Van der Waals Equation
The ideal gas law, PV = nRT, is a powerful tool for describing gas behaviour, but it rests on two assumptions that break down under real-world conditions: gas molecules have zero volume, and there are no intermolecular forces between them. At high pressures and low temperatures these assumptions fail badly — molecules are squeezed together so their own volume matters, and attractive forces between them become significant. Real gases therefore deviate from ideal behaviour, and we need a more accurate equation to describe them. In 1873, Dutch physicist Johannes Diderik van der Waals proposed a modified equation of state that accounts for both deficiencies. He introduced two correction terms into the ideal gas law: a pressure correction that accounts for intermolecular attractions (using constant a), and a volume correction that accounts for the finite size of molecules (using constant b). The resulting van der Waals equation fits experimental data for real gases far better than the ideal gas law across a wide range of temperatures and pressures. Understanding the van der Waals equation gives chemists and engineers insight into why gases liquefy, how refrigerants behave, why noble gases have such different boiling points despite similar structures, and how to design industrial processes involving high-pressure gases such as the Haber process for ammonia synthesis. The constants a and b are fingerprints of a gas's molecular identity — larger, more polarisable molecules have higher values of a, while physically bulkier molecules have higher values of b.
- Starting Point — The Ideal Gas Law
- Volume Correction (the b term)
- Pressure Correction (the a term)
- Significance of Constants a and b
- Limitations of the Van der Waals Equation
Starting Point — The Ideal Gas Law
The ideal gas law is PV = nRT, where P is pressure (Pa), V is volume (m³), n is moles, R = 8.314 J mol⁻¹ K⁻¹, and T is temperature (K). This law treats gas molecules as point masses with no size and no forces between them.
Volume Correction (the b term)
Real molecules occupy space. The volume actually available for molecules to move around in is less than the container volume V, because each molecule excludes neighbouring molecules from its own volume. Van der Waals called this excluded volume nb, where b is the excluded volume per mole (units: L mol⁻¹ or m³ mol⁻¹). The correction replaces V with (V − nb): the 'free' volume is reduced. For example, b for H₂ is 0.0266 L mol⁻¹, while for larger CCl₄ it is 0.1383 L mol⁻¹, reflecting the much larger size of the carbon tetrachloride molecule.
Pressure Correction (the a term)
Intermolecular attractive forces (London dispersion, dipole–dipole, hydrogen bonding) pull molecules toward each other. A molecule about to strike the container wall is pulled slightly backward by its neighbours, so it hits the wall with less force than it would in the absence of attractions. The measured pressure P is therefore lower than the ideal pressure. Van der Waals showed the pressure deficit is proportional to n²/V² (because it depends on the frequency of molecular encounters, which scales with concentration squared). Adding the correction term an²/V² restores the ideal pressure: P_ideal = P + an²/V². Substituting into the ideal gas law gives the van der Waals equation. The constant a (units: L² atm mol⁻² or Pa m⁶ mol⁻²) is larger for gases with stronger intermolecular forces. For He, a = 0.034 L² atm mol⁻²; for water vapour, a = 5.536 L² atm mol⁻², reflecting strong hydrogen bonding.
Significance of Constants a and b
The constant a is a measure of the strength of intermolecular attractions. A high a means the gas is easier to liquefy (attractions are strong), so it has a higher boiling point. The constant b is a measure of molecular size — it equals approximately four times the actual molar volume of the molecules themselves (because two spherical molecules exclude a sphere of volume 4πr³/3 centred on each). Together, a/b gives an approximate measure of the ratio of intermolecular forces to molecular volume, which correlates with critical temperature: T_c = 8a/(27Rb).
Limitations of the Van der Waals Equation
Despite its improvements over the ideal gas law, the van der Waals equation is still an approximation. It predicts three mathematical roots for volume at certain temperatures and pressures (the S-shaped portion of the isotherm), which is unphysical but is actually related to the liquid–vapour phase transition region (Maxwell construction). More sophisticated equations of state (Redlich–Kwong, Peng–Robinson) are used in modern engineering, but van der Waals's insight — that real gas behaviour arises from molecular size and intermolecular forces — remains the conceptual foundation of all of them.
The Key Equation
- P — Measured pressure (atm or Pa): The actual pressure exerted by the real gas on the container walls.
- a — Van der Waals constant a (L² atm mol⁻²): A measure of the strength of intermolecular attractions; larger for polar or polarisable molecules.
- n — Amount of gas (mol): Number of moles of gas molecules present.
- V — Volume of container (L or m³): The total volume of the container holding the gas.
- b — Van der Waals constant b (L mol⁻¹): Excluded molar volume; approximately four times the actual volume of one mole of molecules.
- R — Universal gas constant (J mol⁻¹ K⁻¹ or L atm mol⁻¹ K⁻¹): 8.314 J mol⁻¹ K⁻¹ or 0.08206 L atm mol⁻¹ K⁻¹.
- T — Absolute temperature (K): Temperature measured on the Kelvin scale.
- an²/V² — Pressure correction term (atm or Pa): Accounts for the reduction in pressure caused by intermolecular attractive forces.
- nb — Volume correction term (L or m³): Accounts for the volume physically occupied by the gas molecules themselves.
Worked Example
Problem. Calculate the pressure exerted by 2.00 mol of CO₂ gas in a 1.00 L container at 400 K using (a) the ideal gas law and (b) the van der Waals equation. For CO₂: a = 3.592 L² atm mol⁻², b = 0.04267 L mol⁻¹. Use R = 0.08206 L atm mol⁻¹ K⁻¹.
Answer. P (ideal) ≈ 65.65 atm; P (van der Waals) ≈ 57.41 atm
Origins · 1 minJohannes Diderik van der Waals and how this was discovered
Van der Waals proposed his famous equation of state in his 1873 doctoral thesis 'On the Continuity of the Gas and Liquid State,' recognising that real gas molecules have finite volume and experience attractive forces. His equation was the first to predict the existence of a critical point and to provide a unified description of gases and liquids. He received the Nobel Prize in Physics in 1910 for this work, which also laid the foundation for understanding the forces between molecules now known as 'van der Waals forces.' Van der Waals was largely self-taught and did not receive a formal secondary education in science; he worked as a schoolteacher and passed university entrance examinations independently as an adult. He began his physics doctorate at age 35 — proof that great scientific insights can come at any stage of life.
Related Concepts
- The Ideal Gas Law (PV = nRT) — The van der Waals equation is a direct modification of the ideal gas law; understanding PV = nRT is essential before learning how and why corrections are applied.
- Kinetic Molecular Theory of Gases — The assumptions of kinetic molecular theory (zero volume, no forces) are exactly what the van der Waals correction terms relax; understanding KMT clarifies why real gases deviate.
- Intermolecular Forces (London Dispersion, Dipole–Dipole, Hydrogen Bonding) — The constant a in the van der Waals equation directly measures the strength of intermolecular attractions; knowing the types of IMFs explains why a differs between gases.
- Liquefaction of Gases and Critical Points — The van der Waals equation predicts the critical temperature T_c = 8a/27Rb and is used to explain the conditions under which gases can be liquefied.
- Compressibility Factor (Z) — The compressibility factor Z = PV/nRT quantifies deviation from ideal behaviour; van der Waals gases have Z ≠ 1, and Z vs P plots are a key way to visualise when corrections matter.