An improper rotation axis, 𝑆, arises from which of the following situations? A mirror plane and an identity operation, a mirror plane and an inversion, two mirror planes, a proper rotation axis and a mirror plane, or two proper rotation axes.
In order to answer this question, let’s review what we remember about each of these symmetry operations. Let’s begin with the identity operation. The identity operation is given the letter E, and this refers to the entire object. This means that performing an identity operation actually does nothing to the object itself. This also means that every molecule has an identity operation by default.
Next, let’s consider mirror planes. There are two types of mirror planes, denoted 𝜎 v for vertical or 𝜎 h for horizontal. A 𝜎 v plane is a plane which is parallel to the principal axis. Conversely, a 𝜎 h, or horizontal plane, is perpendicular to the principal axis.
Next, we have the inversion operation, given the letter i. To perform an inversion operation on a molecule, you should imagine that you take each atom and push it through the center of the molecule and out the other side. Let’s illustrate this with an example. Here, we have a molecule, something like methane. When we perform the inversion operation on methane, we’re going to imagine that we take each of the atoms and push them through the center.
In our case, the center of our molecule is the carbon in the middle. As we do this to each hydrogen in turn, we end up inverting our molecule, like this. You can see that atom number one has been pushed through the center of the molecule and out the other side, the opposite side. The same has happened with number two, number three, and number four. The carbon itself stays where it is, since it’s already at the center.
Finally, we come to a proper rotation axis. A proper rotation axis is denoted C subscript 𝑛, and it denotes a rotation by 360 degrees divided by 𝑛. If we imagine a molecule a bit like water, we can find a C two axis down the middle. This means that if we rotate the molecule along this axis by 180 degrees, we end up with something indistinguishable. Molecules can have multiple proper rotation axes, but whichever one has the highest value of 𝑛 is given the name principal axis.
But what about the improper rotation axis 𝑆, which we’re asked about in the question? An improper rotation axis is actually two successive transformations. The first transformation is a rotation through 360 degrees divided by 𝑛. The second is a reflection through a plane, which is perpendicular to the axis that we just performed the rotation about. Note that neither of these transformations has to necessarily be asymmetry operation itself.
Let’s go back to our example of methane. Methane has an improper rotation axis, an 𝑆 four axis. In the first step, we rotate through 360 divided by four, so we rotate by 90 degrees. In the second step, we’re going to reflect through a plane, which is perpendicular to the axis we just rotated about. Here, we have the plane which is perpendicular to the rotation axis, so now all we have to do is reflect our molecule. And here we can see that what we’ve ended up with is the same molecule as we started. Note that although we’ve labeled each of the hydrogens with a number, these are actually all indistinguishable. So, our end result is exactly the same as our starting material.
Now that we’ve illustrated what an improper rotation axis is, we can answer the question. An improper rotation axis is two successive transformations. The first is a rotation through 360 divided by 𝑛. And the second is reflection in a mirror plane. This corresponds to the fourth answer on our list, a proper rotation axis and a mirror plane. So, this is our definition for an improper rotation axis.