Van Der Waals Calculator

What the van der Waals calculator does

Real gases do not quite obey the ideal gas law, and the van der Waals equation fixes that with two corrections. This calculator applies it, working with the pressure, volume, temperature, amount of gas, and the two van der Waals constants a and b. Fill in any five and it solves for the sixth.

Below is why real gases need this equation, what each correction does, and a worked example.

How to use it

1. Enter five of the six quantities: pressure, volume, the constant a, the constant b, the number of moles, and temperature.
2. Leave exactly one field empty, the one you want the calculator to find.
3. Press Calculate for the missing quantity, or Reset to clear it.

Why real gases need a better equation

The ideal gas law rests on two convenient fictions: that gas molecules take up no space, and that they exert no forces on one another. These assumptions work beautifully when a gas is dilute and warm, with its molecules far apart and moving fast. But push a gas to high pressure or cool it toward becoming a liquid, and both fictions break down. The molecules are now crowded close enough that their own size matters and their mutual attractions pull on them noticeably.

The van der Waals equation, proposed by Johannes van der Waals in 1873, keeps the simple form of the ideal gas law but adds a correction for each of these effects. The result fits the real behaviour of gases far better, especially in the conditions where the ideal law goes astray, and it earned van der Waals a Nobel Prize. What makes it so instructive is that each correction has a clear physical meaning, so the equation not only fits the data but explains why real gases differ from ideal ones.

The equation it uses

The van der Waals equation takes the ideal gas law and modifies both the pressure and the volume:

( P + a n² ÷ V² ) × ( V − n b ) = n R T

Here P, V, n, and T are the pressure, volume, amount, and temperature as before, R is the gas constant, and a and b are the two correction constants, each measured for the particular gas. The term added to the pressure accounts for molecular attractions, and the term subtracted from the volume accounts for the space the molecules themselves occupy. Set both a and b to zero and the whole thing collapses neatly back into the familiar PV = nRT, showing that the ideal gas law is just the van der Waals equation with the corrections switched off. The calculator solves this equation for whichever of the six quantities you leave blank, including the cases where it has to untangle a cubic relationship for volume or amount.

The attraction term, a

The constant a measures how strongly the gas molecules attract one another. Real molecules feel a faint pull toward their neighbours, and that pull slightly holds the gas back, so a real gas pushes on its container a little less hard than an ideal one would. The correction adds a term to the pressure to account for this softening, restoring what the attractions take away.

Gases whose molecules attract strongly, like carbon dioxide or water vapour, have larger values of a, while gases of small, weakly interacting molecules, like helium, have very small values. The size of a is, in effect, a measure of how sticky the gas is. It is largest for gases that are easy to liquefy, since the same attractions that pull molecules together in the gas are what let them condense into a liquid.

The volume term, b

The constant b accounts for the fact that molecules are not points but take up room of their own. In the ideal picture, the whole volume of the container is available for molecules to move through. In reality, the molecules themselves occupy some of that space, leaving a little less room than the container's full volume. The correction subtracts this excluded volume, so the equation uses the space actually available rather than the total.

Larger molecules have larger values of b, since they take up more room, while small molecules have smaller values. This correction becomes important at high pressure, where the gas is squeezed so tightly that the molecules' own bulk is no longer negligible against the shrinking space. Together with the attraction term, it lets the van der Waals equation track a real gas through conditions where the ideal law would fail, and the calculator carries both corrections automatically.

Units and precision

The calculator works in SI units: pressure in pascals, volume in cubic metres, amount in moles, temperature in kelvin, the constant a in joule-cubic-metres per mole squared, and the constant b in cubic metres per mole. The gas constant is built in. Values of a and b for common gases are tabulated in physics and chemistry references, and you enter the ones for your gas. The equation is solved exactly for the quantity you leave blank. Results are shown to four decimal places.

A worked example

Take 1 mole of carbon dioxide in a volume of 0.001 cubic metres (1 litre) at 300 kelvin, using carbon dioxide's constants, a of about 0.364 and b of about 0.0000427.

The van der Waals equation gives a pressure of about 2.24 megapascals. The ideal gas law, for the same conditions, would predict about 2.49 megapascals. The real gas pressure is lower because the attractions between carbon dioxide molecules, captured by the term a, hold the gas back and soften its push on the walls. The gap between the two figures is exactly the kind of real-gas deviation the van der Waals equation is built to capture.

Questions people ask

What is the van der Waals equation?

It is (P + an²/V²)(V − nb) = nRT, a correction to the ideal gas law that adds a term for molecular attraction (a) and subtracts one for molecular volume (b).

Why is it better than the ideal gas law?

Because it accounts for two things the ideal law ignores: that molecules attract one another and that they take up space. This makes it far more accurate at high pressures and low temperatures.

What do the constants a and b represent?

The constant a measures the strength of attraction between molecules, larger for stickier gases. The constant b measures the volume the molecules themselves occupy, larger for bigger molecules.

When does it reduce to the ideal gas law?

When both a and b are set to zero, the equation becomes PV = nRT exactly. This happens in effect for dilute, warm gases, where the corrections are negligible.

References

A quick note on where the physics comes from. The van der Waals equation and the meaning of its two constants are standard thermodynamics, set out in OpenStax's Chemistry, in the Chemistry LibreTexts, and in Georgia State University's HyperPhysics. The SI units follow the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, Chemistry 2e, Section 9.6, Non-Ideal Gas Behavior. https://openstax.org/books/chemistry-2e/pages/9-6-non-ideal-gas-behavior
2. Chemistry LibreTexts, Van der Waals' Equation. chem.libretexts.org, Van der Waals' Equation
3. HyperPhysics, Van der Waals Equation of State. http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/waal.html