To determine the number of carbon dioxide ([tex]\( \text{CO}_2 \)[/tex]) molecules produced per octane ([tex]\( \text{C}_8\text{H}_{18} \)[/tex]) molecule burned in a combustion reaction, we need to first balance the given reaction:
[tex]\[
\text{C}_8\text{H}_{18} + \text{O}_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O} + \text{heat}
\][/tex]
Here are the steps:
1. Balance the Carbon Atoms:
- Octane ([tex]\( \text{C}_8\text{H}_{18} \)[/tex]) has 8 carbon atoms.
- To balance the carbons, each of these carbon atoms will form one molecule of carbon dioxide ([tex]\( \text{CO}_2 \)[/tex]).
- Therefore, we'll need 8 [tex]\(\text{CO}_2 \)[/tex] molecules to balance the carbon atoms on both sides.
The equation becomes:
[tex]\[
\text{C}_8\text{H}_{18} + \text{O}_2 \rightarrow 8\text{CO}_2 + \text{H}_2\text{O}
\][/tex]
2. Balance the Hydrogen Atoms:
- Octane has 18 hydrogen atoms ([tex]\( \text{H}_{18} \)[/tex]).
- Water ([tex]\( \text{H}_2\text{O} \)[/tex]) has 2 hydrogen atoms per molecule.
- To balance the 18 hydrogen atoms, we'll need 9 water molecules [tex]\((\text{H}_2\text{O})\)[/tex].
The equation now becomes:
[tex]\[
\text{C}_8\text{H}_{18} + \text{O}_2 \rightarrow 8\text{CO}_2 + 9\text{H}_2\text{O}
\][/tex]
3. Balance the Oxygen Atoms:
- On the product side, we have:
- 8 molecules of [tex]\(\text{CO}_2\)[/tex] contribute [tex]\(8 \times 2 = 16\)[/tex] oxygen atoms.
- 9 molecules of [tex]\(\text{H}_2\text{O}\)[/tex] contribute [tex]\(9 \times 1 = 9\)[/tex] oxygen atoms.
- In total, [tex]\(16 + 9 = 25\)[/tex] oxygen atoms are needed.
- Since [tex]\(\text{O}_2\)[/tex] is diatomic (each molecule contains 2 oxygen atoms), we will need [tex]\(\frac{25}{2} = 12.5\)[/tex] molecules of [tex]\(\text{O}_2\)[/tex] to provide the necessary oxygen atoms.
So, the balanced equation is:
[tex]\[
\text{C}_8\text{H}_{18} + 12.5\text{O}_2 \rightarrow 8\text{CO}_2 + 9\text{H}_2\text{O}
\][/tex]
Now that the reaction is fully balanced, we can clearly see that for every molecule of octane ([tex]\(\text{C}_8\text{H}_{18}\)[/tex]) burned, 8 molecules of carbon dioxide ([tex]\(\text{CO}_2\)[/tex]) are produced.
Thus, the correct answer is:
C. 8
