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A thermodynamic transformation that has a path (in its state space) that lies on the surface of its equation of state (e.g., $PV=NkT$) is always reversible (right?). However, if the path is a continuous quasistatic path in state space but not on the surface of equation of state, it is considered irreversible.

Why is this so? In this case the gas still has a uniform thermodynamic variables (because all the intermediate states are points in the state space). Why can it not be reversed exactly along the same path?

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"However, if the path is a continuous quasistatic path in state space but not on the surface of equation of state, it is considered irreversible." How would you make the system to go into state where $P,V,T$ are defined but do not obey the equation of state? –  Ján Lalinský Feb 2 at 23:16

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Let's look at your first statement:

A thermodynamic transformation that has a path (in its state space) that lies on the surface of its equation of state (e.g., $PV=NkT$) is always reversible

I don't think this is right, but there may be some implicit qualifiers in your statement that I'm missing.

Here's an example to illustrate why. Consider a system consisting of a thermally insulated container with a partition in the middle dividing it into compartments $A$ and $B$. Suppose that we fill each side of the container with an equal number of molecules of a certain monatomic ideal gas. Let the temperature of the gas in compartment $A$ be $T_A$ and the temperature of the gas in compartment $B$ be $T_B$. Suppose further that the partition is designed to very very slowly allow heat transfer between the compartments.

If $T_B>T_A$, then heat will irreversibly transfer from compartment $B$ to compartment $A$ until the gases come into thermal equilibrium. During this whole process, each compartment has a well-defined value for each of its state variables since the whole process was slow ("infinitely" slow in an idealized setup). However, the process was not reversible because heat spontaneously flowed from a hotter body to a colder body. You can also show that the sum of the entropies of the subsystems $A$ and $B$ increased during the process to convince yourself of this.

So we have found a process for which the thermodynamic evolution of each subsystem can be described by a continuous curve in its state space, but the process is irreversible.

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I was going to ask for clarification as to what then are sufficient conditions for reversibility, but then I realized that the relations amongst reversible, isentropic, adiabatic, and equilibrium-maintaining processes might warrant a whole separate question. –  Chris White Sep 24 '13 at 19:38
    
@ChrisWhite Yeah I'd say so especially considering the dizzying array of terminological variation that exists in the thermo literature. –  joshphysics Sep 24 '13 at 20:03
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Hi Josh, I am not sure to agree with you. What is your thermodynamical system? You have here a pair of thermodynamical systems. Each system evolves along a quasi static transformation, but it is reversible if you consider each system separately (it is enough suitably change the corresponding external system). The transformation of the whole system, I agree, is not reversible, but I am not sure that you can consider the whole system as moving through states of equilibrium. Each of these states has no an everywhere definite temperature, as there are two temperatures at each instant. –  Valter Moretti Feb 2 at 19:47
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@pppqqq Yes that's rather vague I agree. I think I originally had in mind some sort of a membrane that can be designed to allow for the heat to flow between two gases as slowly as one wishes so that despite the finite temperature difference, the process is quasi-static. After any finite amount of time, there would be a net flux of heat, but the process would be sufficiently slow that one could still use $\delta Q = T dS$ to determine the entropy change the each subsystem on either side of the membrane. –  joshphysics Feb 2 at 20:23
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@V.Moretti Good points. I agree with everything you're saying. I think I should be more careful about differentiating between systems and subsystems. In particular, I suppose that my claim really is that there exists a system $\Sigma$ and an irreversible process $P$ on $\Sigma$ during which $\Sigma$ is out of equilibrium, yet for which certain of it's subsystems $\Sigma_1$ and $\Sigma_2$ undergo quasistatic processes $P_1$ and $P_2$. Perhaps this is not actually an answer to the original question; I'm not entirely sure now that I look back at it. –  joshphysics Feb 2 at 20:35

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