I would like to better understand the mathematical description of chemical stoichiometry and thermodynamic chemical equilibrium. This problem has many features and I know my description might be too vague.
There are generally two approaches to the problem, the constrained optimization problem (e.g. minimization of gibb's energy with mass balance constraints) or using the $\Delta G_{rx}=-RTln(K_{eq})$ method based upon the equilibrium constant for a set of linearly independent stoichiometric equations. The latter method can only be applied assuming the initial species assumed present are non-negative because having a negative or zero value for a species concentration would violate the functional form of the Keq method.
$N = [\nu_{ij}]$, where $\nu_{ij}$ are the stoichiometric coefficents of the jth species in the ith equation.
$K_{eq,i}=\prod activity_j^{\nu_{ij}}$ for each stoichiometric equation $i$
To address the non-negativity constraint, a sort of model reduction can be applied by generating the power set of the set of possible chemical species under consideration. The equilibrium constant method can be applied to each element of the power set assuming non-negativity. This would require that the element of the power set would correctly represent the actual equilbrium by checking some auxiliary condition such as the activity of the chemical species according to thermodynamic equilibrium of ideal substance.
Suppose the (very simple) chemical equations that can represent chemical reactions (not chemical mechanisms) indicated by the following construct:
General: $(\{Chemical Species\},\{Chemical Elements\})$
Example: $(\{a(c),ab_2(g),b_2(g)\},\{a,b\})$
There now needs to be a choice of ordering for the set of Chemical Species and for the set of Chemical Elements. One choice is to just use the sequence suggested by the listing of elements in the each set (not sure how to mathematically describe this, but i have the feeling it is something related to an indexed set, but dont understand how the ordering arises from a set if a set isn't supposed to have ordering). In most literature, the prior construct is a ordered pair of 'ordered sets' and is called a chemical system.
Chemical Species:
$a(c) = 1*a$
$ab_2(g) = 1*a + 2*b$
$b_2(g) = 2*b$
One can construct each vector representations of the chemical species from the chemical elements once the ordering is chosen.
Chemical Elements: $a=(1,0),b=(0,1)$
And with a chosen ordering of the chemical species the following element abundance matrix can be generated (i.e. a mapping from chemical species quantities to elemental quantities): $ A = \bordermatrix{~ & a(c) & ab_2(g) & b_2(g) \cr a & 1 & 1 & 0\cr b & 0 & 2 & 2\cr} $ $ A = \begin{pmatrix} 1 & 1 & 0\\ 0 & 2 & 2\end{pmatrix} $
The coefficents to the stoichiometric equation can be generated by taking finding the null space (matrix $N$ of the matrix $A$. Once the stoichiometric matrix $N$ is found, it can be put in a 'canonical form' by the operations: $(RREF(N^T))^T$ $
This produces $N=\begin{bmatrix}1 \\ -1 \\ 1\end{bmatrix}$
Where, $AN=0$. (This is essentially the condition of mass balance due to reaction, but the relation doesn't constrain the quantities being balanced to be positive)
The process of finding the stoichiometric equations for a given set of chemical species and set of chemical elements can be constructed in this manner. But the process involved making certain decisions, such as choosing an ordering of the chemical species from the set of allowed species.
If a different ordering is chosen for the chemical species, it is possible to represent the
$A' = \bordermatrix{~ & ab_2(g) & a(c) & b_2(g) \cr a & 1 & 1 & 0\cr b & 2 & 0 & 2\cr} $
$ A' = \begin{pmatrix} 1 & 1 & 0\\ 2 & 0 & 2\end{pmatrix} $
This produces $N'=\begin{bmatrix}1 \\ -1 \\ -1\end{bmatrix}$ for the 'canonical form' of the stoichiometric matrix.
From a thermodynamic viewpoint, this essentially changes what one would consider the 'products' compared to the 'reactants' in a chemical reaction. But for equilibrium this doesn't really matter, because the equilibrium is based upon the quantity of the chemical elements and not the starting quantity of chemical species (the starting quantity of chemical species is only indicative of the starting elemental species quantities). The form of the $\Delta(G_{rx})$ and $K_{eq}$ will change according to whatever ordering is chosen. I often chose specific orderings to force the exponents in the $K_{eq}$ expressions to be $1$ for chemical species listed first in the elemental abundance matrix.
Additionally, there is the problem of assuming that one of the assumed chemical species present before calculating equilibrium might be a limiting reagant (i.e. a solid reacting to form gas and is no longer present). This wouldn't be properly accounted for in the $K_{eq}$ method due to the condition of non-negative. Therefore, one can consider the original construct to be:
$(\{ab_2(g),b_2(g)\},\{a,b\})$
$ A = \begin{pmatrix} 1 & 0\\ 2 & 2\end{pmatrix} $
The stoichiometric matrix $N$ is the zero (trivial) subspace, $A$ is of rank 2. Admittedly, The mass balance would be sufficient constraints to this system without needing to address the functional form of the thermodynamic constraints; most problems are much harder! There is also a thermodynamic condition that would need to be checked (such as activity of a condensed species) to confirm that the assumed chemical species present are actually present or not.
To Conclude,
I have already identified in literature how to generate the stoichiometric equations for a given collection of chemical species and chemical elements. The stoichiometric coefficents are used in thermodynamic theory to calculate equilibrium using certain procedures. However the change of ordering of the matrix columns of $A$ changes the coefficents represented by $N$ and the changes the functional form of the $K_{eq}$ problem. What is the proper mathematical description needed to talk about how changing the ordering of $A$ influences the other constructs.
The non-negativity constraint can be realized by taking subsets of the chemical species assumed to be in some parent set with an additional check that the exclusion was valid (thermodynamic check on species activity). In the most extreme case it would be necessary to have $(# of species)!-1$ different cases, that is all elements of the power set of chemical species other than the null element. (That is the maximal number of cases, almost never realized because if the chemical elements are present then there must be some chemical species that are comprised of them.) How can I discuss the 'model reduction' of choosing an element of the power set of initial chemical species to calculate thermodynamic equilibrium. Just like the re-ordering of the elemental abundance matrix, the functional form of the thermodynamic constraints is affected by the choice of starting species.
I have already solved my problem and can calculate thermodynamic equilbrium, I just dont know how to talk about it and identify when two of the mathematical constructs generated are actually the same or how to relate the sub-cases to the initial problem. I know this relates to convex optimization type problems, but it has a feel of representation of groups. Feel free to give me any suggestions, places where this might already be explained, or if you would like me to further clarify any concepts I only briefly discussed.
