To solve the question, we need to analyze the hypothetical case where the value of \( l \) is defined as \( 0, 1, 2, 3, \ldots, (n+1) \) for the principal quantum number \( n \). We will evaluate each option given in the question.
### Step-by-Step Solution:
1. **Understanding the Quantum Numbers**:
- The principal quantum number \( n \) can take positive integer values (1, 2, 3, ...).
- The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n+1 \). Therefore, for \( n = 1 \), \( l \) can be \( 0, 1, 2 \).
2. **Assigning Orbitals**:
- For \( l = 0 \): This corresponds to the \( s \) orbital.
- For \( l = 1 \): This corresponds to the \( p \) orbital.
- For \( l = 2 \): This corresponds to the \( d \) orbital.
3. **Calculating Electron Configuration**:
- For \( n = 1 \), the hypothetical electron configuration would be:
- \( 1s^2 \) (2 electrons)
- \( 1p^6 \) (6 electrons)
- \( 1d^{10} \) (10 electrons)
- Total electrons = \( 2 + 6 + 10 = 18 \).
4. **Evaluating Each Option**:
- **Option 1**: Atomic number of the first noble gas is 8.
- The first noble gas is Helium (He) with an atomic number of 2, not 8. This option is incorrect.
- **Option 2**: Atomic number of the first alkali metal is 9.
- The first alkali metal is Lithium (Li) with an atomic number of 3. This option is incorrect.
- **Option 3**: Carbon has electrons in 2pz.
- Carbon (C) has an atomic number of 6, with the configuration \( 1s^2 2s^2 2p^2 \). It does not have electrons in \( 2p_z \) specifically, as it has \( 2p_x \) and \( 2p_y \) filled. This option is incorrect.
- **Option 4**: Element with atomic number 13 has a half-filled valence shell.
- The element with atomic number 13 is Aluminum (Al) with the configuration \( 1s^2 2s^2 2p^6 3s^2 3p^1 \). The \( 3p \) subshell has one electron, which is not half-filled. However, if we consider the hypothetical case, we can analyze the \( 1d^5 \) configuration, which is half-filled. This option is correct.
5. **Conclusion**:
- The only correct option is **Option 4**: Element with atomic number 13 has half-filled valence shell.