Mole Concept and Molar Mass

Chemical reactions happen atom-to-atom and molecule-to-molecule, but no laboratory balance can weigh a single

atom. Chemists needed a "counting unit" — a fixed, enormous number of particles bundled into one convenient

unit — so that a number of particles could be converted into a mass that can actually be weighed out. That

counting unit is the mole.

Definition. One mole of any substance is defined as the amount of that substance which contains exactly as

many elementary entities (atoms, molecules, ions, or any other specified particle) as there are atoms in

exactly $12\ \text{g}$ of the carbon-12 isotope. Experimentally, this number has been measured to extraordinary

precision and is called the Avogadro constant (or Avogadro's number):

So $1\ \text{mol}$ of any substance always contains $6.022 \times 10^{23}$ elementary entities of that

substance — $1\ \text{mol}$ of carbon atoms contains $6.022 \times 10^{23}$ carbon atoms; $1\ \text{mol}$ of

water molecules contains $6.022 \times 10^{23}$ water molecules; $1\ \text{mol}$ of sodium ions contains

$6.022 \times 10^{23}$ $\text{Na}^+$ ions. The mole is one of the seven SI base units, used specifically to

measure the physical quantity called amount of substance (see the table of SI base units below).

Molar mass. The molar mass of a substance is defined as the mass, in grams, of one mole of that substance.

Remarkably, the molar mass (in $\text{g mol}^{-1}$) of any element or compound is numerically equal to its

atomic mass, molecular mass, or formula mass (in u), calculated in Section 1.3. This is not a coincidence — it

is a direct consequence of how both the atomic mass unit and the mole are defined relative to carbon-12. So:

• The atomic mass of oxygen is $16.00\ \text{u}$ $\Rightarrow$ the molar mass of oxygen atoms is $16.00\ \text{g mol}^{-1}$.
• The molecular mass of water is $18.02\ \text{u}$ $\Rightarrow$ the molar mass of water is $18.02\ \text{g mol}^{-1}$ — i.e. $18.02\ \text{g}$ of water contains exactly $6.022 \times 10^{23}$ water molecules.

Converting between mass, moles and number of particles. These three quantities — mass ($m$, in grams),

number of moles ($n$), and number of particles ($N$) — are linked by two simple relations:

where $M$ is the molar mass. These two equations are used constantly throughout chemistry: given a mass, they

give the number of moles and hence the number of atoms or molecules; given a number of moles, they give the

mass that must be weighed out.

Molar volume of a gas. For gases specifically, Avogadro's law (Section 1.2) gives a third useful relation: …