Solution Concentration Calculations

What concentration tells us

When a substance dissolves, it becomes the solute; the liquid it dissolves in is the solvent. Together they form a solution. For example, when sodium hydroxide dissolves in water, sodium hydroxide is the solute and water is the solvent.

Concentration tells us how much solute is present in a given volume of solution. Knowing the total mass of solute is not enough: the same mass spread through a larger volume produces a less concentrated solution. Chemists use concentration to quantify substances in samples and to compare solutions fairly.

There are two ways of expressing concentration here: the mass of solute per unit volume, in g/dm³, and the amount in moles per unit volume, in mol/dm³.

Concentration in grams per cubic decimetre

A concentration of 10 g/dm³ means that each cubic decimetre of solution contains 10 g of dissolved solute. It does not mean that the sample must have a volume of exactly 1 dm³: the unit describes the proportion of solute to solution volume.

Calculate mass concentration using:

$$\text{concentration (g/dm}^3\text{)}=\frac{\text{mass of solute (g)}}{\text{volume of solution (dm}^3\text{)}}$$

For a fixed solution volume, doubling the mass of dissolved solute doubles the concentration. For a fixed mass of solute, doubling the solution volume halves the concentration. Adding water therefore dilutes a solution: the mass of solute stays the same, but it is distributed through a greater volume.

Getting the volume into the correct unit

A cubic decimetre is a unit of volume equal to one litre:

$$1\text{ dm}^3=1000\text{ cm}^3$$

To convert cm³ to dm³, divide by 1000. For example, 250 cm³ is $250\div1000=0.250\text{ dm}^3$. To convert in the other direction, multiply by 1000.

Suppose 3.0 g of sodium hydroxide is dissolved to make a total solution volume of 250 cm³. First convert the volume to 0.250 dm³, then calculate:

$$\text{concentration}=\frac{3.0}{0.250}=12\text{ g/dm}^3$$

This means that a 1 dm³ sample of this solution would contain 12 g of sodium hydroxide. The actual smaller sample still contains only 3.0 g.

Concentration in moles per cubic decimetre

A mole is a unit for the amount of a substance, linked to the number of particles present. Chemical equations describe reacting amounts in moles, so molar concentration is especially useful when calculating how much of a dissolved substance will react.

Calculate molar concentration using:

$$c=\frac{n}{V}$$

Here, $c$ is concentration in mol/dm³, $n$ is the amount of solute in mol and $V$ is the total solution volume in dm³. The unit mol/dm³ can also be written as mol dm⁻³.

Suppose a solution contains 0.040 mol of solute in a total volume of 200 cm³. The volume is $200\div1000=0.200\text{ dm}^3$, so:

$$c=\frac{0.040}{0.200}=0.20\text{ mol/dm}^3$$

Each cubic decimetre of this solution contains 0.20 mol of solute. As with mass concentration, increasing the amount of solute at fixed volume increases concentration; increasing volume at fixed amount decreases it.

Finding molar concentration from a mass

If a question gives a mass rather than an amount in moles, first use the substance’s molar mass: the mass of one mole, in g/mol.

$$n=\frac{\text{mass (g)}}{\text{molar mass (g/mol)}}$$

Then divide the calculated number of moles by the solution volume in dm³.

For the sodium hydroxide solution above, the molar mass of NaOH is 40 g/mol. Its 3.0 g of solute corresponds to $3.0\div40=0.075\text{ mol}$. Dividing by the solution volume gives $0.075\div0.250=0.30\text{ mol/dm}^3$.

Thus, 12 g/dm³ and 0.30 mol/dm³ describe the same sodium hydroxide solution. To convert a concentration from g/dm³ to mol/dm³, divide by the molar mass; to reverse the conversion, multiply by it. The volume unit stays the same.

Using the relationships in reverse

The equations can also find how much solute is present in a known volume. Rearranging gives:

$$\text{mass}=\text{mass concentration}\times V$$ $$n=cV$$

In both cases, use volume in dm³. Concentration describes the amount per unit volume; multiplying by the sample volume gives the mass or number of moles in that sample.

Essential equations

$$\text{concentration (g/dm}^3\text{)}=\frac{\text{mass of solute (g)}}{\text{solution volume (dm}^3\text{)}}$$