To find the heat of reaction for the reaction \( \text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s) \), we will manipulate the given reactions and their enthalpy changes (\( \Delta H \)).
### Given Reactions:
1. \( \text{Fe}_3\text{O}_4(s) \rightarrow 3\text{Fe}(s) + 2\text{O}_2(g), \quad \Delta H = z \, \text{kJ} \) (Reaction 1)
2. \( 2\text{Fe}(s) + \text{O}_2(g) \rightarrow 2\text{FeO}(s), \quad \Delta H = -x \, \text{kJ} \) (Reaction 2)
3. \( 4\text{Fe}(s) + 3\text{O}_2(g) \rightarrow 2\text{Fe}_2\text{O}_3(s), \quad \Delta H = -y \, \text{kJ} \) (Reaction 3)
### Desired Reaction:
We want to find the heat of reaction for:
\[ \text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s) \]
### Steps to Solve:
1. **Invert Reaction 1**:
\[
3\text{Fe}(s) + 2\text{O}_2(g) \rightarrow \text{Fe}_3\text{O}_4(s), \quad \Delta H = -z \, \text{kJ} \quad \text{(Equation 4)}
\]
2. **Invert Reaction 2**:
\[
2\text{FeO}(s) \rightarrow 2\text{Fe}(s) + \text{O}_2(g), \quad \Delta H = +x \, \text{kJ} \quad \text{(Equation 5)}
\]
Divide by 2:
\[
\text{FeO}(s) \rightarrow \text{Fe}(s) + \frac{1}{2}\text{O}_2(g), \quad \Delta H = \frac{x}{2} \, \text{kJ}
\]
3. **Invert Reaction 3**:
\[
2\text{Fe}_2\text{O}_3(s) \rightarrow 4\text{Fe}(s) + 3\text{O}_2(g), \quad \Delta H = +y \, \text{kJ} \quad \text{(Equation 6)}
\]
Divide by 2:
\[
\text{Fe}_2\text{O}_3(s) \rightarrow 2\text{Fe}(s) + \frac{3}{2}\text{O}_2(g), \quad \Delta H = \frac{y}{2} \, \text{kJ}
\]
4. **Add Equations**:
Now we add Equations 4, 5, and 6:
\[
\text{FeO}(s) + \text{Fe}_2\text{O}_3(s) + 3\text{Fe}(s) + 2\text{O}_2(g) \rightarrow \text{Fe}_3\text{O}_4(s) + 2\text{Fe}(s) + \frac{3}{2}\text{O}_2(g)
\]
The \( 3\text{Fe} \) cancels with \( 2\text{Fe} \) and \( 2\text{O}_2 \) cancels with \( \frac{3}{2}\text{O}_2 \), leading to:
\[
\text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s)
\]
5. **Calculate Total Enthalpy Change**:
The total enthalpy change is:
\[
\Delta H = -z + \frac{x}{2} + \frac{y}{2}
\]
Simplifying gives:
\[
\Delta H = \frac{x + y - 2z}{2} \, \text{kJ}
\]
### Final Answer:
The heat of reaction for \( \text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s) \) is:
\[
\Delta H = \frac{x + y - 2z}{2} \, \text{kJ}
\]
