Equilibrium curve experimental determination

The mass balance equations can only be solved if the relation between Yi and X are known. We assumed a linear relation for the equilibrium curve. This relation is given by the equation Y, = Ke X, were Ke is a equilibrium constant experimentally determined. 

Simple phase diagrams are produced for ternary systems in which two of the three components are completely miscible. The essential characteristics of such diagrams are shown in Fig. 2.7 in which liquids A and B and A and C are completely miscible whilst B and C are only partially miscible. The region, 2L, under the curve Imnop is an area of immiscibility. Systems with compositions within this region separate into two liquid phases at equilibrium. From experimental determinations it is possible to draw tie lines connecting those systems within the 2L region which separate at equilibrium into two phases with compositions given by the points of intersection of the tie lines and the curve Imnop. In Fig. 2.7 all systems on the tie line, mo, will separate at equilibrium into... 

Specifying P determines the v-l-e relationship (equilibrium) curve from experimental data. 

The curves are calculated with the help of experimentally determined equilibrium constants.)... 

Considering the multistage industrial unit, in any equilibrium stage, the quantity of solution in the underflow may be a function of the concentration of the solution in the thickener, and the concentration of the overflow solution will be the same as that in the underflow. If the curved line EF (Figure 10.18) represents the experimentally determined composition of the underflow for various concentrations, any point f on this line represents the composition of a mixture of pure B with a solution of composition g, and Of/fg is the ratio of solution to solids in the underflow. If the amount of solution removed in the underflow is not affected by its concentration, the fractional composition of the underflow with respect to the insoluble material B (xB) is a constant, and is represented by a straight line, through E, parallel to the hypotenuse, such as EF. Point E represents the composition of the underflow when the solution is infinitely weak, that is when it contains pure solvent. If K is the mass of solution removed in the underflow per unit mass of solids, the ordinate of E is given by ... 

An interesting example of a one-component systems is SiOa, which can exist in five different crystalline forms or as a liquid or a vapor. As C = 1, the maximum number of phases that can coexist at equilibrium is three. Each phase occupies an area on the T P diagram the two-phase equilibria are represented by curves and the three-phase equilibria by points. Figure 13.1 (2, p. 123), which displays the equUi-brium relationships among the sohd forms of Si02, was obtained from calculations of the temperature and pressure dependence of AG (as described in Section 7.3) and from experimental determination of equUibrium temperature as a function of equilibrium pressure. 

S0rensen ° analysed the In O2 vs. Inx experimentally determined functions at different temperatures. The slope of such curves can be shown to he simply related, by equilibrium equations, to the number of atoms and defects composing a cluster. By this method, clusters and clusters aggregations as given in Table 9 are proposed. 

Diffusion-type models are two-parameter models, involving kt or Ds and La, while BDST models are one-parameter models, involving only 0, as gmax is an experimentally derived parameter. The determination of La requires the whole experimental equilibrium curve, and in case of sigmoidal or other non-Langmiur or Freundlich-type isotherms, these models are unusable. From this point of view, BDST models are more easily applied in adsorption operations, at least as a first approximation. 

If one takes into account the reversibility of monomer M2 (Equations 35 and 36), relatively little change is observed in the shape of the curves. The equilibrium concentrations of methyl methacrylate used in these calculations were obtained from older measurements (21). The experimentally determined concentrations of a-methylstyrene are lower than the calculated ones. For a better fit of the curves to the measured... 

Weller s work [5-7] on the kinetics of ESPT brought out the importance of competition between the rates of deactivation of the excited states and the rates of proton transfer. In cases where the deactivation rates are slow enough for a complete establishment of excited-state equilibrium, fluorimetric titrations provide a method for experimental determinations of pK a. However, it has been realized that for a fairly large number of ESPT molecules, there is a frequent mismatch of pA"f, values obtained from Forster cycle and fluorimetric titrations methods. There are also examples of extended fluorimetric titration curves resulting from low proton availability in the mid-pH region (4-10). Various modifications of the Forster cycle and extensions of Weller s original kinetic considerations have been made from time to time and have been reviewed periodically. Some of the earlier important ones include those by Weller [7] in 1961, Vander Donckt [8] in 1970, Schulman [9] in 1974, and Klopffer [10] in 1977. The review by Ireland and Wyatt [11] contains extensive references of experimental results available in the literature until 1976. 

