Grade 10

Ideal and real gases

How pressure, volume, temperature and amount of gas are linked, and why real gases depart from the ideal law.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: particles far apart

In a gas the particles are far apart and move freely. Heat it and the particles move faster and push harder on the walls; squeeze it into a smaller volume and they hit the walls more often.

PV = nRT

Definition: Ideal gas

A gas whose particles have no volume and no attractions. It obeys $PV = nRT$ with $R = 0.08314\ \text{L·bar·mol}^{-1}\text{·K}^{-1}$ and $T$ in kelvin.

Push the volume down or raise

T

and watch the particles: the displayed pressure is computed from

PV = nRT

, while the particles (which do not collide with each other) show why it changes.

Slide

V

T

and

n

. Switch to

\ce{CO2}

\ce{He}

(van der Waals) and compare with the dashed ideal curve.

SchoolSchool level: the special cases

At constant temperature $P \propto 1/V$ (Boyle). At constant pressure $V \propto T$ (Charles), and at constant volume $P \propto T$ (Gay-Lussac), with $T$ in kelvin. At the same $T$ and $P$ , equal volumes contain equal numbers of molecules (Avogadro).

Example: One mole in 10 L

What pressure does 1 mol of an ideal gas exert at 300 K in a 10 L container?

Solution

$P = nRT/V = 1 \times 0.08314 \times 300 / 10 \approx 2.49$ bar, the value shown by the graph.

UndergraduateUndergraduate: real gases

\left(P + \frac{a\,n^2}{V^2}\right)\left(V - nb\right) = nRT

The van der Waals equation adds two corrections: $b$ for the volume of the particles and $a$ for the attractions between them. For $\ce{CO2}$ , $a = 3.64\ \text{L}^2\text{·bar·mol}^{-2}$ and $b = 0.0427\ \text{L·mol}^{-1}$ ; its curve departs from the ideal one at small volume and low temperature.