Pure components, theory

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

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). 

Equatioa-of-state theories employ characteristic volume, temperature, and pressure parameters that must be derived from volumetric data for the pure components. Owiag to the availabiHty of commercial iastmments for such measurements, there is a growing data source for use ia these theories (9,11,20). Like the simpler Flory-Huggias theory, these theories coataia an iateraction parameter that is the principal factor ia determining phase behavior ia bleads of high molecular weight polymers. 

Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent hquid-diffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binaiy diffusion coefficients have significantly limited the utihty of the theories (see Reid et al.). 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, nl =. 3 mol/kg, Kipi = K2P2 = 1. Use the ideal adsorbed-solution theory to determine ni and n. Substituting the pure component Langmuir isotherm... 

A theory of regular solutions leading to predictions of solution thermodynamic behavior entirely in terms of pure component properties was developed first by van Laar and later greatly improved by Scatchard [109] and Hildebrand [110,1 11 ]. It is Scatchard-Hildebrand theory that will be briefly outlined here. Its point of departure is the statement that It is next assumed that the volume... 

It is difficult to point to the basic reason why the average-potential model is not better applicable to metallic solutions. Shimoji60 believes that a Lennard-Jones 6-12 potential is not adequate for metals and that a Morse potential would give better results when incorporated in the same kind of model. On the other hand, it is possible that the main trouble is that metal solutions do not obey a theorem of corresponding states. More specifically, the interaction eAB(r) may not be expressible by the same function as for the pure components because the solute is so strongly modified by the solvent. This point of view is supported by considerations of the electronic models of metal solutions.46 The idea that the solvent strongly modifies the solute metal is reached also through a consideration of the quasi-chemical theory applied to dilute solutions. This is the topic that we consider next. 

To recapitulate, the Flory version of the Prigogine free-volume or corresponding-states polymer solution theory requires three pure-component parameters (p, v, T ) for each component of the solution and one binary parameter (p ) for each pair of components. 

In many cases, one may measure spectra of solutions of the pure components directly, and the above estimation procedure is not needed. For the further development of the theory of multicomponent analysis we will therefore abandon the hat-notation in K. Given the pure spectra, i.e. given K (pxq), one may try and estimate the vector of concentrations (pxl) of a new sample from its measured... 

One major question of interest is how much asphaltene will flocculate out under certain conditions. Since the system under study consist generally of a mixture of oil, aromatics, resins, and asphaltenes it may be possible to consider each of the constituents of this system as a continuous or discrete mixture (depending on the number of its components) interacting with each other as pseudo-pure-components. The theory of continuous mixtures (24), and the statistical mechanical theory of monomer/polymer solutions, and the theory of colloidal aggregations and solutions are utilized in our laboratories to analyze and predict the phase behavior and other properties of this system. 

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

A great many of the difficulties (and sometimes the misunderstandings) arise from point (c). It is however important to notice that the APM describes the properties of solutions as finite differences between suitable composition-dependent averages and the properties of the pure components. Series expansions in powers of 6, p, 6, and a were introduced afterwards for the purpose of qualitative discussion and comparison with other treatments, e.g., the theory of conformal solutions.34>85>36 They introduce artificial difficulties due to their slow convergencef which have nothing to do with the physical ideas of the APM. Therefore expansions of this type should be proscribed for all quantitative applications one should instead use the compact expressions of the excess functions. 

The outer and core radii were determined from optical resonance measurements using Mie theory solutions (Aden and Kerker, 1951 Bohren and Huffman, 1983) for concentric spheres to interpret the resonance spectra. Figure 36 presents some of the data of Ray et al. for a pure component glycerol droplet and for a coated droplet having an initial coating thickness given by yo = 0-321. Here y is a reduced thickness defined by y = (a - aj/a. 

Two different methods have been presented in this contribution for correlation and/or prediction of phase equilibria in ternary or mul> ticomponent systems. The first method, the clinogonial projection, has one disadvantage it is not based on concrete concepts of the system but assumes, to a certain extent, additivity of the properties of individiial components and attempts to express deviations from additivity of the properties of individual components and attempts to express deviations from additivity by using geometrical constructions. Hence this method, although simple and quick, needs not necessarily yield correct results in all the cases. For this reason, the other method based on the thermodynamic description of phase equilibria, reliably describes the behaviour of the system. Of cource, the theory of concentrated ionic solutions does not permit a priori calculation of the behaviour of the system from the thermodynamic properties of pure components however, if a satisfactory equation is obtained from the theory and is modified to express concrete systems by using few adjustable parameters, the results thus obtained are still substantially more reliable than results correlated merely on the basis of geometric similarity. Both of the methods shown here can be easily adapted for the description of multicomponent systems. 

