Melt an ice cube in the pan of a balance.  The weight does not change.  One of the ways we know this melting process was a physical change and not a chemical change is precisely that the weight does not change.  The volume, however, does change.  The ice is larger than the same amount of liquid water.  This suggests weight is a more fundamental measure of amounts than volume.  The case with steam underlines this dramatically.  A gram of water fills 1 cubic centimeter.  A gram of steam fills 1500 cm³.  But they are the same amount of water.

But sometimes, volume is a better measure.  Ask Avogadro.

Amadeo Avogadro argued that the combining ratios of elements by volume were a better measure of the chemical formula.  In short, two volumes of hydrogen gas will combine with a volume of oxygen to produce water.  The chemical formula should therefore be H₂O, and so the relative atomic weights of oxygen and hydrogen would be 16 to 1, not 8 to 1.

The basis for his argument was experimental work that showed gases always united in simple proportions by volume, like the case with water…2 to 1.  This suggested that given volumes had equal numbers of atoms.  Here’s how should Avogadro argued.  Gases are eminently compressible.  Suppose the distance between particles was much greater than the size of the particles themselves. I always think of a playground with kids playing with yo-yos.  Each kid has an effective size much greater than his actual size.  And the effective size is the same whether one kid is bigger than another, as long as the yo-yo strings are the same.  Avogadro thought temperature corresponded to the yo-yo strings.  So at a given temperature, a given volume would have a given number of particles, independent of the kind of particle…an atom of hydrogen or oxygen.

Of course, atoms are so small you could not count the actual number of particles in a given volume.  But you could usefully define a standard unit of volume and connect it to relative atomic weights.  A given volume of oxygen weighs 16 times the same volume of hydrogen.  Since the number of particles is the same in these two volumes, 16 to 1 is also the ratio of atomic weights.  This delightful analysis leads us to Avogadro’s number, a number absurdly large and one which Avogadro himself died without knowing.