Chemistry 1.49 - Solution concentration

Concentration in grams per cubic decimetre

A solution contains a solute dissolved in a solvent. Mass concentration
tells us how many grams of a chosen solute are present per cubic decimetre of the
finished solution.

For example, $20 \,\mathrm{g} \,\mathrm{dm}^{-3}$ means that $1 \,\mathrm{dm}^{3}$ of solution contains $20 \,\mathrm{g}$ of the
solute. It does not mean $20 \,\mathrm{g}$ was added to $1 \,\mathrm{dm}^{3}$ of water: dissolving the
solute can change the final volume, so the denominator is the volume of solution.

The units must match the formula:

$1 \,\mathrm{dm}^{3} = 1000 \,\mathrm{cm}^{3}$

Therefore:

$V\,(\mathrm{dm^3})=\frac{V\,(\mathrm{cm^3})}{1000}$

Worked example

A solution contains $8.40 \,\mathrm{g}$ of sodium chloride in a final solution volume of
$250 \,\mathrm{cm}^{3}$. Calculate its concentration in $\,\mathrm{g} \,\mathrm{dm}^{-3}$.

1. Convert the volume.

$250 \,\mathrm{cm}^{3} / 1000 = 0.250 \,\mathrm{dm}^{3}$

2. Select the relationship and substitute.

$c=\frac{8.40}{0.250}\,\mathrm{g\,dm^{-3}}$

3. Calculate and state the unit.

$c=33.6\,\mathrm{g\,dm^{-3}}$

Sense-check: $250 \,\mathrm{cm}^{3}$ is one quarter of $1 \,\mathrm{dm}^{3}$, so one cubic decimetre
would contain four times as much solute. $4 \times 8.40 \,\mathrm{g} = 33.6 \,\mathrm{g}$, matching the
calculation.

The most common power-of-ten error is dividing by $250$ while still reporting
$\,\mathrm{g} \,\mathrm{dm}^{-3}$. Either convert the volume first, as above, or use the equivalent direct
route $c=\frac{1000m}{V\text{ in }\mathrm{cm^3}}$.