Solve (P + an²/V²)(V − nb) = nRT for a real gas. Pick a gas, enter any three values, and the fourth fills in with full working.
Computed from the other three values.
Pressure
P = 5.65 atm
P = nRT/(V − nb) − an²/V² = (2 × 0.08205736 × 350) / (10 − 0.0854) − (3.59 × 4) / 100 = 5.65 atm = 5.65 atm
The ideal gas law, PV = nRT, assumes gas molecules are point particles that never attract or repel one another. Real molecules do both: they attract each other weakly at a distance and physically occupy space, so no real gas can be compressed to zero volume. Johannes Diderik van der Waals introduced two correction terms in 1873 to account for exactly this, earning the 1910 Nobel Prize in Physics for the work.
Definition
The Van der Waals equation, (P + an²/V²)(V − nb) = nRT, corrects the ideal gas law for intermolecular attraction (the a term) and the finite volume of gas molecules themselves (the b covolume).
When a and b are both set to zero, the equation collapses exactly back to PV = nRT — the Van der Waals equation is a strict generalisation, not a replacement.
Every gas has its own a and b, measured experimentally. Larger, more polar molecules like water vapour and ammonia have a bigger a because they attract each other more strongly.
| Gas | a (L²·atm/mol²) | b (L/mol) |
|---|---|---|
| Helium (He) | 0.0346 | 0.0238 |
| Hydrogen (H₂) | 0.244 | 0.0266 |
| Nitrogen (N₂) | 1.39 | 0.0391 |
| Oxygen (O₂) | 1.36 | 0.0318 |
| Methane (CH₄) | 2.25 | 0.0428 |
| Ammonia (NH₃) | 4.17 | 0.0371 |
| Carbon dioxide (CO₂) | 3.59 | 0.0427 |
| Water vapour (H₂O) | 5.536 | 0.0305 |
Example 1
A cylinder holds n = 2 mol of carbon dioxide in V = 10 L at T = 350 K. Find the pressure, and compare it with what the ideal gas law alone would predict.
CO₂'s attractive term pulls the real pressure about 1.6% below the ideal prediction at this density — small, but measurable.
Example 2
n = 1.5 mol of nitrogen is compressed to P = 50 atm at T = 280 K. What volume does it occupy?
At 50 atm the gap widens to roughly 4% — high pressure is exactly where this equation earns its keep over the plain ideal gas law.
For everyday air at room temperature and roughly atmospheric pressure, the ideal gas law calculator and even the Charles law calculator give answers close enough for practical purposes — this is the same approximation discussed in the limitations of Charles' law. Reach for the Van der Waals equation instead when pressure is high, temperature is low, or the gas is close to condensing, since that's precisely where molecular attraction and finite molecular size stop being negligible.
Converting your inputs first can help sanity-check the result: use the temperature converter to get T into Kelvin, and the volume converter if your volume wasn't already in litres or cubic metres.
It is a correction to the ideal gas law that accounts for two things real gas molecules do that ideal ones don’t: attract each other (the a n²/V² term) and take up physical space (the b covolume term). The full equation is (P + a n²/V²)(V − nb) = nRT.
When pressure is high or temperature is low enough that a gas is far from its ideal behaviour — near a gas’s condensation point, or at pressures well above atmospheric. At everyday temperature and pressure, the ideal gas law is usually close enough.
a measures the strength of intermolecular attraction (larger for gases like water vapour and ammonia that hydrogen-bond); b measures the volume the gas molecules themselves occupy (the "covolume"), which sets a floor below which the gas cannot be compressed further.
The Van der Waals constants a and b are conventionally tabulated in litre-atmosphere units per mole, so the amount of gas is entered directly in moles to keep the arithmetic transparent — no hidden unit juggling.
That means the volume you entered is smaller than n×b, the space the gas molecules themselves would occupy — a physically impossible squeeze. Try a larger volume or a smaller amount of gas.
For a dilute gas at room temperature, the difference is usually under 2%. At high pressure or near condensation, it can be 10% or more — which is exactly the regime where the ideal gas law breaks down and this equation is needed.
The Van der Waals equation is the real-gas correction to the six idealised laws below.