Question

On which quantum numbers does the energy of an electron depend in each of the following? a. a one-electron atom or ion b. an atom or ion with more than one electron

Short answer

For a one-electron atom or ion, electron energy only depends on the principal quantum number (n). While in a multi-electron atom or ion, energy depends on both the principal quantum number (n) and the azimuthal quantum number (l). The magnetic and electron spin quantum numbers do not affect energy in any of the cases.

Step-by-step solution

a. One-electron atom or ion

For a one-electron atom or ion, the energy of the electron only depends on the principal quantum number (n). The principal quantum number essentially determines the energy level the electron occupies in the atom. It is represented by a whole positive number (n=1, 2, 3, ...). The energy of the electron increases as n increases, meaning electrons farther from the nucleus have higher energy. In a one-electron atom or ion, angular, magnetic, and electron spin quantum numbers do not affect the energy level.

b. Atom or ion with more than one electron

In atoms or ions with more than one electron, the energy of an electron depends on both the principal quantum number (n) and the azimuthal quantum number (l). While the principal quantum number determines the energy level, the azimuthal quantum number (also known as the angular momentum or orbital quantum number) determines the shape of the electron cloud. l can have values from 0 to (n-1). The value of l affects the energy of the electron due to the fact that electrons experience both the electrostatic attraction from the nucleus and repulsive forces from other electrons occupying the atom or ion. In a multi-electron atom or ion, the energy increases with increasing n and l values, so the electron's energy is dependent on both of these quantum numbers. It is important to note that magnetic (m_l) and spin (m_s) quantum numbers still do not impact the energy of the electron in an atom with more than one electron.

Key concepts

One-electron atoms

In the world of quantum mechanics, the concept of an electron's energy starts simple with a system known as "one-electron atoms." These are atoms or ions that have just a single electron orbiting the nucleus. The fundamental simplicity lies in the factors that determine the electron's energy. For one-electron atoms, the electron's energy is solely governed by the principal quantum number, denoted by \( n \).
The principal quantum number indicates which energy level the electron resides in, essentially the size and energy of the orbital.

• \( n \) is a positive integer: 1, 2, 3, and so on.
• As \( n \) increases, the electron energy increases.

The energy levels are specific, and as \( n \) grows, the electron is found further away from the nucleus, resulting in higher energy. Interestingly, other quantum numbers such as angular, magnetic, and spin do not affect the energy of these electrons in a one-electron system. It stays simple and is a great starting point for understanding how quantum numbers play their roles in an atom.

Multi-electron atoms

When we shift from a single-electron system to atoms or ions with multiple electrons, things get a tad more complex. In such systems, determining the energy of an electron isn't just about the principal quantum number \( n \) anymore. There's a new player involved: the azimuthal quantum number \( l \).
For these multi-electron atoms, electron energy depends on:

• The principal quantum number \( n \), determining the electron energy level.
• The azimuthal quantum number \( l \), affecting the energy level due to its influence on the electron cloud's shape.

The azimuthal quantum number \( l \) can have integer values ranging from 0 to \( n-1 \), changing how electrons perceive both the attractive forces of the nucleus and the repulsive forces from other electrons.
This interplay of forces is what contributes to the variation in energy levels within such complex systems. However, similar to one-electron atoms, the magnetic quantum number \( m_l \) and electron spin quantum number \( m_s \) still do not alter electron energy in these scenarios.

Principal quantum number

Originally introduced to simplify the description of electron configurations, the principal quantum number \( n \) primarily signifies the main energy shell an electron occupies in an atom. This number is essential in both one-electron and multi-electron atoms.
Here’s what \( n \) does:

• It identifies the size and energy level of an orbital.
• The greater the \( n \), the larger and higher the energy of the orbital.
• Each "step up" in \( n \) means the electron is found further from the nucleus.

Its integer values start from 1 and can go as high as needed based on the atomic or ionic structure. With increasing \( n \), not only does the energy of the electron increase, but it also reflects the electron's positioning farther from the nucleus, contributing to higher energy levels. Thus, \( n \) serves as a critical factor in defining the electron’s position and potential energy on an atomic scale.

Azimuthal quantum number

The azimuthal quantum number \( l \), occasionally referred to as the angular momentum quantum number, adds a layer of refinement in understanding electron behavior within atoms, especially in multi-electron systems. It provides insight into the electron cloud's geometrical shape and distribution.
Here’s what \( l \) influences:

• It defines the shape of the atomic orbitals.
• The values of \( l \) range from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• Each value of \( l \) corresponds to a specific orbital shape (\( l = 0 \) for s-orbital, \( l = 1 \) for p-orbital, etc.).

In multi-electron atoms, the azimuthal quantum number determines energy due to its role in electron cloud shape, impacting electron repulsions.
In combination with \( n \), it plays a critical role in fine-tuning an electron's energy state. Overall, \( l \) provides a nuanced understanding of electron placement and energy beyond just the primary shell level.