The Van der Waals Equation

The ideal gas law assumes negligible intermolecular attractions and negligible molecular volume, assumptions that break down under extreme conditions. At high pressures and low temperatures, gas molecules are forced closer together, making intermolecular forces and molecular size significant. This causes deviation from ideal behavior, requiring corrections to predict real gas behavior accurately.

The pressure correction adjusts for attractive forces between gas molecules that reduce measured pressure below the ideal value. These intermolecular attractions decrease both collision frequency and force with container walls. The pressure reduction is directly proportional to the square of molar concentration, reflecting how attraction strength increases with molecular density.

The volume correction calculates actual available volume for molecular motion by subtracting the volume excluded by molecules themselves. For n moles where each molecule occupies volume b, the excluded volume is nb. The actual free volume becomes total volume minus nb, accounting for the physical space occupied by gas molecules.

Coefficient 'a' represents intermolecular attraction strength; larger values indicate stronger cohesion and greater pressure correction. Coefficient 'b' represents excluded volume from repulsive interactions; larger values mean less free space available. Both are empirical constants unique to each gas and remain temperature-independent within the van der Waals model.

Intermolecular attractions pull molecules together, reducing the force and frequency of collisions with container walls. This causes real gas pressure to be lower than predicted by ideal gas behavior. The magnitude of this pressure reduction depends on molecular concentration squared, making it more significant at higher densities.

Within the van der Waals model, coefficients 'a' and 'b' are treated as empirical constants characteristic of each gas that remain unaffected by temperature changes. This simplification allows the equation to provide reliable corrections across a range of conditions, though real substances may show some temperature dependence in practice.

The virial equation of state is preferred for higher precision, particularly over wide ranges of temperatures and pressures. While the van der Waals equation offers valuable insights into real gas behavior, it is not universal for all substances. Virial coefficients, commonly tabulated at various temperatures, capture deviations from ideal behavior more accurately.