= Ternary tie-line consistency residual
{title2=$\mathbf r(\lambda)=\mathbf z-\mathbf x^\alpha-\lambda(\mathbf x^\beta-\mathbf x^\alpha)$}
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.<Tie-line in IUPAC Gold Book>[Entry 15367: tie line connects compositions of independent coexisting 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> $\lambda$, define
$$
\mathbf r(\lambda)=\mathbf z-\mathbf x^\alpha-\lambda\mathbf d,
\qquad 0\le\lambda\le1.
$$
Exact <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 gives
$$
\widehat\lambda=\frac{\mathbf d\mathbin{\cdot}(\mathbf z-\mathbf x^\alpha)}{\mathbf d\mathbin{\cdot}\mathbf d},
\qquad \mathbf r_* =\mathbf z-\mathbf x^\alpha-\widehat\lambda\mathbf d,
\qquad \mathbf d\mathbin{\cdot}\mathbf r_*=0.
$$
This 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 $\widehat\lambda=0.30$ and zero residual. Changing the overall composition to $\mathbf z=(0.50,0.29,0.21)$ preserves closure but gives
$$
\widehat\lambda=\frac{41}{140}\approx0.292857,
\qquad \mathbf r_*=\frac1{280}(-1,3,-2),
\qquad \|\mathbf r_*\|_2=\frac{\sqrt{14}}{280}\approx0.01336.
$$
No 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>.
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