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Estimation of Very Small Distances: Size of a Molecule

Special methods are used to measure extremely small distances, such as the sizes of molecules and atoms.

Distances of the order of 10-5 m can be measured with optical microscopes. Electron microscope is used for measuring distances of the order of 10-8 m. Distances of the order of 10-7 m can be measured by interference experiments.

Distances between the molecules of crystals (~10-10 m or 1) can be measured by x-ray diffraction experiments.

Subatomic distances can be measured by the alpha-particle scattering experiments. When only a rough estimate of such small distances is desired, the following two methods may be used:
  1. Using Avogadro’s Hypothesis
The actual volume occupied by the atoms in a substance is always less than the volume of that substance because there are empty spaces between atoms.

According to Avogadro’s hypothesis, the actual volume occupied by the atoms in a certain mass of a substance is two-thirds the volume occupied by that mass of the substance.

Let m be the mass of a substance in grams. If M is the molecular mass of the substance in grams, then by definition, the number of atoms in a molecular mass is N, the Avogadro’s number.

Number of atoms in 1 gram = N/M

Number of atoms in m grams = m N/M

Assuming that each atom is a sphere of radius r, the volume of atoms in m grams of the substance is

ρ is the density of the substance, the actual volume is

Now, using Avogadro’s hypothesis,

Knowing M, N and ρ, the value of r is determined.
  1. Molecular Layer Method
The size of a molecule of a liquid can be estimated by this method. The idea is to prepare a layer of one molecule thickness of a liquid on water. This method is suitable for liquids whose molecules have a large size.

One such liquid is oleic acid. Dissolve 1 cc of oleic acid in 400 cc of alcohol so that the concentration of the solution is 1/400 of oleic acid in alcohol.

Pour a few drops of this solution on the surface of water in a flat dish. Let n be the number of drops. A film of the solution is formed on the surface of water. Carefully stretch the film as much as possible.

The alcohol will evaporate leaving behind a film of oleic acid in water. Using a tracing paper, trace the outline of the film and with the help of an mm graph paper, find the area of the film.

Let it be A. If v is the volume of one drop, then the volume of n drops of solution is nv.
Volume of oleic acid in n drops of solution =
If A cm2 is the area of the film and t cm its thickness, then we have

If we assume that the film is stretched such that its thickness equals the size of a molecule, then the value of t obtained above, gives an estimate of the size of a molecule of oleic acid.

Table below gives the values of some measured distances. Length measurements can be made over a vast range.

Some measured lengths


Distance of the most distant detected heavenly object

5 × 1025

Radius of our galaxy

6 × 1019

Distance of the nearest star (Alphas Centauri)

4 × 1016

Mean orbit radius of the planet Pluto

6 × 1012

Radius of the sun

7 × 108

Mean distance from earth to moon

2 × 108

Radius of the earth

6.4 × 106

Height of an average man

1.8 × 100

Thickness of a sheet of paper

1 × 10-4

Radius of a hydrogen atom

5 × 10-11

Radius of a proton

1.2 × 10-15

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