Question

Which of the quantum numbers governs (a) the shape of an orbital, (b) the energy of an orbital, (c) the spin properties of the electron, \((\) d) the spatial orientation of the orbital?

Short answer

The quantum numbers governing the different properties of electrons in orbitals are as follows: (a) the shape of an orbital is governed by the azimuthal quantum number \(l\), (b) the energy of an orbital is governed by the principal quantum number \(n\), (c) the spin properties of the electron are governed by the spin quantum number \(m_s\), and (d) the spatial orientation of the orbital is governed by the magnetic quantum number \(m_l\).

Step-by-step solution

(a) Shape of an orbital

The shape of an orbital is governed by the azimuthal quantum number \(l\). This quantum number determines the angular momentum and the shape of an electron cloud around the nucleus.

(b) Energy of an orbital

The energy of an orbital is governed by the principal quantum number \(n\). This quantum number determines the energy level as well as the relative size of an orbital. As \(n\) increases, the energy level and the average distance from the nucleus to the electron cloud also increases.

(c) Spin properties of the electron

The spin properties of the electron are governed by the spin quantum number \(m_s\). There are two possible spin states for an electron: \(\frac{1}{2}\) (spin-up) and \(-\frac{1}{2}\) (spin-down), represented by \(m_s\).

(d) Spatial orientation of the orbital

The spatial orientation of an orbital is governed by the magnetic quantum number \(m_l\). This quantum number determines the orientation of the electron cloud in space with respect to a magnetic field. It has integer values ranging from \(-l\) to \(l\), where \(l\) is the azimuthal quantum number.

Key concepts

Azimuthal Quantum Number

Understanding the azimuthal quantum number, denoted as \(l\), is essential when studying the structure of atoms. This number defines the shape of an electron's orbital, which is the region around the nucleus where an electron is most likely to be found. The azimuthal quantum number can take on any integer value from 0 to \(n - 1\), where \(n\) is the principal quantum number. Different values of \(l\) correspond to different orbital shapes, known as subshells:

• \(l = 0\) corresponds to an 's' orbital, which is spherical in shape.
• \(l = 1\) defines a 'p' orbital, which has a dumbbell shape.
• \(l = 2\) stands for a 'd' orbital, which is more complex and often described as cloverleaf-shaped.
• \(l = 3\) represents an 'f' orbital with an even more intricate shape.

Each shape allows the electron cloud to occupy a different set of orientations in space, which are critical for the understanding of how atoms can bond and interact with one another.

Principal Quantum Number

The principal quantum number, symbolized by \(n\), is a key determinant of an electron's energy level and the size of its orbital. Simply put, it tells us which shell, or energy level, an electron occupies in an atom. The principal quantum number is a positive integer (\(n = 1, 2, 3, ...\)), and its value increases the further an electron is from the nucleus, implying both higher energy and a larger orbital. As \(n\) gets larger, the difference in energy between adjacent levels decreases. Different energy levels can house a varying number of electrons, critical information when deducing an element’s electron configuration and its position in the periodic table.

Spin Quantum Number

The electron spinning on its own axis gives rise to its spin quantum number, denoted by \(m_s\). This intrinsic property of electrons contributes to their magnetic moment. There are two fundamental spin states:

• Spin-up with \(m_s = +\frac{1}{2}\)
• Spin-down with \(m_s = -\frac{1}{2}\).

An electron can occupy one of these two states which follow the Pauli exclusion principle, stating that no two electrons in the same atom can have identical values for all four quantum numbers. Therefore, for two electrons to share the same orbital, they must have opposite spins.

Magnetic Quantum Number

The magnetic quantum number, \(m_l\), specifies the spatial orientation of an orbital within a subshell, relative to an external magnetic field. It has integral values that range from \(-l\) to \(+l\), including zero. For instance, if the azimuthal quantum number \(l = 1\) for a p-orbital, \(m_l\) can be -1, 0, or +1, meaning there are three different orientations for a p-orbital in three-dimensional space. The magnetic quantum number is connected to the orbital's orientation and helps in determining the magnetic properties of the atom.

Orbital Shape

Orbital shape is intricately linked to the azimuthal quantum number and plays a vital role in the chemical bonding and the spatial arrangement of atoms in molecules. Each shaped orbital - s, p, d, and f - provides a unique environment for electron occupancy, contributing to the molecule's overall geometry and reactivity. Orbital shapes are best visualized using 3D models or probability density functions that depict the statistical locations where an electron is likely to be found.

Orbital Energy

Orbital energy is principally determined by the principal quantum number (\(n\)), with a higher \(n\) corresponding to a higher energy orbital. However, this is not the whole story as electron repulsion and relativistic effects can split energies of orbitals with different azimuthal quantum numbers within the same principal shell. Understanding the relative energy levels is crucial for predicting an element’s reactivity and is foundational in determining the order of electron filling in atomic orbitals - a concept further explained by the Aufbau principle.

Electron Spin Properties

Electron spin properties encompass not only the direction of the electron's spin but also its associated magnetic moment. The spin of electrons leads to phenomena such as the Zeeman effect, where spectral lines split when exposed to a magnetic field due to the different energy levels of spin states. These properties are essential in spectroscopy and are taken into account in quantum mechanical models which accurately describe the behavior of electrons within atoms.

Orbital Spatial Orientation

Orbital spatial orientation is described by the magnetic quantum number and details how orbitals are arranged in three-dimensional space. The specific orientation of orbitals can influence the bonding angles in molecules and thus, the molecular shape. This is vital in understanding stereochemistry and how molecular shapes impact physical and chemical properties of substances. For complex atoms, the overall orientation of electron clouds can account for the magnetic behavior of the atom and the structure of crystalline solids.