Question

List the four possible subshells in the quantum-mechanical model, the number of orbitals in each subshell, and the maximum number of electrons that can be contained in each subshell.

Short answer

The four possible subshells are s (1 orbital, 2 electrons), p (3 orbitals, 6 electrons), d (5 orbitals, 10 electrons), and f (7 orbitals, 14 electrons).

Step-by-step solution

Identify s subshell

The s subshell is the first subshell in any energy level and it has a spherical shape. There is only one orbital in an s subshell, and since each orbital can hold a maximum of two electrons, the s subshell can contain a maximum of 2 electrons.

Identify p subshell

The p subshell is the second type of subshell and is composed of three dumbbell-shaped orbitals. Since each orbital can hold two electrons, the p subshell can contain a maximum of 6 electrons (3 orbitals times 2 electrons per orbital).

Identify d subshell

The d subshell is the third type of subshell and it consists of five orbitals, each of which can hold two electrons. Therefore, the d subshell can contain a maximum of 10 electrons (5 orbitals times 2 electrons per orbital).

Identify f subshell

The f subshell is the fourth type of subshell with seven orbitals. As with the others, each orbital can hold two electrons, so the f subshell has a maximum capacity of 14 electrons (7 orbitals times 2 electrons per orbital).

Key concepts

Electron Configuration

Electron configuration relates to the arrangement of electrons around the nucleus of an atom in atomic orbitals. The configuration adheres to specific rules, such as the Pauli exclusion principle, Hund's rule, and the aufbau principle, ensuring electrons occupy the orbitals in a way that minimizes the energy of the atom. This determines the atom's chemical properties and reactivity.

Electron configuration is normally expressed by notations that consist of the principal quantum number, the subshell letter (s, p, d, or f), and a superscript denoting the number of electrons within that subshell. For example, helium's electron configuration is written as 1s², indicating two electrons in the first energy level's s subshell.

Atomic Orbitals

Atomic orbitals are regions in space around the atom's nucleus where there is a high probability of finding electrons. Each atomic orbital is defined by a unique set of quantum numbers and has a distinctive shape that describes the region of space where an electron is likely to be located. The basic shapes for the s, p, d, and f subshells are spherical, dumbbell, cloverleaf, and complex, respectively.

The number of orbitals increases with the complexity of the subshell: s has 1, p has 3, d has 5, and f has 7. These different orbitals within a subshell are equivalent in energy, a concept known as orbital degeneracy, but they differ in orientation in space.

Quantum Numbers

Quantum numbers are a set of numerical values that describe the unique quantum state of electrons in atomic orbitals. There are four quantum numbers: the principal quantum number (n), angular momentum quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s).

• The principal quantum number (n) describes the energy level and size of the atomic orbital.
• The angular momentum quantum number (l) defines the shape of the orbital, with values ranging from 0 to n-1 corresponding to the s, p, d, and f shapes.
• The magnetic quantum number (m_l) specifies the orientation of the orbital in space and ranges from -l to +l.
• The spin quantum number (m_s) indicates the direction of the electron's spin and can either be +1/2 or -1/2.

These numbers not only dictate the electron's properties but also abide by the Pauli exclusion principle, stating that no two electrons in an atom can have the same set of all four quantum numbers.

Electron Capacity in Subshells

The electron capacity in subshells is determined by the number of orbitals present and the rule that each orbital can contain a maximum of two electrons, which must have opposite spins. Specifically:

• For the s subshell (l=0), with only 1 orbital, the maximum electron capacity is 2 electrons.
• For the p subshell (l=1), with 3 orbitals, the maximum electron capacity is 6 electrons.
• For the d subshell (l=2), with 5 orbitals, the maximum electron capacity is 10 electrons.
• For the f subshell (l=3), with 7 orbitals, the maximum electron capacity is 14 electrons.

The electrons fill these subshells according to the aufbau principle, with lower-energy subshells filling before those with higher energy levels.