Mixtures of gases and real gases

Air is a mixture

The air you breathe is not one gas. By number of molecules it is about $78\ \%$ nitrogen, $21\ \%$ oxygen and $1\ \%$ argon, with a little carbon dioxide and water vapour.

Each of these gases fills the whole room. Each one hits the walls. So what is the pressure of the air? It is the pressure of the nitrogen, plus that of the oxygen, plus that of the argon.

Dalton's law of partial pressures

The partial pressure of a gas in a mixture is the pressure it would have if it filled the whole volume alone, at the same temperature.

Dalton's law. The total pressure of a mixture of ideal gases is the sum of the partial pressures: $p = p_1 + p_2 + p_3 + \dots$

Why does this work? In an ideal gas the molecules do not pull on each other. So a nitrogen molecule does not care whether its neighbours are oxygen or nitrogen. Each gas obeys its own $p_iV = n_iRT$. Add them up:

Divide one gas by the whole. The partial pressure is the mole fraction $x_i = n_i/n$ times the total pressure:

A worked example

A $10$ L cylinder at $300$ K holds $0.312$ mol of nitrogen, $0.084$ mol of oxygen and $0.004$ mol of argon. Find each partial pressure and the total.

1. Nitrogen. First the top line: $nRT = 0.312\ \text{mol} \times 8.31\ \text{J/(mol K)} \times 300\ \text{K} = 778\ \text{J}$. Then $p = \dfrac{nRT}{V} = \dfrac{778\ \text{J}}{0.010\ \text{m}^3} = 77.8\ \text{kPa}$.
2. Oxygen. In the same way, $p = 20.9\ \text{kPa}$.
3. Argon. $p = 1.0\ \text{kPa}$.
4. Total. $77.8\ \text{kPa} + 20.9\ \text{kPa} + 1.0\ \text{kPa} = 99.7\ \text{kPa}$. That is close to the pressure of the air outside, as it should be for air.

Breathing under water

A diver breathes gas at the pressure of the water around them. Under the sea, the pressure grows by about $10$ kPa for each metre. At great depth, even $21\ \%$ oxygen gives a high oxygen pressure. Too much oxygen pressure is harmful. Divers use Dalton's law to choose the mix and the deepest safe depth.

Start with $21\ \%$ oxygen and go deeper. At what depth does the oxygen bar cross the red line? Now raise the oxygen to $32\ \%$. Does the safe depth go up or down?

When is a gas not ideal?

The ideal gas model makes two simplifications.

1. The molecules have no size: they are points.
2. The molecules do not pull on each other.

At low pressure both are nearly true. The molecules are far apart, so their size is small next to the space between them, and they are rarely close enough to pull.

Squeeze the gas hard and both fail. Plot $\dfrac{pV}{nRT}$ against $p$. For an ideal gas the graph is a flat line at $1$.

• When the pull between molecules matters, the gas is easier to squeeze than an ideal gas. $\dfrac{pV}{nRT}$ falls below $1$. Carbon dioxide shows this strongly near room temperature.
• When the size of the molecules matters, at very high pressure, there is less free room. $\dfrac{pV}{nRT}$ rises above $1$.
• At a higher temperature the molecules move faster, the pull matters less, and the gas is closer to ideal.

A real gas cooled and squeezed enough turns into a liquid. An ideal gas never would. That is the clearest sign that real molecules pull on each other.

Practise with new numbers

Try it in Python

The program finds the partial pressures of the gases in the cylinder of the worked example, one at a time, and adds them. Then it checks the oxygen with its mole fraction.

Common mistakes

• Using the mass fraction for the partial pressure. Dalton's law uses the mole fraction.
• Adding the volumes of the gases. Each gas fills the whole container. The pressures add, not the volumes.
• Expecting a real gas to be ideal at all pressures. The ideal gas equation works well only at low pressure and not too low a temperature.
• Thinking the pull between molecules always lowers $pV/(nRT)$. At very high pressure the size of the molecules wins, and it rises above $1$.

Remember these

• Dalton: $p = p_1 + p_2 + \dots$, and $p_i = x_ip$.
• Each gas in a mixture fills the whole volume.
• Ideal gas: point molecules with no pull between them.
• A real gas is nearly ideal at low pressure and high temperature.
• Pull between molecules: $pV/(nRT)$ below $1$. Size of molecules: above $1$.

Check yourself

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

1. A mixture has $3.0$ mol of nitrogen and $1.0$ mol of helium. Its total pressure is $160$ kPa. What is the partial pressure of the helium?

2. A flask of oxygen at $60$ kPa is joined to an empty flask of the same size. The temperature stays the same. What is the new pressure?

3. Why does a gas follow $pV = nRT$ more closely at low pressure?

4. For a real gas at a moderate pressure, $pV/(nRT) = 0.92$. Which effect is winning: the size of the molecules or the pull between them?