📚 Use matrices to balance a chemical equation

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Write a balanced chemical equation for the following reaction (propane combustion):

C_3 H_8+O_2→CO_2+H_2 O

Balancing Chemical Equations: Solving a system of equations using matrices

Example: HClO4 + P4O10 → H3PO4 + Cl2O7

1) We represent each coefficient with a variable (e.g. a-d)
aHClO4 + bP4O10 → cH3PO4 + dCl2O7

2) Write equations to represent the relationship between each element in each species
H: a=3c Cl: a=2d O: 4a + 10b = 4c + 7d P: 4b = c

3) Insert the coefficients from the above equations into an augmented matrix. The vertical line represents an ‘equal sign’. Numbers that are on the wrong side of the ‘equal sign’ must have their sign changed (become negative). Computer programs commonly use matrices to solve equations.

4) Since we are solving for a relationship (the ratio of the coefficients) not specific numbers, we need one less equation than the number of unknowns (4 unknowns, we need 3 equations). The last variable (d) will be set to 1. We can therefore drop the most complex equation.

5) We are attempting to have one 1 in each row and column on the left side, with all the other numbers being 0. The values on the right side correspond to the values of our coefficients.

6) There are 3 valid row operations that we can perform with matrices:

a. Switch two rows
b. Add or subtract two rows from each other
c. Multiply or divide a row by a number

7) We can combine the above operations in some cases (e.g. we can subtract a multiple of a row from another row).

8) Row operations are described above the transition arrows, with the subscript represent the row number (the top row is #1)

9) The value of the coefficients (a, b, and c) can be read from the final matrix:

a. 2 b. 1/6 c. 2/3 d. 1 (implied)

10) We multiply each coefficient by 6 to remove the fractions and sub the numbers into our original equation. Our final, balanced equation is:

12HClO4 + P4O10 → 4H3PO4 + 6Cl2O7

[jduaf]

Why is your second equation not 8a-2d=0 ???

[danielramos]

It's interesting to see it's application in balancing chemical equations. However, it feels excessive for this kind of exercise.

[drakesmith]

Okay, I ended up making sense of his solution. Hell, I like that solution, it makes sense but goddamn, that was a mess of doing elimination. For anyone not in the know this method was referred to Gauss Jordan or Reduced Row Echelon Form (RREF). The echelon bit is French for stairs. This essentially means from the first row, which should have an entry zero, every variable one step right and one step down (diagonal at 45 degrees) should be one. This makes the coefficients one and when written as an eq again just allows the equation of each row to be a variable equals some number, in this case it’s a variable. The variable d is called a parameter. Essentially if the parameter (side measure) takes a step, each variable maintains some relationship to it. This allows for you to model the behavior of the other 3 with only using d. Here you need an integer, so each number being divided by 4 needed a multiple of d that would eliminate the 4, i.e. 4 and so they are all multiplied by the constant accordingly. I’ll make a page on this soon to make it clearer. Back to the problem with elimination, this was how I learned it for the SAT. This is not how I learned it for linear algebra. The reason it looked funky was because the guy took the liberty to start essentially swap the row order which with his notation was more confusing, essentially putting you in a position to lose track. There was also no notation for the progressive execution of row operations (for those not in the know it’s the manipulations of the numbers in the matrix he did). He also kept referring to each row by its old name, adding further confusion. All in all conceptually a good video, but poor execution there. I don’t like being this guy, but I had to say it.

[hanjodurandt]

Thanks, this really helped to reintroduce me to a method I have long since forgotten. It is really thoroughly and well explained, just watch for those typos ;P.

[tbynlogan]

Would this system be considered to have "INFINITELY MANY SOLUTIONS" since the value of "d" can be any number?

[jeffjefferson]

Well up untill the writing all the lines into each other it was pretty straight forward.

I guess, its someting i will need to learn, while its really hard to get some equations done by trial and error.

Thank you!

Greetings,


Jeff

[shetheworkinggirl]

Hi! Why did you interchange the O and H in 3:23..is there a certain reason for it?

[emyhermida]

What if there are more than 3 elements?