Which of the following is the correct order of stability?

Options

A. `H_2gtH_2^(+)gtHe_2gtHe_2^(+)`
B. `H_2gtHe_2^(+)gtH_2^(+)gtHe_2`
C. `H_2gtH_2^(+)gtHe_2^(+)gtHe_2`
D. `H_2gtHe_2gtH_2^(+)gtHe_2^(+)`

Explanation

To determine the correct order of stability among the given molecules (H2, H2+, He2, and He2+), we will analyze their bond orders and the number of electrons in their anti-bonding molecular orbitals.

### Step-by-Step Solution:

1. **Determine the Molecular Orbital Configuration for H2:**
- The molecular orbital configuration for H2 is:
\[ \sigma_{1s}^2 \]
- Here, \(N_b = 2\) (bonding electrons) and \(N_a = 0\) (anti-bonding electrons).
- **Bond Order Calculation:**
\[ \text{Bond Order} = \frac{N_b - N_a}{2} = \frac{2 - 0}{2} = 1 \]

2. **Determine the Molecular Orbital Configuration for H2+:**
- The molecular orbital configuration for H2+ (which has one less electron) is:
\[ \sigma_{1s}^1 \]
- Here, \(N_b = 1\) and \(N_a = 0\).
- **Bond Order Calculation:**
\[ \text{Bond Order} = \frac{N_b - N_a}{2} = \frac{1 - 0}{2} = \frac{1}{2} \]

3. **Determine the Molecular Orbital Configuration for He2:**
- The molecular orbital configuration for He2 is:
\[ \sigma_{1s}^2 \sigma^*_{1s}^2 \]
- Here, \(N_b = 2\) and \(N_a = 2\).
- **Bond Order Calculation:**
\[ \text{Bond Order} = \frac{N_b - N_a}{2} = \frac{2 - 2}{2} = 0 \]

4. **Determine the Molecular Orbital Configuration for He2+:**
- The molecular orbital configuration for He2+ (which has one less electron) is:
\[ \sigma_{1s}^2 \sigma^*_{1s}^1 \]
- Here, \(N_b = 2\) and \(N_a = 1\).
- **Bond Order Calculation:**
\[ \text{Bond Order} = \frac{N_b - N_a}{2} = \frac{2 - 1}{2} = \frac{1}{2} \]

5. **Compare the Bond Orders:**
- H2: Bond Order = 1 (most stable)
- H2+: Bond Order = 1/2
- He2+: Bond Order = 1/2
- He2: Bond Order = 0 (least stable)

6. **Determine Stability Based on Bond Order:**
- The stability of the molecules can be ranked based on their bond orders. Since H2 has the highest bond order, it is the most stable.
- Between H2+ and He2+, both have the same bond order (1/2), but H2+ has 0 electrons in the anti-bonding orbital while He2+ has 1 electron in the anti-bonding orbital. Therefore, H2+ is more stable than He2+.

### Final Order of Stability:
- H2 > H2+ > He2+ > He2