What is the relationship between the principal quantum number 𝑛 and the total number of orbitals?
Quantum numbers are a set of four numbers that help us describe an electron in an atom. The first quantum number is the principal quantum number, which is the topic of this question. This quantum number tells us the energy level or shell the electron occupies, as small as the size of an atomic orbital. The principal quantum number can be any integer from one to seven.
The remaining three quantum numbers further describe an electron in an atom. The next quantum number is the subsidiary quantum number, which describes the orbital type, that is, whether it’s an s-, p-, d-, or f-type atomic orbital. 𝑚 sub 𝐿 is the magnetic quantum number, which describes the orientation of the atomic orbital in space. And finally we have the spin quantum number. This quantum number determines the spin state of an electron in an atomic orbital.
In this question, we need to determine the relationship between the principal quantum number and the total number of orbitals. As the principal quantum number increases, the number of orbitals increases. In the first shell, there is a single s orbital. In the second shell, there is an s orbital as well as three p orbitals. In the third shell, there is an s orbital, three p orbitals, and five d orbitals. So there’s one orbital in the first shell, four orbitals in the second, and nine orbitals in the third. So what’s the relationship between these numbers?
Well, three squared is equal to nine, two squared is equal to four, and one squared is also equal to one. So it seems the relationship between the principal quantum number 𝑛 and the total number of orbitals is 𝑛 squared.