2.1 Molecular model of an ideal gas  (Page 4/19)

Moles and avogadro’s number

It is often convenient to measure the amount of substance with a unit on a more human scale than molecules. The SI unit for this purpose was developed by the Italian scientist Amedeo Avogadro (1776–1856). (He worked from the hypothesis that equal volumes of gas at equal pressure and temperature contain equal numbers of molecules, independent of the type of gas. As mentioned above, this hypothesis has been confirmed when the ideal gas approximation applies.) A mole    (abbreviated mol) is defined as the amount of any substance that contains as many molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. (Technically, we should say “formula units,” not “molecules,” but this distinction is irrelevant for our purposes.) The number of molecules in one mole is called Avogadro’s number     $(N_{\text{A}}),$ and the value of Avogadro’s number is now known to be

We can now write $N=N_{A}n$ , where n represents the number of moles of a substance.

Avogadro’s number relates the mass of an amount of substance in grams to the number of protons and neutrons in an atom or molecule (12 for a carbon-12 atom), which roughly determine its mass. It’s natural to define a unit of mass such that the mass of an atom is approximately equal to its number of neutrons and protons. The unit of that kind accepted for use with the SI is the unified atomic mass unit (u) , also called the dalton . Specifically, a carbon-12 atom has a mass of exactly 12 u, so that its molar mass M in grams per mole is numerically equal to the mass of one carbon-12 atom in u. That equality holds for any substance. In other words, $N_{\text{A}}$ is not only the conversion from numbers of molecules to moles, but it is also the conversion from u to grams: $6.02\times {10}^{23}\text{u}=1\text{g}\text{.}$ See [link] .

Now letting $m_{\text{s}}$ stand for the mass of a sample of a substance, we have $m_{\text{s}}=nM.$ Letting m stand for the mass of a molecule, we have $M=N_{A}m.$