To determine how many moles of hydrogen ([tex]\(H_2\)[/tex]) are needed to produce 13.78 moles of ethane ([tex]\(C_2H_6\)[/tex]), refer to the balanced chemical equation:
[tex]\[C_2H_2 + 2 H_2 \rightarrow C_2H_6\][/tex]
The balanced equation indicates that 1 mole of [tex]\(C_2H_6\)[/tex] is produced from 2 moles of [tex]\(H_2\)[/tex]. This establishes a reaction ratio of:
[tex]\[1 \text{ mole of } C_2H_6 : 2 \text{ moles of } H_2\][/tex]
This means you need 2 moles of hydrogen gas for every mole of ethane produced.
To find out how much [tex]\(H_2\)[/tex] is needed for 13.78 moles of [tex]\(C_2H_6\)[/tex]:
1. Identify the ratio from the balanced equation: [tex]\(2 \, \text{moles of } H_2 \text{ per } 1 \, \text{mole of } C_2H_6\)[/tex].
2. Multiply the given moles of [tex]\(C_2H_6\)[/tex] by the ratio to find the moles of [tex]\(H_2\)[/tex].
[tex]\[
\text{Moles of } H_2 = 13.78 \, \text{moles of } C_2H_6 \times 2 \, \frac{\text{moles of } H_2}{\text{mole of } C_2H_6}
\][/tex]
[tex]\[
\text{Moles of } H_2 = 13.78 \times 2 = 27.56 \, \text{moles of } H_2
\][/tex]
Therefore, 27.56 moles of hydrogen are needed to produce 13.78 moles of ethane.
The correct answer is:
[tex]\[
\boxed{27.56}
\][/tex]
