Question

Give the values for \(n, l\), and \(m_{l}\) for \((a)\) each orbital in the \(2 p\) subshell, (b) each orbital in the \(5 d\) subshell.

Short answer

(a) For each orbital in the \(2p\) subshell, the values are \(n = 2\), \(l = 1\), and \(m_l = -1\), \(0\), \(+1\). (b) For each orbital in the \(5d\) subshell, the values are \(n = 5\), \(l = 2\), and \(m_l = -2\), \(-1\), \(0\), \(+1\), \(+2\).

Step-by-step solution

Determine the values of \(n\) and \(l\) for the \(2p\) subshell.

In the \(2p\) subshell, the given value for the principal quantum number (\(n\)) is \(2\). The letter \(p\) represents the angular momentum quantum number (\(l\)) and each letter corresponds to a numerical value: \(s = 0, p = 1, d = 2, f = 3\), and so on. Therefore, for a \(2p\) subshell, \(l = 1\).

Determine the values of \(m_l\) for the \(2p\) subshell.

Since we know that \(l = 1\) for the \(2p\) subshell, the magnetic quantum number (\(m_l\)) can have values ranging from \(-l\) to \(+l\), i.e., from \(-1\) to \(+1\). Thus, there are three distinct values for \(m_l\): \(-1\), \(0\), and \(+1\). #(b) Each orbital in the \(5d\) subshell#

Determine the values of \(n\) and \(l\) for the \(5d\) subshell.

In the \(5d\) subshell, the given value for the principal quantum number (\(n\)) is \(5\). The letter \(d\) represents the angular momentum quantum number (\(l\)) and each letter corresponds to a numerical value: \(s = 0, p = 1, d = 2, f = 3\), and so on. Therefore, for a \(5d\) subshell, \(l = 2\).

Determine the values of \(m_l\) for the \(5d\) subshell.

Since we know that \(l = 2\) for the \(5d\) subshell, the magnetic quantum number (\(m_l\)) can have values ranging from \(-l\) to \(+l\), i.e., from \(-2\) to \(+2\). Thus, there are five distinct values for \(m_l\): \(-2\), \(-1\), \(0\), \(+1\), and \(+2\).

Key concepts

Principal Quantum Number (n)

The principal quantum number, denoted as n, is a crucial concept in understanding the arrangement of electrons within an atom. This quantum number defines the energy level or shell of an electron within the atom and is always a positive integer, starting with the value of 1. As n increases, the energy of the electron and the size of its orbit also increase, indicating that the electron is located further away from the nucleus. For example, electrons in the 2^nd energy level, denoted as n = 2, are generally farther from the nucleus and possess higher energy than those in the 1^st level (n = 1).

Therefore, if a student is confronting the exercise to determine the quantum numbers of electrons in various subshells, identifying the principal quantum number is the foundational step. This number provides a roadmap to understanding the energy characteristics of the electrons under scrutiny.

Azimuthal Quantum Number (l)

Immediately following the concept of the principal quantum number is the azimuthal (or angular momentum) quantum number, l. This number is derived from the principal quantum number and takes on integer values from 0 to n - 1. Each value of l corresponds to a different subshell or orbital type where electrons reside, characterized by their angular momentum. The standard labels for these subshells are s (sharp), p (principal), d (diffuse), and f (fundamental), which correspond to l values of 0, 1, 2, and 3, respectively.

In the educational example provided, the 2p subshell has an azimuthal quantum number of l = 1, indicating the type of orbital where the electrons are found. Understanding l is essential as it not only informs us about the shape of the electron's orbital but also affects the energy levels within a particular principal quantum number.

Magnetic Quantum Number (ml)

Exploring further into the depths of quantum numbers, we encounter the magnetic quantum number, denoted by m_l. This quantum number can have integer values ranging from -l to +l, inclusive. These values describe the orientation of the electron's orbital in space relative to an external magnetic field. The number of possible m_l values for a given subshell informs us about the number of orbitals present. For instance, in the example given for the 2p subshell with l = 1, there are three possible magnetic quantum numbers: -1, 0, and 1. These correspond to the three p orbitals, each oriented differently in three-dimensional space.

As students progress in their understanding of magnetic quantum numbers, they grasp the concept that for every subshell type, the orbitals are distributed in unique ways, reflecting the diversity and complexity of electron behavior in atoms.

Subshell Notation in Atoms

Subshell notation is a way of representing the distribution of electrons within the sublevels of an atom. This notation consists of the principal quantum number (n), followed by the letter associated with the azimuthal quantum number (l). The letters, as previously mentioned, denote the shape of the orbitals: s, p, d, or f.

Using subshell notation, it is straightforward for students to visualize where electrons are likely to be found. For instance, 3d indicates electrons are in the d-shaped orbitals of the third energy level. This notation also leads naturally to discussions on electronic configuration, and it is the scaffold upon which students can build an understanding of the complex arrangement of electrons within an atom.

Electronic Configuration

The final key topic that draws upon all previously mentioned quantum numbers is the electronic configuration. It describes the precise distribution of electrons in an atom's orbitals and can be thought of as the 'address' of each electron. The configuration is expressed utilizing the principal quantum number (n), subshell notation (s, p, d, f), and the total number of electrons within those subshells, typically denoted as a superscript.

For example, the electronic configuration for Oxygen is 1s² 2s² 2p⁴. This tells us that Oxygen has two electrons in the 1s subshell, two in the 2s subshell, and four in the 2p subshell. This configuration informs chemists and students alike about the atom’s reactivity, bonding potential, and even its physical properties. Understanding electronic configuration is, therefore, fundamental for a deep comprehension of chemistry and materials science.