(a) How many values of the quantum number \(l\) are possible when \(n=7\) ? (b) How many values of \(m_{l}\) are allowed for an electron in a \(6 \mathrm{~d}\)-subshell? (c) How many values of \(m_{l}\) are allowed for an electron in a \(3 \mathrm{p}\)-subshell? (d) How many subshells are there in the shell with \(n=4\) ?

The azimuthal quantum number, often symbolized as \( l \), is an integral part of understanding an atom's electron configuration. Think of it as a way to describe the shape or the subshells of an electron cloud within an atom. For each principal quantum number \( n \), the azimuthal quantum number can take on any integer value ranging from 0 up to \( n-1 \). Each of these values corresponds to a specific type of subshell: \(

• \( l = 0 \) for the s subshell,
• \( l = 1 \) for the p subshell,
• \( l = 2 \) for the d subshell, and so forth.

\)

When working with the principal quantum number \( n=7 \), as in our exercise, there are seven possible azimuthal quantum numbers, which translate into seven electron subshells, each with its unique shape and energy level.

Let's delve into the magnetic quantum number, denoted by \( m_{l} \). This quantum number provides information about the orientation of the electron's orbital in space relative to an external magnetic field. For any given azimuthal quantum number \( l \), the magnetic quantum number can vary from \( -l \) to \( l \), including zero. This results in a total of \( 2l+1 \) possible orientations.

For example, consider an electron in a \( 6d \) subshell, which relates to \( l=2 \). The possible \( m_{l} \)-values for this subshell are five, ranging from \( -2 \) to \( 2 \). Hence, the electron can exhibit one of five different orientations in space within the \( d \) subshell.

Electron subshells are like rooms within a house, where the house is the principal energy level, or the shell, of an atom. The quantum model categorizes electron subshells into \( s, p, d, \) and \( f \), each with varying shapes and numbers of orbitals based on the azimuthal quantum number \( l \).

The \( 3p \) subshell, for instance, corresponds to \( l=1 \), and therefore, can host three different magnetic quantum number values: \( -1 \) , \( 0 \) , and \( 1 \), signifying three possible orientations. These subshells are critical in determining the probable locations of electrons in an atom and play a central role in chemistry and quantum mechanics.

The principal quantum number, represented as \( n \), essentially defines the energy level of an electron in an atom and indirectly its average distance from the nucleus. It can be any positive integer, and the value of \( n \) determines the overall size and energy of an orbital, with higher values indicating greater energy and a larger orbital.

For example, when you're asked to find the number of subshells within the shell with \( n=4 \), you're looking for all possible \( l \) values from 0 up to \( n-1 \), which in this case, yields four subshells. As a rule of thumb, the number of subshells within a shell is always equal to the principal quantum number, providing a structure for holding the electrons that occupy an atom.