= Liquid-liquid equilibrium
{title2=LLE}
{title2=$\mu_i^\alpha=\mu_i^\beta$}
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,
$$
\boxed{\mu_i^\alpha(T,p,\mathbf x^\alpha)=\mu_i^\beta(T,p,\mathbf x^\beta)\quad\text{for every exchanging component }i.}
$$
To see why, transfer an amount $dn_i$ from phase $\alpha$ to phase $\beta$ while holding the total amount of each component fixed. The change in <Gibbs free energy> is $dG=(\mu_i^\beta-\mu_i^\alpha)dn_i$. 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.<Statistical Physics by David Tong/4 Classical Thermodynamics>[§4.4, Gibbs Free Energy, equations 4.173–4.175]{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 $G=(n_A+n_B)g(n_B/(n_A+n_B))$ gives
$$
\mu_A=g-xg',\qquad \mu_B=g+(1-x)g'.
$$
Equality of both component <chemical potentials> between two compositions becomes the <common-tangent construction for phase coexistence>:
$$
g'(x_\alpha)=g'(x_\beta)=\frac{g(x_\beta)-g(x_\alpha)}{x_\beta-x_\alpha}.
$$
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 write
$$
h(x)=\frac{g_{\rm mix}(x)}{RT}=x\ln x+(1-x)\ln(1-x)+\chi x(1-x).
$$
The 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 $h'(x)=\ln(x/(1-x))+3(1-2x)=0$ give the <symmetric regular-solution binodal>:
$$
x_\alpha\approx0.07072018,\qquad x_\beta\approx0.92927982.
$$
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 $\beta$:
$$
z=(1-\lambda)x_\alpha+\lambda x_\beta,\qquad
\lambda=\frac{z-x_\alpha}{x_\beta-x_\alpha}\approx0.26705171.
$$
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 $h''(x)=1/[x(1-x)]-2\chi$, so at $\chi=3$ the two local-stability limits are $(1\mp\sqrt{1/3})/2\approx0.211325,0.788675$. 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 $h''=0$.
\Image[/regular-solution-liquid-coexistence.png]
{title=Original regular-solution model: the supporting tangent gives liquid coexistence, while the spinodal marks loss of local homogeneous stability.}
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.<Hexane with water in the IUPAC-NIST Solubility Database>[Record37_269, Experimental Data and Method/Apparatus/Procedure; no ambiguous table values are used here]{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.<Reference Materials for Phase Equilibrium Studies. 1. Liquid-Liquid Equilibria by Ala Bazyleva and others>[Abstract and Introduction: seven binary reference systems, accepted data and uncertainty estimates]{ref}
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