Gases in liquids and Henry's law

Why a fizzy drink hisses

Open a bottle of a fizzy drink. You hear a hiss, and bubbles rise at once. Before you opened it, the gas was dissolved in the liquid and held there by pressure. Then the pressure fell, and the gas came out.

Warm drinks foam more than cold ones. Fish gather near the surface of a pond on a hot day. Climbers on a high peak feel weak and cannot think clearly. All of these are about gases dissolved in liquids.

What decides how much gas dissolves

The amount depends on three things.

• The gas and the liquid. Most gases dissolve very little in water. Gases that react with water, such as hydrogen chloride and ammonia, dissolve far more.
• The pressure of the gas above the liquid.
• The temperature.

Henry's law

William Henry measured how gases dissolve in water in 1803. His result is Henry's law:

At a fixed temperature, the amount of gas that dissolves in a liquid is proportional to the partial pressure of that gas above the liquid.

Double the pressure and the amount that dissolves doubles. In a mixture of gases, each gas follows its own partial pressure, which is $p = x_{gas} \times P_{total}$ for a gas with mole fraction $x_{gas}$ in the mixture.

We write the law with the mole fraction $x$ of the gas in the solution:

$K_H$ is the Henry's law constant. It depends on the gas, the solvent and the temperature. It has the unit of pressure, such as bar or kbar. When we want $x$, we use $x = \dfrac{p}{K_H}$.

A larger $K_H$ means a smaller solubility at the same pressure. This is easy to read from the picture below: the line with the smaller $K_H$ is steeper.

These values come from standard tables:

• Oxygen in water: $K_H = 34.86\ \text{kbar}$ at $293\ \text{K}$, and $46.82\ \text{kbar}$ at $303\ \text{K}$.
• Nitrogen in water: $76.48\ \text{kbar}$ at $293\ \text{K}$, and $88.84\ \text{kbar}$ at $303\ \text{K}$.
• Helium in water: $144.97\ \text{kbar}$ at $293\ \text{K}$.
• Carbon dioxide in water: $1.67\ \text{kbar}$ at $298\ \text{K}$.

Carbon dioxide has the smallest constant, so it dissolves the most. This is why we can put so much of it into a fizzy drink.

Using Henry's law

In a weak solution, the mole fraction of the gas is $x \approx \dfrac{n_{gas}}{n_{water}}$. One kilogram of water holds $\dfrac{1000}{18} = 55.6\ \text{mol}$. So the moles of gas in $1\ \text{kg}$ of water are $x \times 55.6\ \text{mol}$.

Example. How much nitrogen dissolves in water at $293\ \text{K}$ from air at $1.00\ \text{bar}$, which is $78\ \%$ nitrogen?

• The partial pressure is $p = 0.78 \times 1.00 = 0.78\ \text{bar}$.
• The mole fraction is $x = \dfrac{0.78\ \text{bar}}{76\,480\ \text{bar}} = 1.02 \times 10^{-5}$.
• The moles in $1\ \text{kg}$ of water are $1.02 \times 10^{-5} \times 55.6 = 5.67 \times 10^{-4}\ \text{mol}$. The mass is $5.67 \times 10^{-4} \times 28 = 0.0159\ \text{g}$, about $16\ \text{mg}$.

Check the answer. A very small mole fraction fits a gas that is hardly soluble, and $x$ is much smaller than $1$, so the weak-solution step is fair.

Try it yourself: a gas in water

Temperature and gases

Dissolving a gas gives out heat. A gas that moves into a liquid loses energy. So, by Le Chatelier's principle, heating lowers the solubility of a gas. The constant $K_H$ rises with temperature.

Oxygen shows it well. Its constant rises from $34.86\ \text{kbar}$ to $46.82\ \text{kbar}$ when the water warms from $293\ \text{K}$ to $303\ \text{K}$. The solubility at a fixed pressure becomes $\dfrac{34.86}{46.82} = 0.745$ of its old value. That is a fall of about $25\ \%$ for a $10\ \text{K}$ rise. A warm pond holds less oxygen for its fish.

Try it yourself: temperature

What Henry's law explains

Fizzy drinks. The bottle is sealed with carbon dioxide at a few bar, so a lot of the gas dissolves. When you open it, the pressure above the drink falls to that of the air, where carbon dioxide has a partial pressure of only about $0.0004\ \text{bar}$. Almost all the dissolved gas must leave.

Scuba divers. Every $10\ \text{m}$ of water adds about $1\ \text{bar}$. A diver breathes air at the pressure around the body. At depth, the partial pressure of nitrogen is high, so more nitrogen dissolves in the blood. If the diver rises too fast, the pressure falls quickly, and the extra nitrogen comes out as bubbles in the blood. This causes pain and can be dangerous. It is called the bends. The remedy is to rise slowly, and to stop on the way, so that the gas leaves in small steps. Deep divers breathe air in which helium takes the place of nitrogen, because helium is much less soluble in blood.

High altitude. At a great height, the total pressure of the air is low. The partial pressure of oxygen is low as well, so less oxygen dissolves in the blood. A climber feels weak and cannot think clearly. This condition is called anoxia. Climbers use oxygen cylinders.

When does the law work?

Henry's law works for a gas that does not react with the liquid, at moderate pressure. Carbon dioxide reacts a little with water, so Henry's law describes only the part that stays as dissolved carbon dioxide. Hydrogen chloride reacts fully to give ions, so it does not follow the law.

Raoult's law is the matching rule for a liquid that is the main part of a mixture. Henry's law is the rule for a gas, or for a solute that is very dilute.

Move the depth slider. The line is straight, as Henry's law says. The warmer water has a larger constant, so it holds less gas at every depth.

Try it in Python

The program uses Henry's law to find the nitrogen dissolved at each depth. Then it finds the volume of gas that a fast rise would release.

Common mistakes

• Using the total pressure instead of the partial pressure. Henry's law uses the pressure of that gas alone.
• Forgetting that $K_H$ is in kilobar. Change it to bar before you divide.
• Thinking that a larger $K_H$ means more gas dissolves. It means less gas dissolves at the same pressure.
• Expecting more gas in warm water. Dissolving a gas gives out heat, so warm water holds less.
• Using the law for a gas that reacts with the liquid. Ammonia and hydrogen chloride do not follow it.

Remember these

• Henry's law: $p = K_H\, x$, at a fixed temperature. The amount dissolved is proportional to the partial pressure.
• A larger $K_H$ means a smaller solubility. $K_H$ has the unit of pressure.
• Heating lowers the solubility of a gas, because dissolving a gas gives out heat.
• Uses: fizzy drinks, the bends in divers, anoxia at high altitude, oxygen in warm ponds.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. State Henry's law in words and as one equation.

2. Gas A has $K_H = 10\ \text{kbar}$ in water and gas B has $K_H = 50\ \text{kbar}$, at the same temperature. Which dissolves more at the same pressure?

3. Why does a warm fizzy drink foam more than a cold one when it is opened?

4. Why do deep divers use helium and oxygen, and not air?

5. The pressure on a bottle of air above water doubles, and the composition of the air does not change. What happens to the amount of dissolved oxygen?