The Mole Concept

Why we measure some things by mass, and others by counting.

Stop and notice how we measure quantities in everyday life. Some things we weigh. Some things we count. The choice isn’t random — it depends on the object.

• Apples vary in size and weight, so we measure them by mass: one kilo of apples, not “ten apples.”
• Bananas also vary in size, but they’re easy to grab and count. So we measure them by number: a dozen bananas.
• Grains of rice or wheat are nearly identical in size — but there are thousands in a single fistful. Counting is hopeless. So we measure them by mass: one kilo of rice.
• Potatoes — count, like apples. Five potatoes.

The pattern: if there are too many to count, or sizes vary, we weigh. If we can grab and count, we count.

In a chemistry lab, you face the same dilemma — but at an extreme. When two substances react, they react one atom at a time. So you want to know how many atoms or molecules of each substance you have. But there are billions of trillions of atoms in even a single grain of salt. Counting them is impossible.

The mole is the unit chemists invented to solve exactly this problem. It lets you count by weighing — you measure mass on a scale, and instantly know how many particles you’re holding.

A mole is the amount of substance that contains as many entities (atoms, molecules, ions, or other particles) as there are atoms in exactly 12 g of the $\ce{^{12}C}$ isotope.

This number was determined experimentally using a mass spectrometer. The mass of one $\ce{^{12}C}$ atom was found to be $1.992648 \times 10^{-23}$ g. Therefore:

$N_A = \frac{12 \text{ g/mol}}{1.992648 \times 10^{-23} \text{ g}} = 6.022 \times 10^{23} \text{ mol}^{-1}$

This is Avogadro's constant ($N_A$):

$N_A = 6.022 \times 10^{23} \text{ entities per mole}$

Just like 1 dozen = 12, 1 mole = $6.022 \times 10^{23}$. It's simply a counting number — a convenient-sized chunk of atoms.

• 1 gram-atom = 1 mole of atoms = $6.022 \times 10^{23}$ atoms
• 1 gram-molecule = 1 mole of molecules = $6.022 \times 10^{23}$ molecules

Molar mass is the mass of 1 mole of a substance, expressed in g/mol. Numerically, it equals the atomic or molecular mass in u:

So if the atomic mass of Fe is 56 u, then 1 mole of Fe atoms = 56 g.

Molar Volume is the volume occupied by 1 mole of any gas at a given temperature and pressure.

At STP (Standard Temperature and Pressure: 0 °C, 1 atm):
$\text{Molar volume of any gas} = 22.4 \text{ L} = 22400 \text{ mL}$

This is the same for ALL gases — $\ce{H2}$, $\ce{O2}$, $\ce{CO2}$, $\ce{NH3}$ — at STP, 1 mole of any gas occupies 22.4 L.

The number of moles ($n$) can be calculated three different ways depending on what information is given:

$n = \frac{\text{Given mass}}{\text{Molar mass}} = \frac{w}{M}$

$n = \frac{\text{Number of particles}}{N_A} = \frac{N}{6.022 \times 10^{23}}$

$n = \frac{\text{Volume at STP (L)}}{22.4}$

These three expressions are equivalent — use whichever one matches the information given in the problem.

Key conversions:

• Mass of 1 atom = $\frac{\text{Atomic mass}}{N_A}$ g
• Mass of 1 molecule = $\frac{\text{Molecular mass}}{N_A}$ g
• Number of atoms = $n \times N_A$
• Number of molecules = $n \times N_A$