 Ternary liquid-liquid tie line

A ternary liquid-liquid tie line joins the compositions of two coexisting liquid phases in a three-component mixture at a specified temperature and pressure. IUPAC defines a tie line by its coexisting equilibrium endpoints.^[ref] Each composition is a vector $\mathbf{x}=(x_A,x_B,x_C)$ of mole fractions satisfying $x_A+x_B+x_C=1$. Only two coordinates are independent; a triangular composition diagram displays this constraint.

If $\lambda$ is the mole-based phase fraction in phase $\beta$, conserving every component givesThis is the lever rule in vector form. Wherever $x_i^{\beta }\ne x_i^{\alpha }$, each component must give the same valueA component identical at both endpoints cannot determine $\lambda$, but its overall fraction must equal that common endpoint value.

For an original material-balance example, stipulate endpoint compositions $(0.65,0.25,0.10)$ and $(0.15,0.35,0.50)$. With $\lambda =0.30$, the overall composition is $(0.50,0.28,0.22)$. All three component calculations give $0.30$. By contrast, $(0.50,0.29,0.21)$ still sums to one, but gives apparent phase fractions $0.30$, $0.40$ and $0.275$. Under the stipulated exact endpoints, it does not lie on this tie line. A ternary tie-line consistency residual records the failure of the vector balance; experimental use must account for composition uncertainties.

These chosen endpoints illustrate conservation, not a claim that an actual mixture has these equilibrium compositions. Establish endpoints from equilibrium measurements or a stable Gibbs free energy model. Moving along one liquid-liquid tie line changes phase amounts while preserving its endpoint compositions. Moving elsewhere in the two-phase region can select a different pair of endpoints; all tie lines need not be parallel. Finally, mole-based and mass-based diagrams require consistent fractions: a mole-based $\lambda$ is not automatically a volume or mass fraction.