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In college, I studied that for an ideal gas $PV = nRT$ , that is, at constant temperature and constant number of moles, $P \propto \frac {1}{V}$. Thus, graph between $P$ and $V$ is a rectangular hyperbola when $dT =0$ and $dn=0$. But in my textbook there is this topic called 'Andrews' isotherms' wherein they have given some $P$ vs $V$ curves for gases and at low temperatures and low pressures the graphs seem to be divided into three fragments. How is this possible? Shouldn't the curve be a rectangular hyperbola? How is it possible? Shouldn't ideal gases obey $PV = constant $ at all temperatures and pressure. That is literally the definition of ideal gas : A gas that obeys ideal gas equation at all temperatures and pressure. I also studied that real gases show deviation from ideal gases. But when we draw $P$ vs $V$ curves of a real gas, they deviate from the $P$ vs $V$ curves of ideal gases slightly at high pressures and low temperatures. They almost remain rectagular hyperbolas. But the $P$ vs $V$ curves of real gases don't literally break into three fragements like Andrews' isotherms. The curves in Andrews' isotherms don't even come close to rectangular hyperbolas. What type of curve is being shown in Andrews' isotherms? How are these curves disobeying Boyle's law to such a great extent, not even coming close to a rectangular hyperbola?

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    $\begingroup$ Is the figure in this Physics SE question what you mean? If so, the answer is that these fragmented curves are cases where the the substance is transitioning from a liquid to a gas. $\endgroup$
    – Tyberius
    Apr 16, 2021 at 15:49

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It's been a long time since I studied this. But Andrews' isotherms are not referring to an ideal gas the curves a real system that, at its critical point, the homogeneous substance is exactly balanced with both the liquid and gas form are at the same temperature, pressure and density.

You also might want to read Boltzmann's explanation of the Joule-Thomson effect along with Gibbs' and Maxell's explanation of the {U, S, V} minima where Gibbs developed his famous Gibbs free energy which was the missing potential function in real thermodynamic systems, not ideal analysis of those system. The U,S,V curve is much more important than analyzing Andrews' isotherm by the ideal gas law.

In fact, Maxwell, the most famous physicist in the world realized Gibbs was correct that the potential function minima that determines the direction of a spontaneous reaction is Gibbs free energy, but it was already after Maxwell published his own book erroneously stating that it must be a minimum in the potential function of internal energy, U that determines the spontaneous direction of a reaction.

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