Moles, masses and particle numbers

• One mole of sodium, Na, contains $6.02 \times 10^{23}$ sodium atoms.
• One mole of hydrogen, H₂, contains $6.02 \times 10^{23}$ hydrogen molecules.
• One mole of sodium chloride, NaCl, contains $6.02 \times 10^{23}$ formula units. NaCl is an ionic lattice, not a collection of separate NaCl molecules; its formula represents a 1:1 ratio of sodium ions to chloride ions.
• One mole of sodium ions, Na⁺, contains $6.02 \times 10^{23}$ sodium ions.

Equal amounts in moles contain equal numbers of the stated particles. They do not necessarily have equal masses, because different particles have different masses.

Connecting a particle’s relative mass to a weighable mass

One mole of particles has a mass equal to their relative particle mass in grams. This connects a microscopic particle count to something that can be measured with a balance.

For atoms, use the relative atomic mass, $A_r$. Carbon has $A_r = 12$, so one mole of carbon atoms has a mass of 12 g. Helium has $A_r = 4$, so one mole of helium atoms has a mass of 4 g. Both samples contain the same number of atoms.

For molecules or formula units, use the relative molecular or formula mass, $M_r$. For water, using $A_r(\mathrm{H}) = 1$ and $A_r(\mathrm{O}) = 16$:

$$M_r(\mathrm{H_2O}) = (2 \times 1) + 16 = 18$$

One mole of water molecules therefore has a mass of 18 g. The molar mass, $M$, is the mass per mole: for water it is 18 g mol⁻¹. Notice the distinction: relative mass has no unit, whereas molar mass has units of g mol⁻¹.

Converting between mass and moles

To find the amount in moles, divide the sample’s mass by the mass of one mole:

$$n = \frac{m}{M}$$

Here, $n$ is the amount in mol, $m$ is the mass in g and $M$ is the molar mass in g mol⁻¹.

For example, a 9.0 g sample of water, with molar mass 18 g mol⁻¹, contains:

$$n = \frac{9.0}{18} = 0.50\ \text{mol}$$

This makes sense: 9.0 g is half the mass of one mole of water, so it contains half a mole of water molecules.

To work in the opposite direction, multiply the amount by the molar mass:

$$m = nM$$

For example, 0.25 mol of water has a mass of $0.25 \times 18 = 4.5\ \text{g}$.

Converting between moles and particle numbers

Each mole contains the Avogadro constant number of particles. To find the number of particles, $N$, multiply:

$$N = nN_A$$

For example, 0.20 mol of sodium atoms contains $0.20 \times 6.02 \times 10^{23} = 1.204 \times 10^{23}$ sodium atoms.

To find the amount in moles from a particle count, divide:

$$n = \frac{N}{N_A}$$

A sample containing $3.01 \times 10^{23}$ water molecules therefore contains $\frac{3.01 \times 10^{23}}{6.02 \times 10^{23}} = 0.50\ \text{mol}$ of water molecules. Its particle count is half the number in one mole.

Moving between mass and particle numbers

Moles provide the bridge between a mass and a particle count. First convert to moles, then convert to the quantity required.

For mass to particles, divide by molar mass, then multiply by the Avogadro constant:

$$N = \frac{m}{M}N_A$$

The 9.0 g water sample above contains 0.50 mol of water molecules, so it contains $0.50 \times 6.02 \times 10^{23} = 3.01 \times 10^{23}$ water molecules.

For particles to mass, divide by the Avogadro constant, then multiply by molar mass:

$$m = \frac{N}{N_A}M$$

For example, $1.204 \times 10^{23}$ water molecules represent 0.20 mol. With a molar mass of 18 g mol⁻¹, their mass is $0.20 \times 18 = 3.6\ \text{g}$.

Counting atoms within molecules

The chemical formula tells you how many atoms of each element are present in each molecule. A mole of molecules is not necessarily a mole of atoms.

Each H₂ molecule contains two hydrogen atoms, so one mole of H₂ molecules contains two moles of hydrogen atoms. Likewise, each H₂O molecule contains two hydrogen atoms and one oxygen atom.

The 9.0 g water sample contains $3.01 \times 10^{23}$ water molecules. It therefore contains $2 \times 3.01 \times 10^{23} = 6.02 \times 10^{23}$ hydrogen atoms, but only $3.01 \times 10^{23}$ oxygen atoms. The formula subscripts connect the number of molecules to the number of atoms within them.

With mass $m$ in g, amount $n$ in mol and particle count $N$:

$$n = \frac{m}{M} \qquad m = nM$$ $$N = nN_A \qquad n = \frac{N}{N_A}$$

Mass ↔ particles

Go through moles:

• Mass → particles: divide by $M$, then multiply by $N_A$.
• Particles → mass: divide by $N_A$, then multiply by $M$.

Particle type matters

Use $A_r$ for atoms and $M_r$ for molecules or formula units. To count atoms within molecules, multiply the molecule count by the relevant formula subscript.