Mixing curves

FIG. 14-39 Mixing curves. Pe = Peclet niimher (Npe). (Biihhle-Tray Design Manual, Ameiican Institute of Chemical Engineers, New Yoth, 1958. )... 

Figure 6. ( °Th/ Th) versus for arc lavas. The mixing curve between a nominal MORE...

Pahit source fluids, based on on-site colorimetry (crater lake, thermal acid springs, neutral spring and stream) (a). Mixing curves for the same groups (b). 

Figure 13. Whole-rock Allende chondrules plotted in 8 Mg (SRM 980 0) vs. Mg/Al space showing that a mixing curve (dotted line) between the CAI whole-rock sample and the Mg-rich, Al-poor chondrules (i.e., chondritic chondrules) fits the trend defined by the bulk chondrule data (modified after Galy et al. 2000). The expected evaporation trend is shown for comparison and represents the consequences of free evaporation of chondrules in space. The evaporation curve is described in greater detail by Galy et al. (2000).

Figure 15. Hyperbolic mixing curves between a typical chondritic endmember at high A 0 (SMOW) and A Mg (DSM3) and a typical CAI composition at low A 0 and A Mg compared with existing high-precision data. The curvature of the mixing lines is a function of (Mg/0)cAi/(Mg/0)d, j,fc where CAI and chondrite refers to the two endmember compositions. Each curve is labeled with the corresponding (Mg/0)cAi/(Mg/0Xt i values.

It is clearly seen that the interaction of SDS with C] oE5 is significantly stronger than that of the alkyl ethoxylate sulfate surfactants with C] oE5 In addition, the symmetry in the heat of mixing curves is strikingly different with those for C] 2 2 Cj qEjS showing an asymmetric maximum at about a... 

The Gibbs free energy of mixing curves will have the form shown in figure 3.10A. By application of the above principles valid at equilibrium conditions, we deduce that the minimum Gibbs free energy of the system, at low T, will be... 

Figure 3,10 Solvus and spinodal decomposition fields in regular (B) and subregular (D) mixtures. Gibbs free energy of mixing curves are plotted at various T conditions in upper part of figure (A and C, respectively). The critical temperature of unmixing (or consolute temperature ) is the highest T at which unmixing takes place and, in a regular mixture (B), is reached at the point of symmetry.

Figure 3.10 also shows the fields of spinodal decomposition defined by the loci of the points of inflection in the Gibbs free energy of mixing curves. These points obey the following general conditions ... 

Let us again consider a solid mixture (A,B)N with a solvus field similar to the one outlined in the T-Xplot in figure 3. lOB, and let us analyze in detail the form of the Gibbs free energy of mixing curve in the zone between the two binodes (shaded area in figure 3.10A). 

Figure 3J2 Energy relationships between solvus and spinodal decompositions. (A) Portion of Gibbs free energy of mixing curve in zone between binodal (X ) and spinodal (X ) points. (B) Gibbs free energy variation as a consequence of compositional fluctuations around intermediate points X and X(2). ...

Equation (6.55) is identical to Eq. 11 in Vollmer (1976) and Eq. 2 in Langmuir et al. (1978). The mixing curve resulting from this equation is a hyperbola whose curvature is controlled by the B coefficient. 

The mixing curve is a hyperbola for ratio vs. ratio plot, unless B = 0, that... 

Figure 2.3 requires careful consideration. Figures 2.3a through 2.3e are free energy of mixing curves as a function of concentration (here plotted as Xg, where Xr = 1.0 at pure B and = 0 at pure A) at decreasing temperatures (Js > > T2 > T ). 

Figure 2.3 Free energy of mixing curves for solid and liquid phases at various temperatures (a-e) and resulting temperature-composition phase diagram for a completely soluble binary component system (f). From O. F. Devereux, Topics in Metallurgical Thermodynamics. Copyright 1983 by John Wiley Sons, hic. This material is used by permission of John Wiley Sons, Inc.

That is, there is a different solid and liquid composition, Xs and Xl, resulting from the tangent between the free energy of mixing curves, at all temperatures between T2 and T, as illustrated in Figure 2.3c. We will return to the conditions stipulated by Eq. (2.37) in subsequent sections. 

Phase diagrams from freezing point depressions show true compound formations for simpler amides—e.g., water-N-methylacetamide forms a compound at a mole ratio of 2 to 1, water-N,N-dimethylacetamide at 3 to 2 and 3 to 1, and water-N-methylpyrrolidone at 2 to 1. The heats of mixing and heat capacities at 25°C. of a number of water-amide systems were determined. All mixing curves were exothermic and possess maxima at definite mole ratios, while the heat capacities for the most part show distinct curvature changes at the characteristic mole ratios. Both experimental results point to the stability of the particular complexes even at room temperature. This is further supported by absolute viscosity studies over the whole concentration range where large maxima occur at these same mole ratios for disubsti-tuted amides and N-substituted pyrrolidones. 

The log K values shown in Figure 18.10 are the values that best reproduce all of the heat of mixing curves.v The J1 values are obtained by estimating initial values using the activity coefficients for NaCl(aq).16 These initial values of Jy are then readjusted, as the value for Km is optimized, by adjusting the coefficients of Pitzer s equations, whose form is described in the previous section. Pitzer s equations are, of course, internally consistent so that adjustments to the activity or osmotic coefficient parameters result in adjustments to the thermal parameters (L, L2, 4>J, or J2), and hence, to the heat effects. 

Figure 5.3 3He contents in sediments are plotted against 3He/4He ratios. The curve can be interpreted to be a mixing curve between helium in cosmic dusts (4He = 0.1 cm3STP/g,... 

Fig. 13. Mixing curve between poly(VAd) and RNA in H20 (pH 7.0). Absorbance was determined at 260 nm. A mixed solution was allowed to stand at room temperature for 3 h...

Complex formation between RNA and water soluble copolymers obtained in the copolymerization of methacryloyloxyethyl-type monomers containing nu-cleobases with water soluble monomers was also studied. Mixing curves between copolymers and between copolymers and RNA are shown in Figs. 14-16. The interaction between poly(MAOFU-co-AAm), poly(MAOT-eo-AAm), or poly-(MAOA-cn-AAm) with RNA was observed, as shown in Fig. 14. The overall stoichiometry of the complexes was about 1 1 and the hypochromicity was about 2% for the copolymer-RNA system under the conditions used. The observed interaction was not as strong as for the poly(VAd)-RNA system and poly(MAOA)-poly(MAOT) system [64], since the solubilizer, AAm, in the... 

Figure 16 shows the mixing curve between copolymers containing AA and RNA in water. In aqueous solution, purine and pyrimidine bases in co-... 

Fig. 16a-c. Mixing curve between copolymers and RNA in a 0.1 M phosphate buffer (pH 7.8). Absorbance at 265 nm obtained in a 10-mm cell at 20 °C. (a) Poly-(MAOFU-co-AA)-RNA system, (b) poly-(MAOT-co-AA)-RNA system, (c) poly-(MAOA-co-AA)-RN A system... 

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