1.6: Balancing Chemical Reactions
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The tools of linear algebra can also be used in the subject area of Chemistry, specifically for balancing chemical reactions.
Consider the chemical reaction SnO2+H2→Sn+H2O
An important theory we will use here is the mass balance theory. It tells us that we cannot create or delete elements within a chemical reaction. For example, in the above expression, we must have the same number of oxygen, tin, and hydrogen on both sides of the reaction. Notice that this is not currently the case. For example, there are two oxygen atoms on the left and only one on the right. In order to fix this, we want to find numbers x,y,z,w such that xSnO2+yH2→zSn+wH2O
This is a familiar problem. We can solve it by setting up a system of equations in the variables x,y,z,w. Thus you need Sn:x=zO:2x=wH:2y=2w
We can rewrite these equations as Sn:x−z=0O:2x−w=0H:2y−2w=0
The augmented matrix for this system of equations is given by [10−100200−10020−20]
The reduced row-echelon form of this matrix is [100−120010−10001−120]
The solution is given by x−12w=0y−w=0z−12w=0
which we can write as x=12ty=tz=12tw=t
For example, let w=2 and this would yield x=1,y=2, and z=1. We can put these values back into the expression for the reaction which yields SnO2+2H2→Sn+2H2O
Consider another example.
Potassium is denoted by K, oxygen by O, phosphorus by P and hydrogen by H. Consider the reaction given by KOH+H3PO4→K3PO4+H2O
Balance this chemical reaction.
Solution
We will use the same procedure as above to solve this problem. We need to find values for x,y,z,w such that xKOH+yH3PO4→zK3PO4+wH2O
Finding these values can be done by finding the solution to the following system of equations. K:x=3zO:x+4y=4z+wH:x+3y=2wP:y=z
The augmented matrix for this system is [10−30014−4−10130−2001−100]
The solution is given by x−w=0y−13w=0z−13w=0
Choose a value for t, say 3. Then w=3 and this yields x=3,y=1,z=1. It follows that the balanced reaction is given by 3KOH+1H3PO4→1K3PO4+3H2O
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.