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Application of linear systems to chemistry
An atom is an object that appears in the periodic table of elements. A molecule is some finite combination of atoms written adjacently. A chemical equation is a representation of what happens when molecules come in contact with each other. A balanced chemical equation is a chemical equation that also shows how many of what molecules react and how many of what molecules result. For example, the following is a chemical equation that is not balanced:
H2+O2⟶H2O
because on the left you have a hydrogen molecule and an oxygen molecule consisting of, in total, 2 hydrogen atoms and 2 oxygen atoms respectively but on the right we see one water molecule conssiting of, in total, 2 hydrogen atoms and 1 oxygen atoms. So on the left we have 2 hydrogens, 2 oxygens and on the right 2 hydrogens and 1 oxygen. Because the number of atoms of each type on each side are different, we say the chemical equation is not balanced. To write this reaction balanced we would write
2H2+O2⟶2H2O
because now the total number of atoms on each side of each type are equal.
We will write molecules as vectors in the following way: we will assign rows to atoms and to each molecule we write the number and each type of atoms it has. For example in the above reaction, the atoms are only H and O. The molecules are H2,O2, and H2O. We will use the first row of vectors for H atoms and the second row of vectors for O atoms. In this case we can express each molecule as a vector in the following way:
H2=[20],O2=[02],H2O=[21].
We would write the balanced chemical equation from above as the vector equation
2H2+O2−2H2O=0.
Example: The chemical reaction for the (combustion) reaction of octane and oxygen yielding water and carbon dioxide is
CH3(CH2)6CH3+O2⟶H2O+CO2,
where (CH2)6 denotes 6 copies of CH2. This equation is not balanced because on the left side, in total, there are 8 C atoms, 18 H atoms, and 2 O atoms while on the right there are, in total, 1 C atom, 2 H atoms, and 3 O atoms. To balance this equation we will take vectors →v1,…,→v4 where each vector corresponds to the molecules that appear in the reaction. We will use the first row of each vector for C atoms, the second for H atoms, the third for O atoms. With this interpretation, we have the following four vectors:
→v1=CH3(CH2)6CH3=[8180],
→v2=O2=[002],
→v3=H2O=[021],
→v4=CO2=[102].
To balance the equation is to find weights x1,x2,x3,x4 such that
x1→v1+x2→v2+x3→v3+x4→v4=→0.
(note: the values for x3 and x4 should end up negative because we are "subtracting" everything to the left side of the octane reaction equation -- if you write "−x3" instead of "x3",that is also ok -- all that happens is that x3 will be positive in such a case)
If we substitute the values of →v1,…,→v4 into the vector equation we get
x1[8180]+x2[002]+x3[021]+x4[102]=[000].
Therefore we see that we need to solve the homogeneous system
(∗)[8001180200212][x1x2x3x4]=[000.]
After row operations (I pulled it off in 5 or 6 steps), we can find the matrix that the coefficient matrix is equivalent to:
[8001180200212]∼[100180102516001−98]
Hence the homogeneous equation (∗) becomes
[100180102516001−98][x1x2x3x4]=→0.
If we interpret this equation as a system we get
{x1+18x4=0x2+2516x4=0x3−98x4=0
or equivalently
{x1=−18x4x2=−2516x4x3=98x4
Hence the system (∗) has solution
→x=[x1x2x3x4]=x4[−18−2516981.]
Recall we said that x3 and x4 should be negative. Also it is convention to have only integer coefficients in balanced equations. So we will choose x4=−16 which yields
→x=[225−18−16].
This means that we should balance the equation by writing
2→v1+25→v2−18→v3−16→v4=0
or equivalently
2→v1+25→v2=18→v3+16→v4,
which when interpreted as a chemical equation in the way we defined yields
2CH3(CH2)6CH3+25O2⟶18H2O+16CO2,
which is a balanced chemical equation because on the left-hand-side there are in total 16 C atoms, 36 H atoms, and 50 O atoms on the left and on the right-hand-side the same.
