E. Solutions and Solubility · Solutions and SolubilityConcentration I: Percent, g/100 mL, ppm and ppb
E2.2 E2.1 A1.12 A1.13About 70 minutes
Learning goals- I can define concentration and explain the difference between dilute and concentrated.
- I can calculate concentration as percentage by mass (% m/m), mass/volume (% m/v) and volume (% v/v), and in g/100 mL.
- I can calculate and convert concentrations in parts per million (ppm) and parts per billion (ppb).
- I can rearrange concentration formulas to find the mass or volume of solute, and report answers with correct significant digits.
Words to know
concentration
The quantity of solute in a given quantity of solution (or solvent).
dilute / concentrated
Words for a relatively small or large quantity of solute per unit of solution. They compare; they are not exact.
percentage by mass (% m/m)
(mass of solute ÷ mass of solution) × 100%.
percentage mass by volume (% m/v)
(mass of solute in g ÷ volume of solution in mL) × 100%.
percentage by volume (% v/v)
(volume of solute ÷ volume of solution) × 100%.
parts per million (ppm)
(mass of solute ÷ mass of solution) × 10⁶. For dilute water solutions, 1 ppm ≈ 1 mg/L.
parts per billion (ppb)
(mass of solute ÷ mass of solution) × 10⁹. For dilute water solutions, 1 ppb ≈ 1 µg/L.
How strong is it?
A bottle of rubbing alcohol says 70% isopropyl alcohol. Vinegar says 5% acetic acid. Your town's water report says lead is below 5 µg/L. A fish advisory talks about mercury in parts per million. All of these are ways to state a solution's concentration: how much solute is in a given amount of solution.
The words dilute and concentrated are useful for comparing (a teaspoon of juice crystals in a jug is dilute), but chemists need numbers. In this lesson you learn the units used on labels and in environmental reports. In the next lesson you learn the unit chemists use most: moles per litre.
Always 'solution' on the bottom
In every percent and ppm formula, the bottom of the fraction is the solution (solute + solvent), not just the solvent. If a problem gives the mass of solute and the mass of water, add them first.
Percentage concentrations
Percentage by mass, % m/m (used for solids and for things like ocean salinity):
% m/m = (m_solute ÷ m_solution) × 100%
Percentage by volume, % v/v (used when a liquid is dissolved in a liquid, e.g. rubbing alcohol, wine coolers, windshield washer fluid):
% v/v = (V_solute ÷ V_solution) × 100%
Percentage mass by volume, % m/v (used on many pharmacy and food labels, e.g. saline, vinegar, hydrogen peroxide):
% m/v = (m_solute in g ÷ V_solution in mL) × 100%
A 0.9% m/v saline solution contains 0.9 g of NaCl in every 100 mL of solution.
Grams per 100 mL
Solubility tables often give concentration in g/100 mL, which is the same number as % m/v. A saturated NaCl solution at 20 °C is about 36 g/100 mL of water.
Worked exampleExample 1: Percent concentrations
(a) 12.0 g of sugar is dissolved in 238.0 g of water. Find % m/m. (b) A 500 mL bottle of rubbing alcohol is 70% v/v isopropanol. What volume of isopropanol does it contain? (c) Canadian white vinegar is 5% m/v acetic acid. What mass of acetic acid is in 250 mL?
- (a) m_solution = 12.0 g + 238.0 g = 250.0 g. % m/m = 12.0 ÷ 250.0 × 100% = 4.80%.
- (b) Rearrange: V_solute = (% v/v ÷ 100%) × V_solution = 0.70 × 500 mL = 350 mL (about 3.5 × 10² mL, two significant digits).
- (c) m_solute = (% m/v ÷ 100%) × V_solution = 0.05 g/mL × 250 mL = 12.5 g. (Here we treat the label's '5%' as exact.)
(a) 4.80% m/m; (b) 350 mL; (c) about 12.5 g.
