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Chemical equilibrium at constant T and P is determined by minimizing Gibbs free energy; the equilibrium constant K_p relates partial pressures of reactants and products. K_p depends on temperature via d(ln K)/dT = ΔH_rxn/(RT²). Real combustion products contain incomplete combustion species (CO, OH, NO) in equilibrium, requiring iterative solution for composition.
From thermodynamic properties and equations of state, you know how to characterize the state of a pure substance or a gas mixture — enthalpy, entropy, Gibbs free energy. Now those tools answer a question about *chemical reactions*: given reactants at temperature T and pressure p, which direction does the reaction go, and where does it stop? The organizing principle is Gibbs free energy minimization: at constant T and p, any spontaneous process decreases G, and the system reaches equilibrium when G is minimized over all possible compositions.
For a reaction aA + bB ⇌ cC + dD, the equilibrium constant K_p is defined as K_p = (p_Cc × p_Dd) / (p_Aa × p_Bb), where each partial pressure is measured relative to a standard reference pressure (1 atm or 1 bar). At the G-minimizing composition, thermodynamics requires ΔG° = −RT ln K_p, where ΔG° = ΔH° − TΔS° is the standard Gibbs free energy of reaction, computable from tabulated enthalpies and entropies of formation. A large K_p (K_p >> 1) means ΔG° << 0 — the reaction strongly favors products at temperature T. A small K_p means reactants are favored. K_p = 1 means neither side is preferred, and the mixture composition is near equal partial pressures.
The temperature dependence of K_p follows the van't Hoff equation: d(ln K_p)/dT = ΔH_rxn / (RT²). Integrating: ln(K_p(T₂) / K_p(T₁)) ≈ −(ΔH_rxn/R)(1/T₂ − 1/T₁), valid when ΔH_rxn is approximately constant over the temperature range. For exothermic reactions (ΔH_rxn < 0), K_p decreases as temperature rises — consistent with Le Chatelier's principle: heating an exothermic reaction shifts equilibrium toward reactants. For endothermic reactions, K_p increases with temperature. In combustion engineering, this matters enormously: high-temperature products have K_p values that force significant dissociation of CO₂ and H₂O back into CO, OH, H, and O.
The practical challenge is that real combustion products are not simply CO₂ and H₂O. At temperatures above roughly 1500 K, minor species — CO, OH, H₂, O, NO — exist in thermodynamic equilibrium at concentrations that cannot be ignored for accurate energy and emissions calculations. Finding the mixture composition requires solving a system of simultaneous equilibrium equations (one K_p expression per independent reaction) coupled with atom-balance constraints (carbon, hydrogen, oxygen, and nitrogen atom counts must match the reactant totals). The system is nonlinear and requires iterative solution: assume a composition, evaluate all K_p expressions, check balances, adjust, and repeat until converged.
An equivalent and often more computationally tractable approach is Gibbs free energy minimization subject to atom-balance constraints, using Lagrange multipliers. Instead of writing K_p equations, you minimize G(T, p, n₁, n₂, …) where n_i are the mole numbers of each species. This approach scales naturally to dozens or hundreds of species — it's the method used by NASA's Chemical Equilibrium with Applications (CEA) code and similar thermochemical solvers. Both approaches yield the same equilibrium composition; the choice is architectural, not conceptual. Understanding K_p and the van't Hoff equation gives you intuition for *why* composition shifts with temperature; Gibbs minimization gives you the machinery to *compute* it in complex realistic mixtures.