Comprehension given below is followed by some multiple choice question, Each question has one correct options. Choose the correct option.
Molecular orbitals are formed by the overlap of atomic orbitals. Two atomic orbitals combine to form two molecular orbitals called bonding molecular orbital (BMO) and anti-bonding molecular orbital (ABMO). Energy of anti-bonding orbital is raised above the parent atomic orbitals that have combined and hte energy of the bonding orbital is lowered than the parent atomic orbitals.
energies of various molecular orbitals for elements hydrogen to nitrogen increase in the order
`sigma1s lt sigma^(star)1s lt sigma^(star)2s lt ((pi2p_(x))=(pi2p_(y))) lt sigma2p_(z) lt (pi^(star)2p_(x) = pi^(star)2p_(y)) lt sigma^(star)2p_(z)` and For oxygen and fluorine order of enregy of molecules orbitals is given below.
`sigma1s lt sigma^(star)1s lt sigma2s lt sigma^(star)2s lt sigmap_(z) lt (pi2p_(x) ~~ pi2p_(y)) lt (pi^(star)2p_(x)~~ pi^(star)2py) lt sigma^(star)2p_(z)`
Different atomic orbitalsof one atom combine with those atoms orbitals of the second atom which have comparable energies and proper orientation.
Further, if the overlapping is head on, the molecular orbital is called sigma, `sigma` andif the overlap is lateral, the molecular orbital is called pi, `pi`. The molecular orbitals are filled with electrons according to the same rules as followed for filling of atomic orbitals.
However, the order for filling is not the same for all molecules or their ions. Bond order is one of the most important parameters to compare the strength of bonds.
67) Which of the following pair is expected to have the same bond order?

Options

A. `C_(2)`
B. `B_(2)`
C. `O_(2)`
D. `Be_(2)`

Explanation

To determine which pair of molecules has the same bond order, we will calculate the bond order for each molecule mentioned in the question (C2, B2, O2, and Be2) using the bond order formula:

**Bond Order Formula:**
\[ \text{Bond Order} = \frac{(\text{Number of bonding electrons} - \text{Number of anti-bonding electrons})}{2} \]

### Step 1: Calculate the bond order for C2

1. **Total Electrons:** C2 has 2 carbon atoms, each contributing 6 electrons, so total = 12 electrons.
2. **Electronic Configuration:**
- \(\sigma 1s^2\)
- \(\sigma^* 1s^2\)
- \(\sigma 2s^2\)
- \(\sigma^* 2s^2\)
- \(\pi 2p_x^2\) = \(\pi 2p_y^2\)

3. **Count Bonding and Anti-bonding Electrons:**
- Bonding electrons = 2 (from \(\sigma 1s\)) + 2 (from \(\sigma 2s\)) + 4 (from \(\pi 2p\)) = 8
- Anti-bonding electrons = 2 (from \(\sigma^* 1s\)) + 2 (from \(\sigma^* 2s\)) = 4

4. **Calculate Bond Order:**
\[ \text{Bond Order} = \frac{(8 - 4)}{2} = 2 \]

### Step 2: Calculate the bond order for B2

1. **Total Electrons:** B2 has 2 boron atoms, each contributing 5 electrons, so total = 10 electrons.
2. **Electronic Configuration:**
- \(\sigma 1s^2\)
- \(\sigma^* 1s^2\)
- \(\sigma 2s^2\)
- \(\sigma^* 2s^2\)
- \(\pi 2p_x^1\) = \(\pi 2p_y^1\)

3. **Count Bonding and Anti-bonding Electrons:**
- Bonding electrons = 2 (from \(\sigma 1s\)) + 2 (from \(\sigma 2s\)) + 2 (from \(\pi 2p\)) = 6
- Anti-bonding electrons = 2 (from \(\sigma^* 1s\)) + 2 (from \(\sigma^* 2s\)) = 4

4. **Calculate Bond Order:**
\[ \text{Bond Order} = \frac{(6 - 4)}{2} = 1 \]

### Step 3: Calculate the bond order for O2

1. **Total Electrons:** O2 has 2 oxygen atoms, each contributing 8 electrons, so total = 16 electrons.
2. **Electronic Configuration:**
- \(\sigma 1s^2\)
- \(\sigma^* 1s^2\)
- \(\sigma 2s^2\)
- \(\sigma^* 2s^2\)
- \(\sigma 2p_z^2\)
- \(\pi 2p_x^2\) = \(\pi 2p_y^2\)

3. **Count Bonding and Anti-bonding Electrons:**
- Bonding electrons = 2 (from \(\sigma 1s\)) + 2 (from \(\sigma 2s\)) + 2 (from \(\sigma 2p_z\)) + 4 (from \(\pi 2p\)) = 10
- Anti-bonding electrons = 2 (from \(\sigma^* 1s\)) + 2 (from \(\sigma^* 2s\)) = 4

4. **Calculate Bond Order:**
\[ \text{Bond Order} = \frac{(10 - 6)}{2} = 2 \]

### Step 4: Calculate the bond order for Be2

1. **Total Electrons:** Be2 has 2 beryllium atoms, each contributing 4 electrons, so total = 8 electrons.
2. **Electronic Configuration:**
- \(\sigma 1s^2\)
- \(\sigma^* 1s^2\)
- \(\sigma 2s^2\)
- \(\sigma^* 2s^2\)

3. **Count Bonding and Anti-bonding Electrons:**
- Bonding electrons = 2 (from \(\sigma 1s\)) + 2 (from \(\sigma 2s\)) = 4
- Anti-bonding electrons = 2 (from \(\sigma^* 1s\)) + 2 (from \(\sigma^* 2s\)) = 4

4. **Calculate Bond Order:**
\[ \text{Bond Order} = \frac{(4 - 4)}{2} = 0 \]

### Summary of Bond Orders:
- C2: Bond Order = 2
- B2: Bond Order = 1
- O2: Bond Order = 2
- Be2: Bond Order = 0

### Conclusion:
The pairs that have the same bond order are **C2 and O2**, both with a bond order of 2.