Both DCLS and ICLS are methods that work well on simple systems. We only consider using these methods if the theory describing the analytical tool employed supports the mcidel as.sumptions. The only difference between these two approaches is in how the pure spectra are obtained. Therefore, the choice of direct over indir ect CLS is dictated by the availability of pure-component spectra. 

When two similarly structured anionic surfactants adsorb on minerals, the mixed admicelle approximately obeys ideal solution theory (jUL - Below the CMC, the total adsorption at any total surfactant concentration is intermediate between the pure component adsorption levels. Adsorption of each surfactant component in these systems can be easily predicted from pure component adsorption isotherms by combining ideal solution theory with an empirical correspond ng states theory approach (Z3). ... 

Scamehorn et. al. (20) also presented a simple, semi—empirical method based on ideal solution theory and the concept of reduced adsorption isotherms to predict the mixed adsorption isotherm and admicellar composition from the pure component isotherms. In this work, we present a more general theory, based only on ideal solution theory, and present detailed mixed system data for a binary mixed surfactant system (two members of a homologous series) and use it to test this model. The thermodynamics of admicelle formation is also compared to that of micelle formation for this same system. 

A previously proposed theory to describe mixed adsorption in these systems (20) depended not only on ideal solution theory, but also on the correspond ng states theory to apply to surfactant mixtures. In that model, it was assumed that the adsorption isotherms for the pure components coincided when plotted against a reduced concentration. This occurs when the ratio CACB E/CACrt is the same at any adsorption level. When true, this simplifies the prediction of mixed adsorption isotherms somewhat, but that model is really a special case of the model presented here. 

Over the years, various other theories and models have been proposed for predicting salt effect in vapor-liquid equilibrium, including ones based on hydration, internal pressure, electrostatic interaction, and van der Waals forces. These have been reviewed in detail by Long and McDevit (25), Prausnitz and Targovnik (31), Furter (7), Johnson and Furter (8), and Furter and Cook (I). Although the electrostatic theory as modified for mixed solvents has had limited success, no single theory has yet been able to account for or to predict salt effect on equilibrium vapor composition from pure-component properties alone. 

With the relations given in Table 2.2-1 and the critical exponent values given in Table 2.2-2, the thermodynamic behaviour of a pure component close to the critical point can be described exactly, however further away from the critical point also the mean field contributions have to be taken into account. A theory which is in principle capable to describe... 

For electrolyte solutions such as NaCl + water the critical temperatures of the pure components differ by about a factor of five. From the perspective of nonelectrolyte thermodynamics, the absence of a liquid-liquid immiscibility then comes as a great surprise. It is a major challenge for theory to explain why this salt, as well as similar salts such as KC1 or CaCl2, seems to show a continuous critical line. Perhaps there is a slight indication for a transition toward an interrupted critical curve in Marshall s study [151] of the critical line of NaCl + H20. Marshall observed a dip in the TC(XS) curve some K away from the critical point of pure water, which at first glance seems obscure. It was suggested [152] that the vicinity to an upper critical end point leaves its mark by this dip. 

Physical measurements are directly input to the statistical thermodynamics theory. For example three-phase hydrate formation data, and spectroscopic (Raman, NMR, and diffraction) data were used to determine optimum molecular potential parameters (e,o,a) for each guest, which could be used in all cavities. By fitting only a eight pure components, the theory enables predictions of engineering accuracy for an infinite number of mixtures in all regions of the phase diagram. This facility enables a substantial savings in experimental effort. 

Hydrate experimental conditions have been defined in large part by the needs of the natural gas transportation industry, which in turn determined that experiments be done above the ice point. Below 273.15 K there is the danger of ice as a second solid phase (in addition to hydrate) to cause fouling of transmission or processing equipment. However, since the development of the statistical theory, there has been a need to fit the hydrate formation conditions of pure components below the ice point with the objective of predicting mixtures, as suggested in Chapter 5. 

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