Figure 8.22. Equilibrium curves for the reaction CaC03 + SiC>2 —> CaSiC>3 + Si02 at a CO2 pressure of 1 atm and at a CO2 pressure equal to total pressure on the system. At P-T values between the two CO2 pressure extremes, caicite and quartz are stable. Heavy solid line is experimentally determined and dashed lines are extrapolated or theoretical. (After Barth, 1962.)...

The solubility of N in liquid Ga was experimentally determined [7,8] for the conditions corresponding to the equilibrium pN2 - T curve. In the high pressure experimental system (20 kbar, 2000 K), the nitrogen content in Ga can be increased up to -1 at.% which is sufficient for effective crystallisation from the solution. 

The equilibrium nuclear distance determined from the self-consistent-field curve is 0.897 A which differs by 2.2% from the experimentally determined value of 0.917 A. Curve (e) yields an equilibrium distance of 0.908 A, which differs by 1.0% from the experimentally determined value. The fundamental frequencies of vibration derived from curves (a), (e), and (g) are 4471, 4261, and 4137 cm-1, respectively. 

Figure 9.13. Surface complex formation with ligands (anions) as a function of pH. (a) Binding of anions from dilute solutions (5 x 10 M) to hydrous ferric oxide [TOTFe= 10 M]. Based on data from Dzombak and Morel (1990). I = 0.1. (b) Binding of phosphate, silicate, and fluoride on goethite (a-FeOOH) the species shown are surface species. (6 g FeOOH per liter, Pj = 10 M, Si/ = 8 x 10 M.) The curves are calculated with the help of experimentally determined equilibrium constants (Sigg and Stumm, 1981).

The experimentally determined S-L-V equilibrium data for salicylic acid (2-hydroxy-benzoic acid)-l-propanol-C02 were correlated by using the Stryjek-Vera modification of the Peng Robinson EOS in conjunction with Eq. (35) for the solid state fugacity of the solute (58,62), as described earlier. This procedure also yielded good agreement of the liquid phase compositions of salicylic acid in the temperature and pressure ranges of 273 to 367 K and 1.0 to 12.5 MPa. The P-Ttraces of S-L and L-V equilibria were calculated for a fixed solute concentration on C02-free basis, and subsequently the P-T trace for the S-L-V equilibrium was found from the point of intersection of these two lines. The liquid phase compositions of the solute as a function of pressure at a constant temperature at the condition of S-L-V equilibrium were calculated to assess the effect of pressure or addition of antisolvent on solute crystallization. It was reported that two isobaric points of the CO2 mole fraction could be observed on the curve of the S-L equilibrium temperature vs the CO2 mole fraction at constant temperature as it passes through a mini-... 

For this reason we determined in this study, for the third time and with the highest accuracy of our studies to date, the cloud point curves for PS(100.000)/PVME and PS(1,800,000)/PVME blends. Using the turbidimetric technique described in the Experimental Section, we determined cloud point temperatures for each of the PS molecular weights over the whole concentration range. The results are shown in Figure 4, where the solid points represent the experimentally determined cloud point temperatures. As proof of the closer approach to equilibrium for these measurements as compared to the previous determinations, both sets of data were approximately parallel to each other and did not cross at high PS concentration. 

According to the phase rule, a three-component, two-phase system has three degrees of freedom. Thus, by specifying the temperature, pressure, and concentration of one component in one phase, the state of the system is defined. The component concentration in one phase defines one point on the equilibrium curve, and this point marks one end of a tie line. The other end is determined thermodynamically either from experimental data or on the basis of liquid activity coefficient predictions methods. 

Determinatioti of the Equilibrium Curve.— The method employed for the experimental determination of the equilibrium curve will depend on the nature of the system to be investigated. Where one of the components is a volatile liquid e.g. water) at ordinary temperatures, the composition of the solution may be determined by evaporation of the liquid and weighing of the solid residue. Such systems will receive special consideration later. When both components are solid at ordinary temperatures, the course of the equilibrium... 

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