Binary mixtures, analytical equations

A more comprehensive introduction is Ref. [399], We restrict ourselves to uncharged species and dilute solutions (not binary mixtures). The important subject of polymer adsorption is described in Ref. [400], Adsorption of surfactants is discussed in Ref. [401], Adsorption of ions and formation of surface charges was treated in Chapter 5. In dilute solutions there is no problem in positioning the Gibbs dividing plane, and the analytical surface access is equal to the thermodynamic one, as occurs in the Gibbs equation. For a thorough introduction into this important field of interface science see Ref. [8],... 

The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12], However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much Less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [ 19,20], the available area vanishes only for the unphysical total coverage 9 = 9 +0p = 1 (where the subscripts S and L stand for small and large disk radii, respectively), hence there is no jamming . Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21], The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22], However, there are no such calculations for the RSA model. 

In the above equations denotes the vapor mole fraction of component i, Pf is the vapor pressure of the pure component i, Bu is the second virial coefficient of component i, dn = 2Bn — Bn — B22 and B 2 is the crossed second virial coefficient of the binary mixture. The vapor pressures, the virial coefficients of the pure components and the crossed second virial coefficients of the binary mixtures were taken from [32], The Wilson [38], NRTL [39] and the Van Ness-Abbott [40] equations were used for the activity coefficients in Eq. (17). The expressions for the activity coefficients provided by these three methods were differentiated analytically and the obtained derivatives were used to calculate D = 1 -I- Xj(9 In Yil 2Ci)pj. There is good agreement between the values of D obtained with the three expressions for the systems V,V-dimethylformamide-methanol and methanol-water. For the system V,V-dimethylformamide-water, the D values calculated with the Van Ness-Abbott equation [40] were found in good agreement with those obtained with the NRTL equation, but the agreement with the Wilson expression was less satisfactory. 

The excess Gibbs energy of the ternary mixture was expressed through the Wilson [38], NRTL [39] and Zielkiewicz [32] expressions. Because of the agreement between the latter two expressions, detailed results are presented only for the more simple NRTL expression. The parameters in the NRTL equation were found by htting x-P (the composition of liquid phase-pressure) experimental data [32]. The derivatives (9 i/9xi) c2 ( IX2/dx2)xi and (diX2/dxi)x2 in the ternary mixture were found by the analytical differentiation of the NRTL equation. The excess molar volume (V ) in the binary mixtures (i-j) was expressed via the Redlich-Kister equation... 

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

The performance of cascaded stages may also be described analytically on the basis of material balances and phase equilibrium relations. Consider, for simplicity, a binary mixture of components 1 and 2 and again assume component 1 to be the lower boiling, or more volatile, component. The equilibrium equations at stage j are... 

Note the difference between this method of calculation and the one used in the previous illustration. There we did vapor-liquid equilibrium calculations only for the conditions needed, and then solved the mass balance equations analytically. In this illustration we first had to do vapor-liquid equilibrium calculations for all compositions (to construct the. t- v diagram), and then for this binary mixture we were able to do all further calculations graphically. As shown in the following discussion, this makes it easier to consider other reflux ratios than the one u.sed in this illustration. 

For continuous distillation of a binary mixture of constant relative volatility, the minimum reflux ratio can be determined analytically from the following equation (Treybal, 1980) ... 

The Camahan-Starling equation of state (4.5.4) has been extended by Mansoori et al. [16] to binary mixtures of hard spheres having different diameters. Binary mixtures of hard spheres exhibit fluid-solid phase transitions at packing fractions somewhat larger than that for the pure substance that is, at T) > 0.5. The exact state for the transition depends on composition and on the relative sizes of the spheres. We expect the density of the transition to increase as the size disparity increases the limited computer simulation data available support this expectation [17]. Certain kinds of hard-sphere mixtures are the simplest substances to exhibit a fluid-fluid phase transition [17], but those phase transitions are more like liquid-liquid than vapor-liquid. Analytic representations of the Z(r ) for hard-sphere and other hard-body fluids have been critically reviewed by Boublik and Nezbeda [18]. 

Figure 10.1 When computed from an analytic equation of state using FFF 1, the fugacity vs. composition curve may change significantly with state condition. Top PT diagram for a binary mixture. Filled circles are pure critical points vpl and vp2 are pure vapor-pressure curves cl = critical line mcl = mechanical critical line. Bottom Corresponding fugacity of the more volatile component at 275 K. Broken lines are vapor-liquid tie lines. Isobars at bottom correspond to open circles at top. Bottom same as Figure 8.13. Computed from Redlich-Kwong equation.

If Gm(T, p, x), the appropriate thermodynamic function for a binary mixture, is analytic, the deductions about the behaviour of the various thermodynamic properties are entirely analogous to those for the one-component fluid. The same conclusions arise from any general Taylor series expansion in which all the coefficients are non-zero except those two required to define the critical point [equations (6a, b)]. In particular, the coexistence curve (T vs. x at constant p) should be parabolic, the critical isotherm vs. x at constant T and p) should be cubic, and the molar heat capacity C, ,m should be everywhere finite. 

