Thermodynamics from equations of state

Equations of state relate the pressure, temperature, volume, and composition of a system to each other. In this Chapter, we show how to determine other thermodynamic properties of the system from an equation of state. In a typical equation of state, the pressure is given as an explicit function of temperature, volume, and composition. Therefore, the natural variables are the temperature, volume, and composition of the system. That is, once given the volume, temperature, and composition of the system, the pressure is readily calculated from the equation of state. 

Once one of the free energies of a system is known as a function of its natural variables, then all the other thermodynamic properties of the system can be derived. For these equations of states, the Helmholtz free energy is the relevant quantity. In the following, we demonstrate how to determine the Helmholtz free energy from an equation and then proceed to show how to derive other properties from it. 

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The first part of the chapter is devoted to an analysis of these correlations, as well as to the presentation of the most important experimental results. In a second part the following stage of development is reviewed, i.e. the introduction of more quantitative theories mostly based on bond structure calculations. These theories are given a thermodynamic form (equation of states at zero temperature), and explain the typical behaviour of such ground state properties as cohesive energies, atomic volumes, and bulk moduli across the series. They employ in their simplest form the Friedel model extended from the d- to the 5f-itinerant state. The Mott transition (between plutonium and americium metals) finds a good justification within this frame. 

Because the right-hand sides of Eqs. (30) and (31) can be evaluated from equations of state, we see that such equations plus heat capacity data allow us to completely calculate changes of U and H. Equations (30) and (31) are known as the thermodynamic equations of state. 

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. 

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. 

Separation Processes Equipment for Multiphase Contacting Thermodynamic Equilibrium Diagrams Phase Equilibria from Equations of State Equilibrium Properties from Activity Coefficient relations... 

X-vaiues or distribution coefficients, the accuracy depends upon the veracity of the particular correlations used for the various thermodynamic quantities required. For practical applications, choice of K-value formulation is a compromise among considerations of accuracy, compleAity, and convenience. The more important formulations are (4-27), (4-29), and (4-31). They all require correlations for fugacity coefficients and activity coefficients. The application of (4-27) based on fugacity coefficients obtained from equations of state is presented in this chapter. Equations (4-29) and (4-31) require activity coefficient correlations, and are discussed in Chapter 5. 

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