### Video Transcript

A molecule has the following symmetries only: πΈ, πΆβ, and 2π v. What is the point group of the molecule?

In this question, symmetry refers to symmetry elements, points, lines, or planes, that define the symmetry of the molecule. πΈ signifies the identity symmetry element, meaning we apply no change whatsoever to the molecule. All geometries have this symmetry element. πΆβ indicates a two-fold proper rotation axis indicating that a rotation of 360 degrees divided by two, 180 degrees, produces the same molecule. 2π v signifies that the molecule has two planes of symmetry parallel to the principal rotational axis, our πΆβ axis. These are said to be vertically aligned with the principal axis, thus the letter v.

A molecule of water displays all these symmetry elements and only these symmetry elements. The point group of a molecule is the summary of the symmetry elements that molecule has. In order to workout the point group, we need our point group flow chart. The first point on our flowchart is whether the molecule is linear or not. All linear molecules have a πΆ β axis. Meaning, along their interatomic axis, they can be rotated by an infinitely fine angle and still produce the same molecule.

In the question, weβve been told that this molecule only has πΈ, πΆβ, and 2π v symmetry elements, no πΆ β. After answering no to the question linear, we arrive at the question whether our molecule has two or more πΆ n axes where n is greater than two . So weβd be looking for something like to 2πΆβ axes or 4πΆβ axes. Again, the answer is no. We only have the πΆβ axis. So our n isnβt high enough, and we donβt have enough of them.

The next point on our flowchart is the question, do we have any proper rotational axes at all? We do; we have a πΆβ axis. An answer yes leads us to the next question. Whether we have the same number of πΆβ axes perpendicular to our principal axis. In this case, weβd be looking for 2πΆβ axes. Weβd be looking for configuration, something like this. However, in this case we just have the one πΆβ axis. So, we have to answer no to this question.

The next question is whether we have a plane of symmetry that is horizontal with respect to the principal axis. We only have π vs and no π hs. So, the answer to this question is no.

The last question on this trail is whether we have as many π vs as the n in our principal axis. n is equal to two. So, weβre looking to have 2π vs. So our final answer is πΆ βv. Here, we take the general term πΆ nv and plug in our value for n for our principal axis. This question contain all the information necessary to guide you through the flowchart. You didnβt need to know necessarily that water was an example of a πΆ βv molecule. Weβve demonstrated that any molecule with the symmetry elements πΈ, πΆβ, and 2π v only will have a point group of πΆ βv.