Stability of the species `Li_(2), Li_(2)^(-)` and `Li_(2)^(+)` increases in the order of

Options

A. `Li_(2)^(-)ltLi_(2)^(+)ltLi_(2)`
B. `Li_(2)ltLi_(2)^(-)ltLi_(2)^(+)`
C. `Li_(2)^(-)ltLi_(2)ltLi_(2)^(+)`
D. `Li_(2)ltLi_(2)^(+)ltLi_(2)^(-)`

Answer

To determine the stability of the species \( \text{Li}_2 \), \( \text{Li}_2^- \), and \( \text{Li}_2^+ \), we will calculate the bond order for each species using Molecular Orbital Theory (MOT). The bond order is calculated using the formula:

\[ \text{Bond Order} = \frac{N_B - N_A}{2} \]

where \( N_B \) is the number of electrons in bonding molecular orbitals and \( N_A \) is the number of electrons in anti-bonding molecular orbitals.

### Step 1: Calculate Bond Order for \( \text{Li}_2 \)

1. **Determine the total number of electrons**:
- Each lithium atom has 3 electrons. Therefore, for \( \text{Li}_2 \), the total number of electrons is \( 2 \times 3 = 6 \).

2. **Molecular orbital configuration**:
- The configuration for \( \text{Li}_2 \) is:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \]
- This means:
- \( N_B = 4 \) (from \( \sigma_{1s}^2 \) and \( \sigma_{2s}^2 \))
- \( N_A = 2 \) (from \( \sigma_{1s}^*^2 \))

3. **Calculate bond order**:
\[ \text{Bond Order} = \frac{4 - 2}{2} = 1 \]

### Step 2: Calculate Bond Order for \( \text{Li}_2^+ \)

1. **Determine the total number of electrons**:
- \( \text{Li}_2^+ \) has one less electron than \( \text{Li}_2 \), so it has \( 6 - 1 = 5 \) electrons.

2. **Molecular orbital configuration**:
- The configuration for \( \text{Li}_2^+ \) is:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^1 \]
- This means:
- \( N_B = 3 \) (from \( \sigma_{1s}^2 \) and \( \sigma_{2s}^1 \))
- \( N_A = 2 \) (from \( \sigma_{1s}^*^2 \))

3. **Calculate bond order**:
\[ \text{Bond Order} = \frac{3 - 2}{2} = \frac{1}{2} = 0.5 \]

### Step 3: Calculate Bond Order for \( \text{Li}_2^- \)

1. **Determine the total number of electrons**:
- \( \text{Li}_2^- \) has one more electron than \( \text{Li}_2 \), so it has \( 6 + 1 = 7 \) electrons.

2. **Molecular orbital configuration**:
- The configuration for \( \text{Li}_2^- \) is:
\[ \sigma_{1s}^2 \sigma_{1s}^*^2 \sigma_{2s}^2 \sigma_{2s}^*^1 \]
- This means:
- \( N_B = 4 \) (from \( \sigma_{1s}^2 \) and \( \sigma_{2s}^2 \))
- \( N_A = 3 \) (from \( \sigma_{1s}^*^2 \) and \( \sigma_{2s}^*^1 \))

3. **Calculate bond order**:
\[ \text{Bond Order} = \frac{4 - 3}{2} = \frac{1}{2} = 0.5 \]

### Step 4: Compare the Stability

- The bond orders calculated are:
- \( \text{Li}_2 \): 1 (most stable)
- \( \text{Li}_2^+ \): 0.5
- \( \text{Li}_2^- \): 0.5

Since both \( \text{Li}_2^+ \) and \( \text{Li}_2^- \) have the same bond order, we need to consider the number of electrons in the anti-bonding orbitals. \( \text{Li}_2^+ \) has 2 electrons in anti-bonding orbitals, while \( \text{Li}_2^- \) has 3. Therefore, \( \text{Li}_2^+ \) is more stable than \( \text{Li}_2^- \).

### Conclusion

The stability of the species increases in the order:

\[ \text{Li}_2^- < \text{Li}_2^+ < \text{Li}_2 \]