Question

How many electrons in an atom can have each of the following quantum numbers or sublevel designations? (a) \(n=2, l=1, m_{l}=0\) (b) \(5 p\) (c) \(n=4, l=3\)

Short answer

a) 2 electrons b) 6 electrons c) 14 electrons

Step-by-step solution

Understanding Quantum Numbers and Electron Capacity

Quantum numbers describe the properties of atomic orbitals and the properties of electrons in those orbitals. Specifically, the principal quantum number (). For each set of quantum numbers, determine how many possible electron configurations can exist.

Solve Part (a)

Given: =2, l=1, m_{l}=0. l=1 corresponds to the p sublevel. The possible values of the spin quantum number () are +1/2 and -1/2. Thus, each orbital can hold 2 electrons. Therefore, for =2, l=1, and m_{l}=0, there can be 2 electrons.

Solve Part (b)

Given: 5p. The 'p' sublevel (l=1) can hold a maximum of 6 electrons regardless of the principal quantum number (5). Hence, 5p can hold 6 electrons.

Solve Part (c)

Given: =4, l=3. l=3 corresponds to the f sublevel. For any f sublevel, there are 7 orbitals (=-3 to 3). Each orbital can hold 2 electrons. Therefore, there can be 14 electrons in the 4f sublevel.

Key concepts

Electron Configurations

Electron configurations describe the arrangement of electrons around the nucleus of an atom in its atomic orbitals.
This concept is essential because it dictates an atom's chemical behavior and properties.
Electrons fill orbitals in a specific order, starting from the lowest energy level to the highest, following the Aufbau principle.
The order of filling is determined by the energy levels of orbitals, which generally follow the sequence 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on.

Here are some key points to remember about electron configurations:

• The Aufbau principle: Electrons occupy the lowest energy orbital available.
• Pauli exclusion principle: No two electrons in the same atom can have the same set of four quantum numbers.
• Hund's rule: Electron will fill degenerate orbitals singly as much as possible before pairing.

Understanding electron configurations helps predict chemical reactivity and the types of bonds atoms can form.

Principal Quantum Number

The principal quantum number, denoted by the symbol n, indicates the main energy level occupied by the electron.
It is always a positive integer (n = 1, 2, 3...).
The principal quantum number also determines the size and energy of the orbital.
For example:

• When n = 1, the electron is in the first energy level, which is closest to the nucleus.
• As n increases, the electron's energy and distance from the nucleus increase.

Each principal energy level can contain multiple sublevels, denoted by the azimuthal quantum number (l).
Here is an example:

• For n = 2, there are two possible sublevels: l = 0 (2s) and l = 1 (2p).
• For n = 3, there are three possible sublevels: l = 0 (3s), l = 1 (3p), and l = 2 (3d).

The maximum number of electrons that can fit in a principal energy level is given by the formula: \(2n^{2}\).

Atomic Orbitals

Atomic orbitals are regions in an atom where there is a high probability of finding electrons.
These orbitals are described by the wavefunctions obtained from solving the Schrödinger equation for atoms.
Atomic orbitals come in various shapes and sizes, each corresponding to a different set of quantum numbers.

• s orbitals: These are spherical and centered around the nucleus.
• p orbitals: These are dumbbell-shaped and oriented along the x, y, and z axes.
• d orbitals: These have more complex shapes, including cloverleaf patterns.
• f orbitals: These are even more complex in shape and have multiple lobes.

Each type of orbital can hold a specific number of electrons:

• s orbitals can hold 2 electrons.
• p orbitals can hold 6 electrons.
• d orbitals can hold 10 electrons.
• f orbitals can hold 14 electrons.

The organization of these orbitals and the distribution of electrons influence an atom's overall shape and chemical behavior.