Question

How many orbitals are found in (a) an s subshell, (b) a \(p\) subshell, (c) a \(d\) subshell, and (d) an \(f\) subshell?

Short answer

An s subshell has 1 orbital, a p subshell has 3 orbitals, a d subshell has 5 orbitals, and an f subshell has 7 orbitals.

Step-by-step solution

Identify the s Subshell

An s subshell corresponds to an angular momentum quantum number, l, of 0. For any given subshell, the number of orbitals is given by the formula: number of orbitals = 2l + 1. Plugging in l = 0 for an s subshell gives us 1 orbital.

Identify the p Subshell

A p subshell corresponds to an angular momentum quantum number, l, of 1. Using the formula for the number of orbitals, 2l + 1, and substituting l = 1, we find that there are 2(1) + 1 = 3 orbitals in a p subshell.

Identify the d Subshell

A d subshell corresponds to an angular momentum quantum number, l, of 2. Again using the formula 2l + 1, substitute l = 2 to find 2(2) + 1 = 5 orbitals in a d subshell.

Identify the f Subshell

An f subshell corresponds to an angular momentum quantum number, l, of 3. With the formula 2l + 1, put in l = 3 to determine there are 2(3) + 1 = 7 orbitals in an f subshell.

Key concepts

Understanding the s Subshell in Chemistry

In the realm of chemistry, understanding the s subshell is fundamental to comprehending the structure of atoms and how they bond. The s subshell is characterized by its spherical shape, distinguishable from other subshells by the angular momentum quantum number, denoted as 'l', which is 0 for s subshells.

The shape of the s subshell contributes to its unique property of being able to hold a maximum of two electrons, which both occupy the same spherical space. The number of orbitals in an s subshell is derived using the formula: number of orbitals = 2l + 1. By substituting l with 0 for s subshells, we find that it contains only one orbital. This simplicity in structure and electron capacity makes the s subshell a key starting point for understanding electron configuration and the periodic table's structure.

Exploring the p Subshell in Atoms

Moving on from the s subshell, the p subshell introduces a bit more complexity into an atom's electron configuration. It is designated by an angular momentum quantum number 'l' of 1. With a dumbbell-like shape, the p subshell extends in three dimensions and can accommodate up to six electrons.

To determine the number of orbitals, we again apply the formula: number of orbitals = 2l + 1. Plugging in the value of 1 for l, we find 2(1) + 1 = 3, confirming that there are three orbitals in a p subshell. Each orbital can hold a maximum of two electrons, which is why the p block elements in the periodic table have valence electrons that populate these three orbitals, shaping the chemical properties and reactivity of these elements.

Deciphering the d Subshell's Role

When we venture further into transition metals, we encounter the d subshell with an angular momentum quantum number 'l' of 2. The d subshells become relevant in the middle of the periodic table and are characterized by a more complex, cloverleaf-like shape.

Applying the orbital formula, 2l + 1, with l set to 2, it results in 2(2) + 1 = 5 orbitals within the d subshell. These orbitals accommodate the ten electrons that define the chemical characteristics of transition metals. Because of the increased number of orbitals and electrons, the d subshell plays a crucial role in the formation of metallic bonds and the diverse chemistry of transition elements, including their ability to form colored compounds and display various oxidation states.

The f Subshell's Complexity and Significance

The f subshell is often associated with the inner transition metals and is known for its high complexity. Corresponding to an angular momentum quantum number 'l' of 3, the f subshell has a more intricate shape than the s, p, and d subshells and lies further away from the nucleus.

Using our familiar formula, number of orbitals = 2l + 1, and substituting l with 3 for f subshells, we get 2(3) + 1 = 7 orbitals. With each orbital capable of holding two electrons, the f subshell can accommodate a total of 14 electrons. This large capacity for electrons gives the lanthanides and actinides, the elements that contain f subshells in their electron configurations, unique magnetic and spectroscopic properties. Given their use in advanced technology and materials, understanding the f subshell is vital for modern chemistry and material science.