Consider the following reactions. `DeltaH^(@)` values of the reactions hve been given as `-x, -y` and z kJ.
`Fe_(3)O_(4)(s) to 3Fe(s) + 2O_(2)(g), DeltaH^(@) = z kJ`
`2Fe(s) + O_(2)(g) to 2FeO(s), DeltaH^(@) =-x kJ`
`4Fe(s) + 3O_(2)(g) to 2Fe_(2)O_(3)(s), DeltaH^(@) =-y kJ`
Heat of reaction for the reaction, `FeO(s) + Fe_(2)O_(3)(s) to Fe_(3)O_(4)(s)` is :

Options

A. `(x+y+z)/2`
B. `(2z-x-y)/2`
C. `(x+y-2z)/2`
D. `(2x+2y+z)/2`

Explanation

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 use Hess's law and the given reactions with their respective enthalpy changes.

### Given Reactions:
1. \( \text{Fe}_3\text{O}_4(s) \rightarrow 3\text{Fe}(s) + 2\text{O}_2(g) \), \( \Delta H = z \, \text{kJ} \) (Equation 1)
2. \( 2\text{Fe}(s) + \text{O}_2(g) \rightarrow 2\text{FeO}(s) \), \( \Delta H = -x \, \text{kJ} \) (Equation 2)
3. \( 4\text{Fe}(s) + 3\text{O}_2(g) \rightarrow 2\text{Fe}_2\text{O}_3(s) \), \( \Delta H = -y \, \text{kJ} \) (Equation 3)

### Target Reaction:
We need to find \( \Delta H \) for the reaction:
\[ \text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s) \]

### Step-by-Step Solution:

1. **Reverse Equation 1**:
- To use it in our target reaction, we reverse Equation 1:
\[ 3\text{Fe}(s) + 2\text{O}_2(g) \rightarrow \text{Fe}_3\text{O}_4(s) \quad \Delta H = -z \, \text{kJ} \]

2. **Use Equation 2**:
- We will keep Equation 2 as it is:
\[ 2\text{Fe}(s) + \text{O}_2(g) \rightarrow 2\text{FeO}(s) \quad \Delta H = -x \, \text{kJ} \]

3. **Use Equation 3**:
- We will also keep Equation 3 as it is:
\[ 4\text{Fe}(s) + 3\text{O}_2(g) \rightarrow 2\text{Fe}_2\text{O}_3(s) \quad \Delta H = -y \, \text{kJ} \]

4. **Manipulate Equations**:
- From Equation 2, we can derive \( \text{FeO}(s) \):
\[ \text{FeO}(s) \rightarrow \text{Fe}(s) + \frac{1}{2}\text{O}_2(g) \quad \Delta H = \frac{x}{2} \, \text{kJ} \]
- From Equation 3, we can derive \( \text{Fe}_2\text{O}_3(s) \):
\[ 2\text{Fe}_2\text{O}_3(s) \rightarrow 4\text{Fe}(s) + 3\text{O}_2(g) \quad \Delta H = \frac{y}{2} \, \text{kJ} \]

5. **Combine the Reactions**:
- Now we can add the modified reactions:
\[ \text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s) \]
- The total enthalpy change will be:
\[ \Delta H = \frac{x}{2} + \frac{y}{2} - z \]

6. **Final Expression**:
- Thus, we can express the heat of reaction as:
\[ \Delta H = \frac{x + y - 2z}{2} \, \text{kJ} \]

### Conclusion:
The heat of reaction for \( \text{FeO}(s) + \text{Fe}_2\text{O}_3(s) \rightarrow \text{Fe}_3\text{O}_4(s) \) is given by:
\[ \Delta H = \frac{x + y - 2z}{2} \, \text{kJ} \]