Tie-line endpoint covariance ($C_{ab}$)

Tie-line endpoint covariance records how the measurement errors of the two measured liquid-liquid tie line endpoints vary together. Using independent mole fraction coordinates, represent the two endpoint estimates by column vectors $\mathbf{a},\mathbf{b}$. Their cross-covariance matrix isAn individual off-diagonal block need not be symmetric; the full joint covariance matrix must be a symmetric positive semidefinite matrix. The uncertainties of the two endpoint vectors alone do not determine their cross-covariance matrix.

Near an exactly balanced, interior tie-line phase-fraction fit, let $\mathbf{g}$ be the row-vector sensitivity coefficient defined in tie-line phase-fraction sensitivity. If the overall composition measurement $\mathbf{y}$ is independent of both endpoints, first-order uncertainty propagation givesThe cross term has a positive coefficient because the two endpoint perturbations enter the phase fraction with the same sign. Correlated endpoints can therefore increase the result. The general rule is to propagate the full joint covariance matrix, rather than assume that correlation always increases or always reduces uncertainty.^[ref]

For a stipulated binary mixture example, use $a=0.20$, $b=0.80$, $z=0.44$, so $\lambda =0.40$. Treat $z$ as exact and give each endpoint standard uncertainty $s=0.006$ with correlation coefficient $\rho$. The lever rule yieldsThis gives $0.002$ for $\rho =-1$, about $0.00721$ for $\rho =0$, and $0.010$ for $\rho =1$. These are first-order standard uncertainties under the stipulated joint error model, not bounds on all possible errors.

A shared calibration error affecting the overall composition as well changes the calculation. Suppose the same additive error $c$ affects $a,b,z$ identically. ThenThe shared offset cancels exactly. More generally, a common affine map $x\mapsto mx+c$ with $m\ne 0$ also cancels in this binary ratio. This common affine calibration cancellation in the lever rule concerns only a truly common contribution; separate sample errors remain. If the shared offset has standard uncertainty $0.006$, incorrectly treating its three appearances as independent would assign a spurious contribution $0.01\sqrt{1+0.6^2+0.4^2}\approx 0.01233$ instead of zero.

In general, stack the inputs as $\mathbf{x}=(\mathbf{y},\mathbf{a},\mathbf{b})$ with joint covariance matrix $C$. Let $J$ be the corresponding Jacobian matrix row from tie-line phase-fraction sensitivity, including its residual correction when needed. Then $u^2(\lambda )\approx JC{J}^T$ automatically includes correlations with the overall composition. The measurement guide specifically identifies common instruments, measurement standards and reference data as sources of such correlations.^[ref]

An instructive measurement design analyzes the two equilibrium phases and the overall mixture using one instrument calibration, then repeats the entire procedure with independent instrument calibrations. The within-run contrasts and between-run changes probe different error contributions. Repeated readings under one unchanged instrument calibration cannot by themselves establish the uncertainty of its shared offset. The measurement model must identify which influences are common before the joint covariance matrix can represent them.

Common affine calibration cancellation in the lever rule is the exact invariance of the binary mixture phase fraction under the same affine map applied to the overall composition and both endpoints. For $m\ne 0$,Both a shared offset and a shared scale cancel. The map must be common to all three measurements; separately fitted calibration curves or sample-dependent effects do not satisfy that assumption.

An original example has $a=0.10$, $b=0.70$, $z=0.25$ and $\lambda =0.25$. Applying $x\mapsto 1.08x+0.03$ gives measured values $0.138$, $0.786$ and $0.300$, still yielding $0.162/0.648=0.25$. If only the overall-composition reading then receives an additional $0.002$ error, its inferred fraction changes by $0.002/0.648\approx 0.00309$. Common calibration cancellation leaves independent sample errors intact.

The covariance matrix formulation explains why treating the shared errors as independent gives a false uncertainty. In input order $(z,a,b)$, the Jacobian matrix row isIt annihilates both the common-offset direction $(1,1,1{)}^T$ and the common-scale direction $(z,a,b{)}^T$, since $z=(1-\lambda )a+\lambda b$. A joint covariance matrix representing only these shared affine influences therefore gives $JC{J}^T=0$. The measurement guide explicitly identifies common instruments and reference data as sources of correlation that must be retained.^[ref]

Analyzing the overall sample and separated phases under one unchanged instrument calibration can realize this common contribution; repeating only one sample under another calibration breaks it. The cancellation is a property of this binary ratio. It does not remove nonlinear response, composition closure errors, endpoint misidentification or uncertainty in whether the samples establish liquid-liquid equilibrium.