Lever rule binary system

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... 

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).

Use of tie lines and the lever rule enable one to determine those compositions, as illustrated in Figure 4.6a for a binary system. The composition of the solid (Cs) as a function of the fraction of solid transformed (fs), assuming linear solidus and liquidus lines, is given by... 

When the overall composition is such that the system point falls in the area labeled P=2, two liquid phases are present. The compositions of these phases are given by the positions of the ends of a tie line through the system point. Four representative tie lines are included in the diagram, and these must be determined experimentally. The relative amounts of the two phases can be determined from the lever rule. In the limit of zero mole fraction of ethanol, the tie line falls along the horizontal base of the triangle and displays a miscibility gap for the binary system of benzene and water. (The conjugate phases are very nearly pure benzene and pure water). 

An example of a binary eutectic system AB is shown in Figure 15.3a where the eutectic is the mixture of components that has the lowest crystallisation temperature in the system. When a melt at X is cooled along XZ, crystals, theoretically of pure B, will start to be deposited at point Y. On further cooling, more crystals of pure component B will be deposited until, at the eutectic point E, the system solidifies completely. At Z, the crystals C are of pure B and the liquid L is a mixture of A and B where the mass proportion of solid phase (crystal) to liquid phase (residual melt) is given by ratio of the lengths LZ to CZ a relationship known as the lever arm rule. Mixtures represented by points above AE perform in a similar way, although here the crystals are of pure A. A liquid of the eutectic composition, cooled to the eutectic temperature, crystallises with unchanged composition and continues to deposit crystals until the whole system solidifies. Whilst a eutectic has a fixed composition, it is not a chemical compound, but is simply a physical mixture of the individual components, as may often be visible under a low-power microscope. 

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