Why alkanes containing even number of carbons can be packed more efficiently than alkanes with odd number of carbon atoms?

Question

Variation of boiling and melting points of alkanes with increase in the number of carbons

The odd-numbered alkanes have a lower trend in melting points than even numbered alkanes. This is because even numbered alkanes pack well in the solid phase, forming a well-organized structure, which requires more energy to break apart. The odd-numbered alkanes pack less well and so the "looser" organized solid packing structure requires less energy to break apart.

Source: https://en.wikipedia.org/wiki/Alkane#Melting_points

Why is that the alkanes containing an even number of carbons can be packed more easily compared to the alkanes containing an odd number of carbons?

Both types of alkanes can be laid next to each other; hence, why should the number of carbons even matter?

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I started drawing and noticed something.

The diagram makes it pretty obvious that the alkanes with odd numbers cannot be packed efficiently.

I am not sure if this is the correct answer.

Answer 1

A recent article in PubMed (Angew Chem Int Ed Engl. 2016 Nov 2;55(45):14090-14095.) suggests that this odd-even effect also affects translation diffusion in the liquid state.

In the introduction of the PubMed article, it also addresses your particular question and confirms your intuition that it relates to the symmetry of the molecule. Interestingly, this was only experimentally confirmed in the last decade. To quote the article:

Although this striking odd–even effect was known hundreds of years ago, its molecular origin was only revealed in the last decade. With the help of single-crystal diffraction and controlled crystal growth, the crystalline lattice of n-alkanes was determined.[5] It was confirmed that the intermolecular distances between the end groups, namely, the CH3 groups, are responsible for the alternation in the densities and melting points. In short, n-alkanes with an even number of carbon atoms pack more efficiently into ordered periodic crystalline structures and thus exhibit higher densities and melt at higher temperatures, whereas n-alkanes with an odd number of carbon atoms do not pack as effectively, and thus their densities are lower, and lower temperatures are needed for melting or freezing. A “packing effect” has been established to explain the odd–even effect as well as the related physical properties in the solid state.

Edit: For completeness, this is the article they link to that originally proved the effect experimentally: http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1521-3773(19990401)38:7%3C988::AID-ANIE988%3E3.0.CO%3b2-0/full