Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

1.6: Balancing Chemical Reactions

( \newcommand{\kernel}{\mathrm{null}\,}\)

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+H2Sn+H2O

Here the elements involved are tin (Sn), oxygen (O), and hydrogen (H). A chemical reaction occurs and the result is a combination of tin (Sn) and water (H2O). When considering chemical reactions, we want to investigate how much of each element we began with and how much of each element is involved in the result.

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+yH2zSn+wH2O

where both sides of the reaction have the same number of atoms of the various elements.

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:xz=0O:2xw=0H:2y2w=0

The augmented matrix for this system of equations is given by [101002001002020]

The reduced row-echelon form of this matrix is [10012001010001120]

The solution is given by x12w=0yw=0z12w=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+2H2Sn+2H2O

Observe that each side of the expression contains the same number of atoms of each element. This means that it preserves the total number of atoms, as required, and so the chemical reaction is balanced.

Consider another example.

Example 1.6.1: Balancing a Chemical Reaction

Potassium is denoted by K, oxygen by O, phosphorus by P and hydrogen by H. Consider the reaction given by KOH+H3PO4K3PO4+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+yH3PO4zK3PO4+wH2O

preserves the total number of atoms of each element.

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 [10300144101302001100]

and the reduced row-echelon form is [1001001013000113000000]

The solution is given by xw=0y13w=0z13w=0

which can be written as x=ty=13tz=13tw=t

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+1H3PO41K3PO4+3H2O

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.


This page titled 1.6: Balancing Chemical Reactions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?