To determine how many grams of [tex]\( \text{H}_2\text{O} \)[/tex] will be produced when 354.4 grams of [tex]\(\text{Fe}\)[/tex] are produced, follow these steps:
### Step 1: Identify Molar Masses of All Relevant Substances
- Molar mass of [tex]\(\text{Fe}\)[/tex]: [tex]\( 55.85 \)[/tex] g/mol
- Molar mass of [tex]\(\text{H}_2\text{O}\)[/tex]: [tex]\( 18.01528 \)[/tex] g/mol
- Molar mass of [tex]\(\text{Fe}_3\text{O}_4\)[/tex]: [tex]\( 231.533 \)[/tex] g/mol
- Molar mass of [tex]\(\text{H}_2\)[/tex]: [tex]\( 2.01588 \)[/tex] g/mol
### Step 2: Balanced Chemical Equation
The balanced chemical equation is:
[tex]\[ \text{Fe}_3\text{O}_4 + 4 \text{H}_2 \rightarrow 3 \text{Fe} + 4 \text{H}_2\text{O} \][/tex]
From this equation, you can see the stoichiometric ratios:
- 1 mole of [tex]\(\text{Fe}_3\text{O}_4\)[/tex] produces 3 moles of Fe
- 3 moles of Fe are produced along with 4 moles of [tex]\(\text{H}_2\text{O}\)[/tex]
### Step 3: Calculate Moles of [tex]\(\text{Fe}\)[/tex] Produced
First, determine the number of moles of Fe produced from 354.4 grams of Fe:
[tex]\[ \text{Moles of Fe} = \frac{\text{mass of Fe}}{\text{molar mass of Fe}} \][/tex]
[tex]\[ \text{Moles of Fe} = \frac{354.4 \, \text{grams}}{55.85 \, \text{g/mol}} \][/tex]
[tex]\[ \text{Moles of Fe} \approx 6.346 \, \text{moles} \][/tex]
### Step 4: Determine Moles of [tex]\(\text{H}_2\text{O}\)[/tex] Produced
Use the stoichiometric relationship from the balanced equation to find the moles of [tex]\(\text{H}_2\text{O}\)[/tex] produced. According to the equation:
[tex]\[ 3 \, \text{moles of Fe} \propto 4 \, \text{moles of } \text{H}_2\text{O} \][/tex]
Therefore, the moles of [tex]\(\text{H}_2\text{O}\)[/tex] produced is:
[tex]\[ \text{Moles of } \text{H}_2\text{O} = \left( \frac{4}{3} \right) \times \text{Moles of Fe} \][/tex]
[tex]\[ \text{Moles of } \text{H}_2\text{O} = \left( \frac{4}{3} \right) \times 6.346 \, \text{moles} \][/tex]
[tex]\[ \text{Moles of } \text{H}_2\text{O} \approx 8.462 \, \text{moles} \][/tex]
### Step 5: Convert Moles of [tex]\(\text{H}_2\text{O}\)[/tex] to Grams
Finally, convert the moles of [tex]\(\text{H}_2\text{O}\)[/tex] to grams using its molar mass:
[tex]\[ \text{Mass of } \text{H}_2\text{O} = \text{Moles of } \text{H}_2\text{O} \times \text{Molar mass of } \text{H}_2\text{O} \][/tex]
[tex]\[ \text{Mass of } \text{H}_2\text{O} \approx 8.462 \, \text{moles} \times 18.01528 \, \text{g/mol} \][/tex]
[tex]\[ \text{Mass of } \text{H}_2\text{O} \approx 152.42292407042675 \, \text{grams} \][/tex]
### Step 6: Choose the Closest Option
Given the choices:
a. 1500
b. 150
c. 330
d. 29
The mass of [tex]\( \text{H}_2\text{O} \)[/tex] closest to our calculated value of approximately 152.42 grams is:
Option b. 150
Therefore, the correct answer is:
[tex]\[ \boxed{150} \][/tex] grams of [tex]\(\text{H}_2\text{O}\)[/tex].
