Question 7.PS.6: Quantum Numbers, Subshells, and Atomic Orbitals Consider the......

(a) Four subshells
(b) $4s, 4p, 4d, and  4f; \ell = 0, 1, 2$, and $3$, respectively
(c) One $4s$ orbital, three $4p$ orbitals, five $4d$ orbitals, and seven $4f$ orbitals
(d) 16 orbitals
Strategy and Explanation
(a) There are $n$ subshells in the $n$th level. Thus, the $n = 4$ level contains four subshells.
(b) The number refers to the principal quantum number, $n$; the letter is associated with the $\ell$ quantum number. The four sublevels correspond to the four possible $\ell$ values:
(c) There are a total of $2\ell + 1$ orbitals within a sublevel. Only one $4s$ orbital is possible ($\ell = 0$, so $m_\ell$ must be zero). There are three $4p$ orbitals ($\ell = 1$) with $m_\ell$ values of $1, 0,$ or $– 1$. There are five $4d$ orbitals ($\ell = 2$) corresponding to the five allowed values for $m_\ell : 2, 1, 0, – 1$, and $– 2$. There are seven $4f$ orbitals ($\ell =  3$), each with one of the seven permitted values of $m_\ell : 3, 2, 1, 0, -1, -2$, and $– 3$.
(d) The total number of orbitals in a level is $n^2$. Therefore, the $n = 4$ level has a total of $16$ orbitals.