When studying the equation of state for an ideal gas, it's possible to deviate from it and introduce corrections both to the pressure and to the volume. In doing so, we can obtain the Van der Waals equation of state: $$ \left(P+\dfrac{a}{v^2}\right)(v-b)=T $$ where $k_B$ has been set equal to 1 and $v=V/N$ is the specific volume.

Both the parameters $a$ and $b$ have some physical meaning: namely, $a$ is a measure of the attractive forces among the molecules of the system whereas $b$ measures the "effective space" occupied by the molecules. I also know it's possible to obtain the same equation of state from a mean-field theory of condensation, and again it's possible to interpret physically the parameters $a$ and $b$ introducing for example the excluded volume and characterizing in an adequate way the Lennard-Jones potential.

Nevertheless, if we want to proceed further and carrying on the virial expansion up to third or fourth order we would obtain more and more corrections to the Van der Waals equation.

What will be the physical meaning of this? In other words, when and why are these further corrections needed and what do they represent? In what theory do they appear and are they connected to physical properties of the system?

  • $\begingroup$ Could you elaborate a bit more about the process to derive the Van der Waals equation (e.g. which is the Hamiltonian)? Otherwise, could you provide some references? I think that this would make your question more accessible to a wider audience that is not used to these concepts. $\endgroup$
    – Javi
    Commented Oct 21, 2022 at 10:21
  • $\begingroup$ If you are moving from an ideal gas law into real gas equation, at first, you'll need to decide which real gas law would be most suitable to you,- of which are many. $\endgroup$ Commented Oct 21, 2022 at 11:52
  • $\begingroup$ @AgniusVasiliauskas Of course, I was focusing on the Van der Waals model since from an instructional point of view it is the most used and known. $\endgroup$
    – Gianluca
    Commented Oct 21, 2022 at 14:34

1 Answer 1


The way physics often works, is that you observe experimentally some phenomenon, and ask yourself what it is the simplest possible model that could "explain" this phenomenon (i.e. demonstrate similar behavior).

So van der Waals equation was precisely that. The phenomenon he was after was the gas-liquid phase transition. And he has discovered that already a model of a gas with attracting hard-core particles is sufficient to describe the liquid-gas phase transitions.

You can add more terms, and that will definitely introduce changes. For instance, it will modify the shape of the phase diagram, and you might exploit this to make it (the phase diagram) look more like that of the compound you are interested in. However I doubt it will produce qualitative changes.

One interesting question (in my opinion) is whether you can change vdW equation such that it will introduce more phases. For instance, whether you can add the solid-liquid and solid-gas phase transition into the phase digram by adding new terms? I am inclined to answer this in negative (hence the last sentence of the previous paragraph). The reason is that unlike liquid and gas, solid has a different symmetry. Which means the the liquid-solid phase transition involves a qualitative reconstruction of the ground state. I don't think a single equation of state can describe such phase diagram (this can be seen already from the fact that in contrast to liquid/gas, pressure and density alone are not enough for the equation of state for a solid).


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