stoichiometry - Simplest way to balance any chemical equation

In your opinion, what would be the simplest way to balance any size of the chemical equation?

For example here is an equation: $$\ce{C12H26 + O2 -> CO2 + H2O}$$

I saw some ways by putting letters in front of each molecule, but I just don't get too much how to do this with this method! My teacher is always speaking about another method, but it simply doesn't feel logic/mathematics is enough for me.

Answer

You can do this in the 'guided trial-and-error' method that LordStryker showed which is probably quickest for simple reactions, or approach it in a purely mathematical fashion which is the method I will explain. This method works well for arbitrarily difficult reactions.

Your chemical equation contains 3 atomic species: $\ce{C}$, $\ce{H}$ and $\ce{O}$. This means that you need 3 equations to balance. First I write the chemical reaction as follows: $$\ce{a C12H26 + b O2 -> c CO2 + d H2O}$$ Now I will write the 3 equalities for the 3 atomic species that we have: \begin{align} 12a &= c \tag{for C}\\ 26a &= 2d \tag{for H}\\ 2b &= 2c + d \tag{for O} \end{align} The numerical constants in front of $a$, $b$, $c$ and $d$ come from the number of atoms that are in the molecule for which they are included.

As you may have noticed, this system of equations is ill-defined, because we have 4 unknowns ($a$ to $d$) and only 3 equations. The reason is that we can pick any multiple of the equation without it becoming incorrect,¹ just against convention. This means that we can simply set one of the unknowns to any value. Let's say $a = 1$ for now and revisit this choice after solving the equations.

With $a = 1$ it immediately follows that $c = 12$ and $d = 13$. From those two it then follows that $b = 18.5$. Our balanced reaction becomes:² $$\ce{C12H26 + 18.5 O2 -> 12 CO2 + 13 H2O}$$

In principle this is already correct, but convention is to have only integer numbers in the equation (i.e. no decimal points) therefore we need to revisit our choice of $a = 1$ and pick it such that all numbers will become integers. In this case you can easily see that this will happen for $a = 2$ which then results in $$\ce{2C12H26 + 37 O2 -> 24 CO2 + 26 H2O}$$

If for some reason it is not easy to see which value for $a$ you would need then you can multiply by a large power of 10 to make all the decimals disappear and then check the greatest common divisor of the 4 numbers and divide by that to obtain the same equation.

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Notes

1. 
   
   

   Compare $10=10$ to the multiplied version $30=30$, both are correct, but they are just different by a factor $3$.

   
   


2. 
   

   I'm not sure whether you are familiar with linear algebra, but if you are you will probably have noticed that the set of equations is a linear set so you could solve it through matrix manipulations which makes this method applicable to arbitrarily complex chemical reactions.