To solve this problem, we need to analyze the stoichiometric relationship in the chemical reaction:
[tex]\[
\text{CH}_4 + 2 \text{O}_2 \rightarrow \text{CO}_2 + 2 \text{H}_2\text{O}
\][/tex]
According to the balanced equation:
- 1 mole of [tex]\( \text{CH}_4 \)[/tex] reacts with 2 moles of [tex]\( \text{O}_2 \)[/tex] to produce 1 mole of [tex]\( \text{CO}_2 \)[/tex] and 2 moles of [tex]\( \text{H}_2\text{O} \)[/tex].
The stoichiometric ratio between [tex]\( \text{CO}_2 \)[/tex] and [tex]\( \text{H}_2\text{O} \)[/tex] is 1:2. This means that for every 2 moles of [tex]\( \text{H}_2\text{O} \)[/tex] produced, 1 mole of [tex]\( \text{CO}_2 \)[/tex] is produced.
Given that 10 moles of [tex]\( \text{H}_2\text{O} \)[/tex] are produced, we can use this ratio to determine the moles of [tex]\( \text{CO}_2 \)[/tex] produced:
[tex]\[
\frac{\text{moles of H}_2\text{O}}{2} = \frac{10}{2} = 5 \text{ moles of CO}_2
\][/tex]
Therefore, 5 moles of [tex]\( \text{CO}_2 \)[/tex] are produced.
So, the correct answer is:
[tex]\[
\boxed{5}
\][/tex]
