To determine the mass of [tex]\(H_2\)[/tex] needed to react with [tex]\(8.75 \, \text{g}\)[/tex] of [tex]\(O_2\)[/tex] according to the chemical equation [tex]\(O_{2(g)} + 2H_{2(g)} \rightarrow 2H_2O_{(g)}\)[/tex], we follow these steps:
1. Calculate the amount of moles of [tex]\(O_2\)[/tex]:
- The molar mass of [tex]\(O_2\)[/tex] is [tex]\(32.00 \, \text{g/mol}\)[/tex].
- The mass of [tex]\(O_2\)[/tex] given is [tex]\(8.75 \, \text{g}\)[/tex].
Using the formula [tex]\( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \)[/tex]:
[tex]\[
\text{moles of } O_2 = \frac{8.75 \, \text{g}}{32.00 \, \text{g/mol}} = 0.2734375 \, \text{moles}
\][/tex]
2. Determine the stoichiometric relationship between [tex]\(O_2\)[/tex] and [tex]\(H_2\)[/tex]:
- According to the reaction equation, [tex]\(1 \, \text{mol}\)[/tex] of [tex]\(O_2\)[/tex] reacts with [tex]\(2 \, \text{mol}\)[/tex] of [tex]\(H_2\)[/tex].
Thus, the moles of [tex]\(H_2\)[/tex] required:
[tex]\[
\text{moles of } H_2 = 0.2734375 \, \text{moles of } O_2 \times 2 = 0.546875 \, \text{moles}
\][/tex]
3. Convert the number of moles of [tex]\(H_2\)[/tex] to mass:
- The molar mass of [tex]\(H_2\)[/tex] is [tex]\(2.02 \, \text{g/mol}\)[/tex].
Using the formula [tex]\( \text{mass} = \text{moles} \times \text{molar mass} \)[/tex]:
[tex]\[
\text{mass of } H_2 = 0.546875 \, \text{moles} \times 2.02 \, \text{g/mol} = 1.1046875 \, \text{g}
\][/tex]
4. Compare the calculated mass with the given options:
- The calculated mass is approximately [tex]\(1.1047 \, \text{g}\)[/tex].
There are provided options of [tex]\(0.547 \, \text{g}\)[/tex], [tex]\(1.09 \, \text{g}\)[/tex], [tex]\(4.38 \, \text{g}\)[/tex], and [tex]\(17.5 \, \text{g}\)[/tex].
The closest value is [tex]\(1.09 \, \text{g}\)[/tex].
Therefore, the mass of [tex]\(H_2\)[/tex] needed to react with [tex]\(8.75 \, \text{g}\)[/tex] of [tex]\(O_2\)[/tex] is [tex]\( \boxed{1.09 \, \text{g} \ H_2} \)[/tex].
