Thermodynamics of Ideal Mixing

Chemical potentials find their main use in the interpretation of the equilibria existing when chemical reactions take place. Thus when a substance A reacts with another substance B to produce products, say C and D, chemical potentials (with their ability to handle different phases of material) are especially useful in developing the underlying theory (Frames 5, 27, 28, 29, 35 and 36) and this leads on to supplying answers to a number of key questions, which include  

Before we begin to study reactions in detail we discuss the concept of mixing without a reaction taking place. We now give an example of the use of chemical potentials to study the thermodynamics of mixing for two ideal solutions designated A and B which can be either solid, liquid or gaseous  

In our model, A and B can be solids forming a solid solution, liquids forming a solution or gases forming a mixture. 

The results obtained are general but apply only in cases where the mutual interactions between A and A, B and B and between A and B are virtually identical, meaning that the molecules can mix without showing a preference. 

In real solutions we can anticipate departures from this situation and therefore the thermodynamics presented here needs to have the above caveat imposed. 

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Critical Micelle Concentration. In order to demonstrate the analogy between our treatment of mixed adsorption and earlier treatments of mixed micellization, we will briefly review the thermodynamics of mixed micelles. The thermodynamics of formation of ideal mixed micelles by two surfactants has been treated by Lange and Beck (9 ) and Cling (10). Rubingh ( ) extended the treatment to account for interactions between the surfactants, essentially by writing the cmc in the mixed surfactant solution as. 

Aacaard, P., and H. C. Helceson. 1983. Activity/composi-tion relations among silicates and aqueous solutions n. Chemical and thermodynamic consequences of ideal mixing of atoms on homological sites in montmoril-lonites, illites, and mixed-layer clays. Clays Clay Afiner-fl/.r31 3) 207-17. 

The thermodynamics of relatively ideal mixed films can be approached as follows. It is convenient to define... 

Witlox, H. W. M., 1993, Thermodynamics Model for Mixing of Moist Air with Pollutant Consisting of HF, Ideal Gas, and Water, Shell Research Limited, Thornton Research Center, TNER.93.021,. 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

These assumptions have been expanded upon (Shah and Capps, 1968 Lucassen-Reynders, 1973 Rakshit and Zografi, 1980), especially in regard to the application of the ideal mixing relationship in gaseous films (Pagano and Gershfeld, 1972). It has been pointed out that water may contribute to the energetics of film compression if the molecular structures of the surfactants are sufficiently different (Lucassen-Reynders, 1973). It must be noted that this treatment assumes that the compression process is reversible and the monolayer is truly stable thermodynamically. It must therefore be applied with considerable reservation in view of the hysteresis that is often found for II j A isotherms. 

The implications for films cast from mixtures of enantiomers is that diagrams similar to those obtained for phase changes (i.e., melting point, etc.) versus composition for the bulk surfactant may be obtained if a film property is plotted as a function of composition. In the case of enantiomeric mixtures, these monolayer properties should be symmetric about the racemic mixture, and may help to determine whether the associations in the racemic film are homochiral, heterochiral, or ideal. Monolayers cast from non-enantiomeric chiral surfactant mixtures normally will not exhibit this feature. In addition, a systematic study of binary films cast from a mixture of chiral and achiral surfactants may help to determine the limits for chiral discrimination in monolayers doped with an achiral diluent. However, to our knowledge, there has never been any other systematic investigation of the thermodynamic, rheological and mixing properties of chiral monolayers than those reported below from this laboratory. 

By using a thermodynamic model based on the formation of self-associates, as proposed by Singh and Sommer (1992), Akinlade and Awe (2006) studied the composition dependence of the bulk and surface properties of some liquid alloys (Tl-Ga at 700°C, Cd-Zn at 627°C). Positive deviations of the mixing properties from ideal solution behaviour were explained and the degree of phase separation was computed both for bulk alloys and for the surface. 

Calorimetric measurements can be used to obtain heats of mixing between different surfactant components in nonideal mixed micelles and assess the effects of surfactant structure on the thermodynamics of mixed micellization. Calorimetry can also be successfully applied in measuring the erne s of nonideal mixed surfactant systems. The results of such measurements show that alkyl ethoxylate sulfate surfactants exhibit smaller deviations from ideality and interact significantly less strongly with alkyl ethoxylate nonionics than alkyl sulfates. 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

Calculations including the vapour phase were ftien made to determine the extent of release of various components during the reaction. Two types of calculation were made, one where ideal mixing in the solution phases was considered and the other where non-ideal interactions were taken into account. For elements such as Ba, U and, to a certain extent. Si, the calculations were relatively insensitive to the model adopted. However, the amoimt of Sr in the gas was 24 times higher in the full mo r in comparison to the ideal model. This led to the conclusion that sensitivity analysis was necessary to determine the extent to which accuracy of the thermodynamic parameters used in die model affected die final outcome of the predictions. 

For a binary system of surfactants A and B, the mixed micelle formation can be modeled by assuming that the thermodynamics of mixing in the micelle obeys ideal solution theory. When monomer and micelles are in equilibrium in the system, this results in ... 

The thermodynamics of mixing upon formation of the bilayered surface aggregates (admicelles) was studied as well as that associated with mixed micelle formation for the system. Ideal solution theory was obeyed upon formation of mixed micelles, but positive deviation from ideal solution theory was found at all mixture... 

Scamehorn et. al. (20) also presented a simple, semi—empirical method based on ideal solution theory and the concept of reduced adsorption isotherms to predict the mixed adsorption isotherm and admicellar composition from the pure component isotherms. In this work, we present a more general theory, based only on ideal solution theory, and present detailed mixed system data for a binary mixed surfactant system (two members of a homologous series) and use it to test this model. The thermodynamics of admicelle formation is also compared to that of micelle formation for this same system. 

The mixed admicelle is very analogous to mixed micelles, the thermodynamics of formation of which has been widely studied. If the surfactant mixing in the micelle can be described by ideal solution theory, the Critical Micelle Concentration (CMC) or minimum concentration at which micelles first form can be described by (21) ... 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The changes of the thermodynamic functions on mixing of ideal gases... 

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