Phase lever rule

Hquid—Hquid-phase spHt the compositions of these two feed streams He oa either side of the azeotrope. Therefore, column 1 produces pure A as a bottoms product and the azeotrope as distillate, whereas column 2 produces pure B as a bottoms product and the azeotrope as distillate. The two distillate streams are fed to the decanter along with the process feed to give an overall decanter composition partway between the azeotropic composition and the process feed composition according to the lever rule. This arrangement is weU suited to purifying water—hydrocarbon mixtures, such as a C —C q hydrocarbon, benzene, toluene, xylene, etc water—alcohol mixtures, such as butanol, pentanol, etc as weU as other immiscible systems. 

The overhead vapor of compositionj/gj is totaHy condensed into two equiHbrium Hquid phases, an entrainer-rich phase of composition x and an entrainer-lean phase of composition The relative proportion of these two Hquid phases in the condenser, ( ), is given by the lever rule, where ( ) represents the molar ratio of the entrainer-rich phase to the entrainer-lean phase in the condensate. 

From 160°C to room temperature. The lead-rich phase becomes unstable when the phase boundary at 160°C is crossed. It breaks down into two solid phases, with compositions given by the ends of the tie line through point 4. On further cooling the composition of the two solid phases changes as shown by the arrows each dissolves less of the other. A phase reaction takes place. The proportion of each phase is given by the lever rule. The compositions of each are read directly from the diagram (the ends of the tie lines). 

From 183°C to room temperature. In this two-phase region the compositions and proportions of the two solid phases are given by constructing the tie line and applying the lever rule, as illustrated. The compositions of the two phases change, following the phase boundaries, as the temperature decreases, that is, a further phase reaction takes place. 

Continued compression increases the pressure along the vertical dotted line. The compositions and amounts of the vapor and liquid phases continue to change along the liquid and vapor lines and the relative amounts change as required by the lever rule. When a pressure corresponding to point g is reached, the last drop of vapor condenses. Continued compression to a point such as h simply increases the total pressure exerted by the piston on the liquid. 

The ratio of molar flowrates of the vapor and liquid phases is thus given by the ratio of the opposite line segments. This is known as the Lever Rule, after the analogy with a lever and fulcrum7. 

Q in Figures 4.6a and 4.6b, at equilibrium, two-liquid phases are formed at Points P and R. The line PR is the tie line. The analysis for vapor-liquid separation in Equations 4.56 to 4.59 also applies to a liquid-liquid separation. Thus, in Figures 4.6a and 4.6b, the relative amounts of the two-liquid phases formed from Point Q at P and R follows the Lever Rule given by Equation 4.65. 

Thus, one could expect to find a droplet morphology at those quench conditions at which the equilibrium minority phase volume fraction (determined by the lever rule from the phase diagram) is lower than the percolation threshold. However, the time interval after which a disperse coarsening occurs would depend strongly on the quench conditions (Fig. 40), because the volume fraction of the minority phase approaches the equilibrium value very slowly at the late times. 

If two gas mixtures R and S are combined, the resulting mixture composition lies on a line connecting the points R and S on the flammability diagram. The location of the final mixture on the straight line depends on the relative moles in the mixtures combined If mixture S has more moles, the final mixture point will lie closer to point S. This is identical to the lever rule used for phase diagrams. 

The relative amount of two phases present at equilibrium for a specific sample is given by the lever rule. Using our example in Figure 4.1, the relative amount of Cu(ss) and Ag(ss) at Tj, when the overall composition is xCu, is given by the ratio... 

The relative amount of the different phases present at a given equilibrium is given by the lever rule. When the equilibrium involves only two phases, the calculation is the same as for a binary system, as considered earlier. Let us apply the lever rule to a situation where we have started out with a liquid with composition P and the crystallization has taken place until the liquid has reached the composition 2 in Figure 4.17(a). The liquid with composition 2 is here in equilibrium with a with composition 2. The relative amount of liquid is then given by... 

The temperature-composition diagram can be used to calculate the composition of the two-phase system according to the amount of each solvent present. For example, at temperature T, the composition of the most abundant phase, which consists of liquid A saturated with liquid B, is represented by the point a and the composition of the minor phase, consisting of liquid B saturated with liquid A, is represented by point a. The horizontal line connecting these two points is known as a tie line as it links two phases that are in equilibrium with each other. From this line the relative amounts of the two phases at equilibrium can be calculated, using the lever rule, under the conditions described by the diagram. The lever rule gets its name from a similar rule that is used to relate two masses on a lever with their distances from a pivot, i.e. ... 

