Which of the following orbitals has two spherical nodes?

Options

A. (a) 2s
B. (b) 4s
C. (c) 3d
D. (d) 6f

Answer

To determine which orbital has two spherical nodes, we need to analyze the concept of nodes in atomic orbitals. Here’s a step-by-step solution:

### Step 1: Understand the Concept of Nodes
Nodes are regions in an atomic orbital where the probability of finding an electron is zero. There are two types of nodes:
- **Spherical (Radial) Nodes**: These are spherical surfaces where the probability of finding an electron is zero.
- **Angular Nodes**: These are planar surfaces where the probability of finding an electron is zero.

### Step 2: Identify the Formula for Nodes
The total number of nodes in an orbital is given by the formula:
\[ \text{Total Nodes} = n - 1 \]
where \( n \) is the principal quantum number.

The number of spherical (radial) nodes can be calculated using:
\[ \text{Radial Nodes} = n - l - 1 \]
where \( l \) is the azimuthal quantum number.

### Step 3: Calculate Nodes for Different Orbitals
Now, let’s calculate the number of spherical nodes for various orbitals:

1. **1s Orbital**:
- \( n = 1 \), \( l = 0 \)
- Total Nodes = \( 1 - 1 = 0 \)
- Radial Nodes = \( 1 - 0 - 1 = 0 \)

2. **2s Orbital**:
- \( n = 2 \), \( l = 0 \)
- Total Nodes = \( 2 - 1 = 1 \)
- Radial Nodes = \( 2 - 0 - 1 = 1 \)

3. **3s Orbital**:
- \( n = 3 \), \( l = 0 \)
- Total Nodes = \( 3 - 1 = 2 \)
- Radial Nodes = \( 3 - 0 - 1 = 2 \)

4. **4s Orbital**:
- \( n = 4 \), \( l = 0 \)
- Total Nodes = \( 4 - 1 = 3 \)
- Radial Nodes = \( 4 - 0 - 1 = 3 \)

5. **3d Orbital**:
- \( n = 3 \), \( l = 2 \)
- Total Nodes = \( 3 - 1 = 2 \)
- Radial Nodes = \( 3 - 2 - 1 = 0 \)

6. **6f Orbital**:
- \( n = 6 \), \( l = 3 \)
- Total Nodes = \( 6 - 1 = 5 \)
- Radial Nodes = \( 6 - 3 - 1 = 2 \)

### Step 4: Conclusion
From the calculations above, we find that the **6f orbital** has **two spherical nodes**.

### Final Answer
The orbital that has two spherical nodes is **6f**.

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