= Liquid-liquid tie line
{title2=$z=(1-\lambda)x_\alpha+\lambda x_\beta$}
A \b[liquid-liquid tie line] joins the compositions of two <liquid> phases that coexist at a specified <temperature> and <pressure>. Its endpoints describe the separate phases; an interior point describes an overall composition formed from suitable amounts of them. The <liquid-liquid tie line> therefore connects equilibrium chemistry with material balance rather than describing a spatial boundary between layers.
For a binary <mixture>, use component-B <mole fractions> $x_\alpha$, $x_\beta$ and overall fraction $z$. If $\lambda=n_\beta/(n_\alpha+n_\beta)$ is the mole-based <phase fraction> in phase $\beta$, conservation of component B gives the <lever rule>:
$$
z=(1-\lambda)x_\alpha+\lambda x_\beta,\qquad
\lambda=\frac{z-x_\alpha}{x_\beta-x_\alpha}.
$$
The endpoints must be different and $0\le\lambda\le1$. For invented coexistence endpoints $x_\alpha=0.20$, $x_\beta=0.80$ and overall $z=0.35$, the result is $\lambda=0.25$. In a $4$-mole sample, the phases contain $3$ and $1$ moles; their component-B amounts are $0.60$ and $0.80$ moles, totaling $1.40=4(0.35)$. These mole amounts do not specify the two layer volumes.
A <ternary liquid-liquid tie line> obeys the same vector balance. For chosen endpoint vectors $\mathbf x^\alpha=(0.7,0.2,0.1)$ and $\mathbf x^\beta=(0.1,0.3,0.6)$, a phase fraction $\lambda=0.4$ gives $\mathbf z=(0.46,0.24,0.30)$. Every component must give the same $\lambda$. A normalized composition vector need not lie on this particular line; normalization alone does not establish compatibility with the proposed endpoints.
The <lever rule> determines amounts after equilibrium endpoints have been established; it does not select those endpoints. They can come from measured phase compositions or a stable <common-tangent construction for phase coexistence>. Reliable experiments control temperature and sample purity, measure the separated compositions and distinguish genuine <liquid-liquid equilibrium> from other phase behavior. An IUPAC reference-system assessment specifically warns that turbidity can confuse liquid–liquid and solid–liquid transitions.<Reference Materials for Phase Equilibrium Studies. 1. Liquid-Liquid Equilibria by Ala Bazyleva and others>[Introduction, pp.1–2: composition measurement, emulsions, solid–liquid discrimination and reference-system validation.]{ref}
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