Three-component mixtures

If there is a three-component mixture and simple columns are employed, then the decision is between two sequences, as illustrated in Fig. 5.1. The sequence shown in Fig. 5.1a is called the direct sequence, in which the lightest component is taken overhead in each column. The indirect sequence, shown in Fig. 5.16, takes the heaviest component as the bottom product in each column. There may be... 

For a three-component mixture, there are only two alternative sequences. The complexity increases dramatically as the number of components increases. Figure 5.2 shows the alternative sequences for a five-component mixture. Table 5.1 shows the relationship between the number of products and the number of possible sequences for simple columns. ... 

Most LB-forming amphiphiles have hydrophobic tails, leaving a very hydrophobic surface. In order to introduce polarity to the final surface, one needs to incorporate bipolar components that would not normally form LB films on their own. Berg and co-workers have partly surmounted this problem with two- and three-component mixtures of fatty acids, amines, and bipolar alcohols [175, 176]. Interestingly, the type of deposition depends on the contact angle of the substrate, and, thus, when relatively polar monolayers are formed, they are deposited as Z-type multilayers. Phase-separated LB films of hydrocarbon-fluorocarbon mixtures provide selective adsorption sites for macromolecules, due to the formation of a step site at the domain boundary [177]. 

This type of chromatographic development will only be briefly described as it is rarely used and probably is of academic interest only. This method of development can only be effectively employed in a column distribution system. The sample is fed continuously onto the column, usually as a dilute solution in the mobile phase. This is in contrast to displacement development and elution development, where discrete samples are placed on the system and the separation is subsequently processed. Frontal analysis only separates part of the first compound in a relatively pure state, each subsequent component being mixed with those previously eluted. Consider a three component mixture, containing solutes (A), (B) and (C) as a dilute solution in the mobile phase that is fed continuously onto a column. The first component to elute, (A), will be that solute held least strongly in the stationary phase. Then the... 

Both carbonates decompose to their oxides with the evolution of carbon dioxide. The decomposition temperature for calcium carbonate is in the temperature range 650-850 °C, whilst strontium carbonate decomposes between 950 and 1150°C. Hence the amount of calcium and strontium present in a mixture may be calculated from the weight losses due to the evolution of carbon dioxide at the lower and higher temperature ranges respectively. This method could be extended to the analysis of a three-component mixture, as barium carbonate is reported to decompose at an even higher temperature ( 1300 °C) than strontium carbonate. 

The three-component mixture required a few minutes of shaking to ensure the formation of a homogeneous solution. It is essential that the water be completely dissolved before the solution is used. 

The number of components in a system can change with experimental conditions, and one must exercise care in defining the system. For example, a mixture of H2, 02, and H20(g) at low temperature is a three-component mixture.11 However, heating the mixture to a high temperature causes the three species to be in rapid equilibrium through the reaction... 

Three-component mixtures selected from Na, K, Rb and Cs have been investigated to determine the minimum melting point in each ternary mixture, characterizing low-melting portions of the Na-K-Cs phase diagram and measuring the physical... 

Thermolysis of the thiadiazabicyclo[3.1.0]hexene 30, obtained from reaction of an azide with an isothiazole dioxide affords the 1,2-thiazetidine-1,1-dioxide 31, but in poor yield and as part of a three-component mixture <96T7183>. Isocyanates 32 are available in moderate yield <94ZOR1700>. 

As was reported by Soczewinski, a three-component mixture containing 5 mg of each of the ingredients can be completely separated using a 90 X 100 X 0.5 mm layer of silica [26]. This indicates that the capacity of the chromatographic system can be considerably increased by the apphcation of wider and thicker layers of adsorbent on the chromatoplate [44]. 

Having derived a solution for two-component systems, we could try and extend this solution to three-component systems. A PCA of a data set of spectra of three-component mixtures yields three significant eigenvectors and a score matrix with three scores for each spectrum. Therefore, the spectra are located in a three-dimensional space defined by the eigenvectors. For the same reason, explained for the two-component system, by normalization, the ternary spectra are found on a surface with one dimension less than the number of compounds, in this case, a plane. 

