Question

Determine the number of symmetry operations and assign the symmetry point group to the following molecules: (a) Ruthacene (b) \(\mathrm{N}_{2} \mathrm{O}\) (c) \(\mathrm{PCl}_{5}\) (d) trans \(-\mathrm{N}_{2} \mathrm{~F}_{2}\) (e) \(\mathrm{BF}_{3}\) (f) \(\mathrm{CO}_{2}\) (g) \(\mathrm{N}_{2} \mathrm{O}_{4}\) (h) \(\mathrm{POCl}_{3}\) (i) Acetaldehyde (i) Diborane

Short answer

(a) Ruthacene belongs to the C2h point group. (b) \(N_2O\) belongs to the D∞h point group. (c) \(PCl_5\) belongs to the D3h point group. (d) trans-\(N_2F_2\) belongs to the C2v point group. (e) \(BF_3\) belongs to the D3h point group. (f) \(CO_2\) belongs to the D∞h point group. (g) \(N_2O_4\) belongs to the D4h point group. (h) \(POCl_3\) belongs to the C3v point group. (i) Acetaldehyde belongs to the Cs point group. (j) Diborane belongs to the D2h point group.

Step-by-step solution

Understanding Symmetry

To begin with, it is important to understand the symmetry properties of the molecules. The molecular symmetry is determined primarily by the geometry of the molecule and the spatial position of the atoms.

Identifying Molecular Geometry

To assign the point group, the molecular geometry of each listed molecule is to be known. For example, \(N_2O\) is linear, \(BF_3\) is trigonal planar, \(CO_2\) is linear, and so on.

Determining the Symmetry Operations

The number of symmetry operations for each molecule is determined from its atomic arrangements. For instance, \(CO_2\), being a linear molecule, has unlimited C∞ rotations and σv reflection planes but no inversion i and mirror plane σh.

Assigning the Point Group

Using the identified symmetry operations, the point group of the molecule is assigned. For instance, \(CO_2\) belongs to the D∞h point group while \(BF_3\) belongs to the D3h point group.

Key concepts

Symmetry Operations

Symmetry is a fundamental aspect in chemistry that reveals the harmony within a molecule's structure. Symmetry operations are the actions we can perform on a molecule that result in an identical configuration. Understanding these operations is essential for recognizing patterns in molecular structures.

There are four main types of symmetry operations:

• Rotation (C): Turning the molecule about an axis without changing its location. A subcategory includes proper and improper rotations, denoted by Cn and Sn respectively.
• Reflection (σ): Creating a mirror image across a plane. Reflection can be vertical (σv), horizontal (σh), or dihedral (σd).
• Inversion (i): Moving each point of the molecule through a central point so it comes out equidistant on the opposite side.
• Identity (E): The do-nothing operation which is considered a symmetry operation because it maintains the molecule's appearance.

For students to fully grasp this concept, considering examples from the exercise can be instrumental. For instance, carbon dioxide ($CO\_2$) displays several C∞ rotations due to its linear geometry, whereas molecules with angular configurations might exhibit fewer symmetry operations. The goal is to look for these operations systematically to build understanding and ease in handling molecular symmetries.

Symmetry Point Groups

A symmetry point group is a classification for molecules based on their symmetry operations. It is like a family name that groups together molecules with similar symmetry properties. These groups are described by specific notations that convey information about the symmetries present in the molecule.

Symmetry point groups fall into several categories, including linear, cyclic, dihedral, and cubic, each with their own set of defining features. For instance:

• Linear molecules, like $N\_2O$, belong to the D∞h group if they have a horizontal plane of symmetry (σh), or C∞v if they do not.
• Tetrahedral molecules, such as methane, are placed in the Td group.
• Planar molecules with a central atom and identical surrounding atoms, such as $BF\_3$, are part of the D3h group.

By learning to assign a molecule to a point group, students can later predict numerous properties like the molecule's polarization behavior and its selection rules for spectroscopy. Correctly assigning a molecule to its symmetry point group requires a clear understanding of its symmetry operations, which depend on the molecular geometry.

Molecular Geometry

Molecular geometry is the 3D structure that a molecule takes, defined by the spatial arrangement of its atoms. It not only determines the physical and chemical characteristics of the molecule but also underlies its symmetry. Common molecular geometries include linear, bent, trigonal planar, tetrahedral, trigonal bipyramidal, and octahedral.

Each geometry has distinctive angles and bond distances that maximize the space among electron pairs to minimize repulsion, according to the VSEPR (Valence Shell Electron Pair Repulsion) theory. For example:

• $CO\_2$ is linear, resulting in a molecule that is symmetric along one axis.
• Phosphorus pentachloride ($PCl\_5$) has a trigonal bipyramidal geometry which plays into its specific symmetry properties and resulting point group.

Students often learn about molecular geometry through models or visualization tools, which can help build a vivid understanding of the molecule's spatial configuration. Seeing these structures helps facilitate a better grasp of how symmetry operations apply and why molecules fit into specific point groups. Recognizing the geometry of a molecule, one can anticipate its symmetry related behavior and interactions in a given environment, reinforcing the interconnection between structure and function.