1. Introduction

Linear algebra and chemistry mainly unite in the field of stoichiometry. Here linear algebra has proven to be effective in establishing systematic representations across all types of chemical equations (e.g. combustion, redox, acid/base, etc.) ^{[1, 2, 3, 4]}. In this report, such representations are used specifically for what we call square chemical equations—or reactions that have the same number of compounds as unknown coefficients in their representative equations. The interest in this subset of chemical equations arises from the chemistry-based matrix systems formerly acknowledged. For any square chemical equation, we find a matrix system of the order n X n is always established (hence “square”). In ^[1] I conclude that a proof for why chemical equations can be balanced with linear algebra exists, but never state it. We begin with an extended, explanatory proof.

Consider the characteristic equation

(1)

where A is a square n X n matrix and is an eigenvector for one of n unique eigenvalues λ ^[5]. In ^[1] I describe how a chemical equation A_c, for example, where

(where x₁ through x_r denote both the term and unknown coefficient for the compounds reacting, and x_r+1 through x_p denote likewise for those being produced) can be represented by the matrix setting

(2)

from simple conservation of mass—that is,

(3)

where the above equation refers to the same element ^[6]. Eq 2, however, reformulates to and, if A is n X n, then in eq 1. Thus, if a chemical equation possesses a square coefficient matrix, then there exists at least one .

Definition. n X n coefficient matrices (those derived from square chemical equations), we call chemical matrices.

In ^[1] I continue to explain how the kernel of a chemical matrix (a consequence of its reduced row echelon form) can yield the solution to the chemical equation. This is of no surprise, however, as we have determined that all chemical equations are linear-homogeneous systems; and therefore have a unique null space (and thus chemical solution) equal to the vector ^[6].

Such a solution, however, is only one of n for A_c, as an n X n matrix will have n eigenvalues and thus n eigenvectors associated with it. It is therefore the purpose of this work to examine these other linear algebraic significances.

A majority of chemical equations retain dimensions of m X n where . Below we present two methods—the first for the case of and the second for —on creating chemical matrices that can be analyzed with eq 1.

Chemically, if then the equation in question has more elements reacting than terms in the equation. For example, the reaction

has a total of 5 unique elements, but only 4 terms. We find

where dim(B) = 5 X 4. Similar to adding zero to an algebra problem, we add the term Ag₀N₀O₀K₀P₀ to the product side, yielding the term and coefficient designation x₅.^{1} Such an addition does not alter the chemical meaning—only the dimensions—of a given chemical equation. We then find

where dim(B’) = 5 X 5.

We employ a similar tactic for the case of m < n; however, here we append an additional element to each term in the non-square equation.^{2} For example, consider the reaction

Here

where dim(C) = 2 X 3. Appending to each term established the reaction

where

Again, this method does not modify a chemical equations’ delineation—only its dimensions.

Lemma 1. For any n X n chemical matrix A, .

Proof. By definition chemical matrices denote the changes compounds undergo in a reaction. If there is only a single element in the entire reaction (n = 1), then this element cannot react if the n X n matrix order is to be maintained. If , however, as in the reaction , a reaction does occur, and is therefore the minimum order of a chemical matrix.

Lemma 2. For any n X n chemical matrix A, det(A) = 0.

Proof. By the invertible matrix theorem in ^[7],

(4)

Because , det(A) = 0.

Consequently, ^[7] asserts that the following two conditions are true:

1. the linear transformation is not one-to-one

2. the columns of A do not form a linearly independent set.

Albeit other assertions can be made, these are the few that have chemical significance. We will proceed in respective order.

The linear transformation for chemical matrix A resembles one of the most fundamental laws governing balanced reactions: the law of multiple proportions—that is, “When an element combines with another to form more than one compound the masses of the second element combining with a fixed mass of the first element bear a simple ratio to one another” ^[8]. For example, the reaction 2KClO₃ → 2KCl + 3O₂ is the same as 4KClO₃ → 4KCl + 6O₂, which is the same as 2nKClO₃ → 2nKCl + 3nO₂, where . Modeling this reaction linear algebraically produces the system

where both and , and is the eigenvector for . Because it is the ratio of the elements that must be identical, can be any scalar multiple of itself; therefore, the liner transformation cannot be one-to-one.

Following eq 3, the total mass of a system is conserved for any closed reaction. This implies that, as in the previous example, all reactant columns (positive entries) are dependent on all product columns (negative entries), as the sum of unique scalar multiples of these columns must equal the zero-vector (eq 2). Accordingly, the columns of A do not form a linearly independent set.

Definition. The fact that the commutative property of addition holds for chemical equations, provided the reactants remain on the reactant side and products on the product side, we call chemical commutativity.

