Pivotal condensation and chemical balancing

Abstract

A universal method, called pivotal condensation, for calculating stoichiometric factors of chemical reactions is presented. It is based on our new approach for calculating the basis of the kernel of a matrix over the field of rational numbers. This approach is referred to as kernel pivotal condensation (ker pc) and is presented in detail. It has roughly the same complexity as Gaussian elimination, but can be performed without working with fractions. It is also shown how ker pc can be adapted as a tool to solve inhomogeneous linear systems, invert matrices (this is referred to as inv pc) and determine simultaneously the four subspaces (referred to as 4 pc). Besides, the balancing by inspection method, which is widely used in practice to reduce the size of a linear system arising in chemical balancing, is formulated in a mathematical language. When calculating stoichiometric factors of chemical balancing problems with a non-unique solution the natural question arises how to determine a basis that generates all the solutions over the integers. A method, referred to as smitheration, is introduced that permits to determine such an integer basis from a basis over the rational numbers. If there are few solutions this approach is more efficient than calculating a Smith normal form directly. It is convenient to work over principal ideal domains instead of the ring of integers, so that one can treat balancing problems that depend on one parameter as well.