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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]

A ternary tie-line consistency residual quantifies how far an overall three-component composition is from the mass balance predicted by specified endpoints of a ternary liquid-liquid tie line. It is a geometric consistency check on a proposed two-phase decomposition. A thermodynamic tie line connects coexisting equilibrium compositions; the calculation below checks the lever rule for those endpoints, rather than independently proving that they are equilibrium phases.^[ref]

Use mole fraction vectors ${\mathbf{x}}^{\alpha },{\mathbf{x}}^{\beta }$ for the endpoint liquids and $\mathbf{z}$ for the overall composition, all normalized to sum to one. Put $\mathbf{d}={\mathbf{x}}^{\beta }-{\mathbf{x}}^{\alpha }$. For a proposed mole-based phase fraction $\lambda$, defineExact mass balance requires every component of $\mathbf{r}$ to vanish. Closure gives ${\sum }_i{r}_i=0$ automatically, so this residual has only two independent components. A zero sum alone does not imply that the component balances hold.

When the phase fraction is unknown and the endpoints are distinct, an unweighted least squares tie-line phase-fraction fit minimizes the squared Euclidean norm of $\mathbf{r}$. Differentiating givesThis is an orthogonal projection onto the line through the endpoints. A physically admissible phase fraction also lies in $[0,1]$; if the unconstrained fit lies outside that interval, the closest admissible composition is an endpoint of the line segment. Coincident endpoints do not determine a phase fraction through this formula.

For the hypothetical endpoints ${\mathbf{x}}^{\alpha }=(0.65,0.25,0.10)$ and ${\mathbf{x}}^{\beta }=(0.15,0.35,0.50)$, the composition $(0.50,0.28,0.22)$ gives $\hat{\lambda }=0.30$ and zero residual. Changing the overall composition to $\mathbf{z}=(0.50,0.29,0.21)$ preserves closure but givesNo adjustment of a single phase fraction can eliminate that discrepancy under the stipulated exact endpoints. For experimental use, its significance depends on measurement uncertainty in both the overall composition and endpoint compositions. Unweighted distance is a chosen metric, not an automatic statistical rejection threshold; correlated composition errors require their covariance matrix on the independent composition coordinates. Even a tiny residual establishes only compatibility with this mass balance, not equality of chemical potentials or stability of the proposed liquid-liquid equilibrium.

A tie-line phase-fraction fit estimates how much of each liquid phase is needed to reproduce an overall composition from specified liquid-liquid tie line endpoints. If ${\mathbf{x}}^{\alpha },{\mathbf{x}}^{\beta }$ are endpoint mole fraction vectors and $\lambda$ is the mole-based phase fraction of phase $\beta$, the mass balance predictsThe endpoints must be supplied by equilibrium measurements or a suitable thermodynamic model; fitting $\lambda$ does not establish that an arbitrary pair of compositions is an equilibrium tie line.^[ref]

For a ternary liquid-liquid tie line, use two independent composition coordinates, such as the first two mole fractions, since the third is fixed by their sum. Let $\mathbf{y}$ be these two measured coordinates, $\mathbf{a}$ the corresponding phase-$\alpha$ coordinates, $\mathbf{d}$ the difference between phase-$\beta$ and phase-$\alpha$ coordinates, and $\Sigma$ a positive-definite covariance matrix for $\mathbf{y}$. With exact endpoints, generalized least squares minimizesDifferentiation yields the unconstrained estimateThe second expression imposes the physical bounds. It follows because $J$ is a convex quadratic when the endpoints differ in the measured coordinates. The ternary tie-line consistency residual then records the remaining failure of the mass balance.

For an original hypothetical example, take endpoint compositions $(0.60,0.30,0.10)$ and $(0.20,0.40,0.40)$, with overall measurement $(0.470,0.325,0.205)$. Suppose the first two measured mole fractions have independent standard measurement uncertainties $0.010$ and $0.005$, so $\Sigma =\mathrm{diag}(0.010^2,0.005^2)$. ThenThe predicted composition is $(0.476,0.331,0.193)$, giving residual $(-0.006,-0.006,0.012)$. The separate first- and second-component balances would suggest $0.325$ and $0.250$; the fit combines them according to their stated measurement uncertainties, rather than averaging those two numbers without regard to precision.

With Gaussian composition errors, fixed endpoints and an interior unconstrained fit, the tie-line phase-fraction uncertainty follows directly from the linear estimator:for this example. Endpoint uncertainty, a fit on a boundary, or a nearly vanishing endpoint difference needs additional treatment. In particular, all three normalized mole fractions must not be treated as independent observations with an invertible three-coordinate covariance matrix: their exact closure creates dependence. A small fitted residual is evidence of consistency with the specified mass balance and error model, not a substitute for testing chemical potential equality or phase stability.

Tie-line phase-fraction uncertainty describes the uncertainty of the phase amount inferred from a tie-line phase-fraction fit. It depends on the uncertainties of the overall composition, the endpoint compositions and the model used to fit them. A conditional estimate treating endpoints as exact omits their contribution. All composition vectors below use two independent coordinates of a ternary liquid-liquid tie line; the third mole fraction is fixed by closure.

For fixed endpoints $\mathbf{a},\mathbf{b}$, let $\mathbf{d}=\mathbf{b}-\mathbf{a}$ and let the overall composition measurement $\mathbf{y}$ have covariance matrix ${\Sigma }_y$. With $W={\Sigma }_y^{-1}$ and an interior unconstrained fit,Here $u$ denotes standard uncertainty. The variance formula is exact for this linear estimator with fixed endpoints and known covariance; Gaussian errors are needed for a Gaussian sampling distribution, not for the variance identity itself. This is an instance of covariance-based uncertainty propagation.^[ref]

Endpoint errors enter through tie-line phase-fraction sensitivity. Near an exactly balanced composition $\mathbf{y}=(1-\lambda )\mathbf{a}+\lambda \mathbf{b}$, with the weight matrix held fixed, differentiating the fit givesIf overall and endpoint measurements are mutually independent, their combined first-order contribution isShared instruments or common reference data can correlate these measurements; then cross-covariances, including tie-line endpoint covariance, must also be propagated rather than discarded.^[ref]

For an original example, use endpoints $(0.60,0.30,0.10)$ and $(0.20,0.40,0.40)$ with overall composition $(0.44,0.34,0.22)$, so $\lambda =0.40$. Let the first two overall mole fractions have independent standard uncertainties $0.010$ and $0.005$. Then $\mathbf{g}=(-2,2)$ and $u^2=0.000500$, giving $u\approx 0.02236$ with exact endpoints. If each endpoint instead has independent coordinate uncertainties $0.005$, with all three measurements independent, the endpoint contribution isThe combined value becomes $u\approx \sqrt{0.000604}=0.02458$.

When the endpoints approach one another, their difference becomes less informative about phase fraction; at coincident endpoints this inference is undefined. Near the physical bounds, clipping an unconstrained fit to the interval from zero to one changes its sampling distribution, potentially creating a point mass at an endpoint. A symmetric Gaussian error summary can then be misleading. Propagating a joint input model through the actual bounded fit with a Monte Carlo method is one possible analysis; it must retain composition closure and the specified correlations. Neither a small uncertainty nor a small ternary tie-line consistency residual independently verifies chemical potential equality or phase stability.