Equations of State for Liquids and Solids

EQUATIONS OF STATE FOR LIQUIDS AND SOLIDS WELL BELOW THEIR CRITICAL TEMPERATURES... 

To use equation 2.10 correctly, we need to know how the heat capacities vary in the experimental temperature range. However, these data are not always available. A perusal of the chemical literature (see appendix B) will show that information on the temperature dependence of heat capacities is much more abundant for gases than for liquids and solids and can be easily obtained from statistical mechanics calculations or from empirical methods [11]. For substances in condensed states, the lack of experimental values, even at a single temperature, is common. In such cases, either laboratory measurements, using techniques such as differential scanning calorimetry (chapter 12) or empirical estimates may be required. 

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 thermodynamic equations for the Gibbs energy, enthalpy, entropy, and chemical potential of pure liquids and solids, and for liquid and solid solutions, are developed in this chapter. The methods used and the equations developed are identical for both pure liquids and solids, and for liquid and solid solutions therefore, no distinction between these two states of aggregation is made. The basic concepts are the same as those for gases, but somewhat different methods are used between no single or common equation of state that is applicable to most liquids and solids has so far been developed. The thermodynamic relations for both single-component and multicomponent systems are developed. 

The thermodynamic functions have been defined in terms of the energy and the entropy. These, in turn, have been defined in terms of differential quantities. The absolute values of these functions for systems in given states are not known.1 However, differences in the values of the thermodynamic functions between two states of a system can be determined. We therefore may choose a certain state of a system as a standard state and consider the differences of the thermodynamic functions between any state of a system and the chosen standard state of the system. The choice of the standard state is arbitrary, and any state, physically realizable or not, may be chosen. The nature of the thermodynamic problem, experience, and convention dictate the choice. For gases the choice of standard state, defined in Chapter 7, is simple because equations of state are available and because, for mixtures, gases are generally miscible with each other. The question is more difficult for liquids and solids because, in addition to the lack of a common equation of state, limited ranges of solubility exist in many systems. The independent variables to which values must be assigned to fix the values of all of the... 

The universe as a whole is made up of material with three distinct states, namely the solid state, the liquid state and the gaseous state. In 1662, Robert Boyle showed for the first time the relationship between volume and pressure of a gas under constant temperature to be inverse proportional to one another. In 1802 Gay-Lussac reported his discovery on the relationship between the volume of gas and temperature under constant pressure to be proportional to one another. These two relationships laid the foundation for the equation of state for gaseous state namely,... 

Physical property data for many of the key components used in the simulation for the ethanol-from-lignocellulose process are not available in the standard ASPEN-Plus property databases (11). Indeed, many of the properties necessary to successfully simulate this process are not available in the standard biomass literature. The physical properties required by ASPEN-Plus are calculated from fundamental properties such as liquid, vapor, and solid enthalpies and density. In general, because of the need to distill ethanol and to handle dissolved gases, the standard nonrandom two-liquid (NRTL) or renon route is used. This route, which includes the NRTL liquid activity coefficient model, Henry s law for the dissolved gases, and Redlich-Kwong-Soave equation of state for the vapor phase, is used to calculate properties for components in the liquid and vapor phases. It also uses the ideal gas at 25°C as the standard reference state, thus requiring the heat of formation at these conditions. 

In liquids and solids, particles are very close together and repulsive forces play a much more important role than they do in gases. It is not surprising, therefore, that equations of state, such as those of van der Waals, and Redlich and Kwong, do not do a good job of predicting liquid and solid phase data. Usually, data for liquids and solid are presented in the form of thermal expansion coefficients and isothermal compressibilities. Some data for a and k of selected liquids and solids are given in Table 3. 

Physical properties and laws. Two of the unknown variables may be the mass and volume of a stream material, in which case a tabulated specific gravity for liquids and solids or an equation of state for gases (Chapter 5) would provide an equation relating the variables. In other instances, saturation or equilibrium conditions for one or more of the process streams (Chapter 6) may provide needed relations. 

Given a description of a process system in which a volumetric flow rate is either specified or requested for any process stream, (a) carry out the degree-of-ffeedom analysis, including density estimates for liquid and solid streams and equations of state for gas streams (b) write the system equations and outline the procedure you would use to solve for all requested quantities (c) carry out the calculations (d) list all your assumptions (e.g., volume additivity for liquids or ideal gas behavior) and state whether or not they are reasonable for the given process conditions. 

If a PVT calculation is part of a material balance problem and a volume (or volumetric flow rate) is either given or required for a process stream, label both n (or h) and V (or V) on the flowchart and count a density relationship (for solids and liquids) or an equation of state (for gases) as an additional relation in the degree-of-fireedom analysis. 

The interest in studies of pure ozone, solid or liquid, in explosives lies not only in the momentary production of high temperatures, but primarily in the possibility of determining experimentally the parameters for equations of state for simple molecules in a region of pressures (about 100,000 atm.) and temperatures (5000° to over 10,000° K.) far outside of that which can be investigated by any ordinary method. 

For mixtures the presumption is that the equation of state has exactly the same form as when written for pure species. Equations (4-104) are therefore applicable, with parameters p and q given by Eqs. (4-105) and (4-106). Here, these parameters, and therefore b and a T), are functions of composition. Liquid and vapor mixtures in equilibrium in general have different compositions. The PV isotherms generated by an equation of state for these different compositions are represented in Fig. 4-9 by two similar lines the solid line for the liquid-phase composition and the dashed line for the vapor-phase composition. They are displaced from each other because the equation-of-state parameters are different for the two compositions. 

No satisfactory relationship of this type is known for liquids and solids, but for gaseous systems certain moderately simple equations of state have been proposed. AH gases actually differ in their behavior, and so the problem is approached by postulating the properties of an ideal gas, and then considering deviations from ideal behavior. 

Clarke, M.A. Bishnoi, P.R. Development of a new equation of state for mixed salt and mixed solvent systems, and application to vapour liquid equilibrium and solid (hydrate) vapour liquid equilibrium calculations. Fluid Phase Equilibria 2004, 220, 21-35. 

The equations of state obtained for runs 1,4, and 6 in the region of the melting transition are shown in Fig. 6. The equations of state for all three runs exhibit a plateau as would be expected for a first-order melting transition (the density is a discontinuous function of pressure). This plateau is more distinct for the larger system sizes, but there is no evidence for a strong system-size dependence of the coexistence pressure. Smaller systems generally exhibit metastable extensions of the solid and/or liquid branches of the equation of state, due to the fact that the free energy required to create a solid-liquid interface is comparable to... 

A related thermodynamic method is to fit the temperature and density dependence of simulation data to equations of state for the solid and liquid.[158] The explicit forms of the pressure P(T, p) and internal energy U T, p) equations for each phase are used to calculate the entropy and free energy. The condition AG = 0 determines the solid-liquid equilibrium temperature as a function of pressure. 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

Electrolyte solutions are special and can only be treated by the methods considered in Sec. 9.10. Therefore, electrolyte solutions are not be considered in this discussion. Also, the Henry s law standard state is used only for a component that does not exist as a pure component at the temperature of interest—for example, a dissolved-solid below its melting point, or a dissolved gas much above its critical point. (However, if the liquid mixture can be described by an equation of state—for example, mixtures of hydrocarbons and nitrogen or carbon dioxide—there is no need to use the Henry s law standard state.)... 

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