Q.The partial pressure of $\text{CO}_2$ gas maintained above a bottle of soft drink at $298\ \text{K}$ is $4\ \text{bar}$. If the Henry's law constant for $\text{CO}_2$ dissolved in water at $298\ \text{K}$ is $K_H = 1.67 \times 10^3\ \text{bar}$, calculate the mole fraction of $\text{CO}_2$ in the drink.

Henry's law describes the equilibrium reached when a gas dissolves in a liquid: at a fixed temperature, the amount

of gas dissolved is directly proportional to the partial pressure of that gas maintained above the liquid,

where $p$ is the gas's partial pressure, $x$ is its mole fraction actually dissolved in the liquid, and $K_H$ is

the Henry's law constant — a proportionality constant characteristic of the particular gas-solvent pair and

temperature. A plot of $p$ against $x$ is a straight line through the origin, and a larger $K_H$ corresponds to

lower solubility, since a greater pressure is then needed to dissolve the same mole fraction.

Gas solubility depends on the identity of both the gas and the solvent, and — importantly — falls as temperature

rises, since dissolving a gas is normally exothermic, so by Le Chatelier's principle warming the liquid shifts the

dissolution equilibrium back toward undissolved gas (equivalently, $K_H$ increases with temperature). This

explains why warmer water holds less dissolved oxygen, stressing aquatic life, and why gas solubility calculations

in this chapter always specify a fixed temperature.

Henry's law underlies several practical situations: bottling soft drinks under high $\text{CO}_2$ pressure to

force a large mole fraction of gas into solution, so that the drink fizzes once the pressure is released; deep-sea

divers dissolving unusually large amounts of nitrogen in their blood under pressure, risking decompression

sickness ("the bends") if they ascend too quickly and the dissolved gas suddenly comes out of solution as

bubbles; and the lower dissolved-oxygen intake experienced at high altitude, where the reduced atmospheric partial

pressure of oxygen means less of it dissolves in blood plasma, causing altitude sickness.