Vapor-liquid equilibrium coexistence pressure

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

The application of Eq. (10.3) to specific phase-equilibrium problems requires use of models of solution behavior, which provide expressions for G or for the Hi as functions of temperature, pressure, and composition. The simplest of such expressions are for mixtures of ideal gases and for mixtures that form ideal solutions. These expressions, developed in this chapter, lead directly to Raoult s law, the simplest realistic relation between the compositions of phases coexisting in vapor/liquid equilibrium. Models of more general validity are treated in Chaps. 11 and 12. 

In most industrial processes coexisting phases are vapor and liquid, although liquid/liquid, vapor/solid, and liquid/solid systems are also encountered. In this chapter we present a general qualitative discussion of vapor/liquid phase behavior (Sec. 12.3) and describe the calculation of temperatures, pressures, and phase compositions for systems in vapor/liquid equilibrium (VLE) at low to moderate pressures (Sec. 12.4).t Comprehensive expositions are given of dew-point, bubble-point, and P, T-flash calculations. 

Related Calculations. This illustration outlines the procedure for obtaining coefficients of a liquid-phase activity-coefficient model from mutual solubility data of partially miscible systems. Use of such models to calculate activity coefficients and to make phase-equilibrium calculations is discussed in Section 3. This leads to estimates of phase compositions in liquid-liquid systems from limited experimental data. At ordinary temperature and pressure, it is simple to obtain experimentally the composition of two coexisting phases, and the technical literature is rich in experimental results for a large variety of binary and ternary systems near 25°C (77°F) and atmospheric pressure. Example 1.21 shows how to apply the same procedure with vapor-liquid equilibrium data. 

The vapor-liquid equilibrium curve terminates at the critical temperature and critical pressure (7c nnd Pc). Above and to the right of the critical point, two separate phases never coexist. 

Pure water may coexist as liquid and vapor only at temperature-pressure pairs that fall on the vapor-liquid equilibrium (VLE) curve. At points above the VLE curve (but to the right of the solid-liquid equilibrium curve), water is a subcooled liquid. At points on the VLE curve, water may be saturated liquid or saturated steam (vapor) or a mixture of both. At points below the VLE curve, water is superheated steam. 

The most commonly encountered coexisting phases in industrial practice are vapor and liquid, although liquid/liquid, vaporlsolid, and liquid/solid systems are also found. In this chapter we first discuss the nature of equilibrium, and then consider two rules that give the lumiber of independent variables required to detemiine equilibrium states. There follows in Sec. 10.3 a qualitative discussion of vapor/liquid phase behavior. In Sec. 10.4 we introduce tlie two simplest fomiulations that allow calculation of temperatures, pressures, and phase compositions for systems in vaporlliquid equilibrium. The first, known as Raoult s law, is valid only for systems at low to moderate pressures and in general only for systems comprised of chemically similar species. The second, known as Henry s law, is valid for any species present at low concentration, but as presented here is also limited to systems at low to moderate pressures. A modification of Raoult s law that removes the restriction to chemically similar species is treated in Sec. 10.5. Finally in Sec. 10.6 calculations based on equilibrium ratios or K-values are considered. The treatment of vapor/liquid equilibrium is developed further in Chaps. 12 and 14. 

Remember from Sec. 7.3 that while the the condition dP/dV)j = 0 on the van der Waals loop of an equation of state gave the conditions of mechanical stabiliQ, it did not give the vapor-liquid equilibrium points (that is, the vapor pressure). That had to be determined from the equality of species fugacities in each phase. The situation is much the same here in that the limit of stability from Eq. H.2-9 is not the equilibrium compositions found from the equality of species fugacities in the coexisting liquid phases. 

Figure 11.3-3 shows the vapor-liquid and liquid-liquid equilibrium behavior computed for the system of methanol and n-hexane at various temperatures. Note that two liquid phases coexist in equilibrium to temperatures of about 43°C. Since liquids are relatively incompressible, the species liquid-phase fugacities are almost independent of pressure (see Illustrations 7.4-8 and 7.4-9), so that the liquid-liquid behavior is essentially independent of pressure, unless the pressure is very high, or low enough for the mixture to vaporize (this possibility will be considered shortly). The vapor-liquid equilibrium curves for this system at various pressures are also shown in the figure. Note that since the fugacity of a species in a vapor-phase mixture is directly proportional to pressure, the VLE curves are a function of pressure, even though the LLE curves are not. Also, since the methanol-hexane mixture is quite nonideal, and the pure component vapor pressures are similar in value, this system exhibits azeotropic behavior. 

At the lowest pressure in the figure, P = 0.133 bar, the vapor-liquid equilibrium curve intersects the liquid-liquid equilibrium curve. Consequently, at this pressure, depending on the temperature and composition, we may have only a liquid, two liquids, two liquids and a vapor, a vapor and a liquid, or only a vapor in equilibrium. The equilibrium state that does exist can be found by first determining whether the composition of the liquid is such that one or two liquid phases exist at the temperature chosen. Next, the bubble point temperature of the one or either of the two liquids present is determined (for example, from experimental data or from known vapor pressures and an activity coefficient model calculation). If the liquid-phase bubble point temperature is higher than the temperature of interest, then only a liquid or two liquids are present. If the bubble point temperature is lower, then depending on the composition, either a vapor, or. a vapor and a liquid are present. However, if the temperature of interest is equal to the bubble point temperature and the composition is in the range in which two liquids are present, then a vapor and two coexisting liquids will be in equilibrium. 

