Tie-triangle rule

At fixed T and P, a three-phase ternary has P = 0, which defines a point. On a triangular diagram, a three-phase situation produces three points, each giving the composition of one of the phases. The three points can be connected to form a triangle, and the relative amounts in the three phases can be found by a tie-triangle rule (see Appendix H). 

Analogous results are obtained for N /N and N /N. The result (B.6.17) is a form of the tie-triangle rule for three components in three-phase equilibrium (see H.2). 

When ternary mixtures exhibit three-phase equilibria, a tie-triangle rule can be used to obtain the relative amounts in the three phases. This is analogous to the lever rule for binaries in two-phase equilibria. Consider a ternary mixture of components 1, 2, and 3 in three-phase equilibrium at the overall composition represented by a point P, as in Figure H.2. In the figure, the tie-triangle ABC bounds the three-phase region, the... 

This tie-triangle rule is a consequence of material balances on the system (see (B.6.17) in Appendix B) and therefore it applies to any three-phase equilibrium situation involving ternary mixtures. 

From the properties of similar triangles, the right side of Equation 8.5-6 equals the ratio of distances AC/ AB. We have thus proved the following general rule t A, B, and C are the points on a tie line corresponding to the total mixture, the liquid phase, and the vapor phase, respectively, and if F, L, and V are the corresponding masses, then the liquid-to-vapor mass ratio is... 

If we stop the expansion momentarily at 10 bar (point C), we can use the diagram to obtain the compositions and relative amounts in the two phases. Construct the horizontal through C, then and are given by the intersections of the horizontal (a tie line) with the two-phase curves. These intersections are marked with triangles we find Xi = 0.449 and y = 0.937. On an isothermal Pxy diagram, tie lines are horizontal because at equilibrium both phases have the same pressure. The fraction of the system in the vapor phase can be determined by material balance (a lever rule) ... 

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). 

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