Lever rule

The phase diagram tells you directly the compositions of the two phases that are in equilibrium, but to get the relative amoimts of the material in each of the two phases. 

Illustration of the lever law. The fulcrum represents the starting composition XqN = number of B atoms in the system. is the number of atoms in the a phase and is the number of B atoms in the a phase. Likewise, Np is the number of atoms in the p phase and rpNp is the number of B atoms in the p phase. 

The lever rule also applies to phase diagrams in weight fraction or wt% if the xs in Equation 12.14 are expressed in terms of weight fraction or wt% and mole fractions N /N and Np/N are replaced by weight fractions W /W and Wp/W, respectively. 

Putting these results into Equation 12.13, the free energy of the segregated system becomes 

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Levansu erase Levarternol [51-41-2] Level A suits Level detection systems Level dyeing Leveling power Lever 2000 Lever rule... 

In an undoped, intrinsic semiconductor the equiHbrium concentrations of electrons, and holes,/), are described by a lever rule derived from the law of mass action (eq. 3) ... 

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. 

Furthermore, compositions may be found graphically by a lever rule. All mixtures of pure C02 and pure H2 fall along the ordinate, the distance from pure C02 being inversely proportional to the amount of pure C02... 

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. 

As the mixture freezes, 1,4-dimethylbenzene is removed from solution, the liquid mixture becomes richer in benzene, and the melting temperature falls along line be. For example, when the temperature given by point h is reached, solid 1,4-dimethylbenzene (point i) and a liquid solution with a composition given by point g are present. The lever rule gives the ratio of solid to liquid as... 

Himmelbau (1995) or any of the general texts on material and energy balances listed at the end of Chapter 2. The Ponchon-Savarit graphical method used in the design of distillation columns, described in Volume 2, Chapter 11, is a further example of the application of the lever rule, and the use of enthalpy-concentration diagrams. 

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... 

When component and mixture concentrations of any species i are known, mass proportions can be calculated from the lever rule ... 

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