Parts per million and parts per billion
Many pollutants and nutrients in water are harmful (or helpful) at very small concentrations. Writing 0.0005% is awkward, so we use bigger multipliers:
ppm = (m_solute ÷ m_solution) × 10⁶
ppb = (m_solute ÷ m_solution) × 10⁹
For dilute water solutions, 1 L of solution has a mass of about 1 kg = 10⁶ mg = 10⁹ µg. That gives two very handy shortcuts:
- 1 ppm ≈ 1 mg/L
- 1 ppb ≈ 1 µg/L (µg = microgram = 10⁻⁶ g)
And: 1% = 10 000 ppm, and 1 ppm = 1000 ppb.
ppm by mass also works for solids. Health Canada's standard for mercury in most fish sold in stores is 0.5 ppm, meaning 0.5 mg of mercury per kilogram of fish.
Worked exampleExample 2: ppm and ppb
(a) A 2.0 L sample of well water contains 0.18 mg of lead. Express the concentration in mg/L and in ppb. Health Canada's maximum acceptable concentration (MAC) for lead in drinking water is 0.005 mg/L. Is the water acceptable? (b) A 250 g fish fillet contains 0.10 mg of mercury. Find the concentration in ppm and compare it with 0.5 ppm. (c) Health Canada recommends fluoride in drinking water at 0.7 mg/L. What mass of fluoride is in 1.5 L of this water?
- (a) c = 0.18 mg ÷ 2.0 L = 0.090 mg/L. In µg/L: 0.090 mg/L × 1000 µg/mg = 90 µg/L ≈ 90 ppb.
- Compare: the MAC is 0.005 mg/L = 5 µg/L = 5 ppb. 90 ppb is 18 times the MAC, so the water is not acceptable for drinking.
- (b) Use the same units top and bottom: 0.10 mg ÷ 250 000 mg × 10⁶ = 0.40 ppm. This is below 0.5 ppm.
- (c) m = c × V = 0.7 mg/L × 1.5 L = 1.05 mg, about 1 mg of fluoride ion.
(a) 90 ppb, far above the 5 ppb MAC; (b) 0.40 ppm; (c) about 1 mg.
Significant digits (A1.13)
In multiplication and division, round your answer to the fewest significant digits in the measured data. 0.18 mg ÷ 2.0 L has two significant digits in each, so the answer 0.090 mg/L keeps two (the 9 and the final 0). Keep extra digits during the calculation and round only at the end.
Just for funFinding 1 part per billion is like picking out one particular second in about 32 years: a billion seconds is roughly 31.7 years. Water-testing labs do this kind of detective work every day, without the 32-year wait.
Hands-on activityMake a dilution ladder down to ppm
You need: food colouring, a medicine dropper (or a drinking straw used as a dropper), 7 small clear cups or glasses, water, a white sheet of paper, and a marker.
- Label cups 1 to 7. In cup 1 put 10 drops of food colouring (call this 100% "dye").
- Put 9 drops of water in each of cups 2 to 7.
- Move 1 drop from cup 1 into cup 2 and swirl. Cup 2 is now 1 part in 10 = 10% of the original.
- Move 1 drop from cup 2 into cup 3 (1%), then 3 into 4 (0.1%), and so on to cup 7.
- Put the cups on white paper. Record the last cup in which you can still see colour.
- Work out the concentration of cups 5, 6 and 7 in ppm (cup 7 = 1 part per million of the original dye).
What to look for: cup 5 is 1 part in 10⁴ (100 ppm), cup 6 is 10 ppm, cup 7 is 1 ppm. Most people cannot see colour past about cup 4 or 5. Big idea: a substance can be present, and sometimes harmful, at concentrations far too low to see, which is why water must be tested in a lab. (Food colouring stains: cover the table.)
Your turn
115.0 g of KCl is dissolved in 135.0 g of water. Find % m/m.
m_solution = 150.0 g; 15.0 ÷ 150.0 × 100% = 10.0%.
2What volume of ethanol is in 750 mL of a 12% v/v solution?
3A 0.9% m/v saline bag holds 1000 mL. What mass of NaCl does it contain?
0.009 × 1000 = 9 g (0.9 g per 100 mL × 10).
4Convert 0.025% m/m to ppm.
0.025 × 10 000 = 250 ppm.
5A 500 mL water sample contains 0.0060 mg of arsenic. Find the concentration in µg/L (ppb).
0.0060 mg ÷ 0.500 L = 0.012 mg/L = 12 µg/L ≈ 12 ppb.
