Question

How many electrons in an atom can have each of the following quantum numbers or sublevel designations? (a) \(2 s\) (b) \(n=3, l=2\) (c) \(6 d\)

Short answer

(a) 2 electrons, (b) 10 electrons, (c) 10 electrons

Step-by-step solution

Understanding Sublevel Designations

Each sublevel designation, such as '2s' or '6d', corresponds to a specific combination of quantum numbers.

Calculating Electrons for (a) 2s

An 's' sublevel can hold a maximum of 2 electrons. Therefore, for the 2s sublevel, there can be 2 electrons.

Breakdown of Quantum Numbers for (b) n = 3, l = 2

For the given values of quantum numbers: n = 3 l = 2 (which corresponds to the 'd' sublevel) A 'd' sublevel can hold up to 10 electrons. Therefore, for n = 3, l = 2, there can be 10 electrons.

Calculating Electrons for (c) 6d

The 'd' sublevel can hold a maximum of 10 electrons, regardless of the principal quantum number n. Therefore, for the 6d sublevel, there can be 10 electrons.

Key concepts

sublevel designation

The term 'sublevel designation' refers to the specific combination of quantum numbers used to describe the electron configuration within an atom. These designations tell us which sublevel the electron occupies and provide a shorthand way of expressing this configuration. For example:

• '2s' means that the electron is in the second energy level (n=2) and within the 's' sublevel (l=0).
• '6d' indicates that the electron is in the sixth energy level (n=6) and within the 'd' sublevel (l=2).

The letters s, p, d, and f represent different sublevels and correspond to different shapes and energy levels of the orbitals where electrons are likely to be found.

electron capacity

Each sublevel can hold a specific maximum number of electrons. This depends on the type of orbital associated with that sublevel. The rule to remember is:

• 's' sublevel holds up to 2 electrons.
• 'p' sublevel holds up to 6 electrons.
• 'd' sublevel holds up to 10 electrons.
• 'f' sublevel holds up to 14 electrons.

These capacities are derived from the number of orbitals within each sublevel. Each orbital can hold a maximum of 2 electrons. So for example, since the 'd' sublevel has 5 orbitals (each holding 2 electrons), this gives it a total capacity of 10 electrons.

quantum number n and l

Quantum numbers are essential for understanding where an electron is located in an atom. The principal quantum number, denoted as n, indicates the main energy level of the electron and can be any positive integer (1, 2, 3, ...).
The angular momentum quantum number, denoted as l, indicates the shape of the orbital and is dependent on n. It can take any integer value from 0 to (n-1).
For example:

• For n=3, l could be 0, 1, or 2.

Each value of l corresponds to a different sublevel:
- l=0 corresponds to 's'
- l=1 corresponds to 'p'
- l=2 corresponds to 'd'
- l=3 corresponds to 'f'
Understanding these quantum numbers helps determine the specific sublevel and electron capacity of an atom.