## Chapter 7

## 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 | 1s | 0 | 1 | 1 |

2 | 0 | 2s | 0 | 1 | |

2 | 1 | 2p | 1,0,-1 | 3 | 4 |

3 | 0 | 3s | 0 | 1 | |

3 | 1 | 3p | 1,0,-1 | 3 | |

3 | 2 | 3d | 2,1,0,-1,-2 | 5 | 9 |

4 | 0 | 4s | 0 | 1 | |

4 | 1 | 4p | 1,0,-1 | 3 | |

4 | 2 | 4d | 2,1,0,-1,-2 | 5 | |

4 | 3 | 4f | 3,2,1,0,-1,-2,-3 | 7 | 16 |

## Step-by-Step

## 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 nth 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 |