The correct order of total number of node of atomic orbitals is :

Options

A. `4fgt6sgt5d`
B. `6sgt5dgt4f`
C. `4fgt5dgt6s`
D. `5dgt4fgt6s`

Answer

To determine the correct order of the total number of nodes in atomic orbitals, we will follow these steps:

### Step 1: Understand the Concept of Nodes
Nodes are points or planes in an orbital where the electron density is zero. There are two types of nodes:
- **Radial Nodes**: These are spherical shells where the probability of finding an electron is zero.
- **Angular Nodes**: These are planes where the probability of finding an electron is zero.

### Step 2: Formulas for Calculating Nodes
1. **Radial Nodes**: The number of radial nodes is given by the formula:
\[ \text{Number of Radial Nodes} = n - l - 1 \]
where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.

2. **Angular Nodes**: The number of angular nodes is given by:
\[ \text{Number of Angular Nodes} = l \]

3. **Total Nodes**: The total number of nodes in an orbital can be calculated as:
\[ \text{Total Nodes} = \text{Radial Nodes} + \text{Angular Nodes} = (n - l - 1) + l = n - 1 \]

### Step 3: Calculate Total Nodes for Each Orbital
Now, we will calculate the total number of nodes for the given orbitals: 4f, 5d, and 6s.

1. **For 4f Orbital**:
- \( n = 4 \)
- \( l = 3 \) (since f corresponds to \( l = 3 \))
- Total Nodes:
\[ n - 1 = 4 - 1 = 3 \]

2. **For 5d Orbital**:
- \( n = 5 \)
- \( l = 2 \) (since d corresponds to \( l = 2 \))
- Total Nodes:
\[ n - 1 = 5 - 1 = 4 \]

3. **For 6s Orbital**:
- \( n = 6 \)
- \( l = 0 \) (since s corresponds to \( l = 0 \))
- Total Nodes:
\[ n - 1 = 6 - 1 = 5 \]

### Step 4: Order the Orbitals Based on Total Nodes
Now that we have calculated the total number of nodes for each orbital:
- 4f: 3 nodes
- 5d: 4 nodes
- 6s: 5 nodes

The correct order from least to most nodes is:
1. 4f (3 nodes)
2. 5d (4 nodes)
3. 6s (5 nodes)

### Final Answer
The correct order of total number of nodes of atomic orbitals is:
**4f < 5d < 6s**