Equation of state approach

In this approach, one relates the surface pressure n with the surface excess T2. The surface pressure is defined by Eq. (4.15), 

For an ideal surface film, behaving as a two-dimensional gas the surface pressure n is related to the surface excess F2 by the equation, 

21) is referred to as the Henry s law isotherm, which predicts a linear relationship between F2 and C2. 

Clearly, Eqs. (4.15) and (4.18) are based on an idealized model in which the lateral interaction between the molecules has not been considered. Moreover, in this model the molecules are considered to be dimensionless. This model can only be applied at very low surface coverages where the surfactant molecules are so far apart that lateral interaction may be neglected. Moreover, under these conditions the total area occupied by the surfactant molecules is relatively small compared with the total interfacial area. 

At significant surface coverages, the above equations have to be modified to take into account both lateral interaction between the molecules as well as the area occupied by them. Lateral interaction may reduce n if there is attraction between the chains (e.g. with most nonionic surfactant) or it may increase ti as a result of repulsion between the head groups in the case of ionic surfactants. 

Equation-of-State Approach Although the gamma/phi approach 

A generally applicable alternative to the gamma/phi approach results when both the liquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-204), may be applied to each phase  

This introduces compositions Xt and into the equilibrium equations, but neither is explicit, because the d are functions, not only of T and 

but of composition. Thus, Eq. (4-331) represents N complex relationships connecting T, F, and yi). 

Two widely used cubic equations of state appropriate for VLE calculations, both special cases ofEq. (4-100) [with Eqs. (4-101) and (4-102)], are the Soave-Redlich-Kwong (SRK) equation and the Peng-Robinson (PR) equation. The present treatment is applicable to both. The pure numbers , G, 4, and H and expressions for Ct Tr ) specific to these equations are listed in Table 4-2. The associated expression for is given by Eq. (4-246). 

Differentiating Equation (5.17) at constant temperature, dx = RT dr 2 Using the Gibbs equation, 

We can now use the Redlich-Kwong equation of state [6] and a liquid-phase correlation (or an equation of state) to obtain expressions for and as functions of temperature, pressure and component critical properties. This is the approach taken by the very popular Chao-Seader [6] and Grayson-Streed [6] methods. The only factor that remains undefined is the liquid activity coefficient. The Chao-Seader and Grayson-Streed methods use the regular solution theory to obtain an expression for as follows  

We can now use the K-value expression to calculate various equilibrium properties and perform typical flash calculations. As with the simple thermodynamic approach, we can use the heat capacities, and heats of vaporization to obtain enthalpy balances for vapor and liquid streams. In addition, since we account for vapor- and liquid-phase non-ideality due to component interactions, and temperature and pressure effects, we can also apply standard thermodynamic relationships to compute excess properties for enthalpies, etc. The excess properties account for deviations from an ideal mixing behavior and the resulting deviations in equilibrium behavior. 

The most rigorous approach is the equation of state (EOS) approach. When we use an EOS, both vapor and liquid phases uses the same model. We do not modify the general equilibrium statement from Equation (1.48) because we can calculate the fugacity coefficients directly after we choose a particular EOS. 

There are many types of EOS with a wide range of complexity. The Redlich-Kwong (RK) EOS is a popular EOS that relies only on critical temperatures and critical pressures of all components to compute equilibrium properties for both liquid and vapor phases. However, the RK EOS does not represent liquid phases accurately and is not widely used, except as a method to compute vapor fugacity coefficients in activity-coefficient approaches. On the other hand, the Benedict-Webb-Rubin-Starling (BWRS) EOS [6] has up to sixteen constants specific for a given component This EOS is quite complex and is generally not used to predict properties of mixture with more than few components. 

We give the basic form of the PR EOS in Equation (1.63). The PR-EOS requires three main properties critical temperature, critical pressure and the acentric factor. 

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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 ... 

Equation-of-State Approach Although the gamma/phi approach to X- E is in principle generally applicable to systems comprised of subcritical species, in practice it has found use primarily where pressures are no more than a few bars. Moreover, it is most satisfactoiy for correlation of constant-temperature data. A temperature dependence for the parameters in expressions for is included only for the local-composition equations, and it is at best only approximate. 

Equal probability sampling (EPS), in parameter estimation, 26 1039 Equation of state approach, 24 134. See also Equations of state Equation-oriented approach, 20 729... 

We believe that the PFGC equation of state approach will be the most fruitful new route to predicting phase behavior of the diverse systems encountered in the natural gas/petroleum/coal liquefaction gasification process industry. We commend it to your attention. 

Current thermodynamic theories for polymer systems are combinations of the Flory -Huggins, Guggenheim, and Equations-of-State approaches. All of these theories make use of empirical parameters and are based on assumptions about the underlying molecular model. 

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] ... 

Also, the original Hildebrand approach has been refined to take into account the contribution of polar groups and hydrogen bsolubility parameters. These mndifications of the Flory-Huggins theory and of the solubility parameter concept have made these methods an even more useful tool in the description of solutions, especially of mixtures containing polymer compounds. A comprehensive treatment of these extensions of Flory-Huggins and Hildebrand s theories, as well as the new equation of state approach of Flory (1965), bns re ntly been published (Shinoda, 1978 Olahisi et al 1979). 

It is a logical requirement that equations of state approach the ideal gas equation at the limit of low pressures. As the pressure decreases, the volume increases so that at very low pressures a/V P and V. If these terms are dropped from the van der Waals equation, it reduces to the ideal gas form. The terms and b account for intermolecular forces and molecular volume. The parameters a and b are called the attraction and repulsion parameters, respectively. The parameter b is also referred to as the effective molecular volume. 

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. 

Theoretical approaches have been applied to binary Lennard-Jones 12-6 mixtures by Rick and Haymet, who used DFT [222], and by Cottin and Monson [109], who used their cell theory plus fluid phase equation of state approach. The latter approach gave qualitatively good agreement with experimental results for binary mixtures involving argon, krypton, and methane. Quantitative agreement with experiment was found to be sensitive... 

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