When we draw P-V diagram of a Van der Waals gas, we get loops and since in the centre region of the loops, $\frac{dP}{dV}$ is +ve. Based on these regions we argue that the gas is unstable in that region as suppressing the volume of the gas is not resisted by the gas and it happily decreases its pressure. (Taken from lectures by Prof. David Tong).

However my question is that how can we argue on the basis of Van der Waals potential that the region is an unstable region? Isn't it possible that the said region is just a manifestation of the approximate Partition Function and has nothing to do with real gases ?




1 Answer 1


I am not sure if this will answer your question, but I wonder about a couple of things that might be useful.

First is the equivalent diagram for a real gas.


image from https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Real_Gas_Isotherms.svg/600px-Real_Gas_Isotherms.svg.png Note for a real gas there are horizontal lines where a mixture of gas and liquid exists in equilibrium together. On one of the horizontal lines if the pressure is reduced more gas (or vapour) is formed above the liquid - if the pressure is increased then more liquid is formed from the gas. When all the gas is converted to liquid the pressure rises steeply as the volume is decreased. When all the liquid has vapourized to form the pressure starts to drop as the volume is increased further.

Now the key thing is that in reality the Van Der Waals gas forms a liquid on compression if it is below the critical temperature.

Now I think the most relevant thing to your question now is the following.

  • the gas equations for the Van Der Waals gas do not allow the gas to form a liquid in the way a real gas behaves, but the term $(p + a/Vm^2)$ does mean that the pressure is reduced due intermolecular attractive forces - thus the smaller the volume the closer the gas molecules are together and the lower the pressure that they exert.

  • The attractive forces between molecules mean that $dP/dV$ can be $+ve$ - the physical interpretation is that the attractive forces between the molecules pull the molecules together... this is a bit like a real gas turning into a liquid.

  • The Van der Waals intermolecular potential has a minimum energy at some molecular separation $r$ - for separations greater than $r$ the potential rises... It is simple to argue that there is attraction between the molecules for separations greater than $r$.

Thus, in conclusion the Van der Waals potential tells us that molecules attract each other - and below the critical temperature this can lead to the 'unstable' behaviour you describe in your question.

  • $\begingroup$ On any sub-critical isothermal the substance in state F could be in equilibrium with the substance in state G (same temperature and pressure). So the substance could proceed from F (liquid?) to G (vapour?) by the ratio of state G to state F increasing, rather than proceeding via the unstable region. I've never known how realistic this is! $\endgroup$ Commented Mar 9, 2018 at 14:22
  • $\begingroup$ @PhilipWood - passing along the horizontal line rather than the unstable drop is realistic - I have done/seen people do lab experiments where the horizontal line is observed and you can see the gas turn to liquid... the experiment was with SF6 which has a convenient Tc close to 40 C if I remember correctly $\endgroup$
    – tom
    Commented Mar 9, 2018 at 23:13
  • $\begingroup$ Thank you. I know that this is what real substances do; I'm just a little uneasy about whether we can really predict it from the V der W equation. $\endgroup$ Commented Mar 9, 2018 at 23:21
  • $\begingroup$ @PhilipWood - we cant predict the horizontal line with VdW alone - we would need to include the physics of the liquid vapour equilibrium to get that. Instead we get the graph in the question, but the unstable collapse we do see from VdW eq. hints at the liquefication. But maybe im not understanding your point.... apologies if so - you are thinking deeply about this. $\endgroup$
    – tom
    Commented Mar 9, 2018 at 23:29
  • $\begingroup$ Just to add, horizontal line can be gotten by the Maxwell equal area correction for the V der W equation. $\endgroup$ Commented Mar 10, 2018 at 12:08

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