Interpreting Thermodynamic Plots — Interpret thermodynamic plots (G vs extent, μ vs composition, phase diagrams)

The graphical representation of thermodynamic relationships evolved over more than a century, driven by the need to visualize the abstract quantities that govern chemical and physical transformations. Early thermodynamicists recognized that equations alone could not convey the rich interplay among temperature, pressure, composition, and free energy. Thermodynamic plots became indispensable tools for predicting equilibrium states, identifying phase boundaries, and guiding the design of industrial processes from metallurgy to pharmaceutical crystallization. Understanding these diagrams is not merely an academic exercise—it is the skill that connects theoretical thermodynamics to the practical decisions made in every materials science and chemical engineering laboratory.

The central question that thermodynamic plots address is deceptively simple: Given a set of conditions, which state—or mixture of states—has the lowest Gibbs energy? Whether the answer involves locating the minimum on a G-versus-extent-of-reaction curve, finding the common tangent on a G-versus-composition curve, or reading coexistence regions from a T–x phase diagram, the underlying logic is always the same: nature minimizes Gibbs energy at constant T and P. This lesson develops your ability to extract that information fluently from the three most important classes of thermodynamic plots.

Before diving into specific plot types, it is essential to internalize the foundational principles that unify every thermodynamic diagram you will encounter. The Gibbs energy G = H − TS serves as the master potential at constant temperature and pressure: any spontaneous process decreases G, and equilibrium corresponds to a global or constrained minimum. The chemical potential μ_i = (∂G/∂n_i)_{T,P,$n_j$} quantifies the marginal Gibbs-energy cost of adding one mole of species i while holding everything else constant. These two quantities—G and μ—are the axes and slopes that populate every plot discussed in this lesson.

The most direct way to locate the equilibrium position of a chemical reaction at constant T and P is to plot the total Gibbs energy of the system as a function of the extent of reaction ξ. Because reactant and product mole numbers change linearly with ξ, and because the mixing entropy introduces a logarithmic curvature, the resulting G(ξ) curve is a smooth concave-upward function with a single minimum. The position of this minimum tells you how far the reaction proceeds before reaching equilibrium, and the slope of the curve at any point equals the reaction Gibbs energy ΔG(ξ). The following diagram illustrates a generic gas-phase reaction A ⇌ B at constant temperature and pressure.

Several critical features of the G-versus-ξ plot deserve emphasis. First, the endpoints are not the standard Gibbs energies of formation of A and B; they are the total Gibbs energies of a system consisting entirely of pure A or pure B at the specified T and P. Second, the curvature of the plot arises primarily from the entropy of mixing between reactants and products: even if ΔG° is very large and negative, the minimum never reaches ξ = 1 because the −TΔS_mix term always creates a well. Third, the relationship between the equilibrium extent and the equilibrium constant is given by ΔG° = −RT ln K, which fixes the depth and position of the minimum relative to the endpoints.

G as a Function of Extent of Reaction

For an ideal-gas reaction A ⇌ B starting with one mole of A at constant T and total pressure P, the total Gibbs energy is a function of ξ (where n_A = 1 − ξ and n_B = ξ). The derivative of G with respect to ξ yields the reaction Gibbs energy, and setting this derivative to zero locates equilibrium.

Chemical Potential and the G–x Curve for Binary Solutions

For a binary mixture of components 1 and 2 at fixed T and P, the molar Gibbs energy of mixing can be expressed as a function of the mole fraction x₂. For an ideal solution, the Gibbs energy of mixing arises entirely from the entropy of mixing, and the curve is concave upward everywhere. For a regular solution with a positive interaction parameter, the curve may develop inflection points and the common tangent construction becomes necessary to identify two-phase regions.

A binary phase diagram at constant pressure plots temperature on the vertical axis against composition (mole fraction x_B) on the horizontal axis. It is the physical-space projection of the G-versus-x information computed at many temperatures. At each temperature, the common tangent construction on the G_m-vs-x curve determines which phase(s) are stable; plotting these boundaries as a function of T generates the familiar lens-shaped two-phase regions. Three essential skills for reading binary phase diagrams are identifying the number and nature of phases present, applying the lever rule to determine phase fractions, and recognizing invariant reactions such as eutectics and peritectics.

When interpreting a phase diagram, begin by identifying the single-phase regions (labeled L, α, β, etc.); these are areas where F ≥ 1. Next, locate the two-phase regions (L + α, L + β, α + β); within these regions, the compositions of the coexisting phases are given by the endpoints of a horizontal tie line at the temperature of interest. Finally, the lever rule relates the overall composition to the fractions of each phase: the fraction of a given phase is the length of the opposite arm of the tie line divided by the total tie-line length. This mechanical analogy—imagine the tie line as a seesaw balanced at the overall composition—makes the rule intuitive.

Consider a binary system A–B that exhibits a simple eutectic phase diagram. At T = 900 K, the liquidus intersects the tie line at x_L = 0.65 and the solidus at x_α = 0.20. A sample with overall composition x₀ = 0.40 is quenched at 900 K. Determine the phases present, their compositions, and their mole fractions.

Each thermodynamic plot type has particular strengths and limitations. The G-versus-ξ plot is unparalleled for visualizing reaction equilibria but is restricted to a single reaction coordinate. The G-versus-x plot for binary solutions reveals miscibility gaps and spinodal decomposition boundaries but becomes unwieldy for ternary systems, where Gibbs triangles are needed. Phase diagrams compress a vast amount of free-energy information into an easily readable map, but they hide the underlying Gibbs-energy surfaces and can be misleading if the reader forgets that the diagram applies only at the stated pressure.

The plots discussed in this lesson are the starting point for several advanced topics in physical chemistry and materials science. The G-vs-x framework naturally extends to spinodal decomposition, where the second derivative d²G/dx² becomes negative and the system is unstable against infinitesimal composition fluctuations. In the CALPHAD method, parameterized Gibbs-energy models for each phase are optimized against experimental data, and the common tangent construction is performed computationally to generate phase diagrams for multi-component alloys with dozens of elements. At still higher levels of theory, the Gibbs-energy surface is connected to statistical mechanical partition functions, and molecular simulation techniques (Monte Carlo, molecular dynamics) are used to compute G(T, P, x) from first principles.

As you move into statistical thermodynamics and materials modeling courses, you will encounter these same plots redrawn with computationally derived free energies. The interpretive skills you build now—reading slopes, identifying minima, applying the lever rule—remain exactly the same regardless of how the underlying G function was obtained.