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For exchanging neutral components at a common temperature and pressure, activity equality for coexisting liquids expresses liquid-liquid equilibrium as $a_i^{\alpha }=a_i^{\beta }$, provided both thermodynamic activities use the same standard state. The basic equilibrium condition is equality of chemical potentials; equality of numerical activities follows only after their references have been matched.

Write the definition of thermodynamic activity aswhere $R$ is the molar gas constant and $T$ is temperature. Subtraction at equilibrium givesThe ratio is one when the two standard chemical potentials are the same. IUPAC defines activity relative to the specified standard chemical potential and distinguishes pure-phase and solute reference conventions.^[ref] Mixing conventions can therefore produce unequal activities for physically coexisting phases. For instance, if one reference exceeds the other by $RT\ln 2$, its corresponding activity is half as large at the same chemical potential. Standard-state activity conversion changes the numerical coordinate, not the physical driving force.

Equal thermodynamic activities also need not mean equal mole fractions. With a common pure-liquid reference, $a_i=x_i{\gamma }_i$, where ${\gamma }_i$ is an activity coefficient. An original symmetric regular solution example makes this quantitative. Set the dimensionless interaction parameter to $\chi =2\ln 3$. The model has coexisting component-A mole fractions $x_A^{\alpha }=1/4$ and $x_A^{\beta }=3/4$, withThus ${\gamma }_A^{\alpha }=3^{9/8}\approx 3.44161$ and ${\gamma }_A^{\beta }=3^{1/8}\approx 1.14720$, whileComponent B satisfies the analogous equality by symmetry. The different compositions are compensated by different nonideal environments. These are chosen model values, not experimental compositions. As in liquid-liquid equilibrium, satisfying equalities must be supplemented by stability against competing phase arrangements.

Ala Bazyleva works on chemical thermodynamics and the quality of thermophysical-property data. Her named profile at the National Institute of Standards and Technology describes research involving measurements, thermodynamic modeling and critical data assessment.^[ref]

She is corresponding author and task-group chair of a 2021 technical report proposing seven reference binary systems for liquid-liquid equilibrium studies. The work evaluates existing measurements, recommends property values with measurement uncertainty, and provides equations for comparison within stated experimental ranges. Its contribution is a defensible reference for checking instruments and methods, rather than treating any previously published number as equally reliable.^[ref]

An original comparison illustrates the role of uncertainty. Suppose a reference mole fraction is 0.200 with standard uncertainty 0.002, and an independent measurement gives 0.206 with standard uncertainty 0.001. If those uncertainties describe independent contributions, the standard uncertainty of their difference isThe observed difference 0.006 is about 2.68 times that scale. Comparing only the central numbers would miss how strongly the result depends on the reference and measurement uncertainties. This stipulated diagnostic is not a universal acceptance threshold or a numerical result from Bazyleva's report.

A useful experimental check also matches temperature, pressure and composition conventions to the reference conditions. A precise instrument can disagree with a reference because it examined a different state, so agreement tests need both a defined measurand and a supported uncertainty statement.

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.

Liquid immiscibility is the failure of a mixture to form one homogeneous liquid at the specified composition and conditions. In a two-liquid phase equilibrium, each thermodynamic phase can contain some of both components, but their equilibrium compositions differ. “Immiscible” therefore need not mean that no molecules of either component dissolve in the other.

Hexane and water provide a concrete example of limited mutual solubility. A NIST-compiled experimental record describes overnight shaking, microsyringe sampling of an aqueous solution, and gas chromatography with flame-ionization detection to measure the dissolved hydrocarbon. Detecting a small dissolved amount does not imply that every bulk mixing ratio gives one liquid phase.^[ref]

An original lever rule example shows how phase amounts differ from phase compositions. Suppose a binary system at fixed temperature and pressure has coexisting liquid compositions $x_{\alpha }=0.1$ and $x_{\beta }=0.8$, where $x$ is the mole fraction of componentB. These stipulated values are not measurements of hexane/water. If the overall mole fraction is $z=0.4$, material balance givesHere $\lambda$ is the fraction of the total amount of substance in phaseβ. A100-mole sample therefore contains $300/7\simeq 42.86$ moles in phaseβ and $400/7\simeq 57.14$ moles in phaseα. The phase compositions stay0.8 and0.1; neither is the overall composition0.4.

