Binary point

Considering for instance the formation of compounds, several variants may be observed due to the possible existence of binary (point or line) phases and/or of ternary, stoichiometric or solid solutions phases. Notice that true ternary phases may be formed (that is phases corresponding to homogeneity regions placed inside the diagram and not connected with the components or any binary phases). However within the ternary composition fields, phases are observed which contain all the... 

Calculating D(xq), we first have to multiply xq 2. In binary notation this is merely a shift of the binary point one position to the right. We obtain... 

The calculation of fixed points and periodic obits for the shift map is straightforward. There are exactly two fixed points. Since for a fixed point of period 1 a single shift of the binary point to the right has to reproduce the seed, there are only two possibilities ... 

It is not a diflficult matter to prove sensitivity of the shift map. Since X and y are not identical, the binary expansion of their difference has at least one binary digit 1 in some binary position after the binary point. Iteration of the shift map will bring this 1 closer and closer to... 

It can easily be shown that the maximum deviation from the predictions of the first order model occurs at the binary point ( 4, 6), and that the deviation is 14ft,2- So the predicted equation is ... 

The first-order model 9.1 can be estimated from data 1, 2 and 3 and used to estimate the binary points 4, 5 and 6. The calculated values for test points 4 and 5 (in column H) are much higher than the experimental values. This might lead us to conclude immediately that the first-order model is invalid. If instead we transform the solubility to its logarithm, the predictive model becomes ... 

Add further binary points to be able to use a full cubic model. 

Fig. 5.14. Bandgap energy and lattice constant for binary (points), ternary (connecting lines), and quaternary (region between lines) III-V semiconducting compounds. Breaks in the GaAsP and InGaP lines indicate a transition from indirect to direct band structure [5.15]...

Solid triangles, circles, diamonds, eight-pointed stars and squares are the binary nonvariant points (Z, and Q, N, p, L+M and R) in the subsystems A-B and A-C. It is accepted that temperatures of nonvariant binary points M and L are coincided. Open triangles, circles, diamonds, five-pointed stars, eight-pointed stars and squares are the ternary nonvariant points MQ, EfN, pR, NR, LN and pQ. Shaded circles and squares in binary systems are the metastable points N and R. Shaded circles and five-pointed stars are the metastable points NN and NR. Dashed fine is the monovariant curve Li = L2-G or Li = G-L2. Dash-dotted line is the monovariant curve Li = Lj-S or L = G-S. Solid line is the monovariant curve L1-L2-G-S. Double line is the coincided monovariant curves L1-L2-G-S and Li = Lj-S. Dotted line is the metastable part of critical curve Lj = Lj-G or Lj = G-L2. X is the relative amounts of the nonvolatile component in ternary solutions [X = Xb/(xbH-Xc)] (solvent-free concentration). 

For systems of type II, if the mutual binary solubility (LLE) data are known for the two partially miscible pairs, and if reasonable vapor-liquid equilibrium (VLE) data are known for the miscible pair, it is relatively simple to predict the ternary equilibria. For systems of type I, which has a plait point, reliable calculations are much more difficult. However, sometimes useful quantitative predictions can be obtained for type I systems with binary data alone provided that... 

X, y) for the 1-2 binary and 2-3 binary, respectively. Data points M+1 through N are ternary liquid-liquid equilibrium measurements (T, x, x, x, . The 1-rich phase is indicated... 

Appendix C-6 gives parameters for all the condensable binary systems we have here investigated literature references are also given for experimental data. Parameters given are for each set of data analyzed they often reflect in temperature (or pressure) range, number of data points, and experimental accuracy. Best calculated results are usually obtained when the parameters are obtained from experimental data at conditions of temperature, pressure, and composition close to those where the calculations are performed. However, sometimes, if the experimental data at these conditions are of low quality, better calculated results may be obtained with parameters obtained from good experimental data measured at other conditions. 

ERF error flag, integer variable normally zero ERF= 1 indicates parameters are not available for one or more binary pairs in the mixture ERF = 2 indicates no solution was obtained ERF = 3 or 4 indicates the specified flash temperature is less than the bubble-point temperature or greater than the dew-point temperature respectively ERF = 5 indicates bad input arguments. 

The principal point of interest to be discussed in this section is the manner in which the surface tension of a binary system varies with composition. The effects of other variables such as pressure and temperature are similar to those for pure substances, and the more elaborate treatment for two-component systems is not considered here. Also, the case of immiscible liquids is taken up in Section IV-2. 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

Figure A2.5.11. Typical pressure-temperature phase diagrams for a two-component fluid system. The fiill curves are vapour pressure lines for the pure fluids, ending at critical points. The dotted curves are critical lines, while the dashed curves are tliree-phase lines. The dashed horizontal lines are not part of the phase diagram, but indicate constant-pressure paths for the T, x) diagrams in figure A2.5.12. All but the type VI diagrams are predicted by the van der Waals equation for binary mixtures. Adapted from figures in [3].

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

III-V compound semiconductors with precisely controlled compositions and gaps can be prepared from several material systems. Representative III-V compounds are shown in tire gap-lattice constant plots of figure C2.16.3. The points representing binary semiconductors such as GaAs or InP are joined by lines indicating ternary and quaternary alloys. The special nature of tire binary compounds arises from tlieir availability as tire substrate material needed for epitaxial growtli of device stmctures. 

Figure C2.16.3. A plot of tire energy gap and lattice constant for tire most common III-V compound semiconductors. All tire materials shown have cubic (zincblende) stmcture. Elemental semiconductors. Si and Ge, are included for comparison. The lines connecting binary semiconductors indicate possible ternary compounds witli direct gaps. Dashed lines near GaP represent indirect gap regions. The line from InP to a point marked represents tire quaternary compound InGaAsP, lattice matched to InP.

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... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

In summary, a combination of the plot based on equation (10.6), using any single substance, and determination of the asymptote (10.14), using any pair of substances, provides a sound means of evaluating the parameters K, tC and. Having found these, further experimental points on (10.6) and (10.15), and possibly also (10.7), provide a check on the adequacy of the dusty gas model. Provided attention is limited to binary mixtures, this check can be quite comprehensive. In their published paper Gunn and King... 

Despite the very restricted circumstances In which these equations properly describe the dynamical behavior, they are the starting point for almost all the extensive literature on the stability of steady states in catalyst pellets. It is therefore Interesting to examine the case of a binary mixture at the opposite limit, where bulk diffusion controls, to see what form the dynamical equations should take in a coarsely porous pellet. 

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