Clausius Clapeyron Equation in Phase Equilibria: Gibbs Phase Rule and Vapor Pressure Applications | MIT OpenCourseWare

In this MIT OpenCourseWare lecture, the instructor introduces the Clausius Clapeyron relation in the context of a two-phase coexistence on a phase diagram, emphasizing the condition that mu_alpha equals mu_beta along the line and deriving the slope DP/DT from transformation quantities. The talk also covers practical approximations for vaporization problems and the Gibbs phase rule for unary systems.

• Coexistence line requires equal chemical potentials and one degree of freedom along the ridge
• Clausius Clapeyron equation DP/DT = ΔS/ΔV = ΔH/(T ΔV)
• Three simplifying assumptions for vaporization: ΔH constant, gas volume dominates, and ideal gas behavior
• Gibbs phase rule yields the degrees of freedom in unary phase diagrams

Overview of Clausius Clapeyron in Phase Equilibria

The MIT OCW lecture frames the Clausius Clapeyron equation within a generic phase diagram, using colored example phases alpha and beta and a coexistence line. The core idea is that remaining on the line requires the differential chemical potentials to stay equal, which enforces a single degree of freedom along a curved ridge that represents two-phase equilibrium.

"on the coexistence line, mu of alpha equals mu of beta" - MIT OpenCourseWare

Deriving the Slope of the Coexistence Line

The instructor derives the slope DP/DT by equating the differentials of the molar Gibbs free energies for each phase, Dmu_alpha = Dmu_beta. Using the thermodynamic identity Dmu = -S dT + V dP, this yields (S_alpha - S_beta) dT = (V_alpha - V_beta) dP, or dP/dT = ΔS/ΔV. Recognizing ΔS = ΔH/T, we obtain the famous Clausius Clapeyron form dP/dT = ΔH/(T ΔV). This is framed as a differential equation for the phase boundary with transformation quantities ΔS and ΔV.

"this is a transformation quantity" - MIT OpenCourseWare

Vapor Pressure and Isothermal Transformations

The discussion then specializes to vapor pressure problems, noting that the equation provides a slope for the coexistence line between vapor and a condensed phase, with the sign and magnitude determined by how ΔS and ΔV change with T and P. The lecturer highlights that the same Clausius Clapeyron framework underpins many phase-transformation databases and problems, including the isothermal equilibrium condition where ΔG = 0.

Approximations for Vaporization Problems

To make the problem tractable, three standard approximations are introduced for vaporization over a limited temperature range: ΔH is treated as a constant, the gas molar volume is much larger than that of the condensed phase so ΔV ≈ V_gas, and the gas is treated as ideal. With these, the CC equation reduces to a separable differential equation in P and T, leading to a logP vs T relationship that can be integrated to yield practical vaporization data.

Gibbs Phase Rule and Unary Diagrams

The lecture then transitions to Gibbs phase rule, illustrating how the number of phases coexisting at equilibrium constrains the degrees of freedom in a unary system. For a single phase, two degrees of freedom exist (TP plane), for a two-phase region there is one degree of freedom along the coexistence line, and for a three-phase region the degrees of freedom collapse to zero, pinpointing the triple point. The idea is tied to the algebraic structure of phase equilibrium and helps explain why multi-component systems can exhibit four phases while unary diagrams cannot.

Closing Remarks and Preview

The instructor notes that the saturation vapor pressure concept will be revisited in a detailed extended example and emphasizes the value of phase-diagram thermodynamics for modeling and computation, including the data demands for CP, V, α, and κ across phases and temperatures. The session closes with a reminder to consult the textbook for deeper derivations and the next lecture's focus on Gibbs phase rule in multi-component systems.

"Gibbs phase rule is the answer to the following question" - MIT OpenCourseWare