Balancing chemical equations: form the perspective of Hilbert basis

TL;DR

This paper reinterprets chemical equation balancing through the lens of Hilbert bases, revealing a positive affine monoid structure that allows for a systematic and canonical description of all possible reactions.

Contribution

It introduces a novel mathematical framework using Hilbert basis theory to uniquely characterize elementary reactions in chemical balancing.

Findings

Reveals the positive affine monoid structure underlying chemical equation solutions.

Provides a systematic method to identify all possible reactions.

Defines a canonical set of independent elementary reactions called Hilbert-basis reactions.

Abstract

The balancing of chemical equations is a basic problem in chemistry. A commonly employed method is to convert the task to a linear algebra problem, and then solve the null space of the constructed formula matrix. However, in this method, the directly obtained solution may be invalid, and there is no canonical choice of independent basis reactions. Here, we show that these drawbacks originate from the fact that the fundamental structure of solutions here is not a linear space but a positive affine monoid. This new understanding enables a systematic approach and a complete description of all possible reactions by a unique set of independent elementary reactions, called Hilbert-basis reactions. By clarifying its underlying mathematical structure, our work offers a new perspective on this old problem of balancing chemical equations.