The correct stability order of the following resonance structures is :
x`H_(2)C=overset(o+)underset((I))N=overset(Ɵ)N`
`H_(2)overset(o+)Cunderset((II))-N=overset(Ɵ)N`
`H_(2)overset(Ɵ)C-underset((III))overset(o+)N-=N`
`H_(2)overset(Ɵ)C-underset((IV))N=overset(o+)N`

Options

A. `IgtIIgtIVgtIII`
B. `IgtIIIgtIIgtIV`
C. `IIgtIgtIIIgtIV`
D. `IIIgtIgtIVgtII`

Explanation

To determine the stability order of the given resonance structures, we need to analyze each structure based on the octet rule and the placement of charges. Here’s a step-by-step breakdown of the analysis:

### Step 1: Analyze Each Resonance Structure
1. **Structure I**: \( H_2C = N^+ - N^- \)
- Carbon has 4 bonds (2 from H and 2 from N), fulfilling the octet rule.
- Nitrogen \( N^+ \) has 6 electrons (3 bonds), and \( N^- \) has 8 electrons (4 bonds).
- Overall, all atoms have complete octets.

2. **Structure II**: \( H_2C^+ - N = N^- \)
- Carbon has 3 bonds (2 from H and 1 from N), which means it has only 6 electrons (not fulfilling the octet rule).
- The nitrogen \( N^- \) has 8 electrons (4 bonds), while the other nitrogen has 6 electrons (3 bonds).
- This structure is less stable due to carbon not fulfilling the octet.

3. **Structure III**: \( H_2C - N \equiv N^+ \)
- Carbon has 4 bonds (2 from H and 2 from N), fulfilling the octet rule.
- The terminal nitrogen \( N^+ \) has 6 electrons (3 bonds), while the other nitrogen has 8 electrons (4 bonds).
- This structure is stable as carbon fulfills the octet.

4. **Structure IV**: \( H_2C - N = N^+ \)
- Carbon has 4 bonds (2 from H and 2 from N), fulfilling the octet rule.
- The nitrogen \( N^+ \) has 6 electrons (3 bonds), while the other nitrogen has 8 electrons (4 bonds).
- This structure is stable, but one nitrogen has a positive charge.

### Step 2: Evaluate Stability Based on Octet Rule and Charge Distribution
- Structures that fulfill the octet rule are generally more stable.
- Positive charges should ideally be on less electronegative atoms, while negative charges should be on more electronegative atoms.

### Step 3: Compare the Structures
- **Structure I** is stable with all atoms fulfilling the octet rule and has a positive charge on nitrogen (more electronegative).
- **Structure III** is also stable with fulfilled octets but has a positive charge on nitrogen.
- **Structure IV** has fulfilled octets but has a positive charge on nitrogen.
- **Structure II** is the least stable as carbon does not fulfill the octet rule.

### Conclusion: Stability Order
Based on the analysis:
1. Structure I: \( H_2C = N^+ - N^- \) (most stable)
2. Structure III: \( H_2C - N \equiv N^+ \)
3. Structure IV: \( H_2C - N = N^+ \)
4. Structure II: \( H_2C^+ - N = N^- \) (least stable)

Thus, the correct stability order is:
**I > III > IV > II**