The Ideal Gas Law () is derived from the Kinetic Molecular Theory (KMT), which rests on two key assumptions that are not perfectly satisfied by real gases:
These assumptions work well at high temperatures and low pressures, where molecules are far apart and moving fast. However, at low temperatures and high pressures, real gases deviate significantly from ideal behaviour because:
In 1873, Johannes Diderik van der Waals proposed a modified equation of state to account for these two corrections:
where:
In an ideal gas, the full volume is available for molecular motion. In a real gas, the molecules themselves occupy space. The term represents the excluded volume — the total volume unavailable due to the finite size of moles of molecules.
The corrected (available) volume is:
A larger value of means the molecules are physically larger.
In a real gas, molecules near the container wall experience a net inward attractive force from neighbouring molecules. This reduces the speed and force with which they strike the wall, resulting in a lower observed pressure than an ideal gas would exert.
The ideal pressure would be higher than the measured pressure by :
The term is proportional to the square of the molar concentration ()², because both the molecule hitting the wall and its neighbours pulling it back depend on density. A larger value of means stronger intermolecular attractions.
Real gases approach ideal behaviour under conditions where the two corrections become negligible:
| Condition | Reason |
|---|---|
| High temperature | High kinetic energy overcomes intermolecular attractions |
| Low pressure | Molecules are far apart; their volume is negligible compared to container volume |
Conversely, deviations are largest at low temperature and high pressure.
| Gas | (Pa·m⁶·mol⁻²) | (m³·mol⁻¹) |
|---|---|---|
| He | 0.003 | 2.37 × 10⁻⁵ |
| H₂ | 0.025 | 2.66 × 10⁻⁵ |
| N₂ | 0.137 | 3.87 × 10⁻⁵ |
| CO₂ | 0.364 | 4.27 × 10⁻⁵ |
Gases with stronger intermolecular forces (e.g., CO₂) have larger values. Larger molecules have larger values.
The ideal gas model is the foundation of statistical mechanics. Statistical mechanics connects the microscopic behaviour of individual particles (their positions, momenta, and interactions) to macroscopic thermodynamic quantities (pressure, temperature, volume, entropy).
The van der Waals equation represents the first step beyond the ideal model — it introduces molecular interactions and finite size, making it a more realistic statistical mechanical model. More advanced treatments (virial equations, partition functions) build further on this foundation.