Question

What is the symmetry number for the following molecules? a. \(^{35} \mathrm{Cl}^{37} \mathrm{Cl}\) b. \(^{35} \mathrm{Cl}_{2}\) \(\mathbf{c} .^{16} \mathrm{O}_{2}\) d. \(\mathrm{C}_{6} \mathrm{H}_{6}\) \(\mathbf{e} . \mathrm{CH}_{2} \mathrm{Cl}_{2}\)

Short answer

The symmetry numbers for the given molecules are: a. \(\text{^{35} Cl^{37} Cl}\): 1 b. \(\text{^{35} Cl_{2}}\): 2 c. \(\text{^{16} O_{2}}\): 2 d. \(\text{C_{6}H_{6}}\): 12 e. \(\text{CH_{2}Cl_{2}}\): 3

Step-by-step solution

Determine symmetry operations

Since the two isotopes are different, there are no symmetry operations that will leave the molecule unchanged.

Calculate the symmetry number

The symmetry number for this molecule is 1, as there are no symmetry operations possible. #b. ^{35} Cl_{2}# #Step 1: Identify the structure of the molecule# This molecule is a diatomic molecule consisting of two identical isotopes of chlorine connected by a single bond.

Determine symmetry operations

There is one symmetry operation possible for this molecule: a 180-degree rotation along the bond axis. This operation will maintain the appearance and properties of the molecule.

Calculate the symmetry number

The symmetry number for this molecule is 2, as there are two possible configurations of the molecule that are indistinguishable: the original configuration and the configuration after the 180-degree rotation. #c. ^{16} O_{2}# #Step 1: Identify the structure of the molecule# This molecule is a diatomic molecule consisting of two identical isotopes of oxygen connected by a double bond.

Determine symmetry operations

Similar to ^{35} Cl_{2}, there is one symmetry operation possible for this molecule: a 180-degree rotation along the bond axis.

Calculate the symmetry number

The symmetry number for this molecule is 2, as there are two possible configurations of the molecule that are indistinguishable: the original configuration and the configuration after the 180-degree rotation. #d. C_{6}H_{6}# #Step 1: Identify the structure of the molecule# The molecule mentioned is benzene, a planar, cyclic and hexagonal molecule with alternating single and double bonds between carbon atoms.

Determine symmetry operations

There are several symmetry operations for benzene, including: 1. Six-fold rotation (C6) about its central axis 2. Six-fold rotation in the opposite direction (C6 inverse) 3. Three-fold rotation (180 degrees) (C2) about an axis passing through opposite carbon atoms 4. Reflection in the plane of the molecule (σh) 5. Alternating pairs of carbon atoms can be exchanged (three σv planes)

Calculate the symmetry number

The symmetry number for benzene is 12, as there are 12 independent symmetry operations that can be performed on the molecule. #e. CH_{2}Cl_{2}# #Step 1: Identify the structure of the molecule# This molecule is a tetrahedral molecule with two hydrogen atoms and two chlorine atoms surrounding the central carbon atom.

Determine symmetry operations

There are three C2 axes on this molecule (axes that pass through the middle of the opposite H-Cl bonds). A 180-degree rotation about these axes will result in indistinguishable configurations.

Calculate the symmetry number

The symmetry number for this molecule is 3, as there are three independent symmetry operations that can be performed on the molecule.

Key concepts

Symmetry Operations

When exploring the symmetry of molecules, we focus on symmetry operations. These are movements or operations that you can perform on a molecule, which leave the molecule looking identical to its original state. Common symmetry operations include rotations around an axis, reflections in a plane, and inversions through a point.
In the context of molecules:

• Rotation involves turning the molecule around an axis by a set angle. For instance, turning a molecule by 180 degrees.
• Reflection is flipping the molecule through a plane, akin to looking at the molecule in a mirror.
• Inversion is less common but involves flipping all parts of the molecule through a central point.

A molecule's symmetry operations can reveal much about its structure and properties, offering insight into how it might interact with other molecules.

Symmetry Number

The symmetry number of a molecule represents the number of distinct configurations into which a molecule can be transformed without effecting a change in its spatial perception. Essentially, it counts the number of symmetry operations that can return the molecule to a visually "indistinguishable" state.
For example, consider a diatomic molecule of chlorine, like \(^{35} \text{Cl}_2\):

• Here, a 180-degree rotation along the bond axis makes the molecule identical in appearance, giving it a symmetry number of 2.

The symmetry number is vital in calculating the entropy and partition functions of molecules in thermodynamics. It helps in predicting how molecules behave under different conditions.

Diatomic Molecules

Diatomic molecules, as the name suggests, consist of just two atoms. Examples include oxygen \(^{16} \text{O}_2\) and chlorine \(^{35} \text{Cl}_2\). Such molecules can be made up of two of the same element (homonuclear, like \(\text{O}_2\)) or two different elements (heteronuclear, like \(^{35} \text{Cl}^{37} \text{Cl}\)).
When considering symmetry:

• The simplicity of diatomic molecules allows for limited symmetry operations, usually a rotation around the bond axis.
• In homonuclear diatomics—like \(\text{Cl}_2\) or \(\text{O}_2\)—the symmetry number often reflects this linear simplicity with values such as 2, resulting from the 180-degree axis rotation.

Despite their simplicity, diatomic molecules exhibit clear examples of rotational symmetry useful in illustrating basic symmetry principles.

Benzene Symmetry

Benzene \(\text{C}_6\text{H}_6\) is a staple example in molecular symmetry studies due to its high symmetry. As a planar cyclic molecule, benzene's structure features:

• A hexagonal ring of carbons, with alternating single and double bonds, contributing to its symmetry.
• Multiple symmetry operations include a six-fold rotational symmetry (around central axis), perpendicular reflections, and several mirror planes.
  • This grants benzene a symmetry number of 12.

Benzene's symmetrical nature underpins its stability and reactivity. Understanding its symmetry operations is crucial for predicting its behavior in chemical reactions, particularly substitution reactions due to its aromatic stability.

Tetrahedral Molecules

Tetrahedral molecules, like \(\text{CH}_2\text{Cl}_2\), have four atoms positioned around a central atom, forming a shape similar to a pyramid with a triangular base. This three-dimensional structure offers distinct symmetry operations:

• The molecule boasts three possible C2 rotational axes that pass through opposite bonds, typical for molecules with such geometry.
• These symmetry operations, including the 180-degree rotations about the C2 axes, provide a symmetry number of 3.

Tetrahedral symmetry is significant in predicting the molecule's physical and chemical properties, including its polar/nonpolar nature affecting interactions and solubility. Understanding this symmetry helps in predicting outcomes in synthesis and reactions.