The Lever Rule

Let us now combine the P-xBq diagram with the P-xBap diagram (cf. Fig. 7.7) to obtain the composite P-xB diagram shown in Fig. 7.9. As shown in Fig. 7.9a, there are now three distinct xB composition variables xBap (vapor phase), xBq (liquid phase), and xB (total system). How can we read all three composition values from the P-xB diagram in Fig 7.9b  

As shown in Fig. 7.9, for a given vapor pressure P (dotted line), the compositions vBq, xBap of the coexisting phases are found from the intersections (small circles) with the liquid and vapor boundaries of the hatched two-phase region. These intersections are connected by a horizontal tie-line (heavy solid line) that spans the two-phase hole in the diagram. All points along this tie-line represent the same thermodynamic state (i.e., same temperature, pressure, chemical potentials, and compositions of each phase), but each differs only in the relative amounts of each phase (cf. Sidebar 7.2), whether nearly all vapor (at the extreme left of the tie-line), nearly all liquid (at the extreme right), or roughly equimolar amounts of liquid and vapor (near the middle). 

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 

We can therefore locate xB along the tie-line between Xgap and Xgq, dividing the tie-line into segments of length Lvap and Lliq, respectively, as follows  

In terms of these tie-line segments, the lever rule can now be stated as follows  

If two phases—liquid and vapor—are present in equilibrium, the variance of the system is F = 4 — 2 = 2. Since the temperature is fixed, one other variable, any one of p, Xi, yi, suffices to describe the system. So far we have used x or to describe the system since Xi + X2 = 1, andy + yz = 1, we could equally well have chosen X2 and y2. If the pressure is chosen to describe the two-phase system, the intersections of the horizontal line at that pressure with the liquid and vapor curves yield the values of Xi and y directly. If x is the describing variable, the intersection of the vertical line at Xi with the liquid curve yields the value of p from p the value of y is obtained immediately. 

In any two-phase region, such as L-V in Fig. 14.3(b), the composition of the entire system may vary between the limits x and yi, depending on the relative amounts of liquid and vapor present. If the state point a is very near the liquid line, the system would consist of a large amount of liquid and a relatively small amount of vapor. If a is near the vapor line, the amount of liquid present is relatively small compared with the amount of vapor present. 

Since the derivation of the lever rule depends only on a mass balance, the rule is valid for calculating the relative amounts of the two phases present in any two-phase region of a two-component system. If the diagram is drawn in terms of mass fraction instead of mole fraction, the level rule is valid and yields the relative masses of the two phases rather than the relative mole numbers. 

Consider a single-substance system whose system point is in a two-phase area of a pressure-volume phase diagram. How can we determine the amounts in the two phases  

As an example, let the system contain a fixed amount n of a pure substance divided into liquid and gas phases, at a temperature and pressure at which these phases can coexist in equilibrium. When heat is transferred into the system at this T and p, some of the liquid vaporizes by a liquid-gas phase transition and V increases withdrawal of heat at this T and p causes gas to condense and V to decrease. The molar volumes and other intensive properties of the individual liquid and gas phases remain constant during these changes at constant T and p. On the pressure-volume phase diagram of Fig. 8.9 on page 208, the volume changes correspond to movement of the system point to the right or left along the tie line AB. 

When enough heat is transferred into the system to vaporize all of the liquid at the given 

Thermodynamics and Chemistry, second edition, version 3 2011 by Howard DeVoe. Latest version www.chem.umd.edu/themobook 

T and p, the system point moves to point B at the right end of the tie line. V/n at this point must be the same as the molar volume of the gas, V - We can see this because the system point could have moved from within the one-phase gas area to this position on the boundary without undergoing a phase transition. 

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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. 

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 ... 

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. ... 

The lever rule states in order to balance the lever (i.e. create an equilibrium) then ml = m l where m is the mass of the object and / is the length from the pivot. 

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... 

Morse S. A. (1976). The lever rule with fractional crystallization and fusion. Amer. Jour. Scl, 276 330-346. 

It is not necessary to use zq as a starting point and any composition n can be used which satisfies the lever rule, where... 

For solidification described by the lever rule and assuming linear liquidus and solidus lines, the composition of the solid C, as a function of the fraction solid transformed (/,) is given by the equation... 

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