Calculate the resonance energy of `N_(2)O` from the following data:
`DeltaH_(1)^(@)` of `N_(2)O = 82 kJ "mole"^(-1)`.
Bond energies `N=N = 946 kJ "mole"^(-1)`
`N=N = 418 kJ "mole"^(-1)`
`O=O = 498 kJ "mole"^(-1)`
`N=O = 607 kJ "mol"^(-1)`

Answer

To calculate the resonance energy of \( N_2O \), we can follow these steps:

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

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

### Step 3: Calculate the energy required to break the bonds in the reactants
The total energy required to break the bonds in the reactants \( N_2 \) and \( O_2 \) is:
\[ \text{Energy required} = \text{Bond energy of } N \equiv N + \frac{1}{2} \times \text{Bond energy of } O = O \]
Substituting the values:
\[ \text{Energy required} = 946 \, \text{kJ/mol} + \frac{1}{2} \times 498 \, \text{kJ/mol} = 946 + 249 = 1195 \, \text{kJ/mol} \]

### Step 4: Calculate the energy released during the formation of bonds in \( N_2O \)
The total energy released when forming \( N_2O \) involves breaking one \( N = N \) bond and one \( N = O \) bond:
\[ \text{Energy released} = \text{Bond energy of } N = N + \text{Bond energy of } N = O \]
Substituting the values:
\[ \text{Energy released} = 418 \, \text{kJ/mol} + 607 \, \text{kJ/mol} = 1025 \, \text{kJ/mol} \]

### Step 5: Calculate the enthalpy change (\( \Delta H \))
Using the formula:
\[ \Delta H = \text{Energy required} - \text{Energy released} \]
Substituting the values:
\[ \Delta H = 1195 \, \text{kJ/mol} - 1025 \, \text{kJ/mol} = 170 \, \text{kJ/mol} \]

### Step 6: Calculate the resonance energy
Given that the observed \( \Delta H \) for \( N_2O \) is 82 kJ/mol, we can find the resonance energy using:
\[ \text{Resonance Energy} = \text{Expected } \Delta H - \text{Observed } \Delta H \]
Substituting the values:
\[ \text{Resonance Energy} = 170 \, \text{kJ/mol} - 82 \, \text{kJ/mol} = 88 \, \text{kJ/mol} \]

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

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