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The conditioning of tie-line phase-fraction inference describes how strongly composition perturbations change an inferred phase fraction. For a binary mixture with distinct endpoints, the lever rule gives $\lambda =(z-a)/(b-a)$. Its absolute sensitivity to $z$, with endpoints fixed, is $1/|b-a|$. This is an absolute amplification coefficient; a relative condition number additionally depends on the chosen input and output scales.

For an interior $0\le \lambda \le 1$, the differential in tie-line phase-fraction sensitivity yieldsto first order. If each composition perturbation is bounded by $\epsilon$, this becomes $2\epsilon /|b-a|$. For original stipulated $\epsilon =0.001$, endpoint separations $0.50$ and $0.05$ give local worst-case bounds $0.004$ and $0.040$. Tenfold closer endpoints make the same composition precision tenfold less informative about phase amount. Correlations can change the attainable combinations of errors, as common affine calibration cancellation in the lever rule illustrates.

For fixed multicomponent endpoints and overall-composition covariance matrix $\Sigma$, the tie-line phase-fraction uncertainty isThe denominator is endpoint separation after a whitening transformation: separation must be compared with measurement precision, not only plotted distance. This follows from covariance-based uncertainty propagation and the linear fit.^[ref]

Near coincident endpoints the inverse problem becomes poorly informative, and at coincidence it is undefined. Near the physical bounds, a large propagated uncertainty also warns that the interior linear approximation may fail. More numerical digits in the fitted answer do not resolve this measurement limitation.

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.