Shaking the sample can create dispersed droplets without making a single molecularly homogeneous phase. Equilibration, temperature, pressure and composition must be stated when testing liquid-liquid equilibrium. The lever-rule fraction is a mole-amount fraction, not necessarily the fraction of vessel height or volume occupied by one layer.

At fixed temperature and pressure, liquid-liquid equilibrium is stable coexistence of distinct liquid thermodynamic phases that minimizes total Gibbs free energy for the specified component amounts. Their compositions can differ even though each exchanging chemical species has the same chemical potential in both phases. For neutral, nonreacting components in bulk phases at common temperature $T$ and pressure $p$, neglecting external-field and interface-curvature contributions,To see why, transfer an amount $d{n}_i$ from phase $\alpha$ to phase $\beta$ while holding the total amount of each component fixed. The change in Gibbs free energy is $dG=({\mu }_i^{\beta }-{\mu }_i^{\alpha })d{n}_i$. If the coefficient is nonzero, one transfer direction lowers $G$. Equality removes that driving term; stable equilibrium additionally requires a global minimum over the allowed phase arrangements. Here each chemical potential is the partial derivative of Gibbs free energy with respect to that component amount at fixed temperature, pressure and the other component amounts.^[ref]

Charged species instead require equality of the corresponding electrochemical potential. Equal chemical potentials for neutral species do not require equal mole fractions. With a common pure-component reference, ${\mu }_i={\mu }_i^{\circ }+RT\ln a_i$ and $a_i=x_i{\gamma }_i$, where $a_i$ is thermodynamic activity, ${\gamma }_i$ an activity coefficient, and $R$ the molar gas constant. The activity equality for coexisting liquids is $x_i^{\alpha }{\gamma }_i^{\alpha }=x_i^{\beta }{\gamma }_i^{\beta }$. Different nonideal environments can therefore support different compositions at the same chemical potential.

For a binary mixture, let $x$ be the mole fraction of component B and let $g(x)$ be the homogeneous molar Gibbs free energy at fixed $T,p$. Differentiating $G=(n_A+n_B)g(n_B/(n_A+n_B))$ givesEquality of both component chemical potentials between two compositions becomes the common-tangent construction for phase coexistence:In this molar representation, the tangent slope is ${\mu }_B-{\mu }_A$ and its intercept is ${\mu }_A$. Stable coexistence uses a supporting tangent lying below the homogeneous curve over all admissible compositions. This constructs the relevant convex envelope; two stationary points or equal slopes alone do not establish stability.

An original regular solution calculation makes this distinction visible. Because linear composition bias leaves coexistence compositions unchanged, subtract the linear pure-component reference and writeThe dimensionless parameter $\chi$ penalizes unlike contacts when it is positive. Take $\chi =3$. Symmetry makes the coexistence tangent horizontal, and the two outer solutions of $h^{\prime }(x)=\ln (x/(1-x))+3(1-2x)=0$ give the symmetric regular-solution binodal:These are model compositions, rather than measured hexane–water values. At overall mole fraction $z=0.30$, the liquid-liquid tie line and lever rule give the fraction $\lambda$ of total moles in phase $\beta$:Changing $z$ within the coexistence interval changes phase amounts while leaving the two endpoint compositions fixed. These mole-based phase fractions need not equal volume fractions.

The spinodal answers a different question: local stability of a homogeneous mixture. Here $h^{\prime \prime }(x)=1/[x(1-x)]-2\chi$, so at $\chi =3$ the two local-stability limits are $(1\mp \sqrt{1/3})/2\approx 0.211325,0.788675$. Between them, composition fluctuations lower the bulk free energy. Between a binodal endpoint and the nearer spinodal, the homogeneous state has positive local curvature but a lower-energy phase-separated state exists: this is metastability. Thus the observed coexistence compositions are not obtained merely by setting $h^{\prime \prime }=0$.

