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You can only plot reversible processes in PV diagrams. And I suppose a cycle made of reversible processes should be reversible (am I right?). So every cycle plotted in a PV diagram should be reversible. But that appears not to be the case. Where is the flaw in my logic?

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    $\begingroup$ People generally use pragmatic rather than formal rules when drawing PV diagrams, and will plot anything sensible even if it is not necessarily well defined in formal thermodynamics. We could probably give a more helpful answer if you give us an example of some of these exceptions. $\endgroup$ – By Symmetry Jun 3 '18 at 18:06
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You can plot $P_{ext}$ vs V on a PV diagram, and that would give you the correct answer for the work in both irreversible and reversible processes. And $P_{ext}$ is equal to the force per unit area exerted by the gas on the inside piston face for both. So, if you can specify $P_{ext}$ in advance, you are in business. But, for an irreversible process, you can't use the ideal gas law to calculate the force per unit area that the gas exerts on the piston.

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