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:
- Using Avogadro’s Hypothesis
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
if
ρ is the density of the substance, the actual volume is
Now, using Avogadro’s hypothesis,
or
or
Knowing M, N and ρ, the value of r is determined.
- Molecular Layer Method
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 cm^{2} 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 |
Metres |
Distance of the most distant detected heavenly object |
5 × 10^{25} |
Radius of our galaxy |
6 × 10^{19} |
Distance of the nearest star (Alphas Centauri) |
4 × 10^{16} |
Mean orbit radius of the planet Pluto |
6 × 10^{12} |
Radius of the sun |
7 × 10^{8} |
Mean distance from earth to moon |
2 × 10^{8} |
Radius of the earth |
6.4 × 10^{6} |
Height of an average man |
1.8 × 10^{0} |
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} |