Animated Solution for Chemistry - Atomic Structure: The number of subshells associated with $n=4$ and $m=-2$ quantum numbers is

Imagine an atom as a massive, highly organized city. To find a specific electron, we use a "quantum address" made up of four numbers: $n,l,m,$ and $s$. In this problem, we are given a partial address: the principal quantum number $n=4$ (the city block) and the magnetic quantum number $m=-2$ (the specific apartment number). Our mission is to find out how many "buildings" (subshells) on this block contain an apartment numbered $-2$.

The principal quantum number $n=4$ tells us we are in the 4th shell of the atom. But how many subshells exist here? The rule of quantum mechanics dictates that the azimuthal quantum number, $l$, which defines the subshells, can take integer values from $0$ up to $n-1$.

For $n=4$, the permissible values of $l$ are:

$l=0\to 4s$ subshell

$l=1\to 4p$ subshell

$l=2\to 4d$ subshell

$l=3\to 4f$ subshell

So, we have four distinct subshells in this shell. Now, we need to inspect each one to see if it contains an $m=-2$ orbital.

The magnetic quantum number, $m$, describes the specific spatial orientation of an orbital within a subshell. The crucial rule here is that $m$ is strictly bounded by $l$. It can take any integer value from $-l$ to $+l$, including zero.

Let's evaluate our four subshells against the requirement $m=-2$:

1. The $4s$ subshell ($l=0$): Here, $m$ can only be $0$. It does not go low enough to reach $-2$.

2. The $4p$ subshell ($l=1$): Here, $m$ can be $-1,0,+1$. Still not low enough.

3. The $4d$ subshell ($l=2$): Here, $m$ can be $-2,-1,0,+1,+2$. Bingo! The $4d$ subshell contains an orbital with $m=-2$.

4. The $4f$ subshell ($l=3$): Here, $m$ can be $-3,-2,-1,0,+1,+2,+3$. Bingo again! The $4f$ subshell also contains an orbital with $m=-2$.

Out of the four subshells present in the $n=4$ shell, exactly two of them (the $4d$ and $4f$ subshells) are large enough to accommodate an orbital with a magnetic quantum number of $m=-2$.

Therefore, the number of subshells associated with these quantum numbers is $2$. Always read carefully whether the examiner is asking for the number of subshells, orbitals, or electrons—a single word changes the entire physical meaning of the answer!