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. `2x + z = y`
B. `2y + z = x`
C. `2x - z = y`
D. `2z + x = y`

Answer

To solve the problem, we need to establish the relationships between the heat of formation of \( NO_2 \) (denoted as \( x \)), the heat of reaction for \( N_2 + O_2 \rightarrow 2NO \) (denoted as \( y \)), and the heat of reaction for \( 2NO + O_2 \rightarrow 2NO_2 \) (denoted as \( z \)).

### Step-by-Step Solution:

1. **Write the equations for the reactions:**
- The heat of formation of \( NO_2 \):
\[ \frac{1}{2} N_2(g) + \frac{1}{2} O_2(g) \rightarrow NO_2(g) \quad \Delta H = x \]
- The heat of reaction for \( N_2 + O_2 \rightarrow 2NO \):
\[ N_2(g) + O_2(g) \rightarrow 2NO(g) \quad \Delta H = y \]
- The heat of reaction for \( 2NO + O_2 \rightarrow 2NO_2 \):
\[ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) \quad \Delta H = z \]

2. **Reverse the second reaction:**
To relate these reactions, we will reverse the second reaction:
\[ 2NO(g) \rightarrow N_2(g) + O_2(g) \quad \Delta H = -y \]

3. **Multiply the first reaction by 2:**
To eliminate the fractions, we multiply the first reaction by 2:
\[ N_2(g) + O_2(g) \rightarrow 2NO_2(g) \quad \Delta H = 2x \]

4. **Combine the modified reactions:**
Now, we add the reversed second reaction and the modified first reaction:
\[ (2NO(g) \rightarrow N_2(g) + O_2(g)) + (N_2(g) + O_2(g) \rightarrow 2NO_2(g)) \]
This simplifies to:
\[ 2NO(g) + O_2(g) \rightarrow 2NO_2(g) \]

5. **Combine the enthalpy changes:**
The total enthalpy change for this combined reaction is:
\[ \Delta H = -y + 2x \]

6. **Set the total enthalpy equal to \( z \):**
Since the combined reaction is equivalent to the third reaction, we have:
\[ z = -y + 2x \]

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

### Final Result:
Thus, the relationship between \( x \), \( y \), and \( z \) is:
\[ y = 2x - z \]