In an atom which has 2 K 8 L , 18 M and 2N electrons in the ground state . The total number of electrons having magnetic quantum number m=0 is

Options

A. 6
B. 10
C. 7
D. 14

Answer

To find the total number of electrons in an atom with the given electron configuration (2 K, 8 L, 18 M, and 2 N) that have a magnetic quantum number \( m = 0 \), we will follow these steps:

### Step 1: Determine the total number of electrons
The total number of electrons can be calculated by adding the electrons in each shell:
- K shell: 2 electrons
- L shell: 8 electrons
- M shell: 18 electrons
- N shell: 2 electrons

Total electrons = \( 2 + 8 + 18 + 2 = 30 \) electrons.

### Step 2: Write the electronic configuration
The electronic configuration for the given number of electrons can be written as:
- 1s² (2 electrons)
- 2s² (2 electrons)
- 2p⁶ (6 electrons)
- 3s² (2 electrons)
- 3p⁶ (6 electrons)
- 4s² (2 electrons)
- 3d¹⁰ (10 electrons)

So, the complete electronic configuration is: \( 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} \).

### Step 3: Identify the magnetic quantum number \( m = 0 \)
The magnetic quantum number \( m \) can take values from \( -l \) to \( +l \), where \( l \) is the azimuthal quantum number:
- For \( s \) orbitals (\( l = 0 \)): \( m = 0 \)
- For \( p \) orbitals (\( l = 1 \)): \( m = -1, 0, +1 \) (1 electron has \( m = 0 \))
- For \( d \) orbitals (\( l = 2 \)): \( m = -2, -1, 0, +1, +2 \) (1 electron has \( m = 0 \))

### Step 4: Count the electrons with \( m = 0 \)
Now, we will count the electrons for each subshell that have \( m = 0 \):
- From \( 1s^2 \): 2 electrons (both have \( m = 0 \))
- From \( 2s^2 \): 2 electrons (both have \( m = 0 \))
- From \( 2p^6 \): 2 electrons (1 of the 3 electrons has \( m = 0 \))
- From \( 3s^2 \): 2 electrons (both have \( m = 0 \))
- From \( 3p^6 \): 2 electrons (1 of the 3 electrons has \( m = 0 \))
- From \( 4s^2 \): 2 electrons (both have \( m = 0 \))
- From \( 3d^{10} \): 2 electrons (1 of the 5 electrons has \( m = 0 \))

### Step 5: Total count of electrons with \( m = 0 \)
Now, we can sum the contributions:
- \( 2 (1s) + 2 (2s) + 2 (2p) + 2 (3s) + 2 (3p) + 2 (4s) + 2 (3d) = 14 \) electrons.

### Conclusion
The total number of electrons having a magnetic quantum number \( m = 0 \) is **14**.

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