4.1.3 Balancing Chemical Reactions
Consider the chemical reaction
Here the elements involved are tin Sn oxygen O and Hydrogen H. Some chemical reaction happens and
you end up with some tin and some water. The question is, how much do you start with and how much do
you end up with.
The balance of mass requires that you have the same number of oxygen, tin, and hydrogen on both
sides of the reaction. However, this does not happen in the above. For example, there are two oxygen
atoms on the left and only one on the right. The problem is to find numbers x,y,z,w such
and both sides have the same number of atoms of the various substances. You can do this in a systematic
way by setting up a system of equations which will require that this take place. Thus you
The augmented matrix for this system of equations is then
Row reducing this yields
Thus you could let w = 2 and this would yield x = 1,y = 2, and z = 1. Hence, the description of the
reaction which has the same numbers of atoms on both sides would be
You see that this preserves the total number of atoms and so the chemical equation is balanced. Here is
Example 4.1.21 Potassium is denoted by K, oxygen by O, phosphorus by P and hydrogen by H. The
balance this equation.
You need to have
Equations which preserve the total number of atoms of each element on both sides of the equation
The augmented matrix for this system is
Then the row reduced echelon form is
You could let w = 3 and this yields x = 3,y = 1,z = 1. Then the balanced equation is
Note that this results in the same number of atoms on both sides.
Of course these numbers you are finding would typically be the number of moles of the molecules on
each side. Thus three moles of KOH added to one mole of H3PO4 yields one mole of K3PO4 and three
moles of H2O, water.
Note that in this example, you have a row of zeros. This means that some of the information in
computing the appropriate numbers was redundant. If this can happen with a single reaction, think how
much more it could happen if you were dealing with hundreds of reactions. This aspect of the problem can
be understood later in terms of the rank of a matrix.
For an introduction to the chemical considerations mentioned here, there is a nice site on the web
http://chemistry.about.com/od/chemicalreactions/a/reactiontypes.htm where there is a sample test and
examples of chemical reactions. For names of the various elements symbolized by the various letters, you
can go to the site http://chemistry.about.com/od/elementfacts/a/elementlist.htm. Of course these things
are in standard chemistry books, but if you have not seen much chemistry, these sites give a nice
introduction to these concepts.