In 1811 the Italian physicist Amedeo Avogadro postulated the simple idea that equal volumes of gases (at the same temperature and pressures) must contain the same number of particles. Recall that Dalton's concept of the atom was just being considered and the idea of molecules was still a novel proposal. At first this idea was ignored, and then in 1860 the Italian Chemist Cannizzaro presented a paper at a chemical congress in Karlsruhe, Germany in which he outlined a convincing argument for Avogadro's proposal. In a stroke of genius, Cannizzaro handed out printed copies of his paper to the participants allowing them to go home, read his ideas and draw their own conclusions. The details of Avogadro and Cannizzaro's original ideas loses something in the modern sense since the early debates on the formulas of molecules, atomic weights and even the existence of atoms are not easy to follow. As scientists built up their understanding of the properties of gases it became necessary for them to use Avogadro's hypothesis to make sense of the properties.

Over time it became useful to think in terms of chemical reactions involving reactions between molecules or atoms. The combination of two chemical compounds necessitated a way of thinking that allowed compounds to be mixed so that the numbers of molecules were equal, not necessarily their weights or volumes.

By way of example, let's say you go to the hardware store and buy a pound of 6" long, 1/4" diameter bolts and a pound of 1/4" nuts. Since the individual bolts weigh at least twice as much as the nuts, you will have a good many nuts left over if you combine one bolt with one nut. You've spent good money on nuts you won't use.

Chemical compounds work the same way. If we are going to make a chemical compound like silver(I) iodide, we will probably use the reaction below:

AgNO3 + NaI --------> AgI + NaNO3

If we want to mix together silver (I) nitrate and sodium iodide in exactly a 1:1 ratio, we have to have a way of measuring out these compounds to get exactly the same number of molecules of each.

The way we do this is to return to Avogadro's hypothesis in a modern form. Recall that hydrogen has an atomic weight of 1.0. The mass of an individual hydrogen atom is pretty darn small so we need to use some quantities that make sense on a human scale. Let's assume that in 1.0 gram of hydrogen there is some number of atoms (the exact number doesn't matter for now.) We'll call 1.0 g of hydrogen a gram atomic weight of hydrogen, and the number of atoms to be a mole. The word mole comes from the Latin moles meaning a heap, mass or pile.

If we had a 12.0 g sample of carbon this would be a gram atomic weight of carbon and this sample would contain one mole of carbon. 12.0 g of carbon contains exactly the same number of atoms as 1.0 g of hydrogen because each carbon atom is 12 times the mass of each hydrogen atom. One gram atomic weight of any element contains exactly the same number of atoms.

It's easy to expand this idea to molecules. Carbon dioxide has a molecular weight of 44.0. A 44.0 g sample would be one gram molecular weight and would contain exactly one mole of carbon dioxide molecules.

Calculations on Moles

If you can understand that six eggs is a half dozen eggs then you can easily grasp the concept of how to calculate how many moles of a compound you have in a sample. Let's take some obvious examples.

We know that 1.0 mole of carbon is 12.0 g, therefore we must divide 6.0 g by 12.0 g/mole to get 0.50 moles of carbon. The use of railroad tracks makes this easy to set up and keep track of units.

If uranium has an atomic weight of 238 g/mole, a 40.0 g sample must contain 0.16 moles of uranium atoms.

Sugar is C12H22O11 and has a molecular weight of 342 g/mole. 12.0 g of sugar would be 0.0351 moles. Now this number is getting a bit awkward to write out so let's put it into proper scientific notation as 3.51 x 10-2 moles. Notice that we don't have a problem writing down numbers from say 1,000 to 0.1 in conventional form, but once we get outside of this range it's simply easier to use scientific notation.

The molecular weight of ammonium nitrate is 80.0 g/mole so we must multiply the molecular mass by the number of moles we need to get 3.01 g.

Now, let's have you try some. For each of the following questions, the answer can be viewed by clicking "answer".

1. How many moles are in a 34.6 g sample of Au? answer

2. A 93.1 g sample of salt, NaCl, will represent how many moles? answer

3. How many grams of UF6 are needed if a reaction calls for 9.55 x 10-3 moles? answer

Additional PracticeIf you'd like some additional practice, there is a web page with more examples that may be reached by clicking here.

Avogadro's Number

Avogadro's number is fascinating because it's so huge, in fact, it's so huge it's really meaningless. When Avogadro first proposed his idea of equal numbers of particles in equal volumes of gas he had no idea of just how large the numbers were that he was suggesting. It wasn't until the early twentieth century that Rutherford and his students actually measured the number and came up with an astonishing value of 1023! Avogadro's number is actually quite hard to measure to any level of precision and the very best modern numbers are about 6.023 x 1023. For those of you who like numbers, that's:

602,300,000,000,000,000,000,000

Even the national debt pales in comparison.

Chemists tend to think in two worlds.

When we think in terms of the molecular world we might ask how many atoms are in a molecule. The answer may range from two (the minimum number of atoms for a molecule) up to trillions in huge molecules like the polymers in your clothes or the DNA in your cells.

Alternately, we think of quantities of materials we can actually weigh out in the lab. Here we're talking about samples ranging from micro grams up to kilograms. At these levels we're talking about very large numbers of atoms and the collective term "mole" becomes more useful.

From a practical standpoint, Avogadro's number is an interesting, but not very useful number. No one in their right mind would worry about how many atoms we have in a sample in a bottle on the shelf. For the vast majority of the things we do in the lab we use moles, not Avogadro's number.

Of course, we do have the ability to observe and manipulate individual atoms under the right circumstances. Some years ago IBM researchers used xenon atoms to spell out the IBM logo illustrated in the lovely image below. Yes, these are actual images of atoms on a surface. There are 35 xenon atoms used to spell out the logo "IBM". For fun we could ask how many moles are in this sample. To solve this problem we'd divide 35 by Avogadro's number to get 5.81 x 10-23 moles.