Question

Describe the characteristics of an \(s\) orbital, \(p\) orbital, and \(d\) orbital. Which of the following orbitals do not exist: \(1 p, 2 s, 2 d, 3 p, 3 d, 3 f, 4 g ?\)

Short answer

The orbitals 1p, 2d, 3f, and 4g do not exist.

Step-by-step solution

Characteristics of Orbitals

An **s orbital** has a spherical shape and can hold a maximum of 2 electrons. A **p orbital** is dumbbell-shaped and can be oriented in three different directions (px, py, pz), holding up to 6 electrons in total. A **d orbital** has a more complex cloverleaf shape, with five possible orientations, allowing it to hold up to 10 electrons.

Quantum Number Restrictions for Orbitals

The principal quantum number, denoted by 'n', indicates the energy level of an orbital. For any value of 'n', the possible orbital types are s, p, d, and f, with their angular momentum quantum numbers l being 0, 1, 2, and 3, respectively. The 'l' value must be less than 'n'. Hence, **g orbitals (l = 4)** are not possible unless n is 5 or higher.

Determine Existence of Given Orbitals

Given the restriction that the angular momentum quantum number 'l' must be less than 'n', let's determine which orbitals do not exist: - **1p**: Here, n = 1 while l would be 1 (for p orbital), which is impossible because l must be less than n. Thus, 1p does not exist. - **2s**: With n = 2 and l = 0, this exists. - **2d**: Here, n = 2 and l would be 2, which is not possible since l cannot equal n. So, 2d does not exist. - **3p**: With n = 3 and l = 1, this exists. - **3d**: With n = 3 and l = 2, this exists. - **3f**: Here, n = 3 while l would be 3, which is not allowed because l must be less than n. Thus, 3f does not exist. - **4g**: For g orbitals, l = 4, which is not possible because l must be less than n unless n is 5 or higher. Therefore, 4g does not exist.

Key concepts

s orbital

The s orbital is the simplest and most fundamental among all atomic orbitals. Its most notable feature is its spherical shape, which means it has no directional preference. You can imagine it like a ball surrounding the nucleus at the center.
One s orbital is present in every energy level starting from the first (n=1). As it is spherical, it is labeled as having an angular momentum quantum number (\( l \) = 0).

• It can accommodate a maximum of 2 electrons.
• All electrons in the s orbital occupy the same region of space, which simplifies electron distribution models.

Because of its shape, it has unique properties when combined with other orbitals. The spherical symmetry plays a crucial role in bonding and molecular structures.

p orbital

The p orbital differs significantly from the s orbital in both shape and orientation. It appears as a dumbbell-like structure, offering three different orientation possibilities: the px, py, and pz orbitals.
These orientations allow the p orbitals to orient along the x, y, and z axes, respectively. The p orbitals can be characterized by an angular momentum quantum number (\( l \) = 1), and they appear starting from the second energy level (n=2).

• Each of the three p orbitals can hold up to 2 electrons, totaling 6 electrons for all p orbitals in a given energy level.
• Their shape enables directional bonding, a key feature in forming covalent bonds and molecular geometry.

The versatility and orientation of p orbitals contribute significantly to the electron configuration of atoms, influencing their chemical properties.

d orbital

The d orbital is even more complex than the s or p orbitals, characterized by its cloverleaf shape with five distinct orientations. This intricate shape allows for varied electron density distributions around the nucleus.
D orbitals begin appearing at the third energy level (n=3), with an angular momentum quantum number (\( l \) = 2).

• There are five d orbitals, providing a total capacity of 10 electrons between them in any given energy level.
• These orbitals are heavily involved in the chemical bonding of transition metals, contributing to unique magnetic and electronic properties.

The complex interactions among d orbitals are fundamental to understanding transition metal chemistry, influencing color, magnetism, and catalytic behaviors.

quantum numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. They determine the electron's position and energy, and they collectively form the electron configuration of an atom.
There are four types of quantum numbers:

• Principal quantum number (n): Describes the energy level of the electron. The higher the number, the further the electron from the nucleus.
• Angular momentum quantum number (l): Determines the shape of the orbital. It takes on integer values from 0 to n-1, corresponding to s, p, d, and f orbitals.
• Magnetic quantum number (m_l): Indicates the orientation of the orbital in space. It ranges from -l to +l.
• Spin quantum number (m_s): Describes the spin of the electron, which can be either +1/2 or -1/2.

Quantum numbers are essential to the construction of atomic orbitals, dictating how electrons populate those orbitals and the subsequent properties of elements.

electron configuration

Electron configuration is a way of representing the arrangement of electrons in an atom's orbitals. It reflects how electrons are distributed among the various atomic orbitals and energy levels.
The arrangement follows a specific order based on increasing energy levels, observed as the Aufbau principle, Pauli exclusion principle, and Hund's rule.

• Aufbau principle: Electrons fill orbitals starting from the lowest available energy level.
• Pauli exclusion principle: No two electrons in the same atom can have the same set of all four quantum numbers.
• Hund's rule: Electrons will fill degenerate orbitals singly before pairing up.

A correct electron configuration provides insight into an atom's properties, including its reactivity, magnetism, and color. Understanding this fundamental principle aids in predicting the chemical behavior of the element.