What exactly is a 'chiral centre' in a carbon compound?

Question

The typical progression (in books, YouTube videos) of explanation of chiral carbons in a compound is as follows:

1. Chiral compounds are non-superimposable by their mirror-image;
2. It is easily seen that only carbon atoms with all substituents different are chiral;
3. In a carbon compound with more than one atom, (2.) is exactly how you check for chirality;
4. It isn't necessary for a compound to be chiral despite having chiral carbons.

It seems to me that something between (2) and (3) is being eaten up. How does a chiral carbon in an chiral compound show the non-superimposability property? How does it satisfy the basic definition of chirality?

Answer 1

Skipping axial and helical chirality altogether, no to point #2: there are examples of atom centred chirality without a carbon atom in the centre. The careful oxidation of thioethers may yield chiral sulfoxides:

(Han et al., Chem. Soc. Rev. 2018, 47, 1307-1350, doi 10.1039/C6CS00703A).

Here, chiral requires just the two carbon substitutents to differ, because the third is oxygen, and the fourth the lone electron pair. Thus, applying CIP rules as usual where the lone electron pair has the lowest priority, DMSO is not chiral, contrary to methyl phenylsulfoxide which is.

Just as with stereogenic centres around carbon, deprotonations on the adjacent prochiral carbon atom may proceed with preference for one hydrogen over the other:

(Solladié, Synthesis 1981, 185-196, doi 10.1055/s-1981-29378).

Answer 2

TLDR: They deal with similar, but different, notions of "chirality".

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First, the definitions, from the IUPAC Gold Book:

asymmetric carbon atom
The traditional name (van't Hoff) for a carbon atom that is attached to four different entities (atoms or groups), e.g. Cabcd.

chirality centre
An atom holding a set of ligands in a spatial arrangement which is not superposable on its mirror image. A chirality centre is thus a generalized extension of the concept of the asymmetric carbon atom to central atoms of any element, for example N+abcd, Pabc as well as Cabcd.

chirality
The geometric property of a rigid object (or spatial arrangement of points or atoms) of being non-superposable on its mirror image [...]

chiral
Having the property of chirality. As applied to a molecule the term has been used differently by different workers. Some apply it exclusively to the whole molecule, whereas others apply it to parts of a molecule. For example, according to the latter view, a meso-compound is considered to be composed of two chiral parts of opposite chirality sense; this usage is to be discouraged. [...]

From the first and second, we see that the idea of a "chiral carbon" is unnecessarily restrictive, because other elements such as nitrogen or phosphorus can also have four different substituents (for elements in Period 3 and higher, this includes lone pairs). This isn't the point of your question, but is worth mentioning.

From the second and third, we see that the idea of a "chiral centre" is related to, but is not the same as, the idea of "chirality". The "chiral centre" refers to chirality in the local environment of the atom (it says that the atom is not superimposable on its own mirror image). On the other hand, the term "chirality" is almost exclusively used to refer to an object as a whole, i.e. the molecule and not just one particular atom. In fact, IUPAC discourages the usage of the word "chiral" in the local sense, as seen in the fourth definition.

The idea of "local chirality" (as defined by a "chiral centre") is useful, but one should not read too much into it. The presence of "local chirality" about a carbon atom (for example) can be a quick and easy guide to determining whether the molecule as a whole is chiral, and no doubt you will have seen examples of these. Why does this work? Very loosely speaking, if the "local chirality" isn't "cancelled out" by any other factor, then the molecule as a whole has some net chirality. More formally, if the molecule is to be achiral i.e. superimposable on its own mirror image, then the "local chirality" must be reflected onto something that is its own enantiomer (since it is locally chiral, it can't be reflected onto itself).

At the same time, a compound can have chiral centres but not be chiral: this is the case with meso compounds like meso-tartaric acid. Here there is local chirality at C-2 and C-3, but in a sense it "cancels out": C-2 reflects onto C-3 and vice versa. The opposite is true, too: a compound can be chiral but not have any chiral centres, like an allene.

Answer 3

To deal with just question 2 (which is not a complete picture as chiral centres may be non-carbon atoms and some molecules are chiral even without having an individual chiral centre).

A carbon atom with 4 different things attached to it is necessarily chiral because of its geometry. The bonds around the carbon will form an approximate tetrahedron and any tetrahedron with 4 different things on its corners will be chiral because its mirror-image will be different. This is a fundamental property of the geometry of a tetrahedron and can be tested on any scale by building such a tetrahedron and comparing it to its mirror image.

Try, for example, colouring the corners of a tetrahedral die with 4 different colours. Or build a tetrahedral structure with a piece of plasticine and matchsticks, colouring the match-ends with 4 different colours. With any such structure you will find that its mirror image is different to the original.

To fully understand this you would need to understand the mathematics of symmetry and group theory. But those simple models should be enough to convince you that it is inevitably true.