Write the quantum numbers associated with each of the following. a. the fifth principle energy level b. the 6s sublevel c. an orbital in the 3d sublevel d. the first electron added to the 4f sublevel
The Correct Answer and Explanation is:
Here are the quantum numbers associated with each case:
a. The fifth principal energy level
- n = 5
- l = 0, 1, 2, 3, or 4 (since l ranges from 0 to n − 1)
- ml = −l to +l (depends on the value of l chosen)
- ms = +½ or −½
The principal quantum number, n = 5, defines the energy level. Within this level, the angular momentum quantum number l can range from 0 to 4, corresponding to the s, p, d, f, and g sublevels. Each l value then gives a set of ml values from −l to +l. The spin quantum number ms can be either +½ or −½.
b. The 6s sublevel
- n = 6
- l = 0 (since it is an s sublevel)
- ml = 0 (only one value when l = 0)
- ms = +½ or −½
Here, n = 6 tells us the electron is in the sixth energy level. The s sublevel corresponds to l = 0, and thus only one orientation, ml = 0, exists. Each orbital holds two electrons with opposite spins, so ms is either +½ or −½.
c. An orbital in the 3d sublevel
- n = 3
- l = 2 (d sublevel)
- ml = −2, −1, 0, +1, or +2 (choose any one of these)
- ms = +½ or −½
The d sublevel corresponds to l = 2, giving five possible ml values. Each orbital can hold two electrons with opposite spins.
d. The first electron added to the 4f sublevel
- n = 4
- l = 3 (f sublevel)
- ml = −3 (typically filled starting from lowest ml)
- ms = +½
According to Hund’s rule, electrons fill orbitals singly before pairing. So the first electron added to the 4f sublevel enters the orbital with ml = −3 and has ms = +½ (spin up).
Explanation
In quantum mechanics, electrons in atoms are described by a set of four quantum numbers: n, l, ml, and ms. Each quantum number provides specific information about the electron’s location and behavior.
The principal quantum number (n) defines the main energy level or shell in which the electron resides. As n increases, the electron’s average distance from the nucleus increases, and so does its energy.
The angular momentum quantum number (l) determines the shape of the orbital and is linked to the sublevel. For a given n, l can take integer values from 0 up to n − 1. Each value of l corresponds to a type of sublevel:
- l = 0 → s
- l = 1 → p
- l = 2 → d
- l = 3 → f
The magnetic quantum number (ml) specifies the orientation of the orbital in space and ranges from −l to +l.
The spin quantum number (ms) describes the electron’s spin direction, which can be either +½ or −½.
In each example:
- For a principal level like 5, the full range of sublevels and orbitals are possible.
- In the 6s sublevel, only one orbital exists, with spin being the distinguishing factor.
- In 3d, multiple orbitals are available, and any specific one is defined by an ml value.
- The first 4f electron enters the orbital with the lowest ml (usually −3) and has a positive spin according to Hund’s rule.
This structure helps explain the behavior and arrangement of electrons in atoms and their roles in chemical bonding.
