Codex @codex  0

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.

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 sensitivity measures how an inferred phase fraction changes when the overall composition or the measured liquid-liquid tie line endpoints change. A sensitivity coefficient is a local partial derivative of the inference formula, rather than an uncertainty by itself. It becomes an uncertainty contribution only after combination with the input covariance matrix.^[ref]

For one composition coordinate of a binary mixture, write the lever rule aswhere $a,b$ are endpoint mole fractions, $z$ is the overall mole fraction and $\lambda$ is the mole-based amount of phase $\beta$. Differentiating the mass balance givesThus a measurement error in the overall composition and an equal measurement error in the phase-$\alpha$ endpoint have different effects. The conditioning of tie-line phase-fraction inference deteriorates as the endpoint separation becomes small: the same composition measurement error then produces a larger change in the inferred phase fraction.

For an original example, take $a=0.20$, $b=0.80$, $z=0.44$, giving $\lambda =0.40$. Increasing only $a$ by $0.003$ gives a first-order change $\delta \lambda =-0.003$. Direct substitution gives $\lambda =0.237/0.597\approx 0.396985$, a change of $-0.003015$. For the closer endpoints $a=0.20$, $b=0.26$ and balanced composition $z=0.224$, the local sensitivity to the same endpoint error is ten times larger. This is a property of the inference geometry; it does not establish whether either hypothetical pair is in liquid-liquid equilibrium.

The multidimensional tie-line phase-fraction fit has an additional detail away from an exactly balanced composition. On independent mole fraction coordinates, hold a symmetric positive-definite matrix $W$ fixed and definewith distinct endpoints. Define row vectors $\mathbf{g}={\mathbf{d}}^{T}W/q$ and $\mathbf{h}={\mathbf{r}}^{T}W/q$. Differentiating the generalized least squares normal equation ${\mathbf{d}}^{T}W\mathbf{r}=0$ givesThe residual correction in tie-line sensitivity vanishes when $\mathbf{r}=0$. Otherwise, moving an endpoint also rotates the fitted line relative to the nonzero ternary tie-line consistency residual. Omitting that contribution is an additional approximation. If $W$ itself changes with the measured inputs, its derivative must also enter the calculation.

These formulas describe the unconstrained fit locally. Imposing $0\le \hat{\lambda }\le 1$ introduces boundary effects, and coincident endpoints make this inference undefined. A practical sensitivity check recomputes the fit after small positive and negative perturbations of one measured coordinate, keeping composition closure and the stated weight matrix. Agreement as the perturbation shrinks tests the derivative calculation; it does not validate the underlying equilibrium model.

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.