To determine the correct set of quantum numbers for the last electron of \( \text{Na}^+ \), we need to follow these steps:
### Step 1: Determine the Electron Configuration of \( \text{Na}^+ \)
Sodium (Na) has an atomic number of 11, meaning it has 11 electrons in its neutral state. When sodium loses one electron to become \( \text{Na}^+ \), it has 10 electrons.
The electron configuration for sodium is:
- \( 1s^2 \)
- \( 2s^2 \)
- \( 2p^6 \)
- \( 3s^1 \) (for neutral Na)
For \( \text{Na}^+ \), we remove one electron from the outermost shell, which is the 3s orbital. Therefore, the electron configuration for \( \text{Na}^+ \) is:
- \( 1s^2 \)
- \( 2s^2 \)
- \( 2p^6 \)
### Step 2: Identify the Last Electron
The last electron in \( \text{Na}^+ \) is the one that fills the \( 2p \) orbital. Since \( 2p \) can hold a maximum of 6 electrons, and in this case, it is fully filled with 6 electrons.
### Step 3: Assign Quantum Numbers
The quantum numbers for an electron are defined as follows:
1. **Principal Quantum Number (n)**: Indicates the energy level of the electron. For the \( 2p \) orbital, \( n = 2 \).
2. **Azimuthal Quantum Number (l)**: Indicates the subshell. For \( p \) orbitals, \( l = 1 \).
3. **Magnetic Quantum Number (m)**: Indicates the orientation of the orbital. For \( p \) orbitals, \( m \) can be -1, 0, or +1. The last electron in the \( 2p \) orbital can occupy any of these values, but since we are looking for the last electron in a filled orbital, we can choose \( m = 1 \) (the highest value).
4. **Spin Quantum Number (s)**: Indicates the spin of the electron. Electrons can have a spin of \( +\frac{1}{2} \) or \( -\frac{1}{2} \). We can choose \( s = -\frac{1}{2} \) for the last electron.
### Step 4: Write the Quantum Numbers
Thus, the correct set of quantum numbers for the last electron of \( \text{Na}^+ \) is:
- \( n = 2 \)
- \( l = 1 \)
- \( m = 1 \)
- \( s = -\frac{1}{2} \)
### Final Answer
The correct set of quantum numbers for the last electron of \( \text{Na}^+ \) is:
- \( (n, l, m, s) = (2, 1, 1, -\frac{1}{2}) \)
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