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Step-by-Step Solution
Step 1: State the Hypothetical Scenario
In this hypothetical situation, the azimuthal quantum number (denoted by $l$) does not range from $0$ to $(n-1)$ as in the usual quantum model. Instead, it ranges from $0$ to $(n+1)$, where $n$ is the principal quantum number. That means for a given principal quantum number $n$, the allowed $l$ values are:
$$
l = 0, 1, 2, \dots, n+1.
$$
Step 2: Enumerate Possible $l$ Values for Low $n$
Based on the new rule:
For $n = 1$, $l = 0,1,2$.
For $n = 2$, $l = 0,1,2,3$.
Under normal circumstances, an orbital type is designated as:
$l = 0$: s-orbital
$l = 1$: p-orbital
$l = 2$: d-orbital
$l = 3$: f-orbital
But since $l$ goes up to $n+1$, we have orbitals like $1d$, $2f$, and so on, which do not ordinarily exist in the standard model.
Step 3: Understand the Order of Filling
In this hypothetical scenario, the filling order of subshells begins as:
$$
1s \rightarrow 1p \rightarrow 1d \rightarrow 2s \rightarrow 2p \rightarrow 2d \rightarrow 2f \quad \dots
$$
(The detailed order can be deduced from energy considerations, but the problem statement suggests these subshells are sequentially occupied up to the required number of electrons.)
Step 4: Analyze the Atomic Number 13 Configuration
We want to see why the element with $Z = 13$ has a half-filled valence subshell. According to the hypothetical filling pattern, the orbitals fill as follows:
1s can hold 2 electrons.
1p can hold 6 electrons.
1d can hold 10 electrons, but we fill only up to the needed number of electrons.
After placing 2 electrons in 1s and 6 electrons in 1p, we have used 8 electrons. We have 5 more electrons to reach $Z = 13$, which go into the 1d subshell:
$1s^2 \; 1p^6 \; 1d^5$
A $1d$ subshell can hold a maximum of 10 electrons. Having 5 electrons means it is exactly half-filled. A half-filled subshell often implies extra stability.
Step 5: Conclude with the Correct Answer
Based on the hypothetical model’s electron configurations, the element with atomic number 13 ends up with the configuration $1s^2 \; 1p^6 \; 1d^5$, making its $d$ subshell half-filled. Therefore, the statement "13 has a half-filled valence subshell" is correct.