Chemical commutativity establishes the idea that chemical matrices with will possess and thus chemical solutions. We prove this below.

Let r denote the number of reacting terms in A_c and p the number being produced.

Lemma 3. For any chemical equation A_c, there exists exactly empirical chemical solutions.

Proof. The number of unique arrangements for the reactant side (U_r) in A_c is given by . Similarly, for the product side . Consequently, the total number of unique arrangements is .

This result implies that chemically correct matrices exist, where each has at least one by eq 1.

As trivial as chemical commutativity may seem, it is the basis to obtaining the first inhomogeneous significance we will explore.

2. Main Results

As stated in the introduction, the homogeneous equation is unique to chemical equations as it demonstrates the law of conservation of mass. , however, is one of n eigenvalues for a chemical matrix. In this section, we examine occurrences of and the eigenvectors that correspond.

The following assertions will be based on an n X n chemical matrix D, where

We denote the set of all matrix elements in D that lie upon the main diagonal—that is: {a₁₁, b₂₂, c₃₃, etc.}—as .

Definition. Terms in a chemical equation that consist of only one element in the entire substance, we call singular molecules. For example, O₂ functions as a singular molecule in the reaction x₁CH₄ + x₂O₂ → x₃H₂O + x₄CO₂.

Lemma 4. For any chemical reaction that either reacts or produces a singular molecule , where A is a singular molecule and is A’s corresponding subscript, , provided .

Proof. Because represents a singular molecule in the reaction, all other values in the nth column of D must equal zero ^{[1, 3]}. Thus,

The characteristic polynomial of D, P(D), can be factored into the form , where denotes a unique number (one of n) in the factored form of P(D).^{4} By definition P(D) = 0, and thus each τ denotes an eigenvalue of D. To prove, we find det(D). By Laplace Expansion

Therefore, , or, similarly, .

Lemma 5. For the specific case of in D, , where is the eigenvector for and is the nth unit-vector in .

Proof. Neglecting the trivial solution of , if we assume , then

We find,

Consequently,

We generalize this point below.

Theorem 1. Provided , then .

Proof. We assume , then

We find

respectively. This concludes the proof.

Realize that because of chemical commutativity, obtaining the eigenvector is simply a matter of rearranging a chemical matrix to allow (provided a singular molecule in the reaction exists).

We continue by recalling the mathematical implication of the law of conservation of mass (eq 3). Reformulating eq 3 for each element in A_c yields the linear-homogeneous system

(5)

as shown in ^[1]. Identically,

(6)

where, for instance, for all a elements reacting in A_c and for all A elements being produced.

Definition. The vector that results from the substitution of a random coefficient vector into eq 6, we call residue.^{5}

Formally,

(7)

for any substituted into A_c. For example, in the reaction x₁KClO₃ → x₂KCl + x₃O₂, if we substitute in a random coefficient vector

(Obviously, the substitution of produces , as in eq 6.)^{6}

Let denote a vector with entries in their simplest, whole number ratios. We present the following question: are there any values for that satisify ? We consider solutions of trivial.

Theorem 2. For a chemical matrix A, all satisify .

Proof. Eq 7 is derived from a system of equations similar to eq 5—that is,

where R_n denotes the nth element of . Equivalently,

Let . Then, with equivalent residue of by eq 1. Therefore, , or, indistinguishably, .

The chemical connotation associated with asserts that, on the foundations of eqs 6 and 7, other chemical solutions exist linear algebraically. Albeit inhomogeneous, these solutions associate themselves uniquely with different subsets of chemical equations. They are therefore only idiosyncratic to the structure of chemical equations, with no correspondence to chemical reactivity.

3. Conclusion

Through the use of linear algebra, specialized subsets of chemical equations were analyzed. In this manuscript, we proved correct the calculator-based balancing method explained in ^[1]. Introduced was the notion of chemical commutativity, which eventually inaugurated the mathematical implications of singular molecules. We then proved that all chemical equations will have residue for any , which demonstrates that inhomogeneous solutions to chemical equations exist, but only structurally.

Notes

1. Albeit this term could be appended to either side of the equation, it is simplest to append to the rightmost side of the equation (the nth column).

2. could denote any element in the Periodic Table of the Elements; however, due to the zero subscript, it is non-existent in the equation.

3. Similar to the case of , it is easiest to append the row vector to the bottommost row of a chemical matrix.

4. Note .

5. Note both and .

6. The calculation of residue is commonly used to check chemical equations. If

Competing Interest

The author declares no competing financial interest.

Acknowledgements

This manuscript was supported in part by Janie Mueller, Chuck Hendrick, Dave Hale, Shaun Carmody, Ralph Boedigheimer, and Brad Warner.

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