Does a chiral allene have stereogenic centres?

Question

This a picture of what my professor taught in class.

According the the definition of a stereogenic centre, allene should have two of them. So I don't understand what I am interpreting wrong.

Answer 1

The definition of "stereogenic centre" is unhelpful and can be confusing

The large majority of chiral molecules are chiral because a carbon in them has 4 different things attached to it (atoms or more complicated groups). The tetrahedral arrangement around carbon guarantees that the result will be a chiral molecules (with some complications if there are multiple such centres).

But the underlying cause of the chirality is the overall symmetry of the molecule (or, better, lack of certain types of symmetry like mirror planes). Carbon atoms with 4 different substituents are the commonest way to guarantee that type of symmetry. As a result those carbon atoms are often described as "stereogenic centres". But there are many ways to get chiral molecules (and no mirror planes) without the need for that type of stereogenic centre.

Allenes are an example where the chirality does not result from the arrangement around a single carbon. The Wikipedia definition is confusing and unhelpful and most chemists would not use the term "stereogenic centre" for any atom in the molecule.

A more obvious example of a chiral molecule with no stereogenic centre is hexahelicene (which is made from 6 ortho-fused benzene rings). The terminal benzene units can't exist in the same plane so the whole molecule twists into a spiral to minimise spatial interactions. The 3D structure is shown below.

The molecule is forced by spatial interactions to be a spiral which leads to a strongly chiral spatial structure but one where no single atom cause the chirality and no definition of stereogenic centre makes sense.s

Allene is just a simpler example of the general point that, to understand chirality, the overall symmetry of the molecule needs to be considered and not some count of stereogenic centres.

Answer 2

Chirality is a hypernym for the observation that an object can not be superimposed to its mirror image.

Most frequently, examples of chirality in organic chemistry are caused by the spatial arrangement around a specific atom (hence stereogenic centre). An example are four different substituents around carbon (as in mandelic acid); sulfur (as in methyl phenyl sulfoxide), or conformationally locked amines (as in sparteine)*:

Note that a molecule can be achiral despite presence of stereogenic centers (meso compounds).

The idea of a tetrahedral arrangement of atoms around an other equally was the starting point for chemists to consider asymmetry and stereochemistry in a modern way. In the 1870s, this allowed Jacobus Henricus van't Hoff and Joseph-Achille Le Bel to relate Louis Pasteur's observations on optical activity (macroscopic scale) with underlying structure (microscopic scale) with the now famous models, decades prior e.g., to the introduction of single crystal X-ray diffraction analysis.

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An additional type is the one of axial chirality as seen with allenes. If you orient the molecule model to look along its $\ce{C=C=C}$ axis, the sequence of substituents can be either clockwise (assigned by R), or anticlockwise (S):

(figure from the English Wikipedia)

An other example are biphenyl structures. Large substitutents in 2,2',6,6' position can hinder the free rotation around the C-C single bond between the two arenes and is the axis to consider. Consequently, the two structures below differ from each other (if you can, build a model of both):

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Next is helical chirality about molecules in a shape similar to the one of a screw, or spiral stairs. Beside the popular helix of DNA, helicenes are examples typically shown to demonstrate this type. If six benzene rings are fused together to yield a hexahelicene this way:

(source)

there is steric hindrance where the two ends come close to each other. To get away with this uncomfortable situation, one end gets under the other, yielding a screw turning either clockwise P (for plus), or anticlockwise M (for minus):

(source about heptahelicene, i.e. 7 benzene units)

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Chirality is not limited to organic molecules, let molecules alone. The spacial arrangement of molecules in the solid state can be chiral (chiral space groups, e.g. $P6_1$ and $P6_5$), as well as the habit of crystals (e.g. L- and D-quartz, sodium ammonium ($2R$,$3R$)-tartrate Pasteur worked on [see illustrations in Flack's article]) can be chiral, too.

* The CIP rules assign an electron pair the absolute lowest priority (i.e. even lower than a hydrogen atom).

Flack, H. D. Louis Pasteur’s Discovery of Molecular Chirality and Spontaneous Resolution in 1848, Together with a Complete Review of His Crystallographic and Chemical Work. Acta Cryst. A 2009, 65, 371–389. doi 10.1107/S0108767309024088 and author's copy.