Traditionally, the "redox method" of chemical equation balancing is explained in terms of charge balancing based on the "oxidation states" of the atoms involved but the method can be applied to equations in which all the compounds are neutral. What follows is an attempt to explain the method from a point of view based on ideas from linear algebra. Naturally, the explanation avoids the use of charge balance as the underlying idea which allows it to apply when all compounds are neutral. Balancing chemical equations is thought of in terms of the problem of finding intersections of vector subspaces and the "redox method" is explained in terms of variable reduction based on orthogonal compliments of the vector subspaces.
From a list of the reactants and products of a particular reaction, the problem is to determine the relative amounts of reagents that must be mixed in order for each reagent to be completely consumed in the reaction and further to determine the relative amounts of products that will be formed. In these problems the list of reactants and products is chosen such that once the amount of a one reagent or product has been chosen all the amounts of the other reagents and products are determined. The list is also chosen so a solution can be found in which mass is conserved. That is, a solution can be found such that all reagents and products are completely rather than partially consumed and formed, respectively.
For example, the problem of balancing the chemical equation
+ 2-
H + O -> H O
2
could be considered. The smallest amount is, by convention, normalized to 1 so the balanced equation would be
+ 2-
2 H + 1 O -> 1 H O
2
The reaction
+
H + O -> H O
2
would not be considered because there is no way to balance the equation such that all reagents are completely used up. This example illustrates the property of charge that although it can, in general, be treated like another type of atom, it has the unique ability to be present in negative amounts. It would probably be more accurate to think in terms of atomic nuclei and electrons instead of charge, in which case nothing could have a negative value but this would make many reactions more difficult to solve . The reaction
+ 2- -
H + O -> H O + OH
2
would also not be considered because specifying the amount of one compound does not uniquely determine the amounts of the other compounds.
In many cases it is possible to determine the amounts of all the reagents and products by arbitrarily fixing the amount of one reagent or product and the using the principle of mass conservation to determine the amounts of the other reagents. For example, in the case of the reaction
+ 2-
H + O -> H O
2
the amount of hydrogen ion could be arbitrarily set at 1. In order for there to be the same amount of hydrogen in the products as the reagents the amount of water would be set at 1/2. In order for the amount of oxygen in the reagents to be the same the amount of oxygen in the products the amount of oxygen ion would be set at 1/2. The amounts could then be multiplied by 2 so the that the smallest coefficients would be normalized to 1 and the problem would be solved. In practice normalization is carried out whenever a new smallest amount is determined during balancing but this does not affect the reasoning used to obtain the solution.
There are reactions, however, to which this technique can not be applied. For example, the reaction
MnSO + KMnO + H C O ->
4 4 2 2 4
3- + + 2-
Mn(C O ) + K + H + SO + H O
2 4 3 4 2
can not be solved with this technique although it does have the solution
4 MnSO + 1 KMnO + 15 H C O ->
4 4 2 2 4
3- + + 2-
5 Mn(C O ) + 1 K + 7 H + 4 SO + 4 H O
2 4 3 4 2
The above reaction can be balanced with what I am calling the
"redox method" as follows. Oxidation states are assigned to each atom
so that the sum of the oxidation states of each atom is equal to the
charge of the compound or ion. Where possible the same oxidation
state is assigned to all atoms of the same element. This is done by
assigning all the atoms of one element the same oxidation state (if
possible), then assigning all the atoms of another element the same
oxidation state (again, if possible), and continuing in this fashion
until all the atoms have been assigned an oxidation state. If
possible, the order of assigning oxidation states is chosen such that
oxidation states are not assigned arbitrarily. For example, in the
above reaction it would be desirable to assign an oxidation state to H
before assigning an oxidation state to O. One possible assignment of
oxidation states for the above reaction is
All K = +1
All H = +1
All O = -2
All S = +6
All C = +4
Mn of manganese sulfate = +2
Mn of potassium permanganate = +7
Mn of manganese oxalate ion = +3.
