Binary mixture models

Using ethylammonium nitrate (EAN) as PIL ( . , =0.95, ji = 1.12, a=1.10, P = 0.46), following types of binary mixture models were selected for the analysis and quantification of the microscopic solvent properties (a) [molecular aprotic solvent with HBA ability + PIL cosolvent], (b) [molecular aprotic solvent with both HBD and HBA ability + PIL cosolvent] and (c) [molecular protic solvent + PIL cosolvent] [31]. The molecular solvents included in this analysis were dimethylsul-phoxide (DMSO) ( ., =0.44, 7r = 1.00, a=0.02 and p=0.76) as a polar aprotic HBA solvent, acetonitrile (AN) ( .,. =0.46, 7t =0.75, a=0.19, p=0.40) as polar aprotic HBA/HBD solvent and methanol ( .,. =0.76, 7t =0.60, a=0.98, p=0.66) as a protic solvent. EAN is a N-H-bond donor. In all cases, the pure component part of the mixtures was capable of forming associated species through hydrogen-bonding interactions. For the explored solvent mixtures, empirical parameters . n, a and P were calculated from the wave numbers of the absorbance maxima of the corresponding chemical probes at 25°C. 

The molecular-level assumption underlying the Redlich-Kister expansion is that completely random mixtures are formed, that is, that the ratio of species 1 to species 2 molecules in the vicinity of any molecule is, on the average, the same as the ratio of their mole fractions. A different class of excess Gibbs energy models can be formulated by assuming that the ratio of species 1 to species 2 molecules surrounding any molecule also depends on the differences in size and energies of interaction of the chosen molecule with species 1 and species 2. Thus, around each molecule there is a local composition that is different from the bulk composition. From this picture, the several binary mixture models have been developed. 

It must be emphasized, however, that the treatment presented in Eqs. (68)-(80) is too restrictive since it considers a simple diffusive relaxation only such a treatment applies to solid binary mixtures ( model B in the Hohenberg-Halperin [144] classification), while in a fluid binary mixture it is necessary to include the long range order parameter fluctuations that are transmitted by velocity fluctuations [145]. The resulting model model H in the Hohenberg-Halperin classification leads to a renormalization of the Onsager coefficient A(q) due to mode-coupling effects [146]. Equation [77] remains valid but A(q) is replaced by... 

We can obtain models for three-component (or ternary) miKtures by augmenting the binary mixture models. The hnear model is given by... 

A variant of the two-state model of liquid water was used in a recent work by Patey and co-workers that shows that if we categorize a water molecule in the liquid by using the tetrahedral order parameter into liquid-like (low ih) and ice-like (high h) molecules, and then treat the liquid as a binary mixture, such a binary-mixture model can indeed reproduce most of the anomalies of liquid water [13]. Thus, there does not seem to be any need to invoke the existence of the LDL state. 

In Section 8.2 we discuss the main ideas behind the formalism and illustrate some of the features based on predictions from integral equation calculations involving simple binary mixtures modeled as Lennard-Jones systems (Section 8.2.1), to guide the development of, and provide molecular-based support to, the macroscopic modeling of high-temperature dilute aqueous-electrolyte solutions (Section 8.2.2), as well as to highlight the role played by the solvation effects on the pressure dependence of the kinetic rate constants of reactions in near-critical solvents (Section 8.2.3). 

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

Within this general framework there have been many different systems modelled and the dynamical, statistical prefactors have been calculated. These are detailed in [42]. For a binary mixture, phase separating from an initially metastable state, the work of Langer and Schwartz [48] using die Langer theory [47] gives the micleation rate as... 

Another important class of materials which can be successfiilly described by mesoscopic and contimiiim models are amphiphilic systems. Amphiphilic molecules consist of two distinct entities that like different enviromnents. Lipid molecules, for instance, comprise a polar head that likes an aqueous enviromnent and one or two hydrocarbon tails that are strongly hydrophobic. Since the two entities are chemically joined together they cannot separate into macroscopically large phases. If these amphiphiles are added to a binary mixture (say, water and oil) they greatly promote the dispersion of one component into the other. At low amphiphile... 

One of Che earliest examples of a properly conceived experimental investigation of the flux relations for a porous medium is provided by the work of Gunn and King [53] on the dusty gas model equations, and the following discussion is based largely on their work. Since all their experiments were performed on binary mixtures, the appropriate flux relations are (5.26) and (5,27). Writing... 

In summary, a combination of the plot based on equation (10.6), using any single substance, and determination of the asymptote (10.14), using any pair of substances, provides a sound means of evaluating the parameters K, tC and. Having found these, further experimental points on (10.6) and (10.15), and possibly also (10.7), provide a check on the adequacy of the dusty gas model. Provided attention is limited to binary mixtures, this check can be quite comprehensive. In their published paper Gunn and King... 

They then compared measured and predicted fluxes for diffusion experiments in the mixture He-N. The tests covered a range of pressures and a variety of compositions at the pellet faces but, like the model itself, they were confined to binary mixtures and isobaric conditions. Feng and Stewart [49] compared their models with isobaric flux measurements in binary mixtures and with some non-isobaric measurements in mixtures of helium and nitrogen, using data from a variety of sources. Unfortunately the information on experimental conditions provided in their paper is very sparse, so it is difficult to assess how broadly based are the conclusions they reached about the relative merits oi their different models. 

Though the solution procedure sounds straightforward, if tedious, practice difficulty is encountered immediately because of the implicit nature of the available flux models. As we saw in Chapter 5 even the si lest of these, the dusty gas model, has solutions which are too cumbersc to be written down for more than three components, while the ternary sol tion itself is already very complicated. It is only for binary mixtures therefore, that the explicit formulation and solution of equations (11. Is practicable. In systems with more than two components, we rely on... 

Apart from the Knudsen limit equations (12.12), the only other reasonably compact solution of the dusty gas model equations is that given by equations (5.26) and (5.27), corresponding to binary mixtures. Consequently, if we are to study anything other than the Knudsen limit, attention will... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

In a mixture of / -hexane and benzene (29), the deep catalytic oxidation rates of benzene and / -hexane in the binary mixture are lower than when these compounds are singly present. The kinetics of the individual compounds can be adequately represented by the Mars-VanKrevelen mechanism. This model needs refinements to predict the kinetics for the mixture. 

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

Adsorption of hard sphere fluid mixtures in disordered hard sphere matrices has not been studied profoundly and the accuracy of the ROZ-type theory in the description of the structure and thermodynamics of simple mixtures is difficult to discuss. Adsorption of mixtures consisting of argon with ethane and methane in a matrix mimicking silica xerogel has been simulated by Kaminsky and Monson [42,43] in the framework of the Lennard-Jones model. A comparison with experimentally measured properties has also been performed. However, we are not aware of similar studies for simpler hard sphere mixtures, but the work from our laboratory has focused on a two-dimensional partly quenched model of hard discs [44]. That makes it impossible to judge the accuracy of theoretical approaches even for simple binary mixtures in disordered microporous media. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

Fig, 18a,b, The polymorphic behaviour of the phasmidic-like mesogens a molecular model with six terminal chains b phase diagram for the binary mixture of double swallow-tailed compound (I) with conventional rod-like mesogen (II) (adapted from Letko et al. [123])... 

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. 

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