Linear System involving chemical reaction: Fe + O_2 --> Fe_2 O_3

[thirdeyechai]

I'm having trouble solving this linear system of a chemical reaction.

"Solve a system of linear equations to balance the chemical equation"

Fe + O₂ --> Fe₂O₃

The method I was shown is as follows

x₁Fe + x₂O₂ --> x₃Fe₂O₃

Comparing the number of each type of atom on the two sides of equation I get


Fe x₁ = 2x₃

O 2x₂ = 3x₃


x₁ - 2x₃ = 0
2x₂ -3x₃ = 0

I'm just not really sure where to go from here?

 

[HallsofIvy]

I'm having trouble solving this linear system of a chemical reaction.

"Solve a system of linear equations to balance the chemical equation"

Fe + O₂ --> Fe₂O₃The method I was shown is as follows

x₁Fe + x₂O₂ --> x₃Fe₂O₃

Comparing the number of each type of atom on the two sides of equation I get


Fe x₁ = 2x₃

O 2x₂ = 3x₃


x₁ - 2x₃ = 0
2x₂ -3x₃ = 0

I'm just not really sure where to go from here?

You have 3 unknown values but only 2 equations. The best you can do is solve for 2 of the unknowns in terms of the other. \(\displaystyle x_1- 2x_3= 0\) immediately gives \(\displaystyle x_1= 2x_3\) and \(\displaystyle x_2- 3x_3= 0\) immediately gives \(\displaystyle x_2= 3x_3\). Choose \(\displaystyle x_3\) to be any number (since it is a number of atoms it obviously has to be a non-negative integer) and calculate \(\displaystyle x_1\) and \(\displaystyle x_2\).

For example if you take \(\displaystyle x_3= 3\) then \(\displaystyle x_1= 2(3)= 6\) and \(\displaystyle x_2= 3(3)= 9\). In that case the formula would be \(\displaystyle 6Fe+ 9O_2= 3Fe_2O_3\). But since all of 6, 9, and 3 are divisible by 3 that is just 3 times \(\displaystyle 2F+ 3O_2= Fe_2O_3\). And we could have got that answer just by taking \(\displaystyle x_3= 1\) in the first place so that is what is normally done: choose whatever "free" variables you have (here \(\displaystyle x_3\)) to give the smallest integer values for the others.