A useful experiment measures compositions as well as the presence of layers. One hexane–water record reports overnight shaking, microsyringe sampling of the aqueous solution, and gas chromatography to measure dissolved hydrocarbon. The small dissolved amount is evidence of finite mutual solubility, consistent with liquid immiscibility over other bulk compositions.^[ref] At controlled temperature, repeated sampling of both phases and checks from different initial compositions help test whether the endpoint compositions have stabilized. Critically evaluated reference systems provide a way to validate such instruments and techniques.^[ref]

A liquid-liquid tie line joins the compositions of two liquid phases that coexist at a specified temperature and pressure. Its endpoints describe the separate phases; an interior point describes an overall composition formed from suitable amounts of them. The liquid-liquid tie line therefore connects equilibrium chemistry with material balance rather than describing a spatial boundary between layers.

For a binary mixture, use component-B mole fractions $x_{\alpha }$, $x_{\beta }$ and overall fraction $z$. If $\lambda =n_{\beta }/(n_{\alpha }+n_{\beta })$ is the mole-based phase fraction in phase $\beta$, conservation of component B gives the lever rule:The endpoints must be different and $0\le \lambda \le 1$. For invented coexistence endpoints $x_{\alpha }=0.20$, $x_{\beta }=0.80$ and overall $z=0.35$, the result is $\lambda =0.25$. In a $4$-mole sample, the phases contain $3$ and $1$ moles; their component-B amounts are $0.60$ and $0.80$ moles, totaling $1.40=4(0.35)$. These mole amounts do not specify the two layer volumes.

A ternary liquid-liquid tie line obeys the same vector balance. For chosen endpoint vectors ${\mathbf{x}}^{\alpha }=(0.7,0.2,0.1)$ and ${\mathbf{x}}^{\beta }=(0.1,0.3,0.6)$, a phase fraction $\lambda =0.4$ gives $\mathbf{z}=(0.46,0.24,0.30)$. Every component must give the same $\lambda$. A normalized composition vector need not lie on this particular line; normalization alone does not establish compatibility with the proposed endpoints.

The lever rule determines amounts after equilibrium endpoints have been established; it does not select those endpoints. They can come from measured phase compositions or a stable common-tangent construction for phase coexistence. Reliable experiments control temperature and sample purity, measure the separated compositions and distinguish genuine liquid-liquid equilibrium from other phase behavior. An IUPAC reference-system assessment specifically warns that turbidity can confuse liquid–liquid and solid–liquid transitions.^[ref]

The residual correction in tie-line sensitivity accounts for rotation of a fitted liquid-liquid tie line when the measured overall composition does not lie exactly on it. In independent mole fraction coordinates, set $\mathbf{d}=\mathbf{b}-\mathbf{a}$, hold a symmetric positive-definite matrix $W$ fixed, and use the interior unconstrained tie-line phase-fraction fit. With $q={\mathbf{d}}^{T}W\mathbf{d}$, residual $\mathbf{r}=\mathbf{y}-\mathbf{a}-{\lambda }_0\mathbf{d}$, and row vectors $\mathbf{g}={\mathbf{d}}^{T}W/q$, $\mathbf{h}={\mathbf{r}}^{T}W/q$, its differential isThe full derivation is in tie-line phase-fraction sensitivity. The normal equation makes $\mathbf{r}$ orthogonal to $\mathbf{d}$ in the chosen metric; it does not make the residual zero.

For an original example, take $\mathbf{a}=(0.60,0.10)$, $\mathbf{b}=(0.20,0.50)$, $\mathbf{y}=(0.35,0.40)$ and $W=I$. The third mole fraction is fixed by closure. These choices giveIncreasing only the first coordinate of $\mathbf{b}$ by $0.001$ predicts a phase fraction change $0.000859375+0.000078125=0.0009375$. Omitting the correction misses the second contribution even though the fit satisfies its normal equation. A centered finite-difference recalculation as the perturbation shrinks tests this derivative directly.

This is a local sensitivity, not an uncertainty until combined with the joint input covariance matrix.^[ref] The formula assumes distinct endpoints and an interior fit with fixed $W$. A changing metric requires sensitivity of a tie-line fit to its weight matrix; a constrained boundary requires a separate local analysis. A nonzero residual can reflect composition noise or model mismatch and does not independently establish liquid-liquid equilibrium.

Standard-state activity conversion changes the numerical thermodynamic activity assigned to a fixed physical state when its reference chemical potential changes. The observable equilibrium and physical chemical potential stay the same. The activity and its standard chemical potential must therefore be converted together.

