To calculate the total number of electrons for manganese (Mn) with the condition \( n + l + m = 2 \), we need to follow these steps:
### Step 1: Determine the Atomic Number of Manganese
Manganese (Mn) has an atomic number of 25. This means it has 25 electrons in its neutral state.
### Step 2: Understand Quantum Numbers
In quantum mechanics, each electron in an atom is described by a set of four quantum numbers:
- \( n \): principal quantum number (energy level)
- \( l \): azimuthal quantum number (sublevel)
- \( m \): magnetic quantum number (orientation)
### Step 3: Assign Quantum Numbers to Electrons
We will fill the electrons in the order of increasing energy levels according to the Aufbau principle:
1. **1s**: \( n = 1, l = 0 \) → 2 electrons
2. **2s**: \( n = 2, l = 0 \) → 2 electrons
3. **2p**: \( n = 2, l = 1 \) → 6 electrons
4. **3s**: \( n = 3, l = 0 \) → 2 electrons
5. **3p**: \( n = 3, l = 1 \) → 6 electrons
6. **4s**: \( n = 4, l = 0 \) → 2 electrons
7. **3d**: \( n = 3, l = 2 \) → 5 electrons
### Step 4: Calculate \( n + l + m \)
We will find combinations of \( n, l, m \) that satisfy \( n + l + m = 2 \).
1. For **1s**:
- \( n = 1, l = 0 \) → \( m = 0 \) → \( n + l + m = 1 + 0 + 0 = 1 \) (not valid)
2. For **2s**:
- \( n = 2, l = 0 \) → \( m = 0 \) → \( n + l + m = 2 + 0 + 0 = 2 \) (valid)
3. For **2p**:
- \( n = 2, l = 1 \) → Possible \( m = -1, 0, +1 \) → \( n + l + m = 2 + 1 + m \) (not valid)
4. For **3s**:
- \( n = 3, l = 0 \) → \( m = 0 \) → \( n + l + m = 3 + 0 + 0 = 3 \) (not valid)
5. For **3p**:
- \( n = 3, l = 1 \) → Possible \( m = -1, 0, +1 \) → \( n + l + m = 3 + 1 + m \) (not valid)
6. For **4s**:
- \( n = 4, l = 0 \) → \( m = 0 \) → \( n + l + m = 4 + 0 + 0 = 4 \) (not valid)
7. For **3d**:
- \( n = 3, l = 2 \) → Possible \( m = -2, -1, 0, +1, +2 \) → \( n + l + m = 3 + 2 + m \) (not valid)
### Conclusion
The only valid configuration that satisfies \( n + l + m = 2 \) is from the **2s** orbital.
### Total Number of Electrons
Since we are only considering the valid configuration, the total number of electrons represented by \( n + l + m = 2 \) is **2 electrons** from the **2s** subshell.
