Multicomponent thermodynamic factors

In the theoretical treatment of diffusive reactions, one usually works with diffusion coefficients, which are evaluated from experimental measurements. In a multicomponent system, a large number of diffusion coefficients must be evaluated, and are generally interrelated functions of alloy composition. A database would, thus, be very complex. A superior alternative is to store atomic mobilities in the database, rather than diffusion coefficients. The number of parameters which need to be stored in a multicomponent system will then be substantially reduced, as the parameters are independent. The diffusion coefficients, which are used in the simulations, can then be obtained as a product of a thermodynamic and a kinetic factor. The thermodynamic factor is essentially the second derivative of the molar Gibbs energy with respect to the concentrations, and is known if the system has been assessed thermodynamically. The kinetic factor contains the atomic mobilities, which are stored in the kinetic database. 

If we introduce the (n — l)-dimensional matrix [T] of thermodynamic factors, we can recast the LHS of Eq. 5.35 in terms of the mole fraction gradients. The Maxwell-Stefan diffusion for multicomponent systems is thus [38]... 

It is interesting to note that the thermodynamic factors cancel out of Eqs. 8.8.19 and 8.8.20. The elimination of the thermodynamic factors will prove particularly useful in the estimation of transfer efficiencies in multicomponent distillation (Section 12.3). 

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

Generally speaking, the thermodynamic properties of these complex mixtures (solute-i-multicomponent aqueous solvent) depend on many factors such as the chemical natures of the solute and of the constituents of the mixed solvent, the intermolecu-lar interactions between the components in these mixtures, the mixture composition and the pressure and temperature. In the present paper only low soluble solutes are considered. Therefore, the solutions can be considered as dilute and the intermolecu-lar interactions between the solute molecules can be neglected. Thus, the properties of a solute-free mixed solvent and the activity coefficient of the solute at infinite dilution can describe the behavior of such dilute mixtures. 

A better insight into composition of phases along the separation process is provided by multicomponent process simulation as it can be carried out with commercial process simulating programs, such as ASPEN-h. As usual, the process is separated into theoretical stages. Normally, ASPEN+ provides thermodynamic models and calculates thermodynamic properties such as the distribution coefficients and separation factors. As the accuracy of these results is not sufficient for a design analysis in many cases, distribution coefficients (and if necessary solubilities) can be provided by a user-defined module which uses empirical correlations for these values. 

These preferences usually originate in stereochemical and structural limitations that affect the thermodynamics of the system. As such, they can be considered instructions that direct the self-assembly process. In some cases, the preferences are extreme, so that they dominate the system regardless of alternative influences. In others, they are less intense and must be weighed against other factors. In a complicated multicomponent system, for example, not all of the component preferences will be aligned, i.e., the instmctions may be contradictory. In such a case self-assembly becomes a process in which the preponderance of instructions and their intensities dominate. But how does this take place ... 

When crystallizing from multicomponent systems, kinetic factors often override thermodynamic considerations (the so-called Ostwald rule of stages -section 5.7). The phase which crystallizes is not necessarily the one which is thermodynamically most stable, but the one which crystallizes the fastest. Numerous examples of this sort of behaviour are available. 

In practice, thermodynamic equilibrium can seldom be reached within a single stage. Therefore, some correlation parameters, like tray efficiencies or HETS values, have been introduced to adjust the equilibrium-based theoretical description to the reality. For multicomponent mixtures, however, the application of this concept is often difficult due to diffusional interactions of several components [60, 61], These effects cause an unpredictable behavior of the efficiency factors, which are different for each component, vary along the column height and show a strong dependency on the component concentration [42, 61, 62]. 

For a specified solvent system, water or aqueous solutions for example, there are two variables that must be considered in the solubilization process (1) the molecular nature, purity, and homogeneity of the surfactant and (2) the chemical nature of the additive. From a technological viewpoint, it is important to understand exactly what surfactant structural features serve to maximize the desired solubilizing effect, and the best way to achieve that understanding is through a fundamental knowledge of the molecular and thermodynamic processes involved. In addition, since most technological applications of solubilization involve complex multicomponent systems, such factors as temperature, electrolyte content, and the presence of polymeric species and other solutes must be examined. 

In most commercial process simulators, model parameters for pure component properties and binary parameters can be found for a large number of compounds and binary systems. However, the simulator providers repeatedly warn in their software documentations and user manuals that these default parameters should be applied only after careful examination by the company s thermodynamic experts prior to process simulation. For verification of the model parameters again, a large factual data bank like the DDE is the ideal tool. The DDE allows checking all the parameters used for the description of the pure component properties as a function of temperature and of the binary parameters of a multicomponent system by access to the experimental data stored. On the basis of the results for the different pure component properties and phase equilibria, excess enthalpies, activity coefficients at infinite dilution, separation factors, and so on, the experienced chemical engineer can decide whether all the data and parameters are sufficiently reliable for process simulation. 

The complex distribution system that results from the frontal analysis of a multicomponent solvent mixture on a thin layer plate makes the theoretical treatment of the TLC process exceedingly difficult. Although specific expressions for the important parameters can be obtained for a simple, particular, application, a general set of expressions that can help with all types of TLC analyses has not yet been developed. One advantage of the frontal analysis of the solvent, however, is to produce a concentration effect that improves the overall sensitivity of the technique. The primary parameter used in TLC is the (Rf) factor which is a simple ratio of the distance traveled by the solute to the distance traveled by the solvent front. The (Rf) factor will always be less than unity. If a standard is added to the mixture, then the ratio of the (Rf) factors of the solute to that of the standard is termed the (Rx) factor and is thermodynamically equivalent to the separation ratio (a) in GC or LC. In a similar manner, the capacity ratio (k ) of a solute can be calculated for TLC from its (Rf) factor. Resolution is measured as the distance between the centers of two spots to the mean spot width. Alternative expressions for the resolution can be given in terms of the (Rf) factor and the plate efficiency. The plate efficiency is taken (by analogy to GC and LC) as sixteen times the square of the ratio of the retention distance of the spot to the spot width, but the analogy between TLC and the techniques of GC and LC can only be used with extreme caution. The so called... 

In multicomponent systems, thermodynamic functions such as volume V, Gibbs free energy G, and many other thermodynamic functions that can be expressed as functions of p, Tand Nk are extensive functions of Nk. This extensivity gives us general thermodynamic relations, some of which we will discuss in this section. Consider the volume of a system as a function of p, T and Nk V = V p,T,Nk). At constant p and T, if all the mole numbers were increased by a factor X, the volume V would also increase by the same factor. This is the property of extensivity we have already discussed several times. In mathematical terms, we have... 

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