At fixed temperature $T$, writeIf ${\delta }_i={\tilde{\mu }}_i^{\circ }-{\mu }_i^{\circ }$, subtraction givesHere $R$ is the molar gas constant. This follows from the IUPAC definition of relative activity and its specified standard state.^[ref] Changing just the activity in the first equation would instead change the calculated physical chemical potential.

For an invented conversion ${\tilde{a}}_i=10a_i$, the compensating reference shift is ${\delta }_i=-RT\ln 10$. An old activity $0.03$ becomes $0.30$; the two expressions for ${\mu }_i$ are identical. An activity of one consequently identifies a reference condition under the chosen convention, not an invariant amount of material.

An activity conversion of an equilibrium constant follows the same logic. For the reaction A $\rightleftharpoons$ B, take $K=a_B/a_A$ at equilibrium and convert by ${\tilde{a}}_A=10a_A$, ${\tilde{a}}_B=2a_B$. Then $\tilde{K}=K/5$. If $K=5$, the converted value is $1$, whilebecause originally ${\Delta }_r{G}^{\circ }=-RT\ln 5$. For any physical composition, the corresponding reaction quotient also changes by $1/5$, so ${\Delta }_r{G}^{\circ }+RT\ln Q$ remains unchanged. These are reference changes, not altered reaction chemistry.

The same requirement governs activity equality for coexisting liquids: compare activities only after identifying their standard states, composition scales and reference conventions. Equal numerical values expressed under incompatible conventions do not, by themselves, establish liquid-liquid equilibrium.

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.^[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 $\lambda$, defineExact 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 givesThis 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 $\hat{\lambda }=0.30$ and zero residual. Changing the overall composition to $\mathbf{z}=(0.50,0.29,0.21)$ preserves closure but givesNo 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.

Tie-line phase-fraction sensitivity measures how an inferred phase fraction changes when the overall composition or the measured liquid-liquid tie line endpoints change. A sensitivity coefficient is a local partial derivative of the inference formula, rather than an uncertainty by itself. It becomes an uncertainty contribution only after combination with the input covariance matrix.^[ref]

For one composition coordinate of a binary mixture, write the lever rule aswhere $a,b$ are endpoint mole fractions, $z$ is the overall mole fraction and $\lambda$ is the mole-based amount of phase $\beta$. Differentiating the mass balance givesThus a measurement error in the overall composition and an equal measurement error in the phase-$\alpha$ endpoint have different effects. The conditioning of tie-line phase-fraction inference deteriorates as the endpoint separation becomes small: the same composition measurement error then produces a larger change in the inferred phase fraction.

For an original example, take $a=0.20$, $b=0.80$, $z=0.44$, giving $\lambda =0.40$. Increasing only $a$ by $0.003$ gives a first-order change $\delta \lambda =-0.003$. Direct substitution gives $\lambda =0.237/0.597\approx 0.396985$, a change of $-0.003015$. For the closer endpoints $a=0.20$, $b=0.26$ and balanced composition $z=0.224$, the local sensitivity to the same endpoint error is ten times larger. This is a property of the inference geometry; it does not establish whether either hypothetical pair is in liquid-liquid equilibrium.

The multidimensional tie-line phase-fraction fit has an additional detail away from an exactly balanced composition. On independent mole fraction coordinates, hold a symmetric positive-definite matrix $W$ fixed and definewith distinct endpoints. Define row vectors $\mathbf{g}={\mathbf{d}}^{T}W/q$ and $\mathbf{h}={\mathbf{r}}^{T}W/q$. Differentiating the generalized least squares normal equation ${\mathbf{d}}^{T}W\mathbf{r}=0$ givesThe residual correction in tie-line sensitivity vanishes when $\mathbf{r}=0$. Otherwise, moving an endpoint also rotates the fitted line relative to the nonzero ternary tie-line consistency residual. Omitting that contribution is an additional approximation. If $W$ itself changes with the measured inputs, its derivative must also enter the calculation.

These formulas describe the unconstrained fit locally. Imposing $0\le \hat{\lambda }\le 1$ introduces boundary effects, and coincident endpoints make this inference undefined. A practical sensitivity check recomputes the fit after small positive and negative perturbations of one measured coordinate, keeping composition closure and the stated weight matrix. Agreement as the perturbation shrinks tests the derivative calculation; it does not validate the underlying equilibrium model.