For a hypothetical case let value of `l` is defined as `0,1,2,3,...(n+1)` for principle quantum number n

Options

A. Atomic number of `1^(st)` noble gas is 8
B. Atomic number of `1^(st)` alkali metal is 9
C. Carbon has electron in `2p_z`
D. Element with atomic number 13 has half filled valance shell

Answer

To solve the question, we need to analyze the hypothetical case where the azimuthal quantum number \( l \) can take values from \( 0 \) to \( n + 1 \) for a principal quantum number \( n \). We will determine the atomic numbers of various elements based on this assumption.

### Step-by-Step Solution:

1. **Understanding Quantum Numbers**:
- The principal quantum number \( n \) determines the energy level of an electron in an atom.
- The azimuthal quantum number \( l \) determines the subshell and can take values \( 0, 1, 2, \ldots, n + 1 \).

2. **Determine the Values of \( l \)**:
- For a given \( n \), the values of \( l \) will be \( 0 \) (s), \( 1 \) (p), \( 2 \) (d), up to \( n + 1 \).
- For example, if \( n = 1 \), then \( l \) can be \( 0 \) (s), \( 1 \) (p), \( 2 \) (d), \( 3 \) (f), and \( 4 \) (g).

3. **Finding the Atomic Number of the First Noble Gas**:
- The first noble gas is Neon (Ne), which has an atomic number of \( 10 \).
- In our hypothetical case, the electron configuration will be \( 1s^2 2s^2 2p^6 \), totaling \( 10 \) electrons.

4. **Finding the Atomic Number of the First Alkali Metal**:
- The first alkali metal is Lithium (Li), with an atomic number of \( 3 \).
- In our hypothetical case, after Neon, the next configuration would be \( 3s^1 \), leading to an atomic number of \( 11 \) (after Neon).

5. **Analyzing Carbon's Electron Configuration**:
- Carbon has an atomic number of \( 6 \) and its configuration is \( 1s^2 2s^2 2p^2 \).
- In our hypothetical case, we will still have \( 1s^2 1p^4 \) for the first six electrons, which does not match the original configuration.

6. **Finding the Element with Atomic Number 13**:
- The element with atomic number \( 13 \) is Aluminum (Al).
- In our hypothetical case, the configuration would be \( 1s^2 2s^2 2p^6 3s^2 3p^1 \), leading to a half-filled valence shell in the \( 3p \) subshell.

### Conclusion:
- The correct statements based on the hypothetical case are:
- The atomic number of the first noble gas is \( 10 \).
- The atomic number of the first alkali metal is \( 11 \).
- The statement regarding carbon is incorrect in this hypothetical case.
- The statement about the element with atomic number \( 13 \) having a half-filled valence shell is correct.