With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, Pvap and puq, whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A. 10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. 

The basis of any process design is a knowledge of vapor-liquid equilibrium. It is depicted in Fig. 11,1-1 in the form of vapor pressure lines with constant H2SO4 content in the liquid. Up to liquid concentrations of 75 wt% H2SO4, the coexistent vapor consists of pure water. Therefore, the liquid can be concentrated up by single stage distillation (evaporation). The feasible acid concentration is limited by the temperature level of the steam available at the site. Typically, the acid can be heated up to approximately 170°C, that is, at a total operating pressure of 1 bar, equivalent to an acid concentration of 72 wt%. 

Another technique to determine the vapor-liquid equilibrium of pure substances or mixtures, which has some similarities with what is described in [190, 204-206], is the grand equilibrium method [192]. It is a two-step procedure, where the coexisting phases are simulated independently and subsequently. In the first step, one NpT simulation of the liquid phase is performed to determine the chemical potentials p] and the partial molar volumes v of all components i. These entropic properties can be determined by Widom s test molecule method [207] or more advanced techniques, such as gradual inserticMi [208-210] (see below). On the basis of the chemical potentials and partial molar volumes at a specified pressure po, first order Taylor expansions can be made for the pressure dependence ... 

Could all three phases of water coexist over some finite range of temperatures Could the vapor-liquid equilibrium exist over a range of pressures at one temperature, instead of at just one pressure for any given temperature We have all been told, in previous courses, that the answer is no. But how would you prove that The answer is that we would use the phase rule, often called Gibbs phase rule after Josiah Willard Gibbs (1790-1861). 

For pure water (one component, C = 1) F + P = 3 holds. When three phases are simultaneously in equilibrium with each other, e.g. vapor, liquid and ice, or vapor and two different modifications of ice, then F = 0 there is no degree of freedom, the three phases can coexist only at one fixed pressure and one fixed temperature ( triplepoint ). 

The phase equilibrium for pure components is illustrated in Figure 4.1. At low temperatures, the component forms a solid phase. At high temperatures and low pressures, the component forms a vapor phase. At high pressures and high temperatures, the component forms a liquid phase. The phase equilibrium boundaries between each of the phases are illustrated in Figure 4.1. The point where the three phase equilibrium boundaries meet is the triple point, where solid, liquid and vapor coexist. The phase equilibrium boundary between liquid and vapor terminates at the critical point. Above the critical temperature, no liquid forms, no matter how high the pressure. The phase equilibrium boundary between liquid and vapor connects the triple point and the... 

If the lid is removed, and the external surroundings have partial pressure Ph2o less than 23.8 Torr ( relative humidity < 100% ), then water will evaporate from the beaker into the surroundings until the beaker is empty, because only vapor is stable under these conditions. However, if the external surroundings have partial pressure Ph2o >23.8 Torr, water will condense from the surroundings to fill the beaker, because only liquid is stable under these conditions. Thus, the saturation vapor pressure ( 100% relative humidity ) corresponds to the unique concentration (partial pressure) of water vapor that can coexist at equilibrium in the atmosphere above liquid water at 25°C. Other (T, P) points on the vapor-pressure curve can be interpreted analogously. 

Figure 8.3. Equilibrium constants (log K) for the metastable coexistence of completely ordered and disordered CaMg(C03>2 with dolomite in its stable order/disorder state as a function of temperature at constant pressure. SAT refers to the vapor-liquid curve for pure H2O. (After Bowers et al., 1984.)...

There are three independent variables in coexisting equilibrium vapor/liquid systems, namely temperature, pressure, and fraction liquid (or vapor). If two of these are specified in a problem, the third is determined by the phase behavior of the system. There are seven types of vapor/liquid equilibria calculations in our program, as in Figure 1 under "Single Stage Calculation."... 

The experiment began by charging the equilibrium cell with about 30 cm3 of either phenoPp-cresol or phenol-water solution mixture. The cell was then pressurized with either methane or carbon dioxide until the phenol clathrate formed under sufficient pressure. The systems were cooled to about 5 K below the anticipated clathrate-forming temperature. Clathrate nucleation was then induced by agitating the magnetic spin bar. After the clathrates formed, the cell temperature was slowly increased until the clathrate phase coexisted with the liquid and vapor phases. The nucleation and dissociation steps were repeated at least twice in order to diminish hysteresis phenomenon. The clathrates, however, exhibited minimal hysteresis and the excellent reproducibility of dissociation pressures was attained for all the temperatures and found to be within 0.1 K and 1.0 bar at each time. When a minute amount of phenol or p-cresol clathrate crystals remains and the system temperature was kept constant for at least 8 hours after attaining pressure stabilization, the pressure was considered as an equilibrium dissociation pressure at that specified temperature. 

Liquids Vapor pressure is the most important of the basic thermodynamic properties of fluids. It is the pressure of equilibrium, coexisting liquicf and vapor phases at a specified temperature. The vapor pressure curve is a monotonic function of temperature from its minimum value (the triple point pressure) at the triple point temperature T, to its maximum value (the critical pressure) at T. ... 

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