In order for the the charge to balance, the sum of the oxidation states of the reactants must equal the sum of the oxidation states of the products. For atoms of elements that can be assigned a single oxidation state, the sum of the oxidation states will balance as long as the atoms themselves are balanced. For atoms of elements that can not be assigned a single oxidation state, however, this condition can be used to determine constraints on the amounts they can be assigned relative to each other. In the case of our example we would have the necessary relation
2+ 7+ 3+
4 Mn + 1 Mn -> 5 Mn .
With this constraint any solution in which all atoms are balanced will also be a solution in which charge is balanced. The atoms can usually be balanced by the method of mass conservation described above so at this point it is usually possible to find the solution.
The puzzling fact about the method of balancing equations by assigning oxidation states is that the method is effective even when none of the reactants or product are charged. Consider the reaction
CH + C H -> C H 4 2 2 4 8
The method mass conservation can not be applied because C and H are
components of both reactants. The following oxidation states can be
assigned
All H = +1
C of methane = -4
C of ethyne = -1
C of butene = -2
In order for the charge to balance we must have the relation
4- 1- 2- 1 C + 2 C -> 3 C
which in this case is all we need. The balanced equation is
1 CH + 2 C H -> 3 C H
4 2 2 4 8
This is somewhat confusing because for any amounts of reactants or products the charge will balance and yet we have used constraint that the charge must be balanced in order to obtain our solution.
The link between chemistry and linear algebra can be made by thinking of chemical compounds as vectors. Although vectors are usually thought of as arrows oriented in space, they can also represent chemical compounds. The orientation of an arrow in space can be expressed as a sum of other arrows. The orientation of a certain arrow could be specified in terms of a the sum of an arrow pointing up, an arrow pointing to the side and a arrow pointing forward. The composition of a chemical compound can also be expressed in terms of the sum of other chemical compounds. For example,
3-
Mn(C O ) = 1 Mn + 3 C O + 3 (-)
2 4 3 2 4
Just as a spatial vector can be written as the sum of its components in the x, y, and z directions, a chemical compound can be written as a sum of the its atoms and charges. For example,
3-
Mn(C O ) = 1 Mn + 6 C + 12 O + 3 (-).
2 4 3
Spatial vectors have associated with them concepts that also apply to chemical vectors. The various multiples a specific set of vectors can be added together to form a number of new vectors. The set of all possible vectors that can be formed from a set of vectors can be used to define a space. For example, in three dimensional space one vector defines a line and two vectors that are not scalar multiples of each other define a plane. It is also possible, however, to think of the spaces defined (or spanned) by chemical vectors. For example, the compounds H and O span the space of all compounds of the formula
H O m n
where m and n can be specified arbitrarily.
A second concept associated with spatial vectors that can apply to chemical vectors is the concept of orthogonality. Vectors can be represented by tuples of numbers. A spatial vector in three dimensions can be written as an ordered triple of numbers, the first representing its component in the x direction, the second representing its component in the y direction and the third representing its component in the z direction. The dot product of two vectors represented by tuples is defined as
(X + X + X ) * (Y + Y + Y ) = X * Y + X * Y + X * Y 1 2 3 1 2 3 1 1 2 2 3 3
Two vectors are said to be orthogonal or perpendicular if their dot product is zero. For example, the vector (1,0,0) is orthogonal to all vectors of the form (0,r,s) where r and s are arbitrary. Geometrically, a unit vector in the x direction is orthogonal to all vectors in the y,z plane. Chemical compounds can also be represented by tuples and hence it is possible to think of chemical compounds as being orthogonal to each other. The compound
HO -2
is, for example, orthogonal to the compound
H O . 2
By thinking of chemical compounds as vectors, the problem of balancing chemical equations becomes a question of finding intersections of vector spaces. The reactant compounds will span the space of all compounds that can be created from combinations of the reactants and the product compounds will span the space of all compounds that can be decomposed into combinations of the products. In order for the solution to be such that we may arbitrarily specify the value of exactly one vector, the intersection of the reactant space with the product space will be one dimensional. Because the lists of reactants and products are chosen such that no compound will be assigned an amount of zero and because all the compounds are composed of non-negative amounts of atoms, components (atoms) of the vectors spanning the intersection of the two spaces will all have the same sign.