Assuming that the diffusion coefficients Dj). and Dhi for components in a gas mixture are unchanged from their pure-gas values, the simple expression represented above can be extended to gas mixtures. After substitution of equation 48 into equation 60, the flux equation can be analytically integrated to give the following expression for the permeability of component A in a binary mixture of gases A and B (103) ... 

The possibility of entropy-driven phase separation in purely hard-core fluids has been of considerable recent interest experimentally, theoretically, and via computer simulations. Systems studied include binary mixtures of spheres (or colloids) of different diameters, mixtures of large colloidal spheres and flexible polymers, mixtures of colloidal spheres and rods," and a polymer/small molecule solvent mixture under infinite dilution conditions (here an athermal conformational coil-to-globule transition can occur)." For the latter three problems, PRISM theory could be applied, but to the best of our knowledge has not. The first problem is an old one solved analytically using PY integral equation theory by Lebowitz and Rowlinson." No liquid-liquid phase separation... 

When solid particles are immersed in liquid medium, solid-liquid interfacial interactions will cause the formation of an adsorption layer on their surface. The material content of the adsorption layer is the adsorption capacity of the solid adsorbent, which may be determined in binary hquid mixtures if the so-called adsorption excess isotherm is known. Due to adsorption, the initial composition of the hquid mixture, x , changes to the equilibrium concentration Xj, where n = n +rf2, the mass content of the interfacial phase (e.g., mmol/g). This change, Xj — Xj = Axj, can be determined by simple analytical methods. The relationship between the reduced adsorption excess amount calculated from the change in concentration, = n°(x — xj), and the material content of the interfacial layer is given by the Ostwald-de Izaguirre equation [1-5]. In the case of purely physical adsorption of binary mixtures, the material content of the adsorption layer ( = — n) for component 1 is... 

Although a number of theoretical questions remain (unknown equation of state, non analytic corrections to the third order vertex for 6main result is that the SmA-SmA critical point is expected to belong to a new universality class. Experiments confirm this point a high resolution calorimetric study on a binary mixture gives... 

In the previous section we have described how to implement TPTl for a mixture of Lennard-Jones chains with a FENE bonding potential. Before considering binary mixtures, however, we shall restrict our attention to the particular case of a one component system of polymers. In order to describe the thermodynamic properties of such a system, we will consider two TPTl implementations, which we denote TPTl-MSA and TPTl-RHNC. In TPTl-MSA, we employ the fiilly analytic equation of state described in the previous section. In TPTl-RHNC, the Lennard-Jones reference system is described by means of the Reference Hypernetted Chain theory (RHNC). This is an integral equation theory which can only be solved numerically. 

In this paper, the Kirkwood—Buff formalism was used to relate the Henry s constant for a binary solvent mixture to the binary data and the composition of the solvent. A general equation describing the above dependence was obtained, which can be solved (analytically or numerically) if the composition dependence of the molar volume and the activity coefficients in the gas-free mixed solvent are known. A simple expression was obtained when the mixture of solvents was considered to be ideal. In this case, the Henr/s constant for a binary solvent mixture could be expressed in terms of the Henry s constants for the individual solvents and the molar volumes of the individual solvents. The agreement with experiment for aqueous solvents is better than that provided by any other expression available, including an empirical one involving three adjustable parameters. Even though the aqueous solvents considered are nonideal, their degrees of nonideality are much lower than those of the solute gas in each of the constituent solvents. For this reason, the assumption that the binary solvent behaves as an ideal mixture constitutes a reasonable approximation. 

Let us now turn our attention to n-component mixtures. Exact analytical solutions of the Maxwell-Stefan equations for a film model can be obtained for a mixture of ideal gases for which the binary diffusion coefficients are independent of composition and identical to the diffusivity of the binary gas i-k pair. Solutions of the Maxwell-Stefan equations for certain special cases involving diffusion in ternary systems have been known for a long time (Gilliland (1937) = 0) Pratt (1950) (TV, = 0) Cichelli et al. (1951) Toor (1957),... 

General analysis of the binary solvent mixtures formed by two solvate active components (these solvents are often used in analytical and electrochemistry) was conducted to evaluate their effect on H-acids. The analysis was based on an equation which relates the constant of ion association, K, of the solvent mixture and constants of ion association of the acid Kj and K of each component of the mixed solvent, using equilibrium constants of scheme [9.105] - heteromolecular association constant, ionization constant of the... 

The first systematic approach to a derivation the global phase diagram of ternary fluid mixture using an analytical investigation of the Van der Waals equation of state with standard one-fluid mixing rules was developed by Bluma and Deiters (1999). Eight major classes of ternary fluid phase diagrams were outlined and their relationship to the main types of binary subsystems were established. 

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