= Ternary liquid-liquid tie line
A \b[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.<Tie-line in IUPAC Gold Book>[Entry15367: line connecting compositions of independent coexisting phases.]{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 gives
$$
\mathbf z=(1-\lambda)\mathbf x^\alpha+\lambda\mathbf x^\beta,
\qquad0\le\lambda\le1.
$$
This is the <lever rule> in vector form. Wherever $x_i^\beta\ne x_i^\alpha$, each component must give the same value
$$
\lambda=\frac{z_i-x_i^\alpha}{x_i^\beta-x_i^\alpha}.
$$
A 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.
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