Question

A completely filled \(d\) -orbital \(\left(\mathrm{d}^{10}\right)\) is of (a) Spherical symmetry (b) Octahedral symmetry (c) Tetrahedral symmetry (d) Unsymmetry

Short answer

A completely filled d-orbital \(d^{10}\) is of spherical symmetry.

Step-by-step solution

Understand electron configurations and orbital shapes

Recognize that the shape and symmetry of the electron cloud for an atom is determined by the distribution of electrons in its orbitals. In the case of a completely filled d-orbital \(d^{10}\), all five d-orbitals (dxy, dyz, dxz, dx2-y2, dz2) will be filled. The combined shape of these orbitals is important in understanding the symmetry.

Examine the shape and symmetry of the filled d-orbitals

When all d-orbitals are completely filled with a total of 10 electrons, the distribution of this electron density is uniform in all directions around the nucleus. A completely filled set of d-orbitals contribute to a spherically symmetrical electron cloud because the orbitals are evenly occupied.

Identify the correct symmetry

Knowing that the electron density of the d-orbitals is evenly distributed in three-dimensional space for a completely filled subshell, and taking into account that the other options refer to specific geometries (octahedral and tetrahedral) that are not relevant to a filled d-orbital, we can conclude the symmetry is spherical.

Key concepts

Electron Configurations

Understanding electron configurations is crucial in chemistry, as it explains how electrons are distributed within the orbitals surrounding an atom's nucleus. The configuration follows a set of rules known as the Pauli exclusion principle and Hund's rule, which dictate the order in which electrons fill subshells and orbitals.

For instance, a completely filled d-orbital, denoted \(d^{10}\), indicates that there are ten electrons in the d-subshell. This is significant because the particular filling of these orbitals affects an atom's chemical properties and bonding behavior. The d-orbitals can house up to 10 electrons as they consist of five different orbitals (dxy, dyz, dxz, dx2-y2, and dz2), with each capable of holding two electrons.

When these five d-orbitals are fully occupied, they create a balance that aligns with the concept of spherical symmetry in the electron cloud, which is essential in determining an atom's reactivity and interaction with other atoms.

Orbital Shapes

Orbital shapes are a fundamental aspect of quantum chemistry. They describe the three-dimensional regions where there is a high probability of finding an electron. Each type of orbital—s, p, d, and f—has a unique shape that influences how electrons are distributed in space around an atom's nucleus.

Regarding d-orbitals, they have more complex shapes than s or p orbitals. They exhibit lobes and are oriented differently in three-dimensional space. There are five types of d-orbitals: dxy, dyz, dxz, dx2-y2, and dz2. While their names might seem cryptic, they actually describe the planes and axes along which the lobes of the orbitals are aligned.

When dealing with a completely filled d-orbital, each one of these five orbitals contains two electrons. This results in a balanced distribution that potentially eliminates any directional preference in electron density, hence contributing to an overall spherical symmetry as seen in \(d^{10}\) configurations.

Spherical Symmetry

Spherical symmetry is an elegant concept in chemistry which implies that an object or distribution is symmetrical in all directions from the center point—much like a perfectly round ball. In the context of d-orbitals, when all orbitals are fully occupied as in the \(d^{10}\) scenario, the combination of the filled orbitals leads to an electron cloud that is evenly distributed around the nucleus.

This symmetry is in contrast to other types, such as tetrahedral or octahedral, which have specific geometric arrangements and directional preferences of electron clouds. The spherical symmetry plays a crucial role in an atom's ability to form bonds and its overall chemical behavior. It often indicates a lower energy state where electrons are in their most stable configuration, minimizing repulsive forces among them.