= Tie-line phase-fraction fit
{title2=$\widehat\lambda$}
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> predicts
$$
\mathbf z_{\rm pred}=\mathbf x^\alpha+\lambda(\mathbf x^\beta-\mathbf x^\alpha),\qquad0\le\lambda\le1.
$$
The 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.<Tie-line in IUPAC Gold Book>[Entry15367: endpoints are compositions of coexisting equilibrium phases]{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> minimizes
$$
J(\lambda)=(\mathbf y-\mathbf a-\lambda\mathbf d)^T\Sigma^{-1}(\mathbf y-\mathbf a-\lambda\mathbf d).
$$
Differentiation yields the unconstrained estimate
$$
\lambda_0=\frac{\mathbf d^T\Sigma^{-1}(\mathbf y-\mathbf a)}{\mathbf d^T\Sigma^{-1}\mathbf d},\qquad
\widehat\lambda=\min(1,\max(0,\lambda_0)).
$$
The 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 $\Sigma=\operatorname{diag}(0.010^2,0.005^2)$. Then
$$
\lambda_0=\frac{620}{2000}=0.310.
$$
The 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:
$$
\operatorname{Var}(\lambda_0)=\frac1{\mathbf d^T\Sigma^{-1}\mathbf d},\qquad
u(\lambda_0)=\frac1{\sqrt{2000}}\approx0.02236
$$
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.
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