In an atom, the total number of electrons .having quantum numbers. n = 4 , `|m_(l)|=1` is :

Options

A. 6
B. 12
C. 20
D. 18

Explanation

To find the total number of electrons in an atom with quantum numbers \( n = 4 \) and \( |m_l| = 1 \), we can follow these steps:

### Step 1: Determine the possible values of the azimuthal quantum number \( l \)
The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). For \( n = 4 \):
- Possible values of \( l \) are \( 0, 1, 2, 3 \).

### Step 2: Identify the magnetic quantum number \( m_l \)
The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \). We are interested in the cases where \( |m_l| = 1 \). This means \( m_l \) can be either \( +1 \) or \( -1 \).

### Step 3: Find the values of \( l \) that allow \( m_l = 1 \) or \( m_l = -1 \)
- For \( l = 1 \): \( m_l \) can be \( -1, 0, +1 \) (valid).
- For \( l = 2 \): \( m_l \) can be \( -2, -1, 0, +1, +2 \) (valid).
- For \( l = 3 \): \( m_l \) can be \( -3, -2, -1, 0, +1, +2, +3 \) (valid).
- For \( l = 0 \): \( m_l \) can only be \( 0 \) (not valid).

### Step 4: Count the orbitals that correspond to \( |m_l| = 1 \)
- For \( l = 1 \): 1 orbital (corresponding to \( m_l = +1 \) and \( m_l = -1 \)).
- For \( l = 2 \): 2 orbitals (corresponding to \( m_l = +1 \) and \( m_l = -1 \)).
- For \( l = 3 \): 2 orbitals (corresponding to \( m_l = +1 \) and \( m_l = -1 \)).

### Step 5: Total number of orbitals with \( |m_l| = 1 \)
Adding these up:
- From \( l = 1 \): 1 orbital
- From \( l = 2 \): 1 orbital
- From \( l = 3 \): 1 orbital

Total = \( 1 + 2 + 2 = 5 \) orbitals.

### Step 6: Calculate the total number of electrons
Each orbital can hold a maximum of 2 electrons. Therefore, the total number of electrons is:
\[ \text{Total electrons} = \text{Number of orbitals} \times 2 = 5 \times 2 = 10 \text{ electrons}. \]

### Final Answer
The total number of electrons having quantum numbers \( n = 4 \) and \( |m_l| = 1 \) is **10**.

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