Van der Waals Equation

The ideal gas equation has two significant drawbacks: it ignores the volume occupied by gas molecules themselves, and it neglects attractive intermolecular forces between molecules. These omissions cause inaccuracies, especially at high pressures and low temperatures where molecular interactions become significant. The van der Waals equation addresses both limitations.

In the van der Waals equation, constant b represents the volume of one mole of gas molecules, accounting for their physical size. Constant a quantifies attractive intermolecular forces; it multiplies the square of molar density to correct pressure. Both constants are determined experimentally for each gas type and depend on molecular properties.

The van der Waals equation subtracts the term nb from the total volume, where n is the number of moles and b is the volume per mole of molecules. This gives the remaining volume available for gas molecules to move. At high densities, this correction becomes significant, preventing the unrealistic prediction that molecules can occupy zero volume.

Attractive intermolecular forces reduce the observed pressure proportional to the square of molar density. The van der Waals equation adds a correction term equal to a times the square of molar density to the measured pressure. This accounts for the fact that molecules pull inward on each other, requiring higher external pressure to achieve the same nRT value.

At low densities, the correction terms a and b become negligible, and the van der Waals equation reduces to the ideal gas law. This occurs when gas molecules are far apart and intermolecular forces are weak. The ideal gas equation remains a valid approximation at high temperatures and low pressures where molecular interactions are minimal.

The van der Waals equation successfully predicts liquid-to-vapor phase transitions and the Joule-Thomson effect, which the ideal gas equation cannot describe. These phenomena depend on molecular volume and intermolecular attractions. The equation's ability to model real gas behavior under various conditions makes it essential for understanding condensation and temperature changes during gas expansion.

At high densities, the pressure correction term from intermolecular attractions becomes significant and positive, requiring lower pressure to achieve the same nRT value. However, the volume correction term nb also becomes substantial, partially offsetting this effect. The net result is that pressure increases less than expected in highly compressed gases due to competing molecular volume and attractive force corrections.