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Solutions: Advanced Topics

Molarity is not the only way to describe a solution. When temperature changes, or when you need to talk about very dilute solutions or vapour pressure, other concentration units are more useful.

Concentration Units

Molality ($m$): moles of solute per kilogram of solvent — unlike molarity, it does not change when the solution expands or contracts with temperature: $$m = \frac{\text{mol solute}}{\text{kg solvent}}$$
Mass percent: grams of solute per 100 g of solution: $$\text{mass \%} = \frac{\text{mass solute}}{\text{mass solution}} \times 100\%$$
Parts per million (ppm) and parts per billion (ppb): milligrams of solute per kilogram of solution (ppm) and micrograms per kilogram (ppb) — used for trace amounts like pollutants in water. For dilute aqueous solutions, 1 ppm ≈ 1 mg/L.

Example: Dissolve 10.0 g of NaCl (molar mass 58.44 g/mol) in 200 g of water: $$n = \frac{10.0\ \text{g}}{58.44\ \text{g/mol}} = 0.171\ \text{mol} \;\Rightarrow\; m = \frac{0.171\ \text{mol}}{0.200\ \text{kg}} = 0.856\ m$$ $$\text{mass \%} = \frac{10.0\ \text{g}}{10.0 + 200\ \text{g}} \times 100\% = 4.76\%$$

And a water sample containing 25 mg of lead in 5.0 kg of water is $\frac{25\ \text{mg}}{5.0\ \text{kg}} = 5.0$ ppm lead.

Henry's Law: Gas Solubility

The solubility of a gas in a liquid is directly proportional to the gas's partial pressure above the liquid: $$C = k_H P$$ where $C$ is the molar solubility and $k_H$ is Henry's law constant for that gas–solvent pair at that temperature.

Example: For CO₂ in water at 25 °C, $k_H = 3.4 \times 10^{-2}$ mol/L·atm. At a partial pressure of 1.0 atm: $$C = (3.4 \times 10^{-2})(1.0) = 0.034\ \text{M}$$

Inside a sealed soda bottle at about 4.0 atm of CO₂, the solubility rises to:

$$C = (3.4 \times 10^{-2})(4.0) = 0.14\ \text{M}$$

Open the bottle, the pressure drops to 1 atm, and the excess CO₂ escapes as fizz. Warm soda goes flat faster because $k_H$ decreases as temperature rises.

Raoult's Law and Deviations from Ideality

For an ideal solution, the vapour pressure of each component is its mole fraction times its pure vapour pressure: $$P_A = X_A P_A^\circ$$ The total vapour pressure is the sum of the partial pressures. Real solutions deviate when the attractions between unlike molecules differ from those between like molecules.

Positive deviation: A–B attractions are weaker than A–A and B–B, so molecules escape more easily and the vapour pressure is higher than Raoult's law predicts. Ethanol–water is the classic example — the mixture's vapour pressure is higher than ideal, and it boils at a lower temperature than either pure liquid would suggest.
Negative deviation: A–B attractions are stronger than A–A and B–B, so molecules escape less easily and the vapour pressure is lower than predicted. Acetone–chloroform shows this: hydrogen bonding between the two holds the molecules back, and the mixture's vapour pressure drops below the ideal value.

Example: Pure water has $P^\circ = 23.8$ mmHg at 25 °C. In a solution where the mole fraction of water is 0.90, Raoult's law predicts: $$P = X_{\text{water}} P^\circ = (0.90)(23.8\ \text{mmHg}) = 21.4\ \text{mmHg}$$

Colloids, Solutions, and Suspensions

A true solution has solute particles smaller than 1 nm — they never settle and the mixture is completely transparent. A colloid has particles 1–1000 nm across (milk, fog, gelatin, smoke): they stay dispersed indefinitely, but they are big enough to scatter light — the Tyndall effect, the visible beam of a flashlight through fog or dusty air. A suspension has particles larger than 1000 nm (muddy water, sand in water): they scatter light too, but they settle out over time. The Tyndall effect is the quick test: shine a light through it — a true solution shows no beam, a colloid does.

Practice Problems

Calculate the molality of a solution made by dissolving 15.0 g of glucose (C₆H₁₂O₆, 180.16 g/mol) in 250 g of water.
What is the mass percent of a solution made by dissolving 8.0 g of KCl in 92 g of water?
The Henry's law constant for CO₂ in water at 25 °C is 3.4 × 10⁻² mol/L·atm. What is the solubility of CO₂ at a partial pressure of 2.5 atm?
Pure water has a vapour pressure of 23.8 mmHg at 25 °C. What is the vapour pressure of water in a solution where its mole fraction is 0.80?