Van der Waals Equation of State Calculator

The Van der Waals equation is a real-gas equation of state that improves on the ideal gas law by adding two correction terms: 'a' for intermolecular attraction and 'b' for the finite volume of gas molecules themselves, P=RT/(Vm−b)−a/Vm², where Vm is molar volume.

Constant 'a' (units Pa·m⁶/mol²) measures the strength of attractive forces between molecules, larger values mean stronger attraction and lower resulting pressure. Constant 'b' (units m³/mol) represents the excluded volume per mole due to the molecules' own finite size.

The ideal gas law (PV=nRT) assumes molecules are point particles with no volume and no intermolecular forces. The Van der Waals equation corrects both assumptions, giving more accurate predictions for real gases, especially at high pressure or low temperature where these effects matter most.

Attractive forces between molecules (the 'a' term) pull molecules together, reducing the force they exert on container walls, lowering pressure below the ideal prediction. The excluded volume (the 'b' term) effectively increases the pressure needed to compress the gas into the remaining available space. Which effect dominates depends on the specific gas and conditions.

Gases with strong intermolecular attraction, like water vapor and carbon dioxide (which have relatively large 'a' values), deviate more from ideal behavior than weakly-interacting gases like helium (very small 'a'), especially near their condensation point.

At large molar volume (low density, low pressure), both correction terms become negligible relative to RT/Vm, and the equation smoothly approaches the ideal gas law, real gases behave nearly ideally when they are dilute.

No, the Van der Waals equation is cubic in molar volume, so solving for Vm given P and T requires numerical root-finding (and can have up to three real roots below the critical temperature). This calculator solves directly for pressure given a known molar volume, the well-defined direction of the equation.

The SI units used by this calculator are Pa·m⁶/mol² for 'a' and m³/mol for 'b', though older references sometimes quote 'a' in L²·atm/mol² and 'b' in L/mol, requiring unit conversion before use here.

The Van der Waals equation is a significant improvement over the ideal gas law and correctly predicts qualitative behavior like critical points and phase transitions, but more advanced equations of state (like Redlich-Kwong, Peng-Robinson, or virial expansions) give more quantitatively accurate results for real engineering applications.

It was the first equation of state to successfully predict the liquid-gas phase transition and critical point from a simple molecular model, published by Johannes Diderik van der Waals in 1873, earning him the 1910 Nobel Prize in Physics.