Solubility

Solids and liquids in liquid solvents follow a qualitative rule you already know from making tea or trying (and failing) to mix oil and water: polar solutes dissolve in polar solvents, non-polar solutes dissolve in non-polar solvents, because dissolution needs solute–solvent attractions strong enough to out-compete the solute's own internal bonding and the solvent's own internal bonding. Sugar and salt (polar/ionic) dissolve readily in water (polar); naphthalene (non-polar) dissolves in benzene (non-polar) but barely at all in water. For most solids dissolving in liquids the process absorbs heat, so raising the temperature usually pushes solubility up — though there are well-known exceptions, and the syllabus does not ask you to predict them quantitatively.

Gases in liquids are where the syllabus wants numbers, not just intuition. Shake a closed bottle of soda and gas keeps dissolving until the rate of gas molecules entering the liquid equals the rate leaving it — a dynamic equilibrium. At that equilibrium, English chemist William Henry found a strikingly simple relationship between how much gas is squeezed above the liquid and how much ends up dissolved in it:

where: $p$ is the partial pressure of the gas in the vapour phase above the solution, $x$ is the mole fraction of that gas dissolved in the liquid, and $K_H$ is the Henry's law constant — a number specific to one gas–solvent pair at one temperature, with units of pressure (since $x$ is a dimensionless ratio). Henry's law is a straight line through the origin on a plot of $p$ against $x$: double the pressure, double the dissolved mole fraction. It only applies to gases that dissolve without reacting chemically with the solvent — HCl and NH₃ ionise in water instead of simply dissolving, so Henry's law does not describe them. The figure below shows why $K_H$ is the whole story: it is literally the slope of the $p$-versus-$x$ line, so a steep line (large $K_H$) means you need a lot of pressure to force even a little gas into solution — low solubility — while a shallow line (small $K_H$) means the gas dissolves easily at modest pressure.

Temperature dependence: $K_H$ increases with rising temperature for most gases in water. That matches everyday experience — warm soda goes flat faster, and a lake's dissolved oxygen drops as it warms, stressing fish. A larger $K_H$ at the same fixed pressure forces $x = p/K_H$ smaller, so the gas becomes less soluble.

Applications worth remembering: carbonated drinks are bottled under high CO₂ pressure so more gas dissolves; opening the cap drops the pressure, so by Henry's law the equilibrium mole fraction of dissolved CO₂ falls and the excess escapes as bubbles. Scuba divers who ascend too quickly can suffer the "bends," because dissolved nitrogen in the blood, which stayed in solution under high pressure at depth, suddenly bubbles out as pressure drops. At high altitude the partial pressure of atmospheric O₂ is lower, so less oxygen dissolves in blood, causing altitude sickness.

Problem: The Henry's law constant for CO₂ dissolved in water is $K_H = 1.67 \times 10^{8}$ Pa. A soft drink is bottled under a CO₂ partial pressure of $2.5 \times 10^{5}$ Pa. Find the mole fraction of dissolved CO₂, and then find the new equilibrium mole fraction once the bottle is opened and the CO₂ partial pressure above the liquid falls to atmospheric, about $4.0 \times 10^{1}$ Pa (CO₂'s share of ordinary air).

Step 1 — Apply Henry's law for the sealed bottle. $x = \dfrac{p}{K_H} = \dfrac{2.5\times 10^{5}\ \text{Pa}}{1.67\times 10^{8}\ \text{Pa}} = 1.497\times 10^{-3}$, so about 1.5 CO₂ molecules per thousand molecules of solution while sealed.

Step 2 — Apply Henry's law again for the open bottle. $x_{\text{new}} = \dfrac{4.0\times 10^{1}\ \text{Pa}}{1.67\times 10^{8}\ \text{Pa}} = 2.40\times 10^{-7}$, roughly ten thousand times smaller than the sealed-bottle value.

Step 3 — Interpret the drop. The equilibrium mole fraction the liquid can hold collapses once the cap is off and the CO₂ partial pressure crashes toward atmospheric levels. The drink is now far supersaturated relative to the new, tiny equilibrium value, so the excess dissolved CO₂ escapes as bubbles until the liquid re-equilibrates — this is the fizz. The mole fraction of dissolved CO₂ falls from about $1.5\times 10^{-3}$ (sealed, under pressure) to about $2.4\times 10^{-7}$ (open, at atmospheric CO₂ pressure).

Competitive stretch

Problem: At 298 K, oxygen's Henry's law constant in water is $K_H = 3.30\times 10^{7}$ Pa. Air exerts an O₂ partial pressure of $0.20 \times 1.013\times 10^{5}$ Pa. Find the dissolved mole fraction, then estimate how many O₂ molecules that represents per litre of water (1 L water $\approx$ 55.5 mol).

Step 1 — Partial pressure of O₂. $p_{O_2} = 0.20 \times 1.013\times 10^{5} = 2.026\times 10^{4}$ Pa.

Step 2 — Apply Henry's law. $x_{O_2} = \dfrac{2.026\times 10^{4}}{3.30\times 10^{7}} = 6.14\times 10^{-4}$.

Step 3 — Convert to moles per litre. Since $x_{O_2} \approx n_{O_2}/n_{\text{water}}$ for a dilute solution, $n_{O_2} \approx 6.14\times 10^{-4} \times 55.5\ \text{mol} = 3.4\times 10^{-2}\ \text{mol}$ per litre. About $3.4\times 10^{-2}$ mol (roughly $1.1$ g) of O₂ dissolves per litre of water in equilibrium with air at 298 K — a small but life-sustaining amount for aquatic organisms, and one that shrinks further as $K_H$ rises with warming water.

• Reading "higher $K_H$" as "higher solubility." It is the opposite: $x = p/K_H$, so a larger Henry's constant makes $x$ smaller at fixed pressure — less gas dissolves.
• Using total pressure instead of the gas's own partial pressure when the gas above the solution is part of a mixture, such as O₂ within air rather than pure O₂.
• Applying Henry's law to gases that react with the solvent, like NH₃ or HCl in water — these ionise or react rather than simply dissolving, so the simple proportionality fails.
• Treating $K_H$ as universal for a gas. It is specific to one gas, one solvent, and one temperature — always check whether the given value matches the conditions in the problem.

Ask the tutor: want to try Henry's law numericals with mixed units, or see why Raoult's law turns out to be a special case of Henry's law?