If the heat of formation of `NO_(2) ` is 'x' `[1/2 N_(2)(g) + O_(2)(g) rightarrow NO_(2)(g)]` the heat of reaction `N_(2)(g) + O_(2)(g) rightarrow 2NO(g) ` is y and the heat of reaction `2NO_(g) + O_(2)(g) rightarrow 2NO_(2)(g)` is z, then

Options

A. (a) `2x + z = y`
B. (b) `2y + z = x`
C. (c) `2x - z = y`
D. (d) `2z + x = y`

Answer

To solve the problem, we need to analyze the given reactions and their associated heats of formation and reaction.

### Step-by-Step Solution:

1. **Identify the Reactions and Their Heats**:
- The heat of formation of \( NO_2 \) is given as \( x \):
\[ \frac{1}{2} N_2(g) + O_2(g) \rightarrow NO_2(g) \quad \text{(Heat = } x\text{)} \]
- The heat of reaction for the formation of \( 2NO \) is given as \( y \):
\[ N_2(g) + O_2(g) \rightarrow 2NO(g) \quad \text{(Heat = } y\text{)} \]
- The heat of reaction for the formation of \( 2NO_2 \) from \( 2NO \) and \( O_2 \) is given as \( z \):
\[ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) \quad \text{(Heat = } z\text{)} \]

2. **Manipulate the First Reaction**:
- To find the heat for the formation of \( 2NO_2 \), we can multiply the first reaction by 2:
\[ N_2(g) + 2O_2(g) \rightarrow 2NO_2(g) \quad \text{(Heat = } 2x\text{)} \]

3. **Add the Reactions**:
- Now, we can add the second reaction and the modified first reaction:
\[ N_2(g) + O_2(g) \rightarrow 2NO(g) \quad \text{(Heat = } y\text{)} \]
\[ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) \quad \text{(Heat = } z\text{)} \]
- When we add these two reactions, \( 2NO(g) \) cancels out:
\[ N_2(g) + 2O_2(g) \rightarrow 2NO_2(g) \]
- The total heat for this overall reaction is:
\[ y + z \]

4. **Set Up the Equation**:
- Since the heat for the overall reaction \( N_2(g) + 2O_2(g) \rightarrow 2NO_2(g) \) is also equal to \( 2x \), we can equate the two expressions:
\[ y + z = 2x \]

5. **Rearranging the Equation**:
- Rearranging gives us:
\[ 2x - z = y \]

### Final Relation:
Thus, we have derived the relationship:
\[ 2x - z = y \]