How Chemical Formula Subscripts Scale Atom Counts

Count particles before counting a selected atom

A chemical formula describes one molecule or formula unit. Its subscripts tell how many atoms of each element that particle contains. To convert a macroscopic sample into a count of one element, follow this chain:

$$\text{mass}\rightarrow\text{moles of substance}\rightarrow \text{molecules or formula units}\rightarrow\text{atoms of the chosen element}.$$

For a sample mass $m$ and molar mass $M$,

$$\text{atoms of X}=\frac{m}{M}N_\mathrm A \left(\frac{\text{atoms of X}}{\text{formula unit}}\right),$$

where $N_\mathrm A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$.

Read parentheses before choosing the atom factor

In $\mathrm{Ca(NO_3)_2}$, the outside subscript 2 multiplies every atom inside the parentheses. One formula unit therefore contains one Ca atom, two N atoms, and six O atoms. The oxygen factor is 6, not 3 and not 2.

Example with a new substance

Suppose a sample contains $0.250\,\mathrm{mol\ Ca(NO_3)_2}$. The number of formula units is

$$(0.250\,\mathrm{mol})(6.02214076\times10^{23}\,\mathrm{formula\ units\,mol^{-1}}) =1.51\times10^{23}\,\mathrm{formula\ units}.$$

Each formula unit has six oxygen atoms, so

$$(1.51\times10^{23}\,\mathrm{formula\ units}) \left(\frac{6\,\mathrm{O\ atoms}}{\mathrm{formula\ unit}}\right) =\mathbf{9.03\times10^{23}\ O\ atoms}.$$

Dimensional labels provide a reliable check: moles cancel against $\mathrm{mol^{-1}}$, then formula units cancel in the subscript factor, leaving atoms.

Common counting errors

• Multiplying grams directly by $N_\mathrm A$ counts nothing meaningful; grams must first become moles.
• Stopping after $N_\mathrm A$ gives particles, not the number of a selected atom.
• Ignoring parentheses undercounts atoms.
• A leading coefficient counts whole particles, while a subscript changes each particle's composition.