Mathematically Independent Reactions

1. Mathematically Independent Reactions • If a set of chemical reactions is taking place in a system, a complete mathematically independent sub-set is one where every reaction that is not included in the complete mathematically independent sub-set can be written as a linear combination of the reactions that are included in the complete mathematically independent sub-set • For a given set of reactions, there may be more than one complete mathematically independent sub-sets • The rank of a matrix is equal to the number of mathematically independent rows it contains • Write the chemical reactions in the form of a reaction matrix • Each column is associated with one of the chemical species • Each row is associated with one of the reactions • The entry at row j and column i is the stoichiometric coefficient, νi,j, of the species associated with column i, in the reaction associated with row j • The rank of the full reaction matrix equals the number of reactions in a complete mathematically independent sub-set • A complete mathematically independent sub-set of the reactions can be found • Gaussian elimination starting with the full reaction matrix • Test matrix built up one reaction at a time, testing each reaction for mathematical independence as it is added to the test matrix