Problem 43 - Chemistry and Chemical Reactivity

Question

A particular orbital has \(n=4\) and \(\ell=2 .\) What must this orbital be: (a) \(3 p,\) (b) \(4 p,\) (c) \(5 d,\) or \((\text { d) } 4 d ?\)

Step-by-Step Solution

Answer

The correct answer is (d) 4d.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the electrons in them. Here, \(n\) is the principal quantum number indicating the energy level, and \(\ell\) is the azimuthal quantum number which indicates the subshell type (\(s: \ell=0\), \(p: \ell=1\), \(d: \ell=2\), \(f: \ell=3\)).

Step 2: Identify the Given Quantum Numbers

The problem states that \(n=4\) and \(\ell=2\). This means we are looking for an orbital on the fourth energy level (\(n=4\)) that is a \(d\) subshell (\(\ell=2\)).

Step 3: Matching with Options

We need to find an option where \(n=4\) and the letter corresponds to \(\ell=2\), which is a \(d\) orbital. The options are: (a) \(3p\), (b) \(4p\), (c) \(5d\), and \((d) 4d\).

Step 4: Selecting the Correct Option

From the options given, \(3p\) and \(4p\) correspond to \(\ell=1\) (the \(p\) subshell), so they do not satisfy \(\ell=2\). The option \(5d\) has the right subshell, \(d\), but the wrong principal quantum number, \(n=5\), as we need \(n=4\). The option \(4d\) correctly matches \(n=4\) and \(\ell=2\), making it the correct answer.

Key Concepts

Principal Quantum Number

In the world of quantum mechanics, the principal quantum number, denoted by the letter \( n \), plays a crucial role in understanding atomic structure. This quantum number indicates the primary energy level that an electron occupies in an atom. The values of \( n \) are positive integers \( (1, 2, 3, 4, \ldots) \). Each energy level corresponds to a shell around the nucleus where electrons are likely to be found.
The principal quantum number not only signifies the energy of an electron but also its radial distance from the nucleus. As \( n \) increases, the electron's energy level and its average distance from the nucleus both increase. Thus, higher energy levels are farther from the nucleus and typically hold more electrons. This increase in distance and energy leads to a broader area of influence for these electrons within an atom, explaining why higher shells can accommodate more electrons.
It's essential to grasp the principal quantum number as it lays the foundation for further understanding of other quantum numbers and atomic orbitals.

Azimuthal Quantum Number

The azimuthal quantum number is symbolized by \( \ell \) and it determines the shape of the orbital in which an electron resides. Each principal quantum number, \( n \), can have azimuthal quantum numbers ranging from \( 0 \) to \( n-1 \). This number is vital in defining the subshells within each principal energy level, with specific numbers corresponding to subshell labels:

• \( \ell = 0 \) corresponds to an \( s \) subshell
• \( \ell = 1 \) corresponds to a \( p \) subshell
• \( \ell = 2 \) corresponds to a \( d \) subshell
• \( \ell = 3 \) corresponds to an \( f \) subshell

In the provided exercise, with \( n = 4 \) and \( \ell = 2 \), this indicates a \( d \) subshell on the fourth energy level. Understanding these mappings allows the identification of the specific type of subshell in which electrons are found, which is critical for predicting chemical bonding and electron configurations.

Atomic Orbitals

Atomic orbitals are regions in an atom where there is a high probability of finding electrons. Each atomic orbital is defined uniquely by a set of quantum numbers \( n \), \( \ell \), and \( m_\ell \). The shape and orientation of these orbitals are dictated primarily by the azimuthal quantum number \( \ell \).
Each principal quantum energy level \( n \) can consist of multiple types of subshells, determined by \( \ell \). For instance, an energy level with \( n = 4 \) includes subshells with \( \ell \) values of 0 (4s), 1 (4p), 2 (4d), and 3 (4f). Each subshell contains one or more orbitals, which can hold up to two electrons. The number of orbitals in any given subshell is determined by the number of possible orientations \( m_\ell \) values can take, ranging from \( -\ell \) to \( +\ell \).
Thus, understanding atomic orbitals provides a more detailed picture of how electrons are organized around an atom, laying the groundwork for explaining the electron configuration of an atom.

Electron Configuration

Electron configuration describes the distribution of electrons in the atomic orbitals of an atom. It follows a set of principles including the Aufbau principle, Pauli-exclusion principle, and Hund’s rule, which guides the filling order of electrons in orbitals. Each subshell can hold a specific number of electrons: \( s \) subshell holds a maximum of 2, \( p \) holds 6, \( d \) holds 10, and \( f \) holds 14 electrons.
The typical order of filling starts from the lowest energy level to the highest (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, and so on). Understanding electron configuration is essential, as it determines the chemical behavior of an element, influencing its reactivity, bonding, and magnetic properties.
For example, in the context of the exercise given, recognizing that \( n=4 \) and \( \ell=2 \) corresponds to a 4d subshell allows us to ascertain that the electron filling for this orbital occurs after 4s and 3d, following the established order. Mastery of electron configuration aids in making predictions about atomic and molecular chemistry.