By the early 19th century, scientists had strong evidence that all matter was made up of atoms. For example, Daniel Bernoulli had shown that the pressure exerted by a gas was the result of collisions of the atoms of the gas with the walls of its container.

In 1811, Amadeo Avogadro made a suggestion that eventually allowed for the counting of atoms. He hypothesized that equal volumes of different gases contain exactly the same numbers of atoms when they are under the same conditions of pressure and temperature. We now know that Avogadro's hypothesis is not perfectly true. Nevertheless, it is sufficiently true to have been useful and indeed was used in the quest of the ultimate nature of matter.

Using Avogadro's hypothesis, scientists were able to determine the relative weights of, and to assign a value to, the atoms of all the elements. They discovered that hydrogen was the lightest element and that the weight of an equal amount of any other element could be expressed as the weight of hydrogen multiplied by a number that was close to an integer.

In other words, it was found that pure oxygen placed in a container under specific temperature and pressure weighed about 16 times as much as pure hydrogen placed into an identical container and under the exact same conditions of temperature and pressure. On the assumption that both containers contained the same number of molecules (and atoms), this implied that each oxygen atom weighed 16 times as much as each hydrogen atom.

Thus, if hydrogen is assigned an arbitrary weight of one unit, then oxygen would weigh 16 of those units. This basic unit is referred to as atomic weight.

A hydrogen atom has an atomic weight of 1; an oxygen atom has an atomic weight of 16. Similarly, it was determined that an atom of carbon weighs about 12 atomic units, iron about 56 atomic units, and so on for all the elements. Why this is so can be explained today by our theories of the nature and structure of the atom.

In doing their work, scientists needed to use a measure that would contain the same number of molecules no matter what substance was being measured. They chose something called a "mole," which is essentially an arbitrary quantity, but one that is linked to the atomic weight of molecules.

Precisely what this measure is can best be explained by an example. An atom of oxygen has been assigned the atomic weight of 16. Because a molecule of oxygen consists of two oxygen atoms joined together, the atomic weight of a molecule of oxygen is 32. A mole of oxygen is that quantity of pure oxygen which weighs 32 grams.

A mole is always that amount of a substance which weighs as many grams as the atomic number assigned to its single molecule.

To give another example. The atomic weight of a molecule of water is 18. Each water molecule consists of one atom of oxygen (atomic weight 16) and two atoms of hydrogen (each with an atomic weight of 1). Thus, one mole of water is that quantity of water which weighs 18 grams.

Avogadro's rule says that a mole of any substance always contains the same number of molecules. A mole of oxygen has as many molecules as a mole of hydrogen which has as many molecules as a mole of water.

The number of molecules in a mole is called the Avogadro number even though Avogadro himself never estimated its magnitude. Estimates of its value were made during the 19th century. A more accurate measure of the quantity of molecules in a mole was determined during the early 20th century using a technique that shot x-rays through crystals. According to our present knowledge, the value of the Avogadro number is approximately 6.022 x 10²³.

What this means is that 32 grams of oxygen contains 6.022 x 10²³ molecules of oxygen. As each molecule of oxygen contains two atoms, the total number of atoms in 32 grams of pure oxygen is Avogadro's number multiplied by two.

Take water as an other example. The number of molecules in 18 grams of water is Avogadro's number. Since each molecule consists of three atoms, the total number of atoms is Avogadro's number multiplied by three.

To count atoms, one must know only two things - the actual weight of a substance and its composition, meaning the types and number of atoms in its molecules. (Determining the composition of the molecules allows one to calculate the atomic weight of the molecules. One simply adds up the atomic weight of all the atoms in the molecule.)

Once the composition of a substance is known, the following three steps can be used to compute the total number of atoms in the mass:
(a) determine how many moles in the mass. (To do that simply divide the actual weight of the substance by the weight of a single mole. As explained above, a mole of this substance would be that quantity of this substance which weighs as many grams as the atomic weight of its molecules. This number of grams contains an Avogadro-number of molecules.);
(b) multiply this result by the Avogadro number; and then,
(c) multiply this second result by the number of atoms in each molecule.