The Concept of the Mole and the Avogadro Constant

Starting with Dalton, chemists have recognized the importance of relative numbers of atoms, as in the statement that two hydrogen atoms and one oxygen atom combine to form one molecule of water. Yet it is physically impossible to count every atom in a macroscopic sample of matter. Instead, some other measurement must be employed, which requires a relationship between the measured quantity', usually mass, and some known, but uncountable, number of atoms. Consider a practical example of mass substituting for a desired number of items. Suppose you want to nail down new floorboards on the deck of a mountain cabin, and you have calculated how many nails you will need. If you have an idea of how many nails there are in a pound, then you can buy the nails by the pound.

The SI quantity that describes an amount of substance by relating it to a number of particles of that substance is called the mole (abbreviated mol). A mole is the amount of a substance that contains the same number of elementary entities as there are atoms in exactly 12 g of pure carbon-12. The "number of elementary entities (atoms, molecules, and so on)" in a mole is the Avogadro constant, N_A.

N_A = 6.02214179 X 1023 mol^-1

The Avogadro constant consists of a number, 6.02214179 X 1023, known as Avogadro's number, and a unit, mol-1. The unit mol-1 signifies that the entities being counted are those present in 1 mole.

The value of Avogadro's number is based on both a definition and a measurement. A mole of carbon-12 is defined to be 12 g. If the mass of one carbon-12 atom is measured by using a mass spectrometer (see Figure), the mass would be about 1.9926 X 10-23 g. The ratio of these two masses provides an estimate of Avogadro's number. In actual fact, accurate determinations of Avogadro's number make use of other measurements, not the measurement of the mass of a single atom of carbon-12.

Often the value of NA is rounded off to 6.022 X 1023 mol-1, or even to 6.02 X 1023 mol-1.

If a substance contains atoms of only a single isotope, then

1 mol 12C = 6.02214 x 1023 12C atoms = 12.0000 g 1 mol

If 0 = 6.02214 x 1023 16O atoms = 15.9949g (and so on)

Most elements are composed of mixtures of two or more isotopes so that the atoms in a sample of the element are not all of the same mass but are present in their naturally occurring proportions. Thus, in one mole of carbon, most of the atoms are carbon-12, but some are carbon-13. In one mole of oxygen, most of the atoms are oxygen-16, but some are oxygen-17 and some are oxygen-18. As a result,

1 mol of C = 6.02214 X 1023C atoms

= 12.0107 g 1 mol of O = 6.02214 X 1023O atoms = 15.9994 g, and soon.

The Avogadro constant was purposely chosen so that the mass of one mole of carbon-12 atoms—exactly 12 g—would have the same numeric value as the mass of a single carbon-12 atom—exactly 12 u. As a result, for all other elements the numeric value of the mass in grams of one mole of atoms and the weighted-average atomic mass in atomic mass units are equal. For example, the weighted average atomic mass of lithium is 6.941 u and the mass of one mole of lithium atoms is 6.941 g. Thus, we can easily establish the mass of one mole of atoms, called the molar mass, M, from a table of atomic masses.* For example, the molar mass of lithium is 6.941 g Li/mol Li. The image attempts to portray the distribution of isotopes of an element, and the second image pictures one mole each of four common elements.

Thinking about Avogadro's Number

Avogadro's number (6.02214 X 1023) is an enormously large number and practically inconceivable in terms of ordinary experience. Suppose we were counting garden peas instead of atoms. If the typical pea had a volume of about 0.1 cm3, the required pile of peas would cover the United States to a depth of about 6 km (4 mi). Or imagine that grains of wheat could be counted at the rate of 100 per minute. A given individual might be able to count out about 4 billion grains in a lifetime. Even so, if all the people currently on Earth were to spend their lives counting grains of wheat, they could not reach Avogadro's number. In fact, if all the people who ever lived on Earth had spent their lifetimes counting grains of wheat, the total would still be far less than Avogadro's number. (And Avogadro's number of wheat grains is far more wheat than has been produced in human history.) Now consider a much more efficient counting device, a modern personal computer; it is capable of counting at a rate of about 1 billion units per second. The task of counting out Avogadro's number would still take about 20 million years!

Avogadro's number is clearly not a useful number for counting ordinary objects. However, when this inconceivably large number is used to count inconceivably small objects, such as atoms and molecules, the result is a quantity of material that is easily within our grasp, essentially a "handful."