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A conformational population is the fraction of otherwise specified molecules occupying a particular conformation. It describes how a conformational ensemble is distributed, rather than a single mean structure. For two equilibrium conformations of the same molecular composition, let $G_A,G_B$ be their effective molar Gibbs free energies at fixed temperature and pressure. Their population ratio isThe effective free energies include each state's internal degeneracy. A higher-free-energy state can remain appreciably populated even though it is not the most common structure.

Populations and exchange rates are different observables. A two-state kinetic model $A\rightleftarrows B$ with rates $k_{AB},k_{BA}$ has equilibrium balance $k_{AB}{p}_A=k_{BA}{p}_B$. Multiplying both rates by ten leaves the populations unchanged but speeds exchange tenfold.

In a primary nuclear magnetic resonance study of cyclophilin A, measurements detected coordinated conformational exchange in the free enzyme and related motions during turnover, illustrating why a static structure is insufficient to describe enzyme dynamics.^[ref] Inferring a catalytic mechanism requires the relevant exchange and reaction rates as well as the conformational populations.

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]

The molar Gibbs free energy is the Gibbs free energy per amount of substance, $G_m=G/n$, with units $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}$. For a homogeneous pure substance, extensivity gives $G=n\mu$, so $G_m$ equals its chemical potential $\mu$.

For a mixture, $G/n$ is an average over the components; the separate chemical potentials are the partial derivatives ${\mu }_i=(\partial G/\partial n_i{)}_{T,p,n_{j\ne i}}$. Equality of a component's chemical potential in two phases is a condition for phase equilibrium. This explains why comparing molar free energies predicts which phase of a pure substance is stable at specified temperature and pressure.

For two bulk phases of one substance at common temperature $T$ and pressure $p$, phase coexistence requires equal chemical potentials:For a single component, the molar Gibbs free energy equals $\mu$, so this also equates the two molar free energies. The equality selects a coexistence curve in the $p$–$T$ plane. ^[ref]

Along the curve, $d\mu =-s_{m}dT+v_{m}dp$ giveswhen $\Delta v_m\ne 0$. This is the Clausius-Clapeyron relation, with differences taken from phase $\alpha$ to phase $\beta$ and $L_m=T\Delta s_m$. A positive liquid-to-vapour entropy jump and molar volume jump give a positive coexistence slope. ^[ref]

For a mixture, equilibrium instead requires equality of each exchangeable component's chemical potential between phases. Their compositions can differ, so their bulk molar Gibbs free energies need not be equal. A single-component equality must not be transferred unchanged to different-composition phases.