= Mixture measurements and the Gibbs-Duhem constraint
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A concrete motivating experiment is the <acetone>–<chloroform> <mixture> in Truman State University's Liquid-Vapor Equilibrium of a Binary System, Procedure and Figure 1. A two-neck flask feeds a <distillation> head with a water-cooled <laboratory condenser>, condensate sampling arm and <temperature> probe. Starting with one pure <liquid>, the procedure adds the other component successively and waits for stable <temperature> with returning condensate. Small paired samples from the flask and condensate are read on an <Abbé refractometer>; a tabulated <refractive index>–composition relation gives the two compositions. Controls include three-point <thermometer calibration>, corrected atmospheric <pressure> readings, and agreement of the initial pure-<liquid> and condensate <refractive indices> with the reference value. Small withdrawals and little retained condensate reduce disturbance of composition. Sample <evaporation> can bias the readout. The inferred paired <mole fractions>, <temperature> and <pressure> describe <vapor-liquid equilibrium>, subject to the conversion from <mass fraction> to <mole fraction> and to representative sampling. This teaching protocol reports no measured results of a particular student run.<Liquid-Vapor Equilibrium of a Binary System by James M. McCormick>[Liquid-Vapor Equilibrium of a Binary System, adapted by J. M. McCormick; Procedure, Figure 1, Tables 1–2, pure-solvent checks and repeated additions]{ref}
How does such a measurement expose <activity coefficients>? Let $x_i$ and $y_i$ be the <mole fractions> in the <liquid> and <gas> <thermodynamic phases>, respectively, and let $p_i^*(T)$ be the pure-component <vapor pressure>. Assume the <gas> <thermodynamic phase> is an <ideal gas> <mixture> and neglect the change of the pure-<liquid> reference <chemical potential> between its saturation <pressure> and the experimental <pressure>. Equality of each component's <chemical potential> between the two <thermodynamic phases> then gives
$$
RT\ln(x_i\gamma_i)=RT\ln\frac{y_i p}{p_i^*(T)},
\qquad
\gamma_i\simeq\frac{y_i p}{x_i p_i^*(T)}.
$$
This turns the paired composition readings into estimates of the <activity coefficients>. The approximation needs refinement when the <gas> departs appreciably from <ideal gas> behavior or the neglected <liquid> <pressure> effect matters.
The apparatus measures a boiling-composition series in which <temperature> changes as the <mixture> changes. Its readings therefore cannot be inserted directly into the fixed-<temperature>, fixed-<pressure> <activity coefficient> constraint. They motivate the more general task: test whether the measured mixture properties can arise from a common <Gibbs free energy> while accounting for changing state variables. A failure can reveal incompatible property fits or experimental bias; a pass supplies a consistency check, not proof that every reading is correct.
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