Number of electrons having `l+m=0` in `Mn(z=25)` is

Answer

To find the number of electrons in manganese (Mn, atomic number 25) that have \( l + m = 0 \), we will follow these steps:

### Step 1: Determine the Electronic Configuration of Manganese
Manganese has an atomic number of 25. The electronic configuration can be written as:
- \( 1s^2 \)
- \( 2s^2 \)
- \( 2p^6 \)
- \( 3s^2 \)
- \( 3p^6 \)
- \( 4s^2 \)
- \( 3d^5 \)

So, the complete electronic configuration of manganese is:
\[ 1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^5 \]

### Step 2: Identify Quantum Numbers
- The azimuthal quantum number \( l \) represents the subshell:
- \( l = 0 \) for s subshell
- \( l = 1 \) for p subshell
- \( l = 2 \) for d subshell
- The magnetic quantum number \( m \) can take values from \( -l \) to \( +l \) including zero.

### Step 3: Calculate \( l + m = 0 \)
We need to find combinations of \( l \) and \( m \) such that \( l + m = 0 \).

1. **For the s subshell (\( l = 0 \))**:
- The only value for \( m \) is \( 0 \).
- Thus, \( l + m = 0 + 0 = 0 \).
- There are 2 electrons in the \( 1s \) and 2 in the \( 2s \), giving a total of \( 2 + 2 = 4 \) electrons.

2. **For the p subshell (\( l = 1 \))**:
- Possible values for \( m \) are \( -1, 0, +1 \).
- The only value that satisfies \( l + m = 0 \) is \( m = -1 \) (since \( 1 - 1 = 0 \)).
- Each p subshell can hold 2 electrons, and since there are 3 p orbitals, we have \( 2 \) electrons from \( 3p \).

3. **For the d subshell (\( l = 2 \))**:
- Possible values for \( m \) are \( -2, -1, 0, +1, +2 \).
- The only value that satisfies \( l + m = 0 \) is \( m = -2 \) (since \( 2 - 2 = 0 \)).
- There are 5 electrons in the \( 3d \) subshell, but only 1 electron can be counted for \( m = -2 \).

### Step 4: Total Count of Electrons
Now, we sum the electrons that satisfy \( l + m = 0 \):
- From s subshell: 4 electrons
- From p subshell: 2 electrons
- From d subshell: 1 electron

Total electrons = \( 4 + 2 + 1 = 7 \)

### Final Answer:
The number of electrons having \( l + m = 0 \) in manganese (Mn, \( z = 25 \)) is **7**.

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