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I am starting to learn thermodynamics to get a better understanding of the equations of state for CO$_2$. According to wikipedia there is an inflection point at the critical point for a critical isotherm on a $PV$ diagram: $$ \left( \frac{\partial p}{\partial V} \right)_T = 0\\ \left( \frac{\partial^2 p}{\partial V^2} \right)_T = 0 $$

If I reduce the volume at a critical point at constant temperature, why doesn't the pressure increase? To me it seem logical that the pressure should always increase when I reduce the volume.

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    $\begingroup$ It's because the ideal gas model doesn't work in all cases. It's mostly a good approximation but in some circumstances (like this one, or when you're trying to liquidify a real gas) some other models are used, for instance the Van der Waals gas model. If you look at the PV characteristic below, you can see where the inflection point arises. $\endgroup$ – Soba noodles Sep 6 '16 at 9:57
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    $\begingroup$ The ideal gas law works on "ideal" gases, which do not condense. For CO2 near its critical point, a decrease in volume causes some of the gas to condense into liquid, and this change partially or totally offsets the expected pressure increase. $\endgroup$ – David White Sep 7 '16 at 1:44
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I think you are right when substance is in pure state. When there is a phase change, e.g. from vapor to liquid, the temperature and pressure can be held constant but the volume decreases, i.e. from vapor to liquid.

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