Entropy, excess

Regular Solution Theory. The key assumption in regular-solution theory is that the excess entropy, is zero when mixing occurs at constant volume (3,18). This idea of a regular solution (26) leads to the equations ... 

Solubility parameters are generally tabulated, together with the corresponding liquid molar volumes, only at 25°C. Although solubility parameters are themselves temperature-dependent, the combination of quantities in Eq. 70 is not. Differentiating Eq. 70 with respect to temperature gives — the excess entropy, a quantity which has been assumed to be zero in accord with the definition of a regular solution. Thus only data at 25°C are needed. Solubility parameters may be... 

Using the Flori-Haggins formula [14,15] to estimate the change of mixture excessive entropy ... 

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. 

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.

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... 

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

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... 

Eckart, criteria, 264, 298 procedure, 267 Effective charge, 274, 276 Effective Hamiltonian, 226 Elastic model, excess entropy calculation from, 141 of a solid solution, 140 Electric correlation, 248 Electric field gradient, 188, 189 Electron (s), 200... 

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-... 

Silver-copper, energy of solutions, 142 Silver-gold, excess entropy, 132, 136 excess free energy, 136 Silver-lead, alloy (AgsPb5), calculation of thermodynamic quantities, 136 Silver-zinc, alloy (Ag5Zn5), 129... 

Thalium, excess entropy and partial molar enthalpy of solutions of noble metals in, 133... 

The semiconducting properties of the compounds of the SbSI type (see Table XXVIII) were predicted by Mooser and Pearson in 1958 228). They were first confirmed for SbSI, for which photoconductivity was found in 1960 243). The breakthrough was the observation of fer-roelectricity in this material 117) and other SbSI type compounds 244 see Table XXIX), in addition to phase transitions 184), nonlinear optical behavior 156), piezoelectric behavior 44), and electromechanical 183) and other properties. These photoconductors exhibit abnormally large temperature-coefficients for their band gaps they are strongly piezoelectric. Some are ferroelectric (see Table XXIX). They have anomalous electrooptic and optomechanical properties, namely, elongation or contraction under illumination. As already mentioned, these fields cannot be treated in any detail in this review for those interested in ferroelectricity, review articles 224, 352) are mentioned. The heat capacity of SbSI has been measured from - 180 to -l- 40°C and, from these data, the excess entropy of the ferro-paraelectric transition... 

The density dependence of the entropy can also be studied by introducing fluctuations in volume rather than particle number. Typically the particle number approach is favored the computational demands of volume scaling moves scale faster with system size than do addition and deletion moves. Nevertheless, the Wang-Landau approach provides a means for studying volume fluctuations as well. In this case, the excess entropy is determined as a function of volume and potential energy for fixed particle number one, therefore, calculates (V, U). Here the microstate probabilities follow ... 

For a large number of the more commonly used microscopic solution models it is assumed, as we will see in Chapter 9, that the entropy of mixing is ideal. The different atoms are assumed to be randomly distributed in the solution. This means that the excess Gibbs energy is most often assumed to be purely enthalpic in nature. However, in systems with large interactions, the excess entropy may be large and negative. 

The expressions in the two parentheses can be identified as the surface excess moles and surface excess entropy defined by eqs. (6.2) and (6.5). Equation (6.12) thus reduces to... 

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. 

The solubility parameter concept was established in the 1930s by the work of Hildebrand and Scatchard. The original concept covers regular solutions, i.e., solutions that do not show an excess entropy effect on mixing. The solubility parameter concept offers the following interesting features ... 

Fig. 5. Qualitative comparison between the experimental excess entropies and the APM predictions. White circles sE > 0, black circles sE < 0 (x = 0.5).

Let us now turn to a discussion of the relation of the temperature dependence of the polymer melt s configurational entropy with its glass transition and address the famous paradox of the Kauzmann temperature of glass-forming systems.90 It had been found experimentally that the excess entropy of super-cooled liquids, compared with the crystalline state, seemed... 

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