Method for Multicomponent Solutions

Meissner and Kusik (M2) presented, in 1972, a method of calculating the reduced activity coefficients of strong electrolytes in a multicomponent solution. They based their method on Br0nsted s proposal that, in multicomponent solutions, the activity coefficient of an electrolyte will be influenced most by the interaction of it s cation with all the anions in solution and the interaction of it s anion with all the cations in solution. Ignoring the possible interactions between like charged ions was felt to be valid as such interactions would be very small. Meissner and Kusik proposed that the activity coefficient could then be defined as  

Once again using the notation that- odd number subscripts indicate cations and even number subscripts indicate anions, F] and F2 are expressions for calculating the interaction of cation with anions and anion with cations. They deHned these as  

Xj = —I----- with i = odd nuibers, expresses the ionic strength fraction 

I = ij r m. z with i = odd nxibers to calculate the cationic ionic strength 

Yj = —I— with i = even nunfaers for the anion ionic strength fraction a 

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In 1973, Bromley (B3) disputed Meissner and Kusik s method for multicomponent solutions as follows ... 

FIG. 14-11 Graphical design method for multicomponent systems absorption of butane and heavier components in a solute-free lean oil. 

The UNIQUAC method of Abrams and Prausnitz divides the excess Gibbs free energy into two parts, the combinatorial part and a part describing the inter-molecular forces. The sizes and shapes of the molecule determine the combinatorial part and are thus dependent on the compositions and require only pure component data. As the residual part depends on the intermolecular forces, two adjustable binary parameters are used to better describe the intermolecular forces. The UNIQUAC equations are about as simple for multicomponent solutions as for binary solutions. Parameters for the UNIQUAC equations can be found by Gmehling, Onken, and Arlt. ... 

If Eqs. (20.6-2)-(20.6-8) ate to be used for the case in Fig. 20.6-4rf, an iterative procedure has to be adopted since it involves simultaneous solution of Eqs. (20,6-2)-(20.6-4). Stem and coworkere have extended the above treatment and reported an iteration method for multicomponent mixtures.2 This iteration methnd was developed for solving the arrangement in Fig. 20.6-4d. 

For multicomponent systems, experiments with synthetic methods yield less information than with analytical methods, because the tie lines cannot be determined without additional experiments. A common synthetic method for polymer solutions is the (P-T-m ) experiment. An equilibrium cell is charged with a known amount of polymer, evacuated and thermostated to the measuring temperature. Then flie low-molecular mass components (gas, fluid, solvent) are added and the pressure inereases. These eomponents dissolve into the (amorphous or molten) polymer and the pressure in the equilibrium eell deereases. Therefore, this method is sometimes called pressure-decay method. Pressure and temperature are registered after equilibration. No samples are taken. The composition of the liquid phase is often obtained by weighing and using the material balance. The synthetic method is particularly suitable for measurements near critical states. Simultaneous determination of PVT data is possible. Details of experimental equipment can be found in the original papers compiled for this book and will not be presented here. 

As indicated in the previous chapter, Guggenheim s method for single electrolytes is accurate only to about an ionic strength of. 1 molal this limitation cannot be expected to improve for multicomponent solutions. 

At this point we are not concerned with developing methods for rigorous solution of the above system of equations. Discussion of computational methods for the general case of multicomponent systems is covered in Chapter 13. This chapter considers a simplified model that lends itself to graphical solution and provides a tool for qualitative understanding of the operation of a distillation column. The model has a relatively low level of complexity because of its binary nature and also because of other simplifying assumptions. 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

Smith and Brinkley developed a method for determining the distribution of components in multicomponent separation processes. Their method is based on the solution of the finite-difference equations that can be written for multistage separation processes, and can be used for extraction and absorption processes, as well as distillation. Only the equations for distillation will be given here. The derivation of the equations is given by Smith and Brinkley (1960) and Smith (1963). For any component i (suffix i omitted in the equation for clarity)... 

Almost all methods of chemical analysis require a series of calibration standards containing different amounts of the analyte in order to convert instrument readings of, for example, optical density or emission intensity into absolute concentrations. These can be as simple as a series of solutions containing a single element at different concentrations, but, more usually, will be a set of multicomponent solutions or solids containing the elements to be measured at known concentrations. It is important to appreciate that the term standard is used for a number of materials fulfilling very different purposes, as explained below. 

Sangster, J. Lenzi,F., "On the Choice of Methods for the Predictions of the Water-activity and Activity Coefficients for Multicomponent Aqueous Solutions", Can. J. Chem. Eng.,... 

A variant of the zero average contrast method has been applied on a solution of a symmetric diblock copolymer of dPS and hPS in benzene [331]. The dynamic scattering of multicomponent solutions in the framework of the RPA approximation [324] yields the sum of two decay modes, which are represented by exponentials valid in the short time limit. For a symmetric diblock the results for the observable scattering intensity yields conditions for the cancellation of either of these modes. In particular the zero average contrast condition, i.e. a solvent scattering length density that equals the average of both... 

The general treatment for multicomponent diffusion results in linear systems of diffusion equations. A linear transformation of the concentrations produces a simplified system of uncoupled linear diffusion equations for which general solutions can be obtained by methods presented in Chapter 5. 

Since linear variation of hardness is not always the case, equation (5.7) is approximate. But Glazov and Vigdorovich consider that the production of many very complex solid solution systems does require some method if only rough, for hardness analysis of such systems. They formulate the additivity principle for multicomponent systems as follows the numerical increase in hardness of multi-component solid solutions equals the sum of hardness increments in bi-component solutions... 

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