# When propane burns in oxygen 2

### Balance the given chemical equations. When propane burns in oxygen, it produces carbon dioxide and water:

$C_{3}H_{8}+O_{2} \rightarrow CO_{2}+H_{2}O$

# Solution:

Write equations for each of the elements.

Oxygen: $0a + 2b = 2c + 1d$ (The values of the coefficients are the number of oxygen atoms in the chemical equation)

Carbon: $3a + 0b = 1c + 0d$

Hydrogen: $8a + 0b = 0c + 2d$

Rewrite the equations so that all the variables are on the left side.

$0a+2b-2c-d=0$

$3a+0b-c+0d=0$

$8a+0b+0c-2d=0$

Solve the equations, easiest way is to use a matrix.

$\begin{bmatrix}8 & 0 & 0 & -2 & |0\\3 & 0 & -1 & 0 & |0\\0 & 2 & -2 & -1 & |0\end{bmatrix}$

Put the matrix in reduced row echelon form.

$\begin{bmatrix}1 & 0 & 0 & -\frac{1}{4} & |0\\0 & 1 & 0 & -\frac{5}{4} & |0\\0 & 0 & 1 & -\frac{3}{4} & |0\end{bmatrix}$ (if you do not know how to get to this form, please check this guide.)

Let $s_{1}$ represent a free parameter.

$a = \frac{1}{4}s_{1}$

$b = \frac{5}{4}s_{1}$

$c = \frac{3}{4}s_{1}$

$d = s_{1}$

Because the lowest common denominator is 4, we will let $s_{1} = 4$.

Thus, we get:

$a = 1$

$b = 5$

$c = 3$

$d = 4$

Our final balanced equation becomes:

$C_{3}H_{8}+5O_{2} \rightarrow 3CO_{2}+4H_{2}O$

See a mistake? Comment below so that we can fix it!

## 2 thoughts on “When propane burns in oxygen”

• Julia

How did you get b to equal 5/4 * s1? Why does it not equal 1/2 * s1?

• questionsolutions Post author

The correct answer is 5/4, however, there was a typo in our reduced row matrix, which we fixed. Thank you very much for pointing it out to us! 🙂 To see the full matrix put into the reduced row form, please take a look here: https://goo.gl/jWKsF3