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A binary mixture is a mixture with two specified chemical components. Water mixed with ethanol is a familiar example. “Binary” counts the components used to describe composition, rather than the number of thermodynamic phases: two components can form one homogeneous mixture or coexist in distinct thermodynamic phases.^[ref]

For two nonreacting components with amounts of substance $n_A,n_B$, the amount fractions satisfyOnly one composition coordinate is independent. In an original preparation using1mole of ethanol and3moles of water, $x_{\mathrm{ethanol}}=1/4$. Using rounded molar masses46.07and18.015grams per mole gives the ethanol mass fractionThus25percent by amount of substance and about46percent by mass describe the same preparation. A composition percentage without its basis is ambiguous. The calculation uses component masses, without assuming additive liquid volume.

Hexane and water illustrate why component count and phase count differ. Their limited mutual solubility permits two liquid thermodynamic phases, each containing some of the other component. A primary compiled experiment equilibrated the liquids by overnight shaking, sampled the aqueous portion with a microsyringe, and measured dissolved hydrocarbon by gas chromatography with flame-ionization detection.^[ref] The liquid immiscibility article develops a lever rule calculation distinguishing each phase's composition from the overall composition.

A binary mixture is not automatically an ideal solution. Interactions between unlike molecules can change activities, mixing enthalpy and phase stability. Stating temperature, pressure and the composition basis is therefore part of specifying the system, rather than optional experimental detail.

C. L. De Ligny is listed with H. D. Nelson as an author of 1968 measurements of hydrocarbon solubility in water. The IUPAC-NIST record for the hexane/water system names this publication as its original experimental source and records temperature-dependent measurements. It does not identify which coauthor performed each laboratory operation.^[ref]

This work is relevant to the quantitative meaning of liquid immiscibility. A bulk system can contain separate liquid regions while its aqueous phase still contains a small, measurable amount of hydrocarbon. Equilibrium composition and the total quantity recovered from an aliquot answer different questions.

For an original calculation, suppose a homogeneous solution has hydrocarbon mole fraction $x=3\times 10^{-6}$ and total amount of substance $n=1\mathrm{mol}$. It contains $xn=3\mu \mathrm{mol}$ hydrocarbon. A representative aliquot containing $0.1\mathrm{mol}$ of the solution contains $0.3\mu \mathrm{mol}$ hydrocarbon while retaining the same mole fraction. These chosen values do not reconstruct the NIST table.

The database describes microsyringe sampling and gas chromatography with flame-ionization detection, connecting the small dissolved amount to a measurement procedure.^[ref] Sample size, composition and instrument response must be distinguished before comparing solubility measurements.

H. D. Nelson is credited with C. L. De Ligny as an author of experimental hydrocarbon solubility measurements published in 1968. The NIST hexane/water record identifies their joint work as its original measurement source. The verified byline supports this contribution without supplying a fuller identity or an individual division of laboratory tasks.^[ref]

The recorded method equilibrates the liquid system, samples the aqueous solution with a microsyringe, and uses gas chromatography with flame-ionization detection. It addresses a quantitative question: how much hexane is dissolved in water, rather than simply whether two visible bulk liquid regions remain.^[ref]

An invented instrument calibration example illustrates why a detector reading is not itself a concentration. Suppose peak area obeys $A=A_0+sc$, with blank area $A_0=20$, response slope $s=10$ area units per $\mu \mathrm{mol}{\mathrm{L}}^{-1}$, and sample area $A=200$. Then $c=(200-20)/10=18\mu \mathrm{mol}{\mathrm{L}}^{-1}$. These are teaching values, not measurements or calibration constants from Nelson and De Ligny's paper. The example separates equilibration, sampling and detector response, all of which matter when interpreting a small dissolved signal.

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]