Vapor-liquid equilibrium lever rule

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. 

At any point C on the line BD we may construct a horizontal tie line and obtain the compositions of the liquid and vapor phases. On Txy diagrams, tie lines are horizontal because at equilibrium both phases are at the same T. In the figure, point C lies on the 415 K tie line and the equilibrium compositions are Xj = 0.418 and yj = 0.774. The cooling process in Figure 9.6 is done at a constant overall mole fraction z = 0.6, so we can apply a lever rule to obtain the fraction of material that is vapor at point C ... 

These equations give the fraction of liquid and vapor, if the volume of the two-phase system is known. The two equations in (2 10) are known as the lever rule if LVis viewed as a lever with force Xl acting on point L, force Xv on point V, and the pivot placed at E, the lever would be in mechanical equilibrium (see Example 2.1 1. 

The method described here was originally presented by Ponchon (1921) and Savarit (1922). Besides the enthalpy-composition diagram, the method makes use of the lever rule for relating the rates of vapor and liquid products of a binary equilibrium stage to their compositions and enthalpies, as described below. 

The mass balances that lead to Equations (E8.4D) and (E8.4E) are general and not limited to the vapor and liquid phases thus, the lever rule can be applied to find the relative amounts of any two phases in equilibrium. The fraction of material present in one phase can be computed by taking the length of the tie line from the overall composition to the composition of the other phase and then dividing by the total length of the line. 

The type of information we can get from a Txy plot is similar to that from a Pxy plot. For any given temperature and composition, we can again identify whether we have only a liquid phase, only a vapor phase, or a combination of two phases— liquid in equilibrium with vapor. The liquid phase is now on the bottom of the phase diagram— that is, at low T— while the vapor phase is on the top. For the two-phase region, the composition of each phase can also be determined, and a tie line connects vapor and liquid compositions for a given temperature. For example, the tie line shown in Figure 8.4 indicates that at 358 K and 1 atm, methanol with liquid mole fraction Xa = 0.15 is in equilibrium with a vapor of composition ya = 0.52. Again, the lever rule can be applied to discern the relative amounts of liquid and vapor. 

---