How many of these orbitals have maximum orbital angular distribution is maximum at an angle of `45^(@)` to the axial direction.
`d_(xy), d_(x^(2)-y^(2)) d_(yz), d_(xz), d_(z^(2)), P_(x) P_(y)P_(z), s`

Answer

To determine how many of the given orbitals have maximum orbital angular distribution at an angle of 45 degrees to the axial direction, we can analyze each orbital type listed in the question.

### Step-by-Step Solution:

1. **Identify the Orbitals**: The orbitals given are:
- \( d_{xy} \)
- \( d_{x^2-y^2} \)
- \( d_{yz} \)
- \( d_{xz} \)
- \( d_{z^2} \)
- \( p_x \)
- \( p_y \)
- \( p_z \)
- \( s \)

2. **Understand Orbital Shapes**:
- **d Orbitals**:
- \( d_{xy} \): Has lobes in the xy-plane and is oriented at 45 degrees to the x and y axes.
- \( d_{x^2-y^2} \): Lies along the x and y axes, not at 45 degrees.
- \( d_{yz} \): Has lobes in the yz-plane and is oriented at 45 degrees to the y and z axes.
- \( d_{xz} \): Has lobes in the xz-plane and is oriented at 45 degrees to the x and z axes.
- \( d_{z^2} \): Has a different shape, primarily along the z-axis, not at 45 degrees.

- **p Orbitals**:
- \( p_x \): Lies along the x-axis, not at 45 degrees.
- \( p_y \): Lies along the y-axis, not at 45 degrees.
- \( p_z \): Lies along the z-axis, not at 45 degrees.

- **s Orbital**: The s orbital is spherical and does not have a directional preference, hence not at 45 degrees.

3. **Determine Maximum Angular Distribution**:
- The orbitals that have maximum angular distribution at 45 degrees to the axial direction are:
- \( d_{xy} \)
- \( d_{yz} \)
- \( d_{xz} \)

4. **Count the Orbitals**:
- From the analysis, we find that there are **3 orbitals** (\( d_{xy}, d_{yz}, d_{xz} \)) that meet the criteria.

### Final Answer:
The number of orbitals that have maximum orbital angular distribution at an angle of 45 degrees to the axial direction is **3**.

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