Estimation of thermodynamic mixing parameters

Stoichiometric saturation measurements in carefully controlled laboratory experiments offer perhaps the most promising technique for the estimation of thermodynamic mixing parameters (3 Glynn and Reardon, Am. J. ScL, in press). Unfortunately, the results obtained can usually not be verified by a second independent and accurate method, such as reaction calorimetry or measurement of thermodynamic equilibrium solubilities (4). The conditions necessary in obtaining good stoichiometric saturation data (as opposed to thermodynamic equilibrium data) were discussed earlier. 

Q can be used to estimate thermodynamic mixing parameters in the absence of more... 

Fedders and Muller (213) have derived an estimate of the solid-inter-action parameter from another point of view, which ascribes the mixing enthalpy to bond distortions associated with the alloy formation and relates these distortions to the macroscopic elastic properties of the crystal. They concluded that the results based on elastic-crystal parameters yield a similar form for the thermodynamic properties as those estimated by DLP model based on optical-crystal parameters. 

Field or laboratory observations of miscibility gaps, spinodal gaps, critical mixing points or distribution coefficients can be used to estimate solid-solution excess-free-energies, when experimental measurements of thermodynamic equilibrium or stoichiometric saturation states are not available. As an example, a database of excess-free-energy parameters is presented for the calcite, aragonite, barite, anhydrite, melanterite and epsomite mineral groups, based on their reported compositions in natural environments. 

The extent to which small molecules will be sorbed into a polymer at equilibrium depends on the entropy and enthalpy of mixing and the activity of these molecules in the environment with which the polymer is equilibrated. These points are best illustrated by considering a polymer surrounded by a vapor of component 1 at a partial pressure of pj. If the saturation vapor pressure of pure 1 is P], then the activity in the vapor phase is = Pi/Pi > Flory (3) has developed a thermodynamic model for mixing small molecules of molar volume with large polymer molecules of molar volume 2. This model combines an estimate for the entropy of mixing with a measure of the enthalpy of mixing expressed in terms of an interaction parameter Xj and results in the following expression for phase equilibrium ... 

These effects are part of the solubility parameters concept. This concept allows to predict compatibility qualitatively (see Section 6.2). Within the framework of the general principles of the thermodynamics of solutions, the numerical evaluation of compatibility implies the evaluation of the value of the free energy (the Gibbs energy) of mixing over the whole range of solution concentration. However, sueh evaluations are difBcult and often excessive for most practical purposes. Therefore, for the estimation of compatibility, it is more convenient to know the value of aity numerical criterion mentioned in the previous subchapters. 

The molecular parameters characterizing the pure components and the mixtures in the S-S theory, are taken from reference [6], The pure component parameters were estimated from equation of state data [13,14]. Values for the mixing parameters e i2 and v i2 were adjusted to give quantitative agreement between the computed and experimental critical conditions. Since all the model parameters are available, we are in a position to predict other thermodynamic properties. As an example, spinodal conditions are considered. Details concerning the computational methods have been presented elsewhere [5]. It can be observed in Figure 1 that, in comparison to the experimental spinodals, the predicted spinodals become too narrow with decreasing molar mass. If the flexibility parameter c is allowed to vary with molar mass in a manner dictated by the experimental spinodal data, a quantitative description of these data can be obtained [6]. 

The above approach is empirical. Thermodynamic models for describing solution behavior can also be employed to determine gas solubilities, and these models are amenable to the estimation of gas solubilities in multicomponent systems from sets of single salt data. The thermodynamic approach employed is known as the Pitzer species interaction model, and it is used to determine the activity coefficient of the gas from a summation of interaction terms with anions, cations, and neutral species [3, 10, 11]. These interaction parameters are determined empirically from solubility data in a range of electrolyte solutions and have been tabulated for a wide range of salts, permitting the solubility of oxygen to be determined in mixed electrolyte solutions over a wide range of temperature and concentrations. 

It is to the extension of this approach for pure fluids and the estimation of the thermodynamic properties of mixtures that we now turn. This requires the introduction of mixing rules to provide a(x) and b(x), that are now functions of mole fraction x, from the values of a and b for pure substances. The expressions for a x) and b x) include the interactions between unlike molecules, and methods are then required to determine the parameters Oy and by for molecules i and j from the values of a and b for the pure fluids. This step is achieved using combining rules. 

It is known that the rules of equilibrium thermodynamics may be applied to such metastable states, which gives the possibility to estimate the state of thermodynamic miscibility in the region of thermodynamic instability using values of the free energy of mixing or the interaction parameter. 

The analysis of concentration and temperature dependencies of thermodynamic parameters, calculated using phase diagrams, provides not only explanation of some compatibility features but also the way to estimate structural characteristics of oligomer systems. This approach is rather prospective, that will be illustrated below by examples analyzing mixing thermodynamics of various oligomer system types. 

The mixing rules with a binary interaction parameter or with an excess function are supported by thermodynamic concepts, and may achieve better performance than the others. In any case, the parameters have to be estimated from a wide range of experimental data and blends, so that the application of these mixing rules can be extended to other samples different from those used to derive the parameter values. 

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