To find the number of electrons in manganese (Mn, atomic number 25) that satisfy the condition \( l + m = 0 \), we will follow these steps:
### Step 1: Understand Quantum Numbers
- The azimuthal quantum number \( l \) determines the subshell:
- \( l = 0 \) for s subshell
- \( l = 1 \) for p subshell
- \( l = 2 \) for d subshell
- \( l = 3 \) for f subshell
- The magnetic quantum number \( m \) can take values from \( -l \) to \( +l \), including zero.
### Step 2: Write the Electronic Configuration of Mn
- The electronic configuration of Mn (Z = 25) is:
\[
1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^5
\]
### Step 3: Identify the Electrons in Each Subshell
- **1s**: \( l = 0 \), \( m = 0 \) (2 electrons)
- **2s**: \( l = 0 \), \( m = 0 \) (2 electrons)
- **2p**: \( l = 1 \), \( m = -1, 0, +1 \) (6 electrons)
- **3s**: \( l = 0 \), \( m = 0 \) (2 electrons)
- **3p**: \( l = 1 \), \( m = -1, 0, +1 \) (6 electrons)
- **4s**: \( l = 0 \), \( m = 0 \) (2 electrons)
- **3d**: \( l = 2 \), \( m = -2, -1, 0, +1, +2 \) (5 electrons)
### Step 4: Calculate \( l + m \) for Each Electron
- For electrons in the **1s** subshell: \( l + m = 0 + 0 = 0 \) (2 electrons)
- For electrons in the **2s** subshell: \( l + m = 0 + 0 = 0 \) (2 electrons)
- For electrons in the **2p** subshell:
- \( m = -1 \): \( l + m = 1 - 1 = 0 \) (2 electrons)
- \( m = 0 \): \( l + m = 1 + 0 = 1 \) (1 electron)
- \( m = +1 \): \( l + m = 1 + 1 = 2 \) (1 electron)
- For electrons in the **3s** subshell: \( l + m = 0 + 0 = 0 \) (2 electrons)
- For electrons in the **3p** subshell:
- \( m = -1 \): \( l + m = 1 - 1 = 0 \) (2 electrons)
- \( m = 0 \): \( l + m = 1 + 0 = 1 \) (1 electron)
- \( m = +1 \): \( l + m = 1 + 1 = 2 \) (1 electron)
- For electrons in the **4s** subshell: \( l + m = 0 + 0 = 0 \) (2 electrons)
- For electrons in the **3d** subshell:
- \( m = -2 \): \( l + m = 2 - 2 = 0 \) (1 electron)
- \( m = -1 \): \( l + m = 2 - 1 = 1 \) (1 electron)
- \( m = 0 \): \( l + m = 2 + 0 = 2 \) (1 electron)
- \( m = +1 \): \( l + m = 2 + 1 = 3 \) (1 electron)
- \( m = +2 \): \( l + m = 2 + 2 = 4 \) (1 electron)
### Step 5: Count the Electrons with \( l + m = 0 \)
- From **1s**: 2 electrons
- From **2s**: 2 electrons
- From **2p**: 2 electrons (only \( m = -1 \))
- From **3s**: 2 electrons
- From **3p**: 2 electrons (only \( m = -1 \))
- From **4s**: 2 electrons
- From **3d**: 1 electron (only \( m = -2 \))
### Total Count
Adding these up:
\[
2 + 2 + 2 + 2 + 2 + 2 + 1 = 13
\]
### Final Answer
The number of electrons having \( l + m = 0 \) in Mn is **13**.
