 Liquid-liquid tie line

A 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:The endpoints must be different and $0\le \lambda \le 1$. 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.^[ref]