Which of the following orbital has (have) one spherical node?

Options

A. 1s
B. 2s
C. 2p
D. 3p

Answer

To determine which of the given orbitals has one spherical node, we can follow these steps:

### Step 1: Understand the concept of nodes
Nodes are points in an atom where the probability of finding an electron is zero. There are two types of nodes:
- **Angular nodes**: Related to the shape of the orbital and can be calculated using the angular momentum quantum number (l).
- **Radial (spherical) nodes**: Related to the size of the orbital and can be calculated using the formula:
\[ \text{Number of radial nodes} = n - l - 1 \]
where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.

### Step 2: Identify the quantum numbers for the given orbitals
We will analyze the orbitals provided:
- **1s**: \( n = 1, l = 0 \)
- **2s**: \( n = 2, l = 0 \)
- **2p**: \( n = 2, l = 1 \)
- **3p**: \( n = 3, l = 1 \)

### Step 3: Calculate the number of spherical nodes for each orbital
Using the formula \( n - l - 1 \):

1. **For 1s**:
\[ n - l - 1 = 1 - 0 - 1 = 0 \quad \text{(0 spherical nodes)} \]

2. **For 2s**:
\[ n - l - 1 = 2 - 0 - 1 = 1 \quad \text{(1 spherical node)} \]

3. **For 2p**:
\[ n - l - 1 = 2 - 1 - 1 = 0 \quad \text{(0 spherical nodes)} \]

4. **For 3p**:
\[ n - l - 1 = 3 - 1 - 1 = 1 \quad \text{(1 spherical node)} \]

### Step 4: Conclusion
From the calculations:
- The **2s** orbital has **1 spherical node**.
- The **3p** orbital also has **1 spherical node**.

Thus, the orbitals that have one spherical node are **2s and 3p**.

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