By balancing a chemical reaction, you must first balance the atom that is in smaller amount in the chemical species that you have in the reaction. In the combustion reaction presented in the question that atom is carbon (C). Therefore, the first thing to do is multiply the CO2 by 7, since those are the carbon atoms that are in the C7H16O. The reaction would be like this,
C7H16O + O2 → 7CO2 + H2O
You can make a table of the amount of atoms of C, H and O that you have in the reagents and products after you put a coefficient 7 in front of the CO2, in the following way,
R P
C 7 7
H 16 2
O 3 15
Then you balance the hydrogens. It is better that you leave the last oxygens since there is an oxygen molecule alone, so when adding a coefficient to balance it, the quantities of the rest of the atoms in the equation would not be altered.
To balance the hydrogens you add an 8 in front of the H2O molecule in the reagents, since there are 16 hydrogens in the molecule C7H16O and
8H x 2H = 16H.
The reaction and the table would be like this,
C7H16O + O2 → 7CO2 + 8H2O
R P
C 7 7
H 16 16
O 3 22
Finally, you balance the atoms of O. To do this, you add a coefficient of 21/2 to the O2 molecule, this is because (21/2) x2 = 21 and 21 O coming from the O2 molecule plus an O coming from the C7H16O molecule gives a total of 22 O, which are equal to the amount of O we have in the products.
C7H16O + (21/2)O2 → 7CO2 + 8H2O
To bring the coefficients of the reactants and the products to whole numbers multiply all the coefficients by 2. Then the reaction is like,
2 C7H16O + 21O2 → 14CO2 + 16H2O
As you can see when the given reaction is balanced using the smallest possible integers, the coefficent of oxygen gas is 21
