Consider the statement ltbrlt I. Bond length in `N_(2)^(+)` is `0.02Å` greater than in `N_(2)`.
II. Bond length of `NO^(+)` is `0.09Å` less than in NO.
III `O_(2)^(2_(-))` has shorter bond length than `O_(2)`.
Which of the following statements are true?

Options

A. I and II
B. II and III
C. I , II and III
D. I and III

Answer

To solve the question, we need to analyze the bond lengths and bond orders of the given species: \( N_2^+ \), \( NO^+ \), and \( O_2^{2-} \). Let's break down the statements one by one.

### Step 1: Analyze the bond order of \( N_2 \) and \( N_2^+ \)

1. **Bond Order of \( N_2 \)**:
- The molecular orbital configuration of \( N_2 \) is \( \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^2 \sigma_{2p_z}^2 \pi_{2p_x}^2 \pi_{2p_y}^2 \).
- The bond order is calculated as:
\[ \text{Bond Order} = \frac{(\text{Number of bonding electrons} - \text{Number of antibonding electrons})}{2} \]
- For \( N_2 \), the bond order is \( 3 \) (6 bonding electrons - 0 antibonding electrons).

2. **Bond Order of \( N_2^+ \)**:
- For \( N_2^+ \), one electron is removed from the bonding molecular orbital.
- The bond order becomes \( 2.5 \) (5 bonding electrons - 0 antibonding electrons).

### Conclusion for Statement I:
- Since \( N_2 \) has a bond order of \( 3 \) and \( N_2^+ \) has a bond order of \( 2.5 \), the bond length of \( N_2^+ \) is greater than that of \( N_2 \).
- **Statement I is True**.

### Step 2: Analyze the bond order of \( NO \) and \( NO^+ \)

1. **Bond Order of \( NO \)**:
- The molecular orbital configuration of \( NO \) is similar to \( N_2 \) but has an additional electron.
- The bond order for \( NO \) is \( 2.5 \).

2. **Bond Order of \( NO^+ \)**:
- For \( NO^+ \), one electron is removed from the antibonding orbital.
- The bond order becomes \( 3 \) (5 bonding electrons - 1 antibonding electron).

### Conclusion for Statement II:
- Since \( NO \) has a bond order of \( 2.5 \) and \( NO^+ \) has a bond order of \( 3 \), the bond length of \( NO^+ \) is less than that of \( NO \).
- **Statement II is True**.

### Step 3: Analyze the bond order of \( O_2 \) and \( O_2^{2-} \)

1. **Bond Order of \( O_2 \)**:
- The bond order of \( O_2 \) is \( 2 \) (10 bonding electrons - 6 antibonding electrons).

2. **Bond Order of \( O_2^{2-} \)**:
- For \( O_2^{2-} \), two electrons are added to the antibonding orbital.
- The bond order becomes \( 1 \) (8 bonding electrons - 6 antibonding electrons).

### Conclusion for Statement III:
- Since \( O_2 \) has a bond order of \( 2 \) and \( O_2^{2-} \) has a bond order of \( 1 \), the bond length of \( O_2^{2-} \) is greater than that of \( O_2 \).
- **Statement III is False**.

### Final Answer:
- The true statements are **I and II**. Therefore, the correct answer is that statements I and II are true.

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