Concentration of Solutions

Lessons

Senior 3 · Chemistry

Concentration of Solutions

5 lessons

Lesson 2 of 5

Percentage by Mass and by Volume

20 min

Three labels, three kinds of percentage

A pump bottle of hand sanitiser says "70% alcohol". The bottle of concentrated hydrochloric acid in the school preparation room says "36%". A hospital drip of normal saline says "0.9% sodium chloride". All three are percentages, but each is a percentage of something different: of a volume, of a mass, and of a mass per volume. Reading a label correctly means knowing which.

Definition

Percentage by mass (% w/w): the mass of solute as a percentage of the mass of the whole solution.

$\%\ \text{by mass} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 100$

The mass of solution is the mass of solute plus the mass of solvent. Example: 10 g of salt in 90 g of water makes 100 g of solution that is 10% salt by mass.

Worked example

Problem. 15 g of sodium chloride is dissolved in 135 g of water. (a) Find the percentage by mass of sodium chloride. (b) What mass of sodium chloride is in 40 g of this solution?

Solution. (a) Mass of solution $= 15 + 135 = 150\,\mathrm{g}$ :

$\%\ \text{by mass} = \frac{15}{150} \times 100 = 10.0\%$

(b) $\frac{10.0}{100} \times 40 = 4.0\,\mathrm{g}$ .

Definition

Percentage by volume (% v/v): the volume of a liquid solute as a percentage of the volume of the whole solution.

$\%\ \text{by volume} = \frac{\text{volume of solute}}{\text{volume of solution}} \times 100$

It is used when the solute is itself a liquid, such as ethanol. Example: 70 cm³ of ethanol made up to 100 cm³ with water is 70% ethanol by volume.

Notice the words made up to. Liquid volumes do not simply add: 70 cm³ of ethanol mixed with 30 cm³ of water gives only about 97 cm³ of solution, because the small water molecules fit into the spaces between the ethanol molecules (Figure 2.2). So a percentage by volume is made by measuring the solute and then adding water until the solution reaches the final volume.

Worked example

Problem. (a) A 500 cm³ bottle of hand sanitiser is 70% ethanol by volume. What volume of ethanol does it contain? (b) How would you make 250 cm³ of 60% ethanol by volume from pure ethanol?

Solution. (a) $\frac{70}{100} \times 500 = 350\,\mathrm{cm^3}$ of ethanol.

(b) Volume of ethanol $= \frac{60}{100} \times 250 = 150\,\mathrm{cm^3}$ . Measure 150 cm³ of ethanol into a 250 cm³ volumetric flask (or measuring cylinder) and add water until the solution reaches the 250 cm³ mark. Do not simply add 100 cm³ of water: the volumes do not add exactly.

Percentages in everyday life

Table 2.1: Some common solutions

Solution	Typical strength	Kind of percentage
alcohol-based hand sanitiser	60–80% ethanol; the CDC advises at least 60% alcohol	by volume
concentrated hydrochloric acid	about 36% HCl, density $1.18\,\mathrm{g\,cm^{-3}}$	by mass
concentrated sulfuric acid	about 98% $\mathrm{H_2SO_4}$ , density $1.84\,\mathrm{g\,cm^{-3}}$	by mass
normal saline, for drips	0.9 g of sodium chloride in every 100 cm³	mass per volume

From a percentage to a molarity

To turn a percentage by mass into $\mathrm{mol\,dm^{-3}}$ you need the density of the solution, which gives the mass of $1\,\mathrm{dm^3}$ of it. Then find the mass of solute in that $1\,\mathrm{dm^3}$ , and divide by the molar mass.

Worked example

Problem. Concentrated hydrochloric acid is 36% HCl by mass and has a density of $1.18\,\mathrm{g\,cm^{-3}}$ . Find its concentration in $\mathrm{mol\,dm^{-3}}$ . (H = 1, Cl = 35.5)

Solution. Mass of $1\,\mathrm{dm^3}$ ( $1000\,\mathrm{cm^3}$ ) of the acid $= 1000 \times 1.18 = 1180\,\mathrm{g}$ .

Mass of HCl in it $= \frac{36}{100} \times 1180 = 424.8\,\mathrm{g}$ .

$c = \frac{424.8}{36.5} = 11.6\,\mathrm{mol\,dm^{-3}}$

That is about 116 times as concentrated as the $0.10\,\mathrm{mol\,dm^{-3}}$ acid used in titrations, which is why it must be handled in a fume cupboard and diluted before use.

