Question

Assign the symmetry point group to the following molecules: (a) Benzene (b) \(\mathrm{H}_{2} \mathrm{O}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{4}\) (d) Acetylene

Short answer

The point groups for the given molecules are: (a) Benzene - D6h, (b) Water - C2, (c) Ethene - D2h, and (d) Acetylene - D∞h.

Step-by-step solution

Assigning Point Group for Benzene

Benzene has a six-fold rotation axis along the axis perpendicular to the plane of the molecule, and also has a plane of symmetry in the plane of the molecule. These characterstics make it belong to the D6h point group.

Assigning Point Group for \(\mathrm{H}_{2} \mathrm{O}\)

Water (\(\mathrm{H}_{2} \mathrm{O}\)) is a bent molecule without any symmetry with respect to a plane or inversion center. However, it has a C2 rotation axis. Therefore, it belongs to the C2 point group.

Assigning Point Group for \(\mathrm{C}_{2} \mathrm{H}_{4}\)

Ethene (\(\mathrm{C}_{2} \mathrm{H}_{4}\)) has a plane of symmetry in the plane of the molecule and also has a C2 rotation axis. Therefore, it belongs to the D2h point group.

Assigning Point Group for Acetylene

Acetylene has a linear geometry with a C∞v rotation axis and an infinite number of planes of symmetry. Therefore, it belongs to the D∞h symmetry point group.

Key concepts

benzene D6h point group

Benzene is a classical example of a molecule exhibiting high symmetry due to its unique structure. It consists of six carbon atoms forming a planar hexagonal ring. One central feature is its six-fold rotational axis (\(C_6\)), which means it can be rotated by 60 degrees and appear the same. This axis is perpendicular to the plane of the molecule.
Moreover, benzene has multiple symmetry planes. Specifically, it has:

• A horizontal plane of symmetry (\( ext{σ}_h\)) that lies on the plane of the ring.
• Six vertical planes (\( ext{σ}_v\)) that pass through each carbon atom and the opposite midpoint of the ring.

These symmetry elements classify benzene into the\(D_{6h}\)point group, making it one of the most symmetrical aromatic compounds known.

water C2 point group

The water molecule (\( ext{H}_2 ext{O}\)) is an interesting case when it comes to symmetry point groups because of its angular or bent shape. Unlike linear molecules, water lacks a center of inversion or a significant symmetry plane. Instead, its primary symmetry element is the\(C_2\)axis.
Here's a closer look:

• A\(C_2\)axis allows rotation by 180 degrees around the central oxygen atom, switching the positions of the two hydrogen atoms while maintaining the overall shape of the molecule.

This\(C_2\)symmetry is the most prominent, thereby placing water into the\(C_2\)point group. It's a simple representation indicating limited but essential symmetry of this ubiquitous molecule.

ethylene D2h point group

Ethylene or\( ext{C}_2 ext{H}_4\)is an organic compound that shows higher symmetry due to its planar conformation. The molecule forms a planar double-bonded structure between the two carbon atoms, with each carbon atom bonded to two hydrogen atoms. Its geometrical symmetry elements include:

• A\(C_2\)axis of rotation passing through the double bond, allowing a 180-degree rotational symmetry.
• A plane of symmetry (\( ext{σ}_h\)) that bisects the molecule horizontally.
• Additional perpendicular planes intersecting through the carbon atoms and the center of the double bond (\( ext{σ}_v\)).

These features give ethylene the\(D_{2h}\)point group designation, reflecting its balance of rotational and reflectional symmetry.

acetylene D∞h point group

Acetylene (\( ext{C}_2 ext{H}_2\)) is a simple alkyne with a distinctive linear structure, contributing to its symmetry. Within the context of symmetry point groups, acetylene features an infinite number of symmetry operations due to:

• A\(C_{ ext{∞}}\)axis which is endless and continuous, indicating that the molecule can be rotated by any angle along this axis that stretches along the bond between the two carbon atoms.
• An infinite number of vertical planes (\( ext{σ}_v\)) that can be conceptualized as slicing through the linear structure at any angle.
• A plane of symmetry (\( ext{σ}_h\)) aligned horizontally through the hydrogen atoms.

The combination of these symmetry elements categorizes acetylene into the\(D_{ ext{∞}h}\)point group, illustrating its extensive symmetry simplified by its linear configuration.