How this instrument works
Johannes Diderik van der Waals proposed this equation in his 1873 doctoral thesis, patching two flaws in the ideal gas law PV = nRT: real molecules occupy volume, and real molecules attract each other. The correction (V − nb) replaces the container's volume with the volume actually available for molecules to move through, since the n moles carry molecules that together take up a share of the space equal to nb. The term an²⁄V² is added back onto pressure because molecules near the wall are pulled inward by their neighbors, striking that wall slightly softer than an ideal gas would, so the bare formula understates true pressure until this term restores it.
Both constants are measured, not derived from first principles — a in pascal-cubic-metres-squared per mole squared, b in cubic metres per mole — and every real gas carries its own pair, usually tabulated from critical temperature and pressure or from direct pressure-volume-temperature data. A gas with strong intermolecular attraction, like ammonia or water vapour, carries a large a; a gas built from bulky molecules, like propane, carries a large b even when its attraction is modest. Nitrogen, used in the worked example below, has both traits at modest values — a = 0.1382 Pa·m⁶/mol² and b = 3.186 × 10⁻⁵ m³/mol — small enough that at ordinary pressures nitrogen behaves close to an ideal gas, which is exactly what the STP convention was built to demonstrate.
The equation is a genuine improvement over PV = nRT, not an exact one. Near a gas's critical point, or once the entered volume approaches nb, the algebra can misbehave and return a pressure that falls as volume shrinks — impossible for a stable gas — which is why refrigeration and process engineers switch to a cubic equation of state such as Redlich–Kwong or Peng–Robinson once conditions get close to liquefaction. Well away from that region, at the pressures and temperatures most lab and industrial work actually happens in, van der Waals typically tracks real gas behavior to within a fraction of a percent, which is why it has stayed in use for a century and a half.
- Enter Amount of gas in moles — 1 mol for a single mole of any gas, more for a larger sample.
- Set Volume, in cubic metres or litres, to the space the gas is actually confined to.
- Enter Temperature in Celsius; the instrument adds 273.15 internally before it touches the formula.
- Look up Van der Waals constant a and Van der Waals constant b for your specific gas and enter both — every real gas has its own pair.
- Read Pressure, corrected for the molecules' own size and their pull on each other, beyond what the ideal gas law assumes.
Worked example — 1 mole of nitrogen at STP
Take Amount of gas = 1 mol, Volume = 0.0224 m³ (22.4 litres, the classic textbook STP molar volume), Temperature = 0 °C, Van der Waals constant a = 0.1382 Pa·m⁶/mol², and Van der Waals constant b = 3.186 × 10⁻⁵ m³/mol — the standard pair for nitrogen. Feed those five numbers through P = nRT⁄(V − nb) − an²⁄V² and Pressure reads 101,257 Pa, or 101.257 kPa.
Run just the excluded-volume correction — nRT ⁄ (V − nb) with these same numbers — and pressure alone would climb to about 101,533 Pa: crowding the molecules' own bulk out of the available space forces more frequent, harder collisions with the container wall. The attraction term, an²⁄V² ≈ 275 Pa, then pulls that back down, landing the final reading at 101,257 Pa — about 68 Pa, or 0.07 percent, below 101,325 Pa, the pressure the classic STP definition assigns to an ideal gas under the same conditions. For nitrogen at this density, the pull between molecules outweighs the room they take up, so the real-gas answer lands a hair low, not high.
Questions
What do the a and b constants actually represent?
a measures how strongly molecules of that gas attract each other — larger for polar or heavier molecules such as water or carbon dioxide — and carries units of Pa·m⁶/mol². b measures the volume genuinely occupied by one mole of the gas's own molecules, in m³/mol; it is why (V − nb), not V, is what the formula treats as the space truly available for the gas to move through.
Why does this example come out lower than the ideal gas law would give?
For nitrogen at STP density, the attraction correction, an²⁄V² ≈ 275 Pa, outweighs the extra pressure the excluded-volume correction adds on its own — a rise to only about 101,533 Pa. Subtract the larger term and the net result, 101,257 Pa, lands a little below the classic STP pressure of 101,325 Pa: molecules pulling on their neighbors wins out over molecules taking up room, for this gas at this density.
Where do I find a and b for a gas other than nitrogen?
Standard references tabulate them gas by gas — the NIST Chemistry WebBook and the CRC Handbook of Chemistry and Physics both list Van der Waals constants alongside critical temperature Tc and critical pressure Pc, from which a and b can also be derived directly: a = 27R²Tc²⁄(64Pc) and b = RTc⁄(8Pc).
Does this equation stay accurate at any temperature or pressure?
No. It is a clear improvement over the ideal gas law at moderate density, but near a gas's critical point, or once volume approaches nb, its own algebra can return a pressure that falls as volume shrinks — physically impossible for a stable gas. Process engineers switch to a cubic equation of state such as Redlich–Kwong or Peng–Robinson once conditions near liquefaction.
Why does the Temperature field take Celsius when the formula wants kelvin?
Celsius is the scale most lab thermometers and gas datasheets actually report, so that is what the Temperature field accepts. Internally the instrument adds 273.15 before multiplying by R, exactly as the formula requires, so the conversion never has to be done by hand.
What happens if Volume is smaller than moles times constant b?
The (V − nb) term goes to zero or negative and the formula returns an undefined or unphysical pressure. That is the equation's way of flagging that the entered volume is smaller than the space the gas's own molecules would physically occupy — not an achievable state, and a sign to recheck the Volume or Amount of gas entries.