To determine the mole ratios from the given balanced chemical equation:
[tex]\[
N_2H_4 + 2H_2O_2 \rightarrow N_2 + 4H_2O
\][/tex]
First, let's analyze the mole ratio of hydrazine [tex]\((N_2H_4)\)[/tex] to hydrogen peroxide [tex]\((H_2O_2)\)[/tex]:
- According to the balanced equation, 1 mole of [tex]\(N_2H_4\)[/tex] reacts with 2 moles of [tex]\(H_2O_2\)[/tex].
- This implies a mole ratio of [tex]\(N_2H_4\)[/tex] to [tex]\(H_2O_2\)[/tex] of [tex]\(\frac{1}{2}\)[/tex] or equivalently [tex]\(0.5\)[/tex].
Next, let's analyze the mole ratio of hydrazine [tex]\((N_2H_4)\)[/tex] to water [tex]\((H_2O)\)[/tex]:
- According to the balanced equation, 1 mole of [tex]\(N_2H_4\)[/tex] produces 4 moles of [tex]\(H_2O\)[/tex].
- This implies a mole ratio of [tex]\(N_2H_4\)[/tex] to [tex]\(H_2O\)[/tex] of [tex]\(\frac{1}{4}\)[/tex] or equivalently [tex]\(0.25\)[/tex].
Therefore, considering these ratios and converting to whole number ratios where convenient:
- The mole ratio of [tex]\(N_2H_4\)[/tex] to [tex]\(H_2O_2\)[/tex] is [tex]\(1:2\)[/tex] (1 hydrazine to 2 hydrogen peroxide).
- The mole ratio of [tex]\(N_2H_4\)[/tex] to [tex]\(H_2O\)[/tex] is [tex]\(1:4\)[/tex] (1 hydrazine to 4 water).
Thus, the correct answer among the given choices is:
[tex]\[ \boxed{1:2 \text{ and } 1:4} \][/tex]
