Equilibrium conditions graphical solution

The mathematical solution of Eq. (A.15) is tedious. An elegant graphical solution has been proposed by Stichlmair and Fair [1]. The occurrence of a reactive azeotrope is expressed geometrically by the necessary condition that the tangent to the residue (distillation) curve be collinear with the stoichiometric line. Such points form the locus of potential reactive azeotropes. In order to become a true reactive azeotrope the intersection point must also belong to the chemical equilibrium... 

Graphical presentation of redox equilibria, like the graphical treatment of acid-base, complexation, and precipitation equilibria is helpful in understanding complicated problems and in obtaining approximate solutions to equilibrium questions. For redox systems in natural waters the equilibrium condition is truly a boundary condition. In many cases, natural systems are not at equilibrium from a redox standpoint. The diagrams usually present an idea of what is possible, not necessarily of the existing or imminent situation. The graphical presentations of redox equilibria are seldom simple because redox reactions usually involve... 

Now the concentrations c, F for the stationary diffusion conditions could be calculated from (44) for every x value, if parameter Bi is specified. From (44) it follows that the evaporation of surfactant to the gas phase is the only reason for adsorption-desorption to deviate from equilibrium state which takes place for Bi=0. Thus the posed problem deals with the development of the surfactant masstransfer from the equilibrium conditions at the initial section x — 0. To estimate the accuracy of the approximate solution (44) the comparison of (44) with the exact solution of stationary diffusion problem given in Ji and Setterwall (1994) can be done. For the dimensionless parameters values Pe == 10, Bi = 10, Co = -0.25 exact solution gives x = 1000/, h = 0.09 h is obtained only approximately from graphical dependence c(//)). Th( ap])ropriate values from (44) are r = 1012/, / , = 0,0666. 

If inlet conditions (F, S, xF, and ys) are known, we have four unknown variables (R, E, x, and y). However, since we have only three equations, we need additional information to be able to solve for the unknown variables, which are the equilibrium data of the ternary system solute, solvent, and diluent, which are usually described graphically in triangular coordinates (Treybal, 1980). 

The functions /,(, /) and fi(x,t) allow solutions to be derived for special cases of inlet and initial conditions. Note that 5 is a dummy variable of integration. In the case where the column has been washed with eluent and a sample has not been introduced, initial condition 3 from Table I applies, and f 1 (x,t) = 0. After a sample has been introduced and washed into the column by eluent, boundary condition 1 from Table I with c = 0 applies and f x,t) = 0. The eluent volume is Q — Avta. The two limiting cases mentioned above may be superimposed, offset from one another by A v a t = Q = VF where Q is the feed volume. This superimposition is applicable only in the case of linear equilibrium, which yields symmetric solutions. Figure 4 shows the results of these equations graphically for selected values. These values may be particularly applicable to proteins in size exclusion supports. 

From this follows that the equilibrium concentrations of B and A should be equal. This condition is met in the crossing point of the acid line of the acid corresponding to the anion and the base line of the cation, i.e., in Pj in Figs. 56, 57, and 58. The pH-coordinate of this point is the pH of the solutions. This graphical way of finding the pH is equivalent to the calculations derived for ampholytes which are weakly acidic and weakly basic (see Sect. 4.2.2.2) ... 

Two-Stage Countercurrent Batch Sorption. The sorbent/solution ratio for a fixed set of process conditions can be determined by trial and error using the graphical technique. This is carried out by graphically fitting two steps between the equilibrium curve and the initial and final solute-liquid concentration boundary concentrations. The operating line can then be drawn, and this provides the coordinates, of the intermediate concentration, the... 

We have called these material balance functions the a fractions. They equal the equilibrium a fractions only at one value of H at which equilibrium and material balance conditions are simultaneously satisfied. This can be found graphically or programmed for computer solution. For pure acid and base, Cj, or C zero, a simple, invariant shaped curve results in the logarithmic plots. 

Fig. 6.23 Graphical determination of the amount adsorbed 2 k given injection step in the titration calorimetry run and the related molality of the equilibrium bulk solution from the intersection between the calorimetric line 6.64 passing through two characteristic points PI and P2) and the experimental adsorption isotherm measured under the same conditions...

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