Vapor-liquid equilibrium free energy

Fig. 14. Schematic illustration of a drop ofliquid spreading in contact with a solid surface, showing the relations between the relevant parameters the contact angle, 0 the solid/vapor interfacial free energy, Ysv the liquid/vapor interfacial free energy, yLV and the solid/liquid interfacial free energy, ySL. Young s equation describes the relationship between these parameters for a stationary drop at thermodynamic equilibrium [175]...

In principle, the parameters can be evaluated from minimal experimental data. If vapor-liquid equilibrium data at a series of compositions are available, the parameters in a given excess-free-energy model can be found by numerical regression techniques. The goodness of fit in each case depends on the suitability of the form of the equation. If a plot of GE/X X2RT versus X is nearly linear, use the Margules equation (see Section 3). If a plot of Xi X2RT/GE is linear, then use the Van Laar equation. If neither plot approaches linearity, apply the Wilson equation or some other model with more than two parameters. 

Cukor and Prausnitz (4), however, point out that Wolds general expression precludes other expressions for the composition dependence of the excess free energy, including that of Wilson (5), which has been used by several authors to predict and correlate vapor-liquid equilibrium... 

Wilson, G. M. (1964). Vapor-liquid equilibrium. XI A new expression for the excess free energy of mixing. Journal of the American Chemical Society, 86, 127-130. 

Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of... 

Consider a beaker of liquid (the condensed phase is indicated with the subscript c) in equilibrium with its vapor (indicated with the subscript v). Hold T and p constant. The degree of freedom is the number of particles Nc in the condensed phase, or the number TV, in the vapor phase. The free energy change of the combined system (vapor plus condensed phase) is expressed in terms of the chemical potentials Pc and Pv. Because dT = dp = 0, the free energy depends only on the chemical potentials and the numbers of particles in the tw o phases ... 

For calculation of the equilibrium compositions of the liquid phase either the equilibrium constants of the dissociation and polycondensation reactions have to be known or they can be computed by methods which use the approach of minimizing Gibbs free energy [200-202]. In addition, ab initio modeling techniques such as density functional theory (DFT) in combination with reactive molecular dynamic (MD) simulations could be used. Once the liquid phase system is modeled, there are in principle two options to describe the vapor-liquid equilibrium. Either equations of state (EOS) or excess Gibbs free energy models (g -models) may be used to describe the thermodynamics of the liquid... 

Accurate vapor-liquid equilibrium measurements can be used to compute liquid-phase activity coefficients and excess Gibbs free energies. Consider the data in Table P7.3 for benzene-2,2,4-trimethylpentane (B-TMP) mixtures at constant temperature of 55"C. 

We have seen in Chapter 12 that when a system containing two phases reaches equilibrium - at constant temperature and pressure - the total Gibbs free energy assumes its minimum value and that this, in turn, leads to the equality of fugacities for any component i in the two phases (Section 12.4.4). In the case of vapor-liquid equilibrium, thus ... 

To find how vapor pressure changes with temperature we make use of the fact that, when a liquid and its vapor are in equilibrium, there is no difference in the molar Gibbs free energies of the two phases ... 

The equilibrium conversion can be calculated from knowledge of the free energy, together with physical properties to account for vapor and liquid-phase nonidealities. The equilibrium conversion can be changed by appropriate changes to the reactor temperature, pressure and concentration. The general trends for reaction equilibrium are summarized in Figure 6.8. 

The discussion of moisture uptake by hygroscopic materials must include a description of the thermodynamics of vapor-liquid equilibria. For gas (g) and liquid (1) phases to be in equilibrium, the infinitesimal transfer of molecules between phases (dng and dn ) must lead to a free energy change of zero. 

When the vapor phase has volume Vl and pressure px and is in equilibrium with a liquid phase of volume V2 and pressure p2, then the differential of the free energy of the system is dF = -p1dV1 -p2dV2. The function F + p2 V2 also depends on the present state of the system only, so that d(F+p2 V2) = PiVl + V2dp2 is a complete differential. Hence... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

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