No. of subshells having n=4 & m = (-2) are

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem of finding the number of subshells with principal quantum number \( n = 4 \) and magnetic quantum number \( m = -2 \), we can follow these steps:

### Step 1: Identify the Principal Quantum Number (n)
The principal quantum number \( n \) is given as 4. This indicates that we are dealing with the fourth energy level.

### Step 2: Determine the Possible Values of the Azimuthal Quantum Number (l)
The azimuthal quantum number \( l \) can take values from 0 to \( n-1 \). Therefore, for \( n = 4 \):
\[ l = 0, 1, 2, 3 \]

### Step 3: Determine the Possible Values of the Magnetic Quantum Number (m)
For each value of \( l \), the magnetic quantum number \( m \) can take values ranging from \( -l \) to \( +l \). We will calculate the possible \( m \) values for each \( l \):

- For \( l = 0 \):
- \( m = 0 \) (1 value)

- For \( l = 1 \):
- \( m = -1, 0, +1 \) (3 values)

- For \( l = 2 \):
- \( m = -2, -1, 0, +1, +2 \) (5 values)

- For \( l = 3 \):
- \( m = -3, -2, -1, 0, +1, +2, +3 \) (7 values)

### Step 4: Count the Subshells with \( m = -2 \)
Now, we need to find out how many times \( m = -2 \) appears in the possible values of \( m \):
- For \( l = 0 \): No \( m = -2 \)
- For \( l = 1 \): No \( m = -2 \)
- For \( l = 2 \): Yes, \( m = -2 \) (1 occurrence)
- For \( l = 3 \): Yes, \( m = -2 \) (1 occurrence)

### Step 5: Total Count of Subshells
Adding the occurrences of \( m = -2 \):
\[ \text{Total} = 1 \text{ (from } l = 2\text{)} + 1 \text{ (from } l = 3\text{)} = 2 \]

### Final Answer
The number of subshells having \( n = 4 \) and \( m = -2 \) is **2**.

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