Question

Which of the four quantum numbers \(\left(n, \ell, m_{\ell}, m_{s}\right)\) determine (a) the energy of an electron in a hydrogen atom and in a many- electron atom, (b) the size of an orbital, (c) the shape of an orbital, (d) the orientation of an orbital in space?

Short answer

a) For a hydrogen atom, the energy level is determined by \(n\). But for many-electron atoms, both \(n\) and \(\ell\) play roles. b) The size of an orbital is influenced by \(n\). c) The shape of an orbital is influenced by \(\ell\). d) The orientation of an orbital is determined by \(m_{\ell}\).

Step-by-step solution

Determine the Quantum Number Affecting the Energy Level

The energy of an electron in an atom is determined by the principal quantum number \(n\) in a hydrogen atom. However, in many-electron atoms, the energy depends both on \(n\) and \(\ell\) due to the electron-electron interactions and the penetration effect.

Identify the Quantum Number Affecting the Size of Orbital

The size of an atomic orbital is defined by the principal quantum number \(n\). As \(n\) increases, the electron is further from the nucleus, thus the orbital size increases.

Determine the Quantum Number Affecting the Shape of Orbital

The shape of an atomic orbital is described by the azimuthal quantum number \(\ell\). For instance, when \(\ell = 0\), the shape of the orbital is spherical (s-orbital), when \(\ell=1\), the shape is dumbbell (p-orbital), and so on.

Define the Quantum Number Affecting the Orientation of Orbital

The orientation of an atomic orbital in space is given by the magnetic quantum number \(m_{\ell}\). For each value of \(\ell\), there can be \(-(2\ell+1)\) values of \(m_{\ell}\), each corresponding to a different orientation of the orbital in space.

Key concepts

Hydrogen Atom Energy Levels

In a hydrogen atom, energy levels are primarily determined by the principal quantum number \(n\). This quantum number represents the electron's shell level and is the main factor dictating the energy of an electron within a hydrogen atom. Higher \(n\) values mean the electron is located further from the nucleus, and thus, the energy levels are higher. However, when dealing with many-electron atoms, the azimuthal quantum number \(\ell\) also plays a crucial role. Interactions between electrons, known as electron-electron repulsions, cause a splitting of energy levels that do not exist in hydrogen. In this context, both \(n\) and \(\ell\) influence the precise energy levels of the electrons. This happens due to shielding and penetration effects, where inner electrons shield outer electrons from the full nuclear charge, effectively modifying energy levels as observed in multi-electron systems.

Atomic Orbitals

Atomic orbitals are regions around the nucleus where there is a high probability of finding an electron. Each orbital is characterized by a unique set of quantum numbers which provide significant information about its properties:

• Size: Primarily determined by the principal quantum number \(n\). The larger the \(n\), the larger the orbital, as electrons occupy a greater volume around the nucleus.
• Shape: Determined by the azimuthal quantum number \(\ell\). Each \(\ell\) defines a unique shape: \(\ell = 0\) for spherically shaped "s" orbitals, \(\ell = 1\) for "p" orbitals, which are dumbbell-shaped, and so forth.
• Orientation: Given by the magnetic quantum number \(m_{\ell}\), which describes how these shapes are oriented in three-dimensional space.

These characteristics collectively define how orbitals exist and overlap, influencing the chemical behavior of atoms.

Principal Quantum Number

The principal quantum number \(n\) is a fundamental part of quantum mechanics that tells us about the large-scale properties of an electron's energy and spatial distribution within an atom.

• It determines the electron's overall energy level or shell. The higher the \(n\), the higher the energy level and the further the electron is from the nucleus.
• It also influences the size of the orbital. Larger \(n\) values mean larger orbitals.
• In the quantum mechanics of the hydrogen atom, \(n\) alone determines the energy. But in more complex, many-electron atoms, \(\ell\) is also needed.

Thus, \(n\) provides essential insight into the electronic structure and determines many of an atom's chemical and physical characteristics.

Azimuthal Quantum Number

The azimuthal quantum number \(\ell\), also known as the angular momentum quantum number, is crucial in defining the shape of an atomic orbital. Each possible value of \(\ell\) indicates a different shape:

• \(\ell = 0\): Describes "s" orbitals, which are spherical.
• \(\ell = 1\): Represents "p" orbitals, which have a dumbbell shape.
• \(\ell = 2\): Corresponds to "d" orbitals, with more complex shapes.
• \(\ell = 3\): Marks "f" orbitals, with even more intricate forms.

By influencing an orbital's shape, \(\ell\) also affects the atomic and chemical properties, dictating how atoms bond and interact with each other. In multi-electron systems, \(\ell\)’s influence extends to energy level splitting, a product of its role in electron-electron interactions.