How many atomic orbitals of the following have more than one node? `1s, 2s, 3p_(x), 3d_(xy), 3d_(z^(2)), 4p_(z), 4d_(x^(2)-y^(2))`

Explanation

To determine how many atomic orbitals have more than one node from the given list, we will calculate the total number of nodes for each orbital using the formula for nodes. Nodes can be classified into two types: radial nodes and angular nodes.

### Step-by-Step Solution:

1. **Understand the Formula for Nodes:**
- The total number of nodes in an orbital can be calculated using the formula:
\[ \text{Total Nodes} = n - 1 \]
where \( n \) is the principal quantum number.

2. **Identify the Orbitals and Their Quantum Numbers:**
- The orbitals given are:
- \( 1s \) (n=1, l=0)
- \( 2s \) (n=2, l=0)
- \( 3p_x \) (n=3, l=1)
- \( 3d_{xy} \) (n=3, l=2)
- \( 3d_{z^2} \) (n=3, l=2)
- \( 4p_z \) (n=4, l=1)
- \( 4d_{x^2-y^2} \) (n=4, l=2)

3. **Calculate Nodes for Each Orbital:**
- **For \( 1s \):**
\[ n = 1 \implies \text{Total Nodes} = 1 - 1 = 0 \quad (\text{Not more than one}) \]
- **For \( 2s \):**
\[ n = 2 \implies \text{Total Nodes} = 2 - 1 = 1 \quad (\text{Not more than one}) \]
- **For \( 3p_x \):**
\[ n = 3 \implies \text{Total Nodes} = 3 - 1 = 2 \quad (\text{More than one}) \]
- **For \( 3d_{xy} \):**
\[ n = 3 \implies \text{Total Nodes} = 3 - 1 = 2 \quad (\text{More than one}) \]
- **For \( 3d_{z^2} \):**
\[ n = 3 \implies \text{Total Nodes} = 3 - 1 = 2 \quad (\text{More than one}) \]
- **For \( 4p_z \):**
\[ n = 4 \implies \text{Total Nodes} = 4 - 1 = 3 \quad (\text{More than one}) \]
- **For \( 4d_{x^2-y^2} \):**
\[ n = 4 \implies \text{Total Nodes} = 4 - 1 = 3 \quad (\text{More than one}) \]

4. **Count the Orbitals with More Than One Node:**
- The orbitals with more than one node are:
- \( 3p_x \)
- \( 3d_{xy} \)
- \( 3d_{z^2} \)
- \( 4p_z \)
- \( 4d_{x^2-y^2} \)
- Thus, there are a total of **5 orbitals** that have more than one node.

### Final Answer:
There are **5 atomic orbitals** that have more than one node.