The intersection of a set of vector spaces can be found by considering the sets of vectors orthogonal to each vector space. The intersection of a set of vector spaces can not contain any element not contained in all spaces. A vector that is orthogonal to a vector space is not contained in that space and can therefore not be contained in the intersection of the vector space with other vector spaces. Given a set of vectors defining a space, it is a relatively simple matter to find a set of vectors orthogonal to that space. This is because a set of vectors that are orthogonal to vectors spanning a given space will be orthogonal to that entire space. Suppose, for example, we have the vectors (1,0,0) and (0,1,1). The problem of finding the set of vectors orthogonal to the space they span reduces to the problem of finding the set of vectors that satisfy the equations
1x + 0y + 0z = 0 0x + 1y + 1z = 0
which can be readily seen to be the vectors contained in the space spanned by the vector (1,1,-1). Applying this approach to the problem of balancing chemical equations, it can be seen that a compound spanning the intersection of the reactant space with the product space is orthogonal to the compounds orthogonal to the reactant space and product space.
It is possible to understand the equation balancing techniques of mass conservation and oxidation state balancing in terms of linear algebra. If the all the reactant compounds are orthogonal to each other and all the product compounds are orthogonal to each other then the equation may be balanced as follows. The value of one compound in either the reactants or the products is arbitrarily assigned an amount. The components of this compound are then balanced on the other side of the equation by assigning the appropriate amount to the appropriate compounds. This can be done by inspection because orthogonality requires that a given component occurs in only one compound on each side of the equation. This component balancing process is repeated until all values are determined. This method is exactly analogous to the method of equation balancing by mass conservation. It is therefore clear that the mass conservation method of balancing equations is successful only when both the reactant compounds and the product compounds are orthogonal to each other.
In cases where the compounds are not orthogonal, the oxidation state balancing method can be used to derive constraints that then make it possible to apply the mass conservation method. The link between the oxidation state balancing method and linear algebra is the realization that the problem of balancing oxidation states is analogous to the problem of finding a vector orthogonal to the reactant and product compounds.
Many equation balancing problems have the property that the reactant and product compounds are orthogonal to each other except with respect to one component. If it were possible to know the constraints, if any, that the ``non-orthogonal component'' placed on solutions to the equation, it would be possible to eliminate that component which would result in a chemical equation with orthogonal reactants and products. Such a system could then be solved by the method of mass conservation. A result of linear algebra is that if two sets of vectors can be found to be perpendicular to a vector containing a certain component, both sets of vectors can be projected onto the vector space perpendicular to that component without changing the relative amount of each vector necessary to form a vector spanning the intersection. The model of this in three dimensional space is that the vectors of an arbitrary plane could be projected onto the y,z plane provided the vector orthogonal to the arbitrary plane was not orthogonal to the x axis.
In the case of the reaction
MnSO + KMnO + H C O ->
4 4 2 2 4
3- + + 2-
Mn(C O ) + K + H + SO + H O
2 4 3 4 2
for example, it can be seen that if manganese sulfate and potassium permanganate are replaced with the compound
KMn S O = 4 MnSO + 1 KMnO 5 4 20 4 4
then the compound
Mn S O K H C (+) 2 6 -2 1 1 4 -1
is orthogonal to all the reactant and product compounds. This creates the option of projecting the reactant and product compounds on a space orthogonal to any one of the nonzero components of the orthogonal vector. Because charge is one of the most frequently occurring species, it is in this case desirable to eliminate it leaving the resulting equation
KMn S O + H C O -> Mn(C O ) + K + H + SO + H O 5 4 20 2 2 4 2 4 3 4 2
which can be solved by the method of mass conservation. If it were not possible to solve the equation by the method of mass conservation at this point, the process of reducing the number of variables could be repeated until it became possible to apply mass conservation.
In the case of the reaction
CH + C H -> C H 4 2 2 4 8
replacing the compounds methane and ethyne with
C H 4 8
Because H occurs in the orthogonal vector we can project all the compounds onto the space orthogonal to H which gives the equation
C -> C 4 4
which can be solved by mass conservation.
In general, the method of oxidation state balancing can be seen as way of eliminating components of the reactant and product compounds that prevent them from being orthogonal. In the terminology of linear algebra certain compounds are being replaced by linear combination of themselves in order to make it possible to map the reactant and product compounds to a lower dimension subspace in which the reactants and products are orthogonal to themselves. If there are n reactant and product compounds then they will span a subspace (of all possible chemical compounds) of dimension n-1 and there will be n-2 independent vectors spanning the the space orthogonal to their intersection.