For a three-component mixture to be split into three relatively pure products, there are only two alternative... 

Figure 13.1 A shows a conventional high performance reversed-phase separation of a three-component mixture of aromatic acid esters obtained with a standard 4.6 mm x 250 mm octadecyl column and methanol water as the eluent. From the view of chromatographic resolution and ruggedness, this is an excellent separation. However, from a practical standpoint, an assay based on this particular separation would not be satisfactory since it wastes large amounts of time between elutions of the individual components.

FIGURE 2.3 Electrical conductivity, ae, at 1000°C versus weight fraction of coarse YSZ, c/wt%, for 40 vol% Ni-60 vol% YSZ cermets made with a three-component mixture of NiO and fine YSZ (particle size of 0.6 pm) and coarse YSZ (particle size of 27 pm). The numbers 24 and 144 mean that the sintered cermets were reduced for 24 and 144 h, respectively. (From Itoh, H. et al., J. Electrochem. Soc., 144, 641-646, 1997. Reproduced by permission of ECS-The Electrochemical Society.)... 

None of these matrices satisfies both (5.96) and (5.97). We can thus conclude that no mixture-fraction basis exists for this set of initial and inlet conditions. Since Win = 3, a three-component mixture-fraction vector exists,75 but is of no practical interest. 

The upper left panel of Figure 12.28 shows the nine factor combinations (design points) of a 3 full factorial design in two independent mixture components, Xj and Xj. The third component of the three-component mixture is not independent, and is obtained from the mixture constraint (Equation 12.87) by difference ... 

Figure 12.34 Pseudo-three-dimensional plot of response as a function of composition for a three-component mixture.

The results of three-component mixture designs are often presented as response surfaces over the triangular mixture space as shown in Figure 12.34. The Scheffe model parameters are seen to be equivalent to the responses at the vertexes. 

ACE50 experiment for a three-component mixture. Dashed lines indicate variation of that component s concentration by +3-fold (an overall 9-fold difference) highlighting that the method is insensitive to ligand concentration. See text for details. Reprinted from [39] with permission from the American Chemical Society. 

According to Wohl (1953), the excess Gibbs free energy of mixing for a three-component mixture is represented by... 

The checkers attempted chromatography of the three-component mixture on a 150-mmol scale using 55- and 75-mm diameter columns. However, mixed fractions were obtained even with seemingly large differences for the components. 

FIGURE 11.12 Hypothetical displacement separation of a three-component mixture containing two trace components loaded at 0.1 mg each and a major component loaded at 20mg. Panel a, main component band and displacer front panel b, peaks of the impurities panel c, peak of the second component under analytical (linear elution) conditions. (Reprinted with permission from Elsevier from Viscomi, G.C. et ah, J. Chromatogr., 440, 157, 1988. Copyright.)... 

In Figure 4.6 a number of iso-Mahalanobis distance contours of a three component mixture have been depicted. The square of the variation coefficient (v) is constant. The ellipses drawn at Figure 4.5 are each contour lines with the same probability density value. This means that if a mixture is set to the centre point of one of the ellipses in the figure, than the probability that the composition of the mixture is present inside the drawn ellipse is in all the cases the same. 

Using the rules of the propagation of errors [33,34] a measure of the robustness of the partition coefficient (C, ) and the robustness of the selectivity (C ) can be obtained. Below, a derivation of robustness of the partition coefficient P, of a compound i and the selectivity Uij for two compounds i and j with respect to variation in extraction liquid composition is given. The general form of a (Special Cubic) mixture model for three-component mixtures is given by ... 

Three-component mixtures can be represented on triangular coordinates and the reasoning used in three-component designs can be extended to four or more components. Some typical three-component designs follow. 

Linear constraint a condition imposed on certain variables which is linear in form. For example, in a three-component mixture, cone, r + cone. + cone, r - 1 is a linear constraint. 

---