I previously introduced the concept of chirality and how it is a property of any molecule with only 1 stereogenic centre. (A molecule with stereogenic centres may or may not be chiral, depending on its stereochemistry.) I also defined an enantiomer as a molecule with a non-superimposable mirror image. (Recall that chirality in inorganic chemistry can arise in 2 different ways.)
It is possible for a stereoisomer to NOT have a superimposable mirror image; in fact, such stereoisomers are called diastereomers. Yes, I recognize that defining something as the negation of something else is unusual. If you have learned set theory or probability - as I did in my mathematical statistics classes – then consider the set of all pairs of the stereoisomers of one compound with the same chemical formula – this is the sample space. The enantiomers form a proper subset within this sample space, and the diastereomers are the complement of the enantiomers.
It is important to note that, while diastereomers are not mirror images of each other, they are still non-superimposable. Diastereomers arise from stereoisomers with 2 or more stereogenic centres; here is an example of how they can arise.
1) Consider a stereoisomer with 2 stereogenic centres and no meso isomers*. This isomer has stereoisomers, where denotes the number of stereogenic centres.
2) Find one pair of enantiomers based on one of the stereogenic centres.
3) Find the other pair enantiomers based on the other stereogenic centre.
4) Take any one molecule from Step #2 and any one molecule from Step #3. These cannot be mirror images of each other. (One molecule cannot have 2 different mirror images of itself.) These 2 molecules are diastereomers.
Think back to my above description of enantiomers as a proper subset within the sample space of the pairs of one set of stereoisomers. You can now see why I emphasized that the sample space consists of pairs, since multiple different pairs of stereoisomers can form enantiomers. In my example above, Steps #2 and #3 produced 2 subsets of enantiomers. It should be clear by now that enantiomers and diastereomers are defined as pairs. To further illustrate this point,
a) call the 2 molecules in Step#2 A and B.
b) call the 2 molecules in Step #3 C and D.
A and B are enantiomers. A and C are diastereomers. Thus, it is entirely possible for one molecule to be an enantiomer with a second molecule and a diastereomer with a third molecule.
Here is an example of 2 diastereomers. Notice that they have the same chemical formula but different 3-dimensional orientations – i.e. they are stereoisomers. These stereoisomers are not mirror images of each other, but they are non-superimposable – i.e. they are diastereomers.
Images courtesy of Popnose, DMacks and Edgar181 on Wikimedia. For brevity, I direct you to the Wikipedia entry for diastereomers showing these 4 images in one panel.
*I will discuss meso isomers in a separate lesson.