Equation, Prausnitz

For the solubility of a solid substance (solute, component 2) in a mixed solvent 1-3, one can write the following equation (Prausnitz et al., 1986) ... 

Assuming that the entire system of water-organism-target is at equilibrium we can state that the chemical s f ugacity (f) in the water (W) and the target (T) site are equal. Applying the conventional (corrected Raoult s Law) equation (Prausnitz 1969) gives... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

Equation (16-36) with y = 1 provides the basis for the ideal adsorbed-solution theoiy [Myers and Prausnitz, AIChE J., 11, 121 (1965)]. The spreading pressure for a pure component is determined by integrating Eq. (16-35) for a pure component to obtain... 

Equation 10.96 does not apply to either electrolytes or to concentrated solutions. Reid, PRAUSNITZ and Sherwood"7 discuss diffusion in electrolytes. Little information is available on diffusivides in concentrated solutions although it appears that, for ideal mixtures, the product /xD is a linear function of the molar concentration. 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

Several activity coefficient models are available for industrial use. They are presented extensively in the thermodynamics literature (Prausnitz et al., 1986). Here we will give the equations for the activity coefficients of each component in a binary mixture. These equations can be used to regress binary parameters from binary experimental vapor-liquid equilibrium data. 

Renon used the concept of local composition to develop a non-random, two-liquid (NRTL) three parameter (al2, tp, ti() equations given below (Prausnitz et al., 1986). 

Alternatively, one may use implicit LS estimation, e.g., minimize Equation 14.23 where liquid phase fugacities are computed by Equation 15.5 whereas vapor phase fugacities are computed by an EoS or any other available method (Prausnitz et al., 1986). 

The NRTL equation developed by Renon and Prausnitz overcomes the disadvantage of the Wilson equation in that it is applicable to immiscible systems. If it can be used to predict phase compositions for vapour-liquid and liquid-liquid systems. 

The UNIQUAC equation developed by Abrams and Prausnitz is usually preferred to the NRTL equation in the computer aided design of separation processes. It is suitable for miscible and immiscible systems, and so can be used for vapour-liquid and liquid-liquid systems. As with the Wilson and NRTL equations, the equilibrium compositions for a multicomponent mixture can be predicted from experimental data for the binary pairs that comprise the mixture. Also, in the absence of experimental data for the binary pairs, the coefficients for use in the UNIQUAC equation can be predicted by a group contribution method UNIFAC, described below. 

Oellrich L, Plocker U, Prausnitz J.M and Knapp H (1981) Equation-of-state Methods for Computing Phase Equilibria and Enthalpies, Int Chem Eng, 21(1) 1. 

Example 6.5 Repeat the calculations from Example 6.4 taking into account vapor-phase nonideality. Fugacity coefficients can be calculated from the Peng-Robinson Equation of State (see Poling, Prausnitz and O Connell6 and Chapter 4). 

VAN AKEN et al. 0) and EDWARDS et al. (2) made clear that two sets of fundamental parameters are useful in describing vapor-liquid equilibria of volatile weak electrolytes, (1) the dissociation constant(s) K of acids, bases and water, and (2) the Henry s constants H of undissociated volatile molecules. A thermodynamic model can be built incorporating the definitions of these parameters and appropriate equations for mass balance and electric neutrality. It is complete if deviations to ideality are taken into account. The basic framework developped by EDWARDS, NEWMAN and PRAUSNITZ (2) (table 1) was used by authors who worked on volatile electrolyte systems the difference among their models are in the choice of parameters and in the representation of deviations to ideality. 

Prausnitz (16) discusses this and related equations as well as the contributions of Margules, Hildebrand, Scatchard, Guggenheim, and others to this topic. For the activity of either component, referenced to the pure liquid, one has... 

The most common model for describing adsorption equilibrium in multi-component systems is the Ideal Adsorbed Solution (IAS) model, which was originally developed by Radke and Prausnitz [94]. This model relies on the assumption that the adsorbed phase forms an ideal solution and hence the name IAS model has been adopted. The following is a summary of the main equations and assumptions of this model (Eqs. 22-29). 

The methods most generally used for the calculation of activity coefficients at intermediate pressures are the Wilson (1964) and UNIQUAC (Abrams and Prausnitz, 1975) equations. Wilson s equation was used by Sato et al. (1985) to predict the composition of fhe condensate gas stripped from a packed bed fermenter at 30°C, whilst Beck and Bauer (1989) used the UNIQUAC equation, with temperature-dependent parameters given by Kolbe and Gmehling (1985), for their model of an anaerobic gas-solid fluidized bed fermenter at 36°C. In this case it was necessary to go beyond the temperature range of fhe source data down to 16°C in order to predict the composition of the fluidizing gas leaving the condenser. 

Lue L, Prausnitz JM. Structure and thermodynamics of homogeneous-dendritic-polymer solutions computer simulation, integral-equation, and lattice-cluster theory. Macromolecules 1997 30 6650-6657. 

To compile quantitatively reliable information, we need a source of experimental measurements. One way to determine the nature of inter-molecular forces between biopolymer molecules in a solvent medium is to measure the so-called osmotic second virial coefficient A2. Expressed in molar (biopolymer) terms, the quantity A2 can be related to the two-body potential of mean force W(r) by the following equation (Vrij, 1976 de Kruif, 1999 Prausnitz, 2003 de Kruif and Tuinier, 2005) ... 

General Method. The effects of composition of mixtures and of pressure on key properties such as enthalpy and entropy are deduced from PVT equations of state. This process is described in books on thermodynamics, for example, Reid, Prausnitz, and Sherwood (Properties of Liquids and Gases, McGraw-Hill, New York, 1977) and Walas (Phase Equilibria in Chemical Engineering, Butterworths, Stoneham, MA, 1985). Only the simplest correlations of these effects will be utilized here for illustration. 

Method of Macknick and Prausnitz The method of Macknick and Prausnitz [11] allows estimation of vapor pressures for liquid hydrocarbons in the range 10 to 1500 mmHg. The method is based on the following equation proposed by Miller [12] ... 

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