The Equation of State Approach

The methodology for solving the vapor-liquid equilibrium problem using an equation of state has been outlined in Section 13.4 and involves expressing the fugacity coefficients of the mixture components in the liquid (/) and vapor (v) phases through an equation of state (EoS). The equilibrium ratio Kj is then calculated from  

Cubic EoS are the most commonly used equations of state for high pressure VLE calculations. They are relatively simple to use and provide reasonable accuracy. We confine, therefore, our discussion to such EoS and proceed to  

examine the information needed to carry out VLE calculations using them  

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The second common procedure for VLE calculations is the equation-of-state approach. Here, fugacity coefficients replace the fugacities for both Hquid and vapor phases, and equation 220 becomes equation 226 ... 

The advancing contact angles, measured on smooth silicon surfaces coated with the polymer brushes or the silane, were used to calculate the solid surface tension ysv according to the equation of state approach (EQS) [39] ... 

There are generally two approaches for treating surfactant adsorption at the A/L and L/L interfaces. The first approach, adopted by Gibbs, treats adsorption as an equilibrium phenomenon whereby the Second Law of Thermodynamics may be applied using surface quantities. The second approach, referred to as the equation of state approach, treats the surfactant film as a two-dimensional layer with a surface pressure jt that may be related to the surface excess F (the amount of surfactant adsorbed per unit area). These two approaches are summarised below. 

Hydrogen Bonding in Solutions The Equation-of-State Approach... 

We now discuss how predictions of solubility and phase behavior can be carried out. Before proceeding with the equation of state approach, we present a brief discussion of the solubility parameter concept because solubility parameters continue to be presented in the literature as one means of correlating SCF-solute behavior. 

The equation of state approach is very attractive for the calculation of VLE But it requires an equation of state and reliable mixing rules, which are able to describe the PvTbehavior not only of the vapor but also of the liquid phase with the required accuracy. In spite of the progress achieved in the last 20 years, up to now there is no universal equation of state and mixing rule which can be successfully applied to all kind of systems in a wide temperature and pressure range for pure compounds and mixtures. 

While there is no difference for the calculation of gas solubilities in comparison to VLE in the case of the equation of state (approach A), for approach B, there is the problem that the standard state (pure liquid at system temperature and system pressure) used for VLE calculations cannot be used anymore, since supercritical compounds are not existent as liquid. This means an alternative standard fugacity is required for the y —y)-approach. 

Corresponding states models have been described in detail by Patterson and more recently, in the context of polymer mixtures, by Patterson and Robard who develop the similarity with the equation-of-state approach. They are less often used in current literature than lattice-graph models, but are useful in systems where volume changes are not negligible, e.g. when pressure (see also... 

Note that the limiting value at low pressure, RT dBp/dT)/Cp, i. is not necessarily equal to zero even though the equation of state approaches that of an ideal gas in this limit. 

Panayiotou, C. Hydrogen Bonding in Solutions The Equation of State Approach, In Handbook of Surface and Colloid Chemistry, Birdi, K.S. (Ed.), CRC Press Boca Raton, FL, 2003. 

These experimental results are consistent with the equation-of-state approach for interfacial tensions (6) A recent formulation of this can be written as follows ... 

By tentatively equating yc and ysv, contact angles can be evaluated by the equation-of-state approach. Other indirect methods for determining contact angles will be described below. 

Enthalpy models are also necessary in VLB design calculations. The procedures followed for vapor and liquid enthalpy calculations are specified in Figure 14. The equation of state approach l-(i) is the most popular for vapor enthalpy. For liquid enthalpy, the equation of state approach l-(i), the Yen-Alexander correlation l-(li), and method 2 of mole fraction averaging the pure component enthalpies are used with about equal frequency. Method 3 for liquid enthalpy includes a heat of mixing term,... 

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