## Q. 7.PS.6

Quantum Numbers, Subshells, and Atomic Orbitals
Consider the $n = 4$ principal energy level.
(a) Without referring to Table 7.3, predict the number of subshells in this level.
(b) Identify each of the subshells by its number and letter designation (as in 1s) and give its $\ell$ values.
(c) Use the $2\ell + 1$ rule to calculate how many orbitals each subshell has and identify the $m_\ell$ value for each orbital.
(d) What is the total number of orbitals in the $n = 4$ level?

 Table 7.3    Relationships Among $n, \ell ,$ and $m_\ell$  for the First Four Principal Energy Levels $n$ Value $\ell$  Value Subshell Designation $m_\ell$  Values Number of Orbitals in Subshell, $2\ell + 1$ Total Number of Orbitals in Shell, $n^2$ 1 0 1$s$ 0 1 1 2 0 2$s$ 0 1 2 1 2$p$ 1,0,-1 3 4 3 0 3$s$ 0 1 3 1 3$p$ 1,0,-1 3 3 2 3$d$ 2,1,0,-1,-2 5 9 4 0 4$s$ 0 1 4 1 4$p$ 1,0,-1 3 4 2 4$d$ 2,1,0,-1,-2 5 4 3 4$f$ 3,2,1,0,-1,-2,-3 7 16

## Verified Solution

(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.

 Sublevels $4s$ $4p$ $4d$ $4f$ $\ell$ value 0 1 2 3