Deriving Gas-Phase Equilibrium Constants from Gibbs Free Energy (MIT OpenCourseWare)

Overview

MIT OpenCourseWare presents a thermodynamics focused lecture that derives the gas-phase equilibrium condition for A + B → 2C at fixed temperature and pressure. The talk uses Gibbs free energy and its differential with respect to temperature, pressure and mole numbers to show that equilibrium corresponds to an extremum of G.

Key takeaways

• Equilibrium is governed by chemical potentials and a single reaction extent in a three-component system.
• By choosing a product as the independent variable, all species’ mole changes are expressed through a single univariate variable.
• ΔG° and the equilibrium constant KP connect thermodynamics to observable gas compositions through partial pressures and mole fractions.
• Dalton’s law links partial pressures to mole fractions when using a standard reference pressure.

Introduction

In this MIT OCW lecture, the gas-phase reaction A + B → 2C is analyzed at fixed T and P. The instructor emphasizes that the condition for equilibrium is that Gibbs energy G is at an extremum with respect to the independent variables T, P and the mole numbers, and for fixed T and P the relevant differential reduces to the sum over μi dNi.

System setup and degrees of freedom

The system has three components A, B, C and three mole numbers NA, NB, NC. The reaction balance imposes two constraints, leaving one degree of freedom. The extent of reaction ξ is introduced so dNi can be linked to dξ. The instructor then chooses the product C as the independent variable and expresses the other dn’s in terms of dNC: dNA = νA/νC dNC, dNB = νB/νC dNC, dNC = dNC. For A + B → 2C, νA = νB = 1 and νC = 2, giving dNA = (1/2)dNC and dNB = (1/2)dNC.

Then the differential for G is written as DG = ∑i μi dNi, and substituting the expressions for dNi in terms of dNC yields an expression DG in terms of dNC. The equilibrium condition requires the coefficient of dNC to vanish, i.e., (dG/dNC) = 0, which is the univariate optimization condition that defines equilibrium in this constrained system. The speaker stresses that even at equilibrium NC can fluctuate (it is unconstrained), reflecting the fact that reactions continue to occur but Gibbs energy is minimized with respect to the constraint.

"For one reaction, univariant, it could be called we have only one degree of freedom."

ΔG° and the equilibrium constant

With expressions for the chemical potentials μi for an ideal gas mixture, the differential is substituted and reorganized. A standard Gibbs energy change ΔG° is defined as the weighted sum of reference potentials: ΔG° ≡ ∑i νi μi° (for the reaction A + B → 2C). This quantity corresponds to the free energy change when all components are in their standard states. The algebra leads to a compact definition of the equilibrium constant KP as KP = ∏i (pi/P0)^{νi}, i.e., a product of partial pressures normalized by the standard pressure raised to the appropriate stoichiometric powers.

"Define delta G naught equals this thing in parentheses, equals the sum over I of the NU of I times the reference potentials."

Using Dalton's law and fixing the total pressure to the reference pressure, the partial pressures are related to mole fractions via pi/P0 = xi. This simplifies KP to the familiar form for A + B → 2C: KP = xC^2/(xA xB). The derivation then connects KP to the standard free energy change through KP = exp(-ΔG°/(RT)), a relation that underpins how thermodynamic data informs equilibrium compositions.

"KP equals exp(-ΔG°, naught over RT)."

Three-component caveat and Dalton's rule

The lecturer notes that when three or more components are present, there is no unique equilibrium composition. For a tri-component system with KP = xC^2/(xA xB), the variables xi are constrained by xA + xB + xC = 1, leaving one degree of freedom. This implies multiple compositions can satisfy equilibrium, and one can parameterize by fixing xC and solving for xA and xB accordingly. This mirrors the general thermodynamics behavior where a single reaction in a multi-component mixture defines a family of equilibrium states rather than a single point.

"For three or more components, there is no unique equilibrium condition."

Temperature dependence and the Van’t Hoff equation

The derivative of the natural log of KP with respect to temperature at fixed pressure is derived from fundamental thermodynamics. The result, known as the Van’t Hoff equation, links the slope d(ln KP)/dT to standard reaction enthalpy ΔH° and, in the most general form, to heat capacity changes ΔCp. In the simple constant-ΔH° case, d(ln KP)/dT = ΔH°/(R T^2). The sign of ΔH° determines how KP changes with temperature: for endothermic reactions (ΔH° > 0), KP increases with T and the equilibrium shifts toward products as T rises; for exothermic reactions (ΔH° < 0), KP decreases with T and the equilibrium shifts toward reactants. The instructor emphasizes that the general form includes ΔCp and is widely used to analyze temperature effects in reacting gas systems.

"This has a name, this is called the Van't Hoff equation."

Near the end, the lecturer ties these ideas back to standard data, remarks on how standard-state data and CP differences control the temperature dependence, and briefly notes how standard data are tabulated in databases for practical reactor design and analysis.