According to molecular orbital theory, which of the following is true with respect to `Li_(2)+"and"Li_(2)-?`

Options

A. 1.Both are stable
B. 2.`Li_(2)^(+)` is stable and `Li_(2)^(-)` is unstable
C. 3.`Li_(2)^(+)` is unstable and `Li_(2)^(-)` is stable
D. 4.Both are unstable

Explanation

To determine which of the statements regarding \( \text{Li}_2^+ \) and \( \text{Li}_2^- \) is true according to molecular orbital theory, we need to calculate the bond order for both species and analyze their stability.

### Step 1: Determine the number of electrons in \( \text{Li}_2^+ \)

- Lithium (Li) has 3 electrons.
- For \( \text{Li}_2 \), there are a total of 6 electrons (3 from each Li atom).
- \( \text{Li}_2^+ \) has one less electron, so it has 5 electrons.

### Step 2: Fill the molecular orbitals for \( \text{Li}_2^+ \)

The molecular orbital configuration for \( \text{Li}_2^+ \) is:
- \( \sigma_{1s}^2 \)
- \( \sigma_{1s}^*^2 \)
- \( \sigma_{2s}^2 \)
- \( \sigma_{2s}^*^0 \)
- \( \sigma_{2p}^0 \)

So, the configuration is:
- \( \sigma_{1s}^2 \) (2 electrons in bonding)
- \( \sigma_{1s}^*^2 \) (2 electrons in anti-bonding)
- \( \sigma_{2s}^2 \) (2 electrons in bonding)
- \( \sigma_{2s}^*^0 \) (0 electrons in anti-bonding)

### Step 3: Calculate the bond order for \( \text{Li}_2^+ \)

Using the bond order formula:
\[ \text{Bond Order} = \frac{\text{Number of electrons in bonding orbitals} - \text{Number of electrons in anti-bonding orbitals}}{2} \]
For \( \text{Li}_2^+ \):
- Number of electrons in bonding orbitals = 4 (2 from \( \sigma_{1s} \) and 2 from \( \sigma_{2s} \))
- Number of electrons in anti-bonding orbitals = 2 (2 from \( \sigma_{1s}^* \))

Thus,
\[ \text{Bond Order} = \frac{4 - 2}{2} = 1 \]

### Step 4: Determine the number of electrons in \( \text{Li}_2^- \)

- \( \text{Li}_2^- \) has one extra electron compared to \( \text{Li}_2 \), so it has 7 electrons.

### Step 5: Fill the molecular orbitals for \( \text{Li}_2^- \)

The molecular orbital configuration for \( \text{Li}_2^- \) is:
- \( \sigma_{1s}^2 \)
- \( \sigma_{1s}^*^2 \)
- \( \sigma_{2s}^2 \)
- \( \sigma_{2s}^*^2 \)
- \( \sigma_{2p}^0 \)

So, the configuration is:
- \( \sigma_{1s}^2 \) (2 electrons in bonding)
- \( \sigma_{1s}^*^2 \) (2 electrons in anti-bonding)
- \( \sigma_{2s}^2 \) (2 electrons in bonding)
- \( \sigma_{2s}^*^1 \) (1 electron in anti-bonding)

### Step 6: Calculate the bond order for \( \text{Li}_2^- \)

Using the bond order formula:
For \( \text{Li}_2^- \):
- Number of electrons in bonding orbitals = 4 (2 from \( \sigma_{1s} \) and 2 from \( \sigma_{2s} \))
- Number of electrons in anti-bonding orbitals = 3 (2 from \( \sigma_{1s}^* \) and 1 from \( \sigma_{2s}^* \))

Thus,
\[ \text{Bond Order} = \frac{4 - 3}{2} = 0.5 \]

### Step 7: Compare the stability of \( \text{Li}_2^+ \) and \( \text{Li}_2^- \)

- \( \text{Li}_2^+ \) has a bond order of 1, while \( \text{Li}_2^- \) has a bond order of 0.5.
- Since stability is directly proportional to bond order, \( \text{Li}_2^+ \) is more stable than \( \text{Li}_2^- \).

### Conclusion

Thus, the correct statement is that \( \text{Li}_2^+ \) is more stable than \( \text{Li}_2^- \).

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