Worked example

Problem. Normal saline contains 0.9 g of sodium chloride in every $100\,\mathrm{cm^3}$ . Find its concentration in $\mathrm{g\,dm^{-3}}$ and in $\mathrm{mol\,dm^{-3}}$ . (NaCl = 58.5)

Solution. $1\,\mathrm{dm^3}$ is ten times $100\,\mathrm{cm^3}$ , so it holds $0.9 \times 10 = 9.0\,\mathrm{g}$ : the mass concentration is $9.0\,\mathrm{g\,dm^{-3}}$ .

$c = \frac{9.0}{58.5} = 0.154\,\mathrm{mol\,dm^{-3}}$

Warning

Common mistakes and safety. (1) In percentage by mass, divide by the mass of the solution, not the mass of water. (2) A percentage by volume is made by making the solution up to the final volume. (3) Concentrated acids are corrosive. Dilute them by adding the acid slowly to water, never water to acid, and wear eye protection. If acid touches your skin or eyes, rinse with plenty of running water for at least 15 minutes and get help. Ethanol and sanitiser are flammable: keep them away from flames.

Tip

Part over whole, times 100. Every percentage is (amount of solute) ÷ (amount of the whole solution) × 100, with both amounts in the same kind of unit: grams over grams, or cm³ over cm³.

ChemSimplified (Dr K): How to calculate percent concentration. Watch the percent by mass and percent by volume parts; mL is the same as cm³, and g/mL is the same as g cm⁻³.

✦

Practice

Define percentage by mass and percentage by volume.
8.0 g of sugar is dissolved in 72.0 g of water. Find the percentage by mass of sugar.
What mass of sodium hydroxide is in 250 g of a solution that is 4.0% sodium hydroxide by mass?
A 750 cm³ bottle of sanitiser is 80% ethanol by volume. What volume of ethanol does it hold?
A drink is 12% alcohol by volume. What volume of ethanol is in a 250 cm³ glass of it?
A glucose drip contains 5.0 g of glucose in every 100 cm³. Find its concentration in $\mathrm{g\,dm^{-3}}$ and in $\mathrm{mol\,dm^{-3}}$ . ( $\mathrm{C_6H_{12}O_6} = 180$ )
Concentrated sulfuric acid is 98% $\mathrm{H_2SO_4}$ by mass, with density $1.84\,\mathrm{g\,cm^{-3}}$ . Calculate its concentration in $\mathrm{mol\,dm^{-3}}$ . ( $\mathrm{H_2SO_4} = 98$ )
A student mixes 60 cm³ of ethanol with 40 cm³ of water and says the solution is exactly 60% by volume. Explain why this is not quite right, and how to make the solution properly.

Show answersHide answers

Percentage by mass: mass of solute ÷ mass of solution × 100. Percentage by volume: volume of solute ÷ volume of solution × 100.
$\frac{8.0}{80.0} \times 100 = 10\%$ .
$\frac{4.0}{100} \times 250 = 10\,\mathrm{g}$ .
$\frac{80}{100} \times 750 = 600\,\mathrm{cm^3}$ .
$\frac{12}{100} \times 250 = 30\,\mathrm{cm^3}$ .
$5.0 \times 10 = 50\,\mathrm{g\,dm^{-3}}$ ; $\frac{50}{180} = 0.28\,\mathrm{mol\,dm^{-3}}$ .
Mass of $1\,\mathrm{dm^3}$ $= 1840\,\mathrm{g}$ ; mass of $\mathrm{H_2SO_4} = 0.98 \times 1840 = 1803\,\mathrm{g}$ ; $c = \frac{1803}{98} = 18.4\,\mathrm{mol\,dm^{-3}}$ .
The volumes do not add exactly: the mixture is slightly less than 100 cm³, so the ethanol is slightly more than 60% of it. Measure 60 cm³ of ethanol into a 100 cm³ volumetric flask and add water up to the mark.

Sources and credits

The advice on sanitiser strength is from the US Centers for Disease Control and Prevention, About Hand Sanitizer.
Photo: Hand sanitizer bottle.jpg, Wikibantu, licensed under CC BY-SA 4.0, via Wikimedia Commons.
Video: How to calculate percent concentration, ChemSimplified, embedded from YouTube.
Written by Ganzaa. The diagrams, table, worked examples and Practice items were drawn and written by Ganzaa.

1Three labels, three kinds of percentage

2Percentages in everyday life

3From a percentage to a molarity

Three labels, three kinds of percentage

Percentages in everyday life

From a percentage to a molarity