Among the following molecular orbitals how many have at least one nodal plane? `sigma^ast1s`,`sigma2s`,`sigma^ast2p_z`,`pi2p_x`,`pi^ast2p_y`, `sigma 2p_z`

Options

A. 4
B. 5
C. 2
D. 6

Answer

To determine how many of the given molecular orbitals have at least one nodal plane, we first need to understand the concept of nodal planes in molecular orbitals.

### Step-by-Step Solution:

1. **Identify the Molecular Orbitals**: The given molecular orbitals are:
- \( \sigma^* 1s \)
- \( \sigma 2s \)
- \( \sigma^* 2p_z \)
- \( \pi 2p_x \)
- \( \pi^* 2p_y \)
- \( \sigma 2p_z \)

2. **Understanding Nodal Planes**: A nodal plane is a region in a molecular orbital where the probability of finding an electron is zero. The number of nodal planes can be determined based on the type of orbital:
- **s orbitals** (like \(1s\) and \(2s\)) have no nodal planes.
- **p orbitals** (like \(2p_x\), \(2p_y\), and \(2p_z\)) have one nodal plane.
- **Antibonding orbitals** (like \( \sigma^* \) and \( \pi^* \)) have one additional nodal plane compared to their bonding counterparts.

3. **Analyzing Each Orbital**:
- \( \sigma^* 1s \): This is an antibonding orbital and has 1 nodal plane.
- \( \sigma 2s \): This is a bonding orbital and has 0 nodal planes.
- \( \sigma^* 2p_z \): This is an antibonding orbital and has 2 nodal planes (1 from being a p orbital and 1 additional from being antibonding).
- \( \pi 2p_x \): This is a bonding orbital and has 1 nodal plane.
- \( \pi^* 2p_y \): This is an antibonding orbital and has 2 nodal planes (1 from being a p orbital and 1 additional from being antibonding).
- \( \sigma 2p_z \): This is a bonding orbital and has 1 nodal plane.

4. **Count the Orbitals with Nodal Planes**:
- \( \sigma^* 1s \): 1 nodal plane
- \( \sigma 2s \): 0 nodal planes
- \( \sigma^* 2p_z \): 2 nodal planes
- \( \pi 2p_x \): 1 nodal plane
- \( \pi^* 2p_y \): 2 nodal planes
- \( \sigma 2p_z \): 1 nodal plane

5. **Total Count**: Now, we count the orbitals that have at least one nodal plane:
- \( \sigma^* 1s \) (1)
- \( \sigma^* 2p_z \) (2)
- \( \pi 2p_x \) (1)
- \( \pi^* 2p_y \) (2)
- \( \sigma 2p_z \) (1)

This gives us a total of 5 orbitals that have at least one nodal plane.

### Final Answer:
**5 molecular orbitals have at least one nodal plane.**