Property from equation of state

Schmidt, M., and Maurer, R H. J., Isotropic pressure-densihed atactic poly(methyl methacrylate) glasses free-volume properties from equation-of-state data and positron aannihilation lifetime spectroscopy. Macromolecules, 33, 3879-3891 (2000a). 

Critical Properties From Equations of State. In industrial work calculating the critical locus of a mixture is frequently necessary. Because this is done at perhaps 0.05 mole fraction intervals for a binary system, the repetitive nature of the calculation demands as direct and clear cut a method of solution as possible. This precludes solving simultaneous equations, such as setting the second and third derivatives of the free energy with respect to composition equal to zero as indicated below ... 

Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate Redlich-Kwong-type equations such as the Soave and Peng-Robinson equations are often used. Both require only T, and (0 as inputs. For organic compounds, the Lee-Erbar-EdmisteF" equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. 

As explained in Sec. 13.3, residual properties and fugacity coefficients readily calculated from equations of state. By Eq. (11.20), applicable to const" composition fluids,... 

The great attraction of this equation is that it contains just properties of the pure species and therefore expresses K-values as functions of T and P, independent of the compositions of the liquid and vapor phases. Moreover, and 4> can be evaluated from equations of state for the pure species or from generalized correlations. This allows K-values for light hydrocarbons to be calculated and correlated as functions of T and P. However, the method is limited for any species to subcritical temperatures, because the vapor-pressure curve terminates at the critical point. 

Equations of state have a much wider application. In this chapter we first present a general treatment of the calculation of thermodynamic properties of fluids and fluid mixtures from equations of state. Then the use of an equation of state for VLE calculations is described. For this, the fugacity of each species in both liquid and vapor phases must be determined. These calculations are illustrated with the Redlich/Kwong equation. Provided that the equation of state is suitable, such calculations can extend to high pressures. 

Before preceding, it is useful to consider the form of the force-force correlation function, which is given in Equation (21), with Equations (22), (24), (25), (26) and (27). The form of the force-force correlation function, derived using density functional formalism, is employed because it permits the use of very accurate equations of state for solvents like ethane and CO2 to describe the density dependence and temperature dependence of the solvent properties. These equations of state hold near the critical point as well as away from it. Using the formalism presented above, we are able to build the known density and temperature-dependent properties of the... 

For a pure supercritical fluid, the relationships between pressure, temperature and density are easily estimated (except very near the critical point) with reasonable precision from equations of state and conform quite closely to that given in Figure 1. The phase behavior of binary fluid systems is highly varied and much more complex than in single-component systems and has been well-described for selected binary systems (see, for example, reference 13 and references therein). A detailed discussion of the different types of binary fluid mixtures and the phase behavior of these systems can be found elsewhere (X2). Cubic ecjuations of state have been used successfully to describe the properties and phase behavior of multicomponent systems, particularly fot hydrocarbon mixtures (14.) The use of conventional ecjuations of state to describe properties of surfactant-supercritical fluid mixtures is not appropriate since they do not account for the formation of aggregates (the micellar pseudophase) or their solubilization in a supercritical fluid phase. A complete thermodynamic description of micelle and microemulsion formation in liquids remains a challenging problem, and no attempts have been made to extend these models to supercritical fluid phases. 

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