Vapor-liquid coexistence, Gibbs-Duhem

Figure 3. Vapor-liquid coexistence line for the Lennard-Jones model. Solid line is presently the best determination of the phase boundary [50] triangles [86] and circles [59] are Gibbs ensemble data. Dashed line is obtained from Gibbs-Duhem integration beginning with the low-temperature (rightmost) Gibbs ensemble datum. Error bars on the true line are the stability analysis prediction of how the error in the initial datum propagates through the integration series.

Mathematical Consistency. Consistency requirements based on the property of exact differentials can be applied to smooth and extrapolate experimental data (2,3). An example is the use of the Gibbs-Duhem coexistence equation to estimate vapor mole fractions from total pressure versus liquid mole fraction data for a binary mixture. 

The n= 12 soft sphere model is the high-temperature limit of the 12-6 Lennard-Jones (LJ) potential. Agrawal and Kofke [182] used this limit as the starting point for another Gibbs-Duhem integration, which proceeded to lower temperatures until reaching the solid-liquid-vapor triple point. The complete solid-fluid coexistence line, from infinite temperature to the triple point, can be conveniently represented by the empirical formula [182]... 

---