Calculate the resonance enegry of `N_(2)O` form the following data
`Delta_(f)H^(Theta) of N_(2)O = 82 kJ mol^(-1)`
Bond enegry of `N-=N, N=N, O=O,` and `N=O` bond is `946, 418, 498`, and `607 kJ mol^(-1)`, respectively.

Options

A. `-88kJ mol^(-1)`
B. `-178 kJ mol^(-1)`
C. `-252 kJ mol^(-1)`
D. `-36 kJ mol^(-1)`

Explanation

To calculate the resonance energy of \( N_2O \) using the provided data, we will follow these steps:

### Step 1: Write the reaction for the formation of \( N_2O \)
The formation reaction can be represented as:
\[ N_2(g) + \frac{1}{2} O_2(g) \rightarrow N_2O(g) \]

### Step 2: Identify the bond energies
The bond energies given are:
- \( N \equiv N \) (triple bond) = 946 kJ/mol
- \( N = N \) (double bond) = 418 kJ/mol
- \( O = O \) (double bond) = 498 kJ/mol
- \( N = O \) (double bond) = 607 kJ/mol

### Step 3: Calculate the total bond energy of the reactants
The total bond energy for the reactants \( N_2 \) and \( \frac{1}{2} O_2 \) is:
\[ \text{Bond energy of } N_2 + \text{Bond energy of } \frac{1}{2} O_2 = 946 + \frac{1}{2} \times 498 \]
Calculating this gives:
\[ 946 + 249 = 1195 \text{ kJ/mol} \]

### Step 4: Calculate the total bond energy of the products
For the product \( N_2O \), we have one \( N = N \) bond and one \( N = O \) bond:
\[ \text{Bond energy of } N_2O = \text{Bond energy of } N = N + \text{Bond energy of } N = O = 418 + 607 \]
Calculating this gives:
\[ 418 + 607 = 1025 \text{ kJ/mol} \]

### Step 5: Calculate the enthalpy change (\( \Delta H \)) for the reaction
Using the bond energies calculated:
\[ \Delta H = \text{Bond energy of reactants} - \text{Bond energy of products} \]
\[ \Delta H = 1195 - 1025 = 170 \text{ kJ/mol} \]

### Step 6: Calculate the resonance energy
The resonance energy can be calculated using the formula:
\[ \text{Resonance Energy} = \Delta H_f^{\circ} (N_2O) - \Delta H \]
Where \( \Delta H_f^{\circ} (N_2O) \) is given as 82 kJ/mol:
\[ \text{Resonance Energy} = 82 - 170 = -88 \text{ kJ/mol} \]

### Final Answer
The resonance energy of \( N_2O \) is \( -88 \text{ kJ/mol} \).

---