Are molecules as inconceivably small
as moles are inconceivably large?


Avogadro's number (the mole) is 6 x 1023, so the number of atoms in a gram of hydrogen is already about 1000 times larger than the number of different chains of 16 amino acids ( 2016 ~ 7 x 1020), which is about 1058 times smaller than estimates of the number of atoms in the universe.

The point of telling this story is not to ridicule a famous chemist so that we can feel superior. It is rather to show that even for very smart scientists such numbers are too big to deal with comfortably.

About 150 years ago chemists began to get an idea of the size of atoms and how very large Avogadro's number is. They quickly became accustomed to assuming that molecules were too small to deal with individually. It was natural for them to switch to speaking of molecules collectively as moles, although this posed a serious problem for teaching students, who might have learned arithmetic by laying two toothpicks next to three toothpicks and counting to five. Since practical chemistry is a science of measuring and weighing, rather than counting, teachers have had to spend a lot of time making frustrated students comfortable with moles and how to use them.

With this background, it is hardly surprising that chemists traditionally assumed intuitively that individual molecules and atoms were too small to observe or measure (except maybe for fabulously large molecules, like a strand of DNA). The prejudice that it was pointless to go after individual molecules was reinforced by faulty understanding of probability and uncertainty concepts in quantum-mechanics that made individual atoms seem ill-defined and impossible to measure.

But think of this. When we consider molecular dimensions, we are typically more interested in linear dimensions (diameter) than in volume (the cube of linear dimension). Although a molecular volume is a fabulously small number, the linear dimension is the cube root of volume (about 3 x 10-8 cm / molecule of H20 - see right).

1 mole of H2O is 18 g or 18 mL
dividing by 6 x 1023
3 x 10-23 mL / molecule
or 30 x 10-24 cm3 / molecule
3 x 10-8 cm, or 3 ┼ngstroms(┼), or 0.3 nanometers (nm) is not nearly as incomprehensible as 10-23.

Our lecture room is about 10 m wide, one of my hairs is of the order of 100 μm thick. It would take about 105 of my hairs side-by-side to span the room.

A typical molecular dimension of 1 nm (about 6 C-C bond lengths) is 10-5 times 100 μm. So it takes about as many molecule to span one of my hairs as it takes hairs to span the room.

room : hair :: hair : molecule

Molecules are small, but not inconceivably small and chemists need to adjust their intuition.

It turns out to be pretty straightforward, using ceramics that expand or contract very slightly with changes in applied voltage, to make a manipulator that can be reliably positioned within 10-8 cm or 0.1 nm. These manipulators are the basis of Scanning Probe Microscopy, which can measure individual molecules and even atoms.

Nuclei are much smaller

Coincidentally, the factor of 10-5, which takes one from room to hair, or from hair to molecule, also takes one from atom to nucleus. Nuclear radii are approximately 1.2 x 10-5 ┼ multiplied by the cube root of the atomic number. Thus the nuclear radius of 14C is about 2.8 x 10-5┼. The van der Waals radius of carbon is 1.7 ┼. That is to say that if a carbon atom were scaled to just fit into our lecture hall, the nucleus would be about twice as thick as a hair.

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latest revision 7/31/2008

Comments on this page are welcomed by the author.

J. Michael McBride
Department of Chemistry, Yale University
Box 208107, New Haven, CT 06520-8107


copyright ę 2004 J.M.McBride