Describe methods for calculating chemical equilibrium compositions in complex reacting systems.

Two main approaches: stoichiometric method - write reactions explicitly, express compositions in terms of extent of reaction, solve K expressions simultaneously. Becomes intractable for many reactions/species. Non-stoichiometric (Gibbs minimization) - minimize total Gibbs energy subject to element balance constraints: min(sum n_i*mu_i) subject to atom balances. Solved using Lagrange multipliers; no need to specify reactions. Algorithms: RAND method (successive substitution), Newton-Raphson with line search. Handling: condensed phases included with appropriate activity, ideal gas or EOS for vapor. Process simulators use element-balance approach (RGibbs in Aspen). Provides compositions, phase distribution, and equilibrium temperature/pressure.