If more of one of the liquids is added, the effect, according to the lever rule, is to shift the point of the pivot until balance is regained. Thus, at the given temperature the composition of the phases remains the same, i.e. each saturated with the other liquid, but the relative amount alters if more of liquid B is added, then in the lever diagram the pivot point will shift to the right and thus more of this phase will form at the expense of the other phase (which is mainly liquid A). This rule applies to all partially miscible liquids. 

Note that the resulting fractional amounts are in weight percent, because the abscissa axis of the phase diagram reports the fractional weights of the two components (similar application of baricentric coordinates to a molar plot of type 7.2 would have resulted in molar fractions of phases in the system). Applying the lever rule at the various T, we may quantitatively follow the crystallization behavior of the system (i.e., atT = 1350 °C, = 0.333 and Xl = 0.666 atT = 1300... 

Apply the Lever Rule to a two-phase field in a binary phase diagram. 

Equations (2.40) and (2.41) are the lever rule and can be used to determine the relative amounts of each phase in any two-phase region of a binary component phase diagram. For the example under consideration, the amount of liquid present mrns out to be... 

Answer Application of the lever rule in the 1 -2-3 -F Liquid region shows that even in the best case (at the left edge of the hatched region) only about 3% of the 1-2-3 phase is formed, the rest being liqnid of different composition. 

Fig. 1.8 Lever rule in phase diagrams. In Fig. 1.7, the composition X5 shows two phase mixtures, the compositions of which are Xi (M(O) phase) and Xj (MO phase). The mixing ratio of the two phases equals (x2 — X5)/(x5 — Xj), (b), which is...

More precisely, we can determine the relative molar amounts / liq, / vap of the two phases (and therefore the remaining composition variable xBx of the total system) by means of a simple lever rule that expresses the overall mass balance of the system. Intuitively, we can see from Fig. 7.9 that xBl must be intermediate between %Bap and xBq, as expressed... 

Figure 7.9 P-xB diagram (b) for a binary two-phase system (a), showing the compositions of coexisting vapor Oeap) and liquid (xgq) phases for a particular vapor-pressure value (dotted line), and the connecting tie-line (heavy solid line) that connects vapor and liquid compositions at this pressure. Varying amounts (rcvap, nhq) of the two phases correspond to different positions along the tie-line, as determined by the lever rule (see text).

Equation (7.52) is equivalent to the simple rule for balancing a schoolyard seesaw if the fulcrum divides the board into lengths L1 L2, then the masses mi, m2 at the two ends should satisfy m Li = m2L2 (i.e., the heavier weight should be at the shorter end) to balance the seesaw. The proof of (7.52) is presented in Sidebar 7.9. The lever rule makes it easy to determine the relative amounts of each phase present from the tie-line ratio, and thus to determine the final unknown composition variable Xg1 from (7.51). 

In contrast, a heterogeneous solution of noncritical composition (e.g., v < xc, as shown by the arrow and dashed line in Fig. 7.11) shows a qualitatively different behavior as it is rises through the coexistence boundary and into the homogeneous region near and above Tc. For each increase in temperature along the dashed line in Fig. 7.11, a horizontal tie-line yields both the compositions of the A-rich and B-rich liquids (from the two ends of the tie-line), as well as the relative amounts of each phase (from the lever rule). Clearly, the critical composition xc remains near the middle of the tie-line as T increases toward Tc, whereas a noncritical composition x xc moves toward one or other terminus of the tie-line as the temperature is raised. 

In each hatched two-phase region, the lever rule (Section 7.3.2) can be used as usual to determine the relative amounts of the two phases at opposite ends of the tie-line. However, the quantity of precipitated solid a and/or /3 is usually of less interest than the composition of the melt, so the principal focus is on the two liquidus lines that meet at the eutectic point. These liquidus lines are also called solubility curves or freezing-point depression curves, in that they map both the saturation-solubility limits (horizontal variations) as well as the freezing-point depression of the liquid (vertical variations). 

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