Equilibrium curve course

Figure 8.3 is a graphical construction that represents a series of equations such as Eq. (8.6), for the case = 1 and the isotherm of Eq. (8.4). From [Ap]i, a line of slope —1/0 = -1 intercepts the equilibrium curve at [A]i. A perpendicular from this intercept cuts the x axis at = [Ap 2, from which a further line of slope -1 is drawn. Of course, may be varied from... 

Usually this minimum reflux ratio can be determined using the point of intersection between the q-line and the equilibrium curve. On some peculiar xy diagrams the R.O.L. or S.O.L. constructed in this way might cross the equilibrium curve at points other than where the g-line crosses. Then the reflux ratio must be increased until neither operating line crosses equilibrium curve before reaching the q-line. Of course, the operating line can be tangent to the equilibrium curve. 

In the limit that R approaches infinity, the r-intcrcepl of the R.O.L. becomes zero. The R.O.L. becomes the 45° line. Of course, the S.O.L. and R.O.L. always intersect along the -line, which means that the S.O.L. also becomes the 45° line. This is the furthest the operating lines can ever be from the equilibrium curve, so this condition gives the minimum number of stages ... 

Of course, if the equilibrium curve is straight (as it will be in dilute solutions), then m is its slope. This is similar to the expression for overall heat transfer coefficients for a double-pipe heat exchanger ... 

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... 

For ideal binary mixtures the course of the equilibrium curve can be computed by eni])loying a combination of Raoult s and Dalton s laws ... 

It is particularly important to know the course of the equilibrium curve accurately near its beginning and end for this reason as many points as possible should be determined between 0 and 10 mol% and between 90 and 100 niol%. In general the equilibrium vapour concentration y is measured at the following compositions of the liquid ... 

Figure 5 demonstrates the kinetic evaluation of these results. On the left-hand side, the initial velocity Vp (intersection points with the ordinate) is plotted versus the concentration of the Al component while the Ti concentration was kept constant. The result is the Al isothermal curve of the second equilibrium reaction generating the active species. It demonstrates in the initial part the demanded sigmoid curve course according to the location of the two successive equilibria. This induction period is enlarged in Fig. 6. It can be clearly seen how sensitively the initial polymerization rate responds to the ratio Al/Ti. With a ratio >1 (i.e., here [AlEtCl2] > 3x10 mol L ), the formation of the active species increases drastically.

Experience with absorbers of this type has indicated that actual trays have a Muiphree vapor-phase efficiency on the order of 40%. This can be taken into account by drawing a pseudoequilibrium curve 40% of the distance (vertically) from the operating line to the equilibrium curve and stepping off the trays as before. Results indicate that six actual trays will be required. Heat effects, of course, must be consisted in establishing the final position of the equilibrium curve but, in this case, the effects are negligible because the large volume of gas carries away the heat of reaction with only a very small temperature increase. 

Tie lines, of course, cannot cross in the two-phase region within the binodal curve of Fig. 7.3-4. Furthermore, a line from A must not coincide with a tie line in the region between lines L,A and LiA, since this would cause contact (pinch) between the operating and equilibrium curves of Fig. 7.3-5, signifying zero driving force for the transfer of A. Such coincidence occurs in the use of a minimum solvent-to-feed ratio. The condition is avoided by using solvent in excess of the minimum solvent-to-feed ratio for the prescribed separation, which is established as follows with reference to Fig. 7.3-ti. 

The course of the equilibrium curve is fixed by the relationship given in Chapter 1.4.3.3... 

Determination of the number of theoretical stages for desorption using countercurrent solvent flow stripping, by means of a stripping gas is analogous. In this case, the fact that Xg is the entry concentration and Xg the exit concentration of the solvent to the stripper must be considered. According to the operating conditions, the course of the desorption equilibrium curve is different from that for the absorption equilibrium curve. The balance line is now below the equilibrium curve and so the reciprocal value of the absorption coefficient is required. 

With the known course of the equilibrium curve for fixed extraction conditions, the required number of theoretical stages N, is determined by drawing steps between the equilibrium curve and the balance line, analogous to the McCabe-Thiele method (see Chapter 2.S.2.4 and Fig. 6-18). 

In distillation, the liquid reflux returned to the top of the column plays, in a sense, very much the role of a solvent. In the course of its downward flow, it dissolves residues of the heavy component contained in the vapor phase and thereby contributes to its enrichment in the lighter, more volatile component. A reduction of reflux, or of the reflux ratio, may therefore be expected to result in an increase in the required number of stages in much the same way as happens in the case of the gas scrubber. Ultimately, when the operating lines simultaneously intersect the equilibrium curve, a pinch results and the number of stages goes to infinity. This is shown in Figure 7.19a. The reflux ratio at which this occurs can be read from the intercept Xpi/ R +1) and represents the minimum required to achieve the desired separation. 

This course of the curves of the thermal transition between the isotropic and anisotropic phases not only derives from the specific features of polymers in comparison to low-molecular-weight substances, but also from the fact that the firee volume in the system increases with an increase in the temperature, and this results in an increase in the probability of independent arrangement of the macromolecules in solution. However, this also simultaneously means broadening of the concentration regions of the isotropic-anisotropic phase transition. The inflection of the phase equilibrium curves discussed in the studies cited above thus not only follows from the formal topological analysis but also from the thermodynamic concepts of the structure of liquids. 

The potential energy 0(z) depends not only on the distance z hut also on the position of the gas molecule in the xy plane parallel to the surface of the solid and distant z from it. For any given position, the adsorption energy will be equal to the value of 0 = 0o minimum of the potential curve (cf. Fig. 1.2), which of course represents the equilibrium position. 

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