Three-component

In the next three sections we discuss calculation of liquid-liquid equilibria (LLE) for ternary systems and then conclude the chapter with a discussion of LLE for systems containing more than three components. 

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

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. ... 

There are three components in a life-cycle analysis ... 

The first equation (1) is the equation of state and the second equation (2) is derived from the measurement process. Finally, G5 (r,r ) is a row-vector that takes the three components of the anomalous ciurent density vector Je (r) = normal component of the induced magnetic field. This system is non hnear (bilinear) because the product of the two unknowns /(r) and E(r) is present. 

The new test system was developed in order to largely eliminate the human factors for manual ultrasonic testing as described above. The system consists of three components ... 

Two alternative means around the difficulty have been used. One, due to Pethica [267] (but see also Alexander and Barnes [268]), is as follows. The Gibbs equation, Eq. III-80, for a three-component system at constant temperature and locating the dividing surface so that Fi is zero becomes... 

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]. 

Explicit expressions for the fluxes can also be found in the case of a ternary mixture, though they are appreciably more complicated than those for a binary mixture. The best starting point is equations (5.7) and (5.8). When there are three components in the mixture it is easy to check that equations (5,8) and the condition = 0 are satisfied by... 

For more than three components extremely heavy algebra is generated in attempting to solve the implicit flux relations, and in general no usefully compact explicit solution is obtained. However, there are two interesting special cases in which explicit flux relations can be obtained with an arbitrary nutr er of components in the mixture. Neither would be expected to correspond accurately with physical situations of practical interest, but they may provide useful qualitative, or semi-quantitative pointers to the behavior of more accurate 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... 

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

Three-component coupling with vinylstannane. norbornene (80). and bro-mobenzene affords the product 91 via oxidative addition, insertion, transme-tallation, and reductive elimination[85]. Asymmetric multipoint control in the formation of 94 and 95 in a ratio of 10 1 was achieved by diastereo-differ-entiative assembly of norbornene (80), the (5 )-(Z)-3-siloxyvinyl iodide 92 and the alkyne 93, showing that the control of four chiralities in 94 is possible by use of the single chirality of the iodide 92. The double bond in 92 should be Z no selectivity was observed with E form[86]. 

In an efficient diastereo-differentiative assembly of three components of norbornene, tv. v-alkenyl iodide, and KCN, the isomerization of the cis to the trans double bond takes place to give the coupled product 224. The isomerization is explained by the formation of the cyclopropane 222. its rearrangement to give a irans double bond in 223, and trapping with CN anion to give 224[168],... 

The system combines having three components and having constant composition, since the copolymer has the same composition as the homopolymer... 

In practice, such a fractionation experiment could be carried out by either lowering the temperature or adding a poor solvent. In either case good temperature control during the experiment is important. Note that the addition of a poor solvent converts the system to one containing three components, so it is apparent that the two-component Flory-Huggins model is at best only qualitatively descriptive of the situation. A more accurate description would require a... 

Neglecting the higher-order terms, we can write the osmotic pressure for this three-component system in terms of the van t Hoff equation ... 

The accompanying sketch qualitatively describes the phase diagram for the system nylon-6,6, water, phenol for T > 70°C.f In this figure the broken lines are the lines whose terminals indicate the concentrations of the three components in the two equilibrium phases. Consult a physical chemistry textbook for the information as to how such concentrations are read. In the two-phase region, both phases contain nylon, but the water-rich phase contains the nylon at a lower concentration. On this phase diagram or a facsimile, draw arrows which trace the following procedure ... 

Owing to the effects of spin-orbit coupling all the triplet terms, except S, are split into three components. For example, in the case of a term, with L = I and S = l,J can take the values 2, 1, 0 (Equation 7.15). 

The model of the plate considered in this section actually corresponds to a shallow shell having zeroth curvatures. The gradient of the punch surface is assumed to be rather small, so that the nonpenetration condition imposed in the domain is the same as in the usual case for a plate. Meanwhile, the restriction imposed on the crack faces contains three components of the displacement vector. 

We use the same notation as in the previous subsection. The boundary of flc consists of three components r,r+,Tj, where T correspond to the positive and negative directions of the normal n, respectively. We introduce the space... 

---