Animated Solution for Chemistry - Atomic Structure: Consider the hypothetical situation where the azimuthal quantum number, $l$ takes values $0,1,2,\dots n+1$, where $n$ is the principle quantum number. Then, the element with atomic number

Imagine a universe where the fundamental rules of quantum mechanics are slightly tweaked. In our standard reality, the azimuthal quantum number $l$ dictates the shape of an orbital and takes integer values from $0$ up to $n-1$. But what if the rules changed? What if $l$ could go all the way up to $n+1$?

This is exactly the thought experiment presented in this fascinating problem. Let's break down how this single modification completely rewrites the periodic table.

Let's look at the very first shell, where the principal quantum number $n=1$.

According to the new hypothetical rule, the allowed values for $l$ are $0,1,2,\dots ,(n+1)$. Substituting $n=1$, we find that $l$ can take the values $0,1,$ and $2$.

This is a massive departure from reality! In our world, the first shell only has an $s$ subshell ($l=0$). But in this hypothetical universe, the first shell contains: - $1s$ subshell ($l=0$) - $1p$ subshell ($l=1$) - $1d$ subshell ($l=2$)

While the allowed subshells have changed, the rules governing the orbitals within those subshells remain the same. The number of orbitals is still given by $2l+1$, and each orbital holds a maximum of 2 electrons.

- The $1s$ subshell holds 2 electrons. - The $1p$ subshell holds 6 electrons. - The $1d$ subshell holds 10 electrons.

This means the first shell ($n=1$) can hold a whopping total of $2+6+10=18$ electrons before it is completely filled!

Now, let's test the options provided in the question. Consider an element with an atomic number of $13$ ($Z=13$). We need to distribute 13 electrons into our newly defined subshells following the Aufbau principle.

1. The first 2 electrons fill the $1s$ subshell: $1s^2$ 2. The next 6 electrons fill the $1p$ subshell: $1p^6$ 3. We have placed $2+6=8$ electrons. We have $13-8=5$ electrons remaining. 4. These final 5 electrons will enter the $1d$ subshell: $1d^5$

The complete electronic configuration for $Z=13$ is $1s^{2}1p^{6}1d^5$.

Look closely at the outermost subshell, which is the $1d$ subshell. It contains exactly 5 electrons. Since a $d$ subshell has a maximum capacity of 10 electrons, having 5 electrons means it is exactly half-filled.

This perfectly aligns with option (b), which states that the element with atomic number 13 has a half-filled valence subshell.

By carefully applying the modified rules to the quantum numbers, we successfully navigated this hypothetical universe. Always remember to read the constraints of a problem carefully—sometimes, the universe you are solving for isn't the one you live in!