Phase Diagrams Demystified: Gibbs Phase Rule Across Unary, Binary and Ternary Systems (Ouzo Case Study)

In this MIT OpenCourseWare lecture, phase diagrams for unary, binary, and especially ternary systems are introduced with a real focus on ternaries. The instructor revisits Gibbs phase rule to understand degrees of freedom and demonstrates how phase equilibria are depicted on flat diagrams. The session covers unary iron, a binary eutectic (lead-tin), and then develops the Gibbs triangle for three-component systems, explaining how to read composition and fixed-parameter subspaces. A tangible example is provided with the ouzo effect, showing how a ternary diagram can predict when a single phase becomes a two-phase mixture as composition changes, and how temperature contours add the third dimension to the flat diagram.

• Gibbs phase rule and degrees of freedom across unary, binary and ternary systems
• From binary tying lines to ternary Gibbs triangle for multicomponent diagrams
• Ouzo case study illustrating two-phase regions in a ternary system
• How temperature affects phase boundaries and the interpretation of 3D information on a 2D diagram

Introduction and Gibbs Phase Rule

The lecture opens with a reinforcement of Gibbs phase rule, stating that the number of degrees of freedom in a system is the number of components minus the number of phases plus two. This rule guides how phase equilibria can be represented on flat diagrams. For unary systems, there are two independent intensive variables (pressure and temperature), and two axes on the diagram, allowing the full phase diagram to be captured in a plane. In a unary diagram, two-phase coexistence appears as lines, each line representing a single degree of freedom. The rule also clarifies that while three-phase coexistence in binaries would reduce apparent degrees of freedom on the 2D image, the complete information exists in higher dimensions than what a flat plot can show.

"the number of degrees of freedom in a system is the number of components minus the number of phases plus two." - MIT OpenCourseWare

Unary and Binary Phase Diagrams

Moving to binary systems, the instructor draws a eutectic diagram, for example lead-tin, with temperature as the vertical axis and the second component (tin) on the horizontal axis. In a binary system, there are three independent intensive parameters (temperature, pressure, and composition X2). On a flat image representing a subspace at fixed pressure (usually 1 atm), phase regions are bound by tie lines, which connect coexisting phases and reflect fixed compositions. Gibbs phase rule for two-phase coexistence yields two degrees of freedom, which are picked up by moving along the tie-line. In a three-phase coexistence area, the one apparent degree of freedom on the 2D diagram is a consequence of fixing temperature and pressure; the actual degrees of freedom lie in the out-of-plane dimensions.

"two phase coexistence implies two components minus two phases plus two." - MIT OpenCourseWare

Ternary Phase Diagrams and the Gibbs Triangle

The discussion then generalizes to ternaries, which have four independent intensive parameters: temperature, pressure, and two independent compositions. Since the sum of mole (or weight) fractions equals one, the composition space can be represented on a 2D map by fixing pressure and showing x1 and x2 as coordinates. To treat all three components equally, the composition map is deformed into a Gibbs triangle, where each corner corresponds to a pure component. Lines of fixed x3 emerge from the x3 corner, and lines of constant x2 emerge from its corner as well, forming a network of lines that delineate compositions across the triangle. The three sets of lines cross to define all three-component compositions on the triangular diagram.

"This construction is called the Gibbs triangle." - MIT OpenCourseWare

Phase Fractions and Lever Rule in Ternaries

In a ternary diagram, phase fractions in three-phase regions are analyzed with the lever rule generalized to a triangular phase diagram. For a given overall composition, the three-phase equilibrium can be visualized as balancing a triangular sheet on a pedestal, with the fulcrum placed to level the sheet. The corner allocations determine the phase fractions, and the proportions reflect the lever arms to balance the system. In two-phase regions, tie lines connect coexisting phases with fixed compositions, and the fraction of each phase is given by the lever rule measured along the tie line. In three-phase regions, the compositions are fixed at the temperature and pressure, and the lever-arm approach in the Gibbs triangle yields the phase fractions at equilibrium.

"the lever rule" - MIT OpenCourseWare

Ouzo Case Study: A Real-World Ternary System

To make the discussion tangible, the instructor introduces the ouzo effect, a ternary system consisting of water, ethanol, and anise essential oil. Ethanol acts as a common solvent for both water and the oil, while water and oil are immiscible. The triangle is labeled with water at one corner, ethanol at another, and the essential oil at the third. Lines of constant ethanol to oil ratio emanate from the water corner, and a line representing roughly 50-50 ethanol-water (100 proof) is used to illustrate how moving along a fixed ratio line can take the system from a single-phase solution to a two-phase suspension as water is added. A proposed phase boundary is drawn so that the two coexisting phases align with the observed cloudiness. The two-phase region is a cloudy azeotrope-like region in the ternary diagram, where a water-rich phase coexists with a middle-phase oil-rich region.

"ouzo is a spirit made up of water, ethanol and anise essential oil." - MIT OpenCourseWare

Temperature Effects and 3D Perspective

Finally, the lecture shows how temperature, while not directly drawn on the 2D Gibbs triangle, is crucial in determining phase stability. Contours representing boundary temperatures can be drawn on the ternary diagram to mimic the 3D nature of phase behavior. The demonstration explains that as temperature increases, the two-phase region can shrink and eventually disappear, returning the system to a single-phase region. This provides a way to interpret how temperature modifies phase boundaries and how the flat diagram encodes information about the third dimension.

"as we increase the temperature, our overall system composition moves from being within the two phase region to being back within the one phase region." - MIT OpenCourseWare