The left plot is the temperature–composition (T–x) phase diagram for a binary system of two components, A (naphthalene) and B (biphenyl), that are completely miscible as a liquid but completely immiscible as solids (no solid solubility). The right plot is the cooling curve — temperature versus time — that results when a liquid mixture of a chosen overall composition is cooled at a slow, steady rate.
1. Choose one of the six overall compositions (mole fraction of B, xB) below the plots. The vertical dashed line on the phase diagram marks that composition.
2. Click Play to animate the cooling process. A marker descends the vertical line on the phase diagram while the matching cooling curve is traced out on the right in real time. Click Pause to stop, and Reset to return to the top.
3. You may also click or drag anywhere on the cooling-curve axis to move directly to that point in time; the phase-diagram marker updates to match.
4. Watch for two features on every curve: a change in slope where the curve first meets the liquidus (freezing begins) and a flat plateau at the eutectic temperature (the eutectic halt, where the last liquid solidifies at constant temperature). Compare how the length of the plateau changes as the chosen composition approaches or moves away from the eutectic composition.
System. Components A and B form an ideal liquid solution and are completely immiscible as pure solids (no solid solution forms). This is the simplest type of solid–liquid equilibrium (SLE) phase diagram, a binary eutectic system.
Liquidus curves. Assuming an ideal liquid solution and negligible solid solubility, each liquidus branch is given by the ideal-solubility (Schröder–van Laar / van’t Hoff) equation, obtained by equating the fugacity of pure solid i to the fugacity of i in the liquid solution:
where $x_i^{L}$ is the mole fraction of component $i$ in the liquid in equilibrium with pure solid $i$, $T_{m,i}$ is the normal melting point of pure $i$, and $\Delta H_{\text{fus},i}$ is its molar enthalpy of fusion. This curve runs from the pure-component melting point down to the eutectic point.
Eutectic point. The two liquidus branches intersect at the eutectic composition $x_E$ and eutectic temperature $T_E$, the only point where liquid can be in equilibrium with both solids simultaneously:
Lever rule. For an overall composition $x_0$ cooled into the two-phase (liquid + solid) region, the fraction of the sample that has solidified, $f_s$, follows from a material balance between the liquid (at the liquidus composition $x_L(T)$) and the pure solid that is forming:
Shape of the cooling curve. A steady rate of heat removal, $Q$, is assumed. The slope of $T$ versus time depends on what heat is being removed at each instant:
Four stages result for a composition away from $x_E$:
1. single-phase liquid cools with a constant, relatively steep slope.
2. first solid ($A$ or $B$) forms at the liquidus temperature; the latent heat released slows the cooling
rate, bending the curve.
3. at $T_E$ the remaining liquid (now at the eutectic composition) freezes entirely at constant
temperature — a thermal arrest, or eutectic halt.
4. once all liquid is gone, the solid mixture of A and B cools as a single phase.
If $x_0 = x_E$ exactly, stage 2 vanishes: the liquid cools until it reaches $T_E$ and then freezes completely at constant temperature, exactly as a pure component would — the defining feature of the eutectic composition. The length of the plateau in stage 3 is set by the fraction of the original liquid that remained until $T_E$ was reached, $f_{L,E}$:
Assumptions used in this simulation: ideal liquid solution, no solid–solid solubility, constant heat capacities $C_p^{L}$ and $C_p^{S}$ (averaged over composition) and constant $\Delta H_{\text{fus}}$, and a constant rate of heat removal. Time is reported in arbitrary units set by $Q$.
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.