6Health Canada's MAC for arsenic is 0.010 mg/L. Is the sample in the last question acceptable?
0.010 mg/L = 10 ppb. 12 ppb is above the MAC, so not acceptable.
7Why do we use ppm and ppb instead of percent for pollutants?
Pollutant concentrations are tiny; ppm and ppb give convenient whole-ish numbers instead of long decimals (5 ppb instead of 0.0000005%).
More practice: warm-up
1In % m/m, what goes on the bottom of the fraction?
The mass of the solution (solute + solvent).
31 ppm in water is approximately how many mg/L?
4A drink is 5% v/v. How many mL of solute in 100 mL?
5Which is more concentrated: 30 ppb or 0.5 ppm?
0.5 ppm = 500 ppb, so 0.5 ppm is more concentrated.
More practice: on your own
1What mass of sugar is needed to make 400.0 g of a 7.50% m/m sugar solution? What mass of water?
Sugar: 0.0750 × 400.0 g = 30.0 g. Water: 400.0 − 30.0 = 370.0 g.
2A creek has 0.24 ppm of phosphate. What mass of phosphate is in 5.0 × 10⁶ L of the creek water?
0.24 ppm ≈ 0.24 mg/L. m = 0.24 mg/L × 5.0 × 10⁶ L = 1.2 × 10⁶ mg = 1.2 × 10³ g (1.2 kg).
3Hydrogen peroxide from a drugstore is 3% m/v. What volume contains 4.5 g of H₂O₂?
V = m ÷ (%/100) = 4.5 g ÷ 0.03 g/mL = 150 mL.
4A 20.0 g soil sample contains 0.0036 g of lead. Find its concentration in ppm.
0.0036 ÷ 20.0 × 10⁶ = 180 ppm.
5Road salt runoff gives a stream 350 mg/L of chloride. A Canadian guideline for long-term protection of aquatic life is 120 mg/L. By what factor is the stream above the guideline?
350 ÷ 120 ≈ 2.9 times the guideline.
Challenge
1Ocean water is about 3.5% m/m dissolved salts and has a density of 1.025 g/mL. What mass of salts is in 1.00 L of ocean water?
Mass of 1.00 L = 1000 mL × 1.025 g/mL = 1025 g. Salts = 0.035 × 1025 g ≈ 36 g.
2A swimming pool holds 60 000 L of water. The target free chlorine level is 2 ppm (2 mg/L). What mass of chlorine (as Cl₂ equivalent) must be present? If the liquid pool chlorine product contains 12% m/v available chlorine, what volume of product gives this amount (ignore chlorine already present)?
Mass = 2 mg/L × 60 000 L = 120 000 mg = 120 g. Product: 12 g per 100 mL, so V = 120 g ÷ 0.12 g/mL = 1000 mL (1.0 L).
3A student says 'a 10% m/m solution means 10 g of solute in 100 g of water.' Explain the error and give the correct mass of water for 10 g of solute.
The 100 g must be the solution, not the water. 10 g solute in 90 g of water makes 100 g of solution, which is 10% m/m. (10 g in 100 g water would be 10 ÷ 110 = 9.1%.)
Quiz
1. 15 g of salt in 85 g of water gives a % m/m of
2. A 40% v/v antifreeze solution has a volume of 2.0 L. The volume of antifreeze is
3. For dilute water solutions, 1 ppm is about
5. A 5% m/v solution contains
6. Which concentration is the largest?
Big idea (E2.2): concentration is a ratio of solute to solution, and the unit chosen depends on how concentrated the solution is: percent for everyday products, ppm and ppb for trace pollutants.
How to help: Read labels together (vinegar, rubbing alcohol, hydrogen peroxide, a sports drink) and ask what the percent means in grams or millilitres. Look up your town's drinking-water report and find a value in mg/L or µg/L.
Common difficulties:
- Dividing by the mass of solvent instead of the mass of solution.
- Mixing units in ppm (mg on top, g on the bottom). Convert to the same unit first.
- Converting mg and µg: 1 mg = 1000 µg.
- Rounding too early, or keeping too many digits at the end.
Safety: the dye ladder uses only food colouring and water.