Chemistry – Nihal Uppugunduri's Website

We balance chemical equations by making sure the number of atoms of each element is the same on both sides. However, with this as the only criterion, it’s actually not immediate that a particular balancing we find would have to be the only one (up to equivalence.) In other words, we ask whether the set of coefficients we could balance with is one-dimensional (since scaling results in equivalency: $O_{2} + 2H_{2} \rightarrow 2H_{2}O$ and $2O_{2} + 4H_{2} \rightarrow 4H_{2}O$ are equivalent.) Does this require actual chemistry, or is it a more general mathematical fact (under maybe generally mild assumptions?) Can we show this to still be true for all the chemical equations we’d ever actually balance?