Uniqueness of Chemical Equation Balancing

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?

Here’s the formulation of the problem. We’re talking about finitely many molecules, so finitely many elements. Let’s say all the elements are $e_{1},\ldots,e_{n}$ , then we can express every molecule in the equation by $e_{1}^{b_{1}}\ldots e_{n}^{b_{n}}$ , where $b_{i}$ is the number of occurrences of $e_{i}$ in the molecule (so basically taking what is normally a subscript in chemistry and making it a superscript here, since we already have the $i$ subscript), and allowing $b_{i}$ to be 0. Let’s say there are $r$ molecules on the left and $s$ on the right, so they all look like

$\displaystyle e_{1}^{b_{11}}\ldots e_{n}^{b_{1n}} + \ldots + e_{1}^{b_{r1}}\ldots e_{n}^{b_{rn}}$

and similarly on the right. So with coefficients, basically the problem of balancing is: find coefficients $k_{1},\ldots,k_{r},l_{1},\ldots,l_{s}$ such that

$\displaystyle k_{1}e_{1}^{b_{11}}\ldots e_{n}^{b_{1n}} + \ldots + k_{r}e_{1}^{b_{r1}}\ldots e_{n}^{b_{rn}} = l_{1}e_{1}^{d_{11}}\ldots e_{n}^{d_{1n}} + \ldots + l_{s}e_{1}^{d_{s1}}\ldots e_{n}^{d_{sn}},$

given $b_{ij},d_{ij}$ . Our assertion of uniqueness is that the set of solution vectors $\left( k_{1},\ldots,k_{r},l_{1},\ldots,l_{s} \right)$ is one-dimensional.

This reduces to $n$ equations, one for each element (conserving the number of appearances of that element on either side.) But we’re looking for $r + s$ unknowns. If $r + s > n + 1$ , then there’s no possible way that the solution set could be one-dimensional, so this isn’t true without some chemistry. Or … since we know the coefficients must be positive, does that change things here? Can we come up with a simply verifiable condition that guarantees uniqueness, one that looks like it’d be true for pretty much any chemical equation we’d actually want to balance?

Our $n$ equations look like, for each $i \in \lbrack 1,n\rbrack$ ,

$\displaystyle b_{1i}k_{1} + \ldots + b_{ri}k_{r} = d_{1i}l_{1} + \ldots + d_{si}l_{s},$

$\displaystyle b_{1i}k_{1} + \ldots + b_{ri}k_{r} - d_{1i}l_{1} - \ldots - d_{si}l_{s} = 0.$

To investigate this further, let’s say that $r + s \leq n + 1$ . Let’s say that this happens to be true for most of the equations we deal with.

EDIT: This looks like something people have actually published research papers on fairly recently. Actually, a link I had visited showed that there are “common” equations which cannot be uniquely balanced with just the equations above, and that additional chemistry is needed for such equations. This is something chemists have thought about and there are results here if we study this enough. An interesting topic for sure!

First written June 1, 2022. I had originally included a link to one of my sources, but that link doesn’t work anymore.

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