Van de Waals Equation of State
The y axis shows the pressure P in pascal, and the x-axis shows the volume V_m of one mole of a substance in cubic metres. Three curves are shown.

• The blue curve shows the ideal gas equation: P=RT/V_m
• The red curve shows the effect of the finite volume of the molecules: P=RT/(V_m-b)
• The purple curve shows the Van der Waals equation. It incorporates the effects of both the finite volume of the molecules, and the interaction between the molecules P=RT/(V_m-b)-a/V_m².

The parameters of the curve shown do not correspond closely with any real gas, but are similar to those for argon. The curves above correspond to a temperature of around 50 K. Notice that:

• All the curves are qualitatively similar: the pressure rises as the molar volume is reduced.
• The region of the curves I have shown corresponds to high pressures and small molar volumes.
• For the ideal gas the molar volume can be reduced without limit. This is clearly unphysical. For both the red and purple (VderW) curves there is an irreducible value (given by the parameter b) below which the volume cannot be reduced. This is marked on the graph as a green vertical line. This molar volume corresponds roughly with molar volume of the solid substance.
• For the red curve (which incorporates the effects of molecular volume but not molecular interaction) the pressure is always greater than it would be for an ideal gas. This difference grows much larger as the molar volume is reduced towards b.
• For the VderW curve shown, the attractive molecular interactions always reduce the pressure as compared with the red curve which incorporates only the effects of finite molecular volume.

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The animation below shows the effect of varying the temperature of the substance from around 15 K up to 100 K. This spans the region in which this argon-like substance would be expected to condense. Notice that:

• At high temperatures are the curves are qualitatively and quantititively similar.
• At low temperatures, the Van der Waals curve develops a "kink". The kink separates two regions. At large molar volumes the curve is quite gas like (flat: easily compressible). However at small molar volumes, the curve rises extemely rapidly: indeed, it becomes nearly vertical on the scale shown. This is similar to the behaviour of a solid (highly imcompressible).
• The ability to qualitatively model both solid-like and gas-like regions using such a simple equation of state makes the Van der Waals Equation useful and interesting. Indeed, by modifying the curve yet further, it is possible to use it to model liquids!

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The Graphing Calculator Equations for the above equations are shown below