and implications for two feldspar thermometry. Amer. Mineral, 68 398-413. 

Solid-Solution Models. Compared with the liquid phase, very few direct experimental determinations of the thermochemical properties of compound-semiconductor solid solutions have been reported. Rather, procedures for calculating phase diagrams have relied on two methods for estimating solid-solution model parameters. The first method uses semiem-pirical relationships to describe the enthalpy of mixing on the basis of the known physical properties of the binary compounds (202,203). This approach does not provide an estimate for the excess entropy of mixing and thus... 

The importance of the excess entropy of mixing in aqueous mixtures explains why many of these systems show phase separation with a lower critical solution temperature (LCST). This phenomenon is rarer—though not unknown—in non-aqueous mixtures (for an example, see Wheeler, 1975). The conditions for phase separation at a critical temperature can be expressed in terms of the excess functions of mixing (Rowlinson, 1969 Copp and Everett, 1953). 

AS 00 partial excess entropy of mixing of solute i at infinite dilution in solvent j... 

Figure 4.19. The surface excess entropy of aqueous solutions for three electrolytes.. Temperature 25°C. (Redrawn from Matubayashi et al., loc. cit.)...

The above method can be apphed to a calculation of other excess functions in terms of intermolecular forces. We may note that the excess entropy of mixing is closely related to the excess volume and to the change of the free volume of the solution with composition. We shall not, however, go into any further details here. 

Equation (25.37) enables us to interpret the existence of a positive excess entropy of mixing in the solutions mentioned in paragraph 3. Similarly, for solutions of 7i-heptane + 7i-hexadecane (c/. chap, XXIV, 4) or benzene + diphenyl II the order of magnitude of the excess entropy is in agreement with (25.37). 

This value is of the same order of magnitude as the number of nearest neighbours, and gives a reasonable interpretation of the entropy change. It seems therefore worthwhile attempting a more detailed discussion of the excess entropy of an associated solution. 

The term regular solutions was first coined by Hildebrand (1929). It was characterized phe-nomenoligically in terms of the excess entropy of mixing. It was later used in the context of lattice theory of mixtures mainly by Guggenheim (1952). It should be stressed that in both the phenomenological and the lattice theory approaches, the regular solution concept applies to deviations from SI solutions, (see also Appendix M). 

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