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Van der Waals P-V graph

  1. Nov 12, 2012 #1
    (the graph of Maxwell equal area rule)

    When the temperature is under the critical temperature, then at certain pressure, there will be three possible point for the volume of the system.
    What does that mean? Do they mean the volume of the system will change between that 3 volume?
    From the above link, they say they will be phase for liquid and gas, what do they mean?
    Thank you.
  2. jcsd
  3. Nov 12, 2012 #2

    Philip Wood

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    Gold Member

    It's usual (but not mandatory) to think of volume as the independent variable. Suppose we have a real fluid at a given temperature in a cylinder fitted with a piston. We then slowly pull out the piston and plot the isothermal on a p – V diagram as the pressure changes. If we start with a real liquid below the critical temperature, we find a portion of the graph where the pressure does't change, even though we're increasing the volume. This is when the liquid is evaporating, so the cylinder has both liquid and vapour in it. Then, at a large enough cylinder volume, all the liquid has evaporated and we're left with vapour - whose pressure falls when we increase the volume further.

    The V der W equation results in a smooth graph with no flat portion. There is no hint of liquid and vapour being present together. Indeed, there is no real hint of distinguishable liquid and vapour states at all. But, as you say, below the critical temperature there are 3 different volumes on the same isothermal at which the substance has the same pressure (and temperature). There's nothing to stop the substance co-existing in the two states furthest apart, as they'd be in mechanical and thermal equilibrium. [We leave out the middle one because it's on the unstable bit of graph.] The equation doesn't tell us that this co-existence will happen, but it doesn't rule it out either. Thus we can magic a flat portion of graph by constructing it from a variable proportion mixture of the fluid in the two extreme equal volume states, rather than from the peak-and-trough bit of the V der W curve.

    I believe that there's never been consensus on whether this approach is valid.
    Last edited: Nov 13, 2012
  4. Nov 12, 2012 #3
    The van der Waals equation does not describe the behavior of any real material. It is only a rough approximation to the behavior of real materials, which, in the limit of large volume per mole, approaches the ideal gas equation. So don't be too surprised if there are three volumes at a given pressure and temperature. The approach described by Philip Wood improves the aesthetic comparison between the van deer Waals equation and the vapor-liquid phase transition of real gases, and it may in some cases also improve the quantitative comparison.
  5. Nov 15, 2012 #4
    Seem like everything in thermodynamics are approximation and prediction.
    Thank you guys for answering.
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