Liquid-vapor coexistence curve

The order of a transition can be illustrated for a fixed-stoichiometry system with the familiar P-T diagram for solid, liquid, and vapor phases in Fig. 17.2. The curves in Fig. 17.2 are sets of P and T at which the molar volume, V, has two distinct equilibrium values—the discontinuous change in molar volume as the system s equilibrium environment crosses a curve indicates that the phase transition is first order. Critical points where the change in the order parameter goes to zero (e.g., at the end of the vapor-liquid coexistence curve) are second-order transitions. 

A key role in this debate was played by experiments by Bischoff and Rosenbauer [153], who reported accurate data on isothermal vapor-liquid coexistence curves as a function of pressure near the critical line of NaCl + H20. Far from the critical point of pure water, one expects the compositions... 

Figure 8. (a) Vapor-liquid coexistence curves for a mixture of Lennard-Jones molecules with parameters chosen to approximate acetone (1) - carbon dioxide (2) ... 

De Pablo et al. [83] find that on the vapor-liquid coexistence curve, p is consistently underestimated while p is overestimated at all temperatures. The same Authors obtain a better agreement, also for T, with the experimental data using SPC for the liquid and a modified SPC for the gas, with charges scaled down to reproduce the gas phase dipole moment. The alternative to simple charge scaling, as suggested by Strauch and Cummings [84], is the use of a polarizable model. 

As the pressure of the system decreases, the boiling temperature decreases, in general. Therefore, we expect the vapor-liquid coexistence envelope to drop to lower temperatures as the pressures decrease. Changes in pressure, however, do not have a strong influence on the phase behavior of liquids. As a result, we do not expect the liquid-liquid phase envelope to change much with pressure. Consequently, we expect that at a low enough pressure, the vapor-liquid coexistence curve will intersect the liquid-liquid coexistence curve. When this occurs, we can have vapor-liquid-liquid equilibria, which is shown in Figure 3.9. 

Figure 10.1 Vapor-liquid coexistence curves for neon (open squares), argon (filled circles), krypton (filled squares), xenon (open down-triangles), methane (open circles), nitrogen (open diamonds), and oxygen (open up-triangles). Data taken from the NIST Chemistry Webbook http //webbook.nist.gov.

PY theory for atomic fluids, the energy equation of state predicts the higher critical temperature. Thus, to construct the vapor-liquid coexistence curve from the energy equation of state requires solutions of the SSOZ-PY equation for states where the compressibility equation of state may predict thermodynamic instability. Such solutions often cannot be found numerically and, where they can be found, are surely of dubious physical significance. It seems from these observations that accurate thermodynamic properties over a wide range of fluid states, suitable, for example, for the study of vapor-liquid equilibria, can not be obtained from solutions of the SSOZ-PY or SSOZ-HNC equations. The SSOZ-MSA theory seems to be an exception to this situation as we shall see below. 

Of particular interest is the application of Eq. 7.7-4 to the vapor-liquid coexistence curve because this gives the change in vapor pressure with temperature., 4t temperatures for which the vapor pressure is not very high, it is found that V " and AY y. If, in addition, the vapor phase is ideal, we have AvapY = = RTIP,... 

An isochoric equation of state, applicable to pure components, is proposed based upon values of pressure and temperature taken at the vapor-liquid coexistence curve. Its validity, especially in the critical region, depends upon correlation of the two leading terms the isochoric slope and the isochoric curvature. The proposed equation of state utilizes power law behavior for the difference between vapor and liquid isochoric slopes issuing from the same point on the coexistence cruve, and rectilinear behavior for the mean values. The curvature is a skewed sinusoidal curve as a function of density which approaches zero at zero density and twice the critical density and becomes zero slightly below the critical density. Values of properties for ethylene and water calculated from this equation of state compare favorably with data. 

Figure 2. Finite-size effects in the vapor-liquid coexistence curve of -octane (TraPPE model) obtained from simulations in the Gibbs ensemble [89]. Open circles and crosses depict results for simulations with N = 200 and 1600, respectively. The upper sets of points use a mean-field exponent (j3 = 0.5) and the lowers ones use an Ising-like exponent (fi — 0.32). The estimated critical temperatures for the Ising-like exponent are also shown.

A supercritical fluid is defined as one that is beyond its critical point and thus cannot be liquefied by a change in pressure. On a P-T plot, the supercritical region lies at the end of the vapor-liquid coexistence curve and is bounded by the lines P = Pc and T = Tc. In most applications, however, the compressed liquid region above the P = Pc line and to the left of the T = Tc line is also useful. Table 1 gives critical parameters for commonly used supercritical fluids. 

Hydrogen fluoride presents another example of a fluid in which hydrogen bonding is crucial for the structural and thermodynamic properties. Figure 1.20 draws the HF vapor-liquid coexistence curve obtained from the RISM/KH theory in comparison with that following from the molecular OZ/RHNC approach [109] and the simulation data... 

Retrieve the vapor pressure and the liquid density data for methane from the DDBSP Explorer Version and export the values to Excel. Implement a liquid vapor pressure curve calculation for the van der Waals equation of state in Excel-VBA and compare the results along the vapor-liquid coexistence curve to the experimental data. 

PI 3.2 Estimate the heat capacity of acetic acid along the vapor-liquid coexistence curve for both phases using the vapor pressure and ideal gas heat capacity correlations given in Appendix A and the dimerization and tetramerization constants from Table 13.6. 

Calculate the saturated vapor pressure and fugacity along the vapor-liquid coexistence curve for benzene, water and acetic acid between 25 "C and the critical temperature of the components. For the real vapor phase behavior, use the virial equation truncated after the second virial coefficient in case of water and benzene and the chemical theory in case of acetic acid. Discuss the results. Inside which temperature range are the results reliable Do the calculations lead to under- or overprediction of the fugacity outside the reliable temperature range ... 

Mattin MG (2006) Comparison of the AMBER. CHARMM, COMPASS, GROMOS, OPLS, TraPPE and UEF fince fields for prediction of vapor-liquid coexistence curves and liquid densities. Fluid Phase Equilib 248 50-55... 

Abstract Configurational-bias Monte Carlo simulations in the Gibbs ensemble have been carried out to determine the vapor-liquid coexistence curve for a pentadecanoic acid Langmuir monolayer. Two different force fields were studied (i) the original monolayer model of Karaborni and Toxvaerd including anisotropic interactions between alkyl tails, and (ii) a modified version of this model which uses an isotropic united-atom description for the methylene and methyl groups and includes dispersive interactions between the tail segments and the water surface. 

MC simulation Vapor-liquid coexistence curve [271] Singh et al., 1990... 

Bolhuis and Frenkel have studied the phase behavior of a mixture of hard spheres and hard rods. In particular, Bolhuis and Frenkel used Gibbs-ensemble simulations to determine the vapor-liquid coexistence curve. In a Gibbs-ensemble simulation one simulates two boxes that are kept in equilibrium with each other via Monte Carlo rules. In this case the gas box has as a low density of hard spheres and the liquid box has a high density of spheres. Similarly to the phase equilibrium calculation of linear alkanes, the exchange step, in which particles are exchanged between the two boxes, is the bottleneck of the simulation. For example, the insertion of a sphere into the gas phase would almost always fail because of overlaps with some of the rods. Bolhuis and Frenkel have used the following scheme to make this exchange possible ... 

For fluids, the point on the phase diagram where the vapor-liquid coexistence curve ends. 

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