= Tie-line phase-fraction uncertainty
{title2=$u(\widehat\lambda)$}
<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,
$$
\lambda_0=\frac{\mathbf d^TW(\mathbf y-\mathbf a)}{\mathbf d^TW\mathbf d},\qquad
\mathbf g=\frac{\mathbf d^TW}{\mathbf d^TW\mathbf d},\qquad
u^2(\lambda_0)=\mathbf g\Sigma_y\mathbf g^T=\frac1{\mathbf d^TW\mathbf d}.
$$
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>.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, sensitivity coefficients, combined variance and correlated input quantities]{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 gives
$$
\delta\lambda=\mathbf g\bigl[\delta\mathbf y-(1-\lambda)\delta\mathbf a-\lambda\delta\mathbf b\bigr].
$$
If overall and endpoint measurements are mutually independent, their combined first-order contribution is
$$
u^2(\widehat\lambda)\approx\mathbf g\left[\Sigma_y+(1-\lambda)^2\Sigma_a+\lambda^2\Sigma_b\right]\mathbf g^T.
$$
Shared instruments or common reference data can correlate these measurements; then cross-covariances, including <tie-line endpoint covariance>, must also be propagated rather than discarded.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2, equation(13), and §§5.2.4–5.2.5, significant covariance from common influences]{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 $u\approx0.02236$ with exact endpoints. If each endpoint instead has independent coordinate uncertainties $0.005$, with all three measurements independent, the endpoint contribution is
$$
(0.6^2+0.4^2)(0.005^2)(2^2+2^2)=0.000104.
$$
The 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.
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