In his paper, Einstein found expressions for the viscosity and the diffusion coefficient for a hard sphere in a continuous medium. Using these expressions and some experimental data based on diluted solutions of sugar in water, he obtained a value for Avogadro's number that was very close to the currently accepted value.
Einstein had an error in one of his equations in the 1905 paper on molecular dimensions, and it was not until he revised his work five years later that he calculated the correct value of Avogadro's number. Even Einstein didn't always get it right the first time! In fact, Einstein had to get help from one of his students to find the error in his original work.
Avogadro's number is defined as the number of molecules in a gram mole of a particular elemental substance. Avogadro's number is named after the chemist Amedeo Avogadro (1776—1856), who first suggested the idea that elements had particular weights. Avogadro did not actually calculate the value of the number, which has been honorarily named after him. In fact, the term “Avogadro's number” was first used by Jean Baptiste Perrin in 1909 in his paper that followed Einstein's theoretical result and calculated the size of molecules.
So what is Avogadro's number good for? Avogadro's original theory, in 1811, suggested that a particular volume of any gas, at the same temperature and pressure, contained the same number of molecules no matter what gas it was. Experiments were done, and eventually it was concluded that one cubic centimeter of gas contained Avogadro's number of gas molecules, or about 6 × 1023 molecules.
The current value of Avogadro's number is 6.022 × 1023, as determined by experiments using X ray diffraction. Avogadro's number is very difficult to determine, and many experiments over the years have refined this current value.
Avogadro's number is a very, very big number. It's so big that it is hard to comprehend. In fact, Avogadro's number of pennies placed next to each other in a line would be more than a million light years long.
Avogadro's number is also used to define the mole. A mole, in addition to a small furry mammal, is defined in chemistry as the amount of a substance that contains Avogadro's number of molecules (or other units). A mole of oxygen contains 6.022 × 1023 oxygen molecules. A mole of sandwiches contains 6.022 × 1023 sandwiches. That's a lot of peanut butter and jelly!
Avogadro's number can also be used to convert between number and mass. Chemists defined the “atomic mass unit (amu)” as a relative measurement of mass. Since atoms and molecules are difficult to see even with the best microscope, it is nearly impossible to measure the mass of an individual atom. So scientists defined the atomic mass unit as
The definition of Avogadro's number can seem confusing, but just think of it as another special name for a certain number of items. For example, a dozen contains twelve items. A gross contains 144 items. Avogadro's number of items contains 6.022 × 1023 items.
The atomic weights of elements, in atomic mass units, are used to line them up sequentially in the periodic table. The atomic mass of carbon-12, for example, is 12 amu, while the atomic mass of oxygen is 16 amu. Due to how atomic masses are defined, then, 12 grams of carbon-12 will contain the same number of atoms as 16 grams of oxygen.
But remember the definition of the mole as 6.022 × 1023 units. A mole of carbon, for example, will contain 6.022 × 1023 atoms of carbon, which will weigh 12 grams! The conversion from moles to grams depends on the molecular mass of the substance in question. To convert from moles to grams, just multiply by the molecular mass in grams per mole.
In coming up with a theoretical way to calculate Avogadro's number, Einstein provided essential support to the atomic theory of matter, which was still in question at the time that Einstein wrote his paper. Einstein's theoretical result spurred Perrin to measure Avogadro's number experimentally, thus providing solid experimental proof for the existence of atoms and molecules.