To determine how many moles of hexane ([tex]\(C_6H_{14}\)[/tex]) are needed to produce 18.4 moles of carbon dioxide ([tex]\(CO_2\)[/tex]), we can use the stoichiometric relationship given by the balanced chemical equation:
[tex]\[
2 \, C_6H_{14} + 19 \, O_2 \rightarrow 12 \, CO_2 + 14 \, H_2O
\][/tex]
From the balanced equation, we see that 2 moles of hexane produce 12 moles of carbon dioxide.
To find out how many moles of hexane are required to form 18.4 moles of carbon dioxide, we can set up the proportion based on the stoichiometry of the reaction:
[tex]\[
\frac{2 \text{ moles } C_6H_{14}}{12 \text{ moles } CO_2} = \frac{x \text{ moles } C_6H_{14}}{18.4 \text{ moles } CO_2}
\][/tex]
Here, [tex]\(x\)[/tex] represents the number of moles of hexane needed. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{2 \text{ moles } C_6H_{14} \times 18.4 \text{ moles } CO_2}{12 \text{ moles } CO_2}
\][/tex]
[tex]\[
x = \frac{2 \times 18.4}{12}
\][/tex]
[tex]\[
x = \frac{36.8}{12}
\][/tex]
[tex]\[
x = 3.0666666666666664
\][/tex]
Rounding to two decimal places, the answer is approximately [tex]\(3.07\)[/tex] moles.
Therefore, the number of moles of hexane ([tex]\(C_6H_{14}\)[/tex]) required to form 18.4 moles of carbon dioxide ([tex]\(CO_2\)[/tex]) is:
[tex]\[
\boxed{3.07 \text{ moles} }
\][/tex]
