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