I would like to ask a specific conceptual question which bothers me for quite some time! First of all i do know the difference in between reversible and irreversible processes. What is thought in Thermodynamics courses is the maximum work can be obtained via reversible processes. Now please consider the following example.

Air (assume as ideal gas) at 5 bar and 298.15K (25℃) is expanded to 1 bar and 298.15K by a mechanically reversible processes: Heating at constant pressure followed by cooling at constant volume.

When one considers the corresponding PV diagram , the work is calculated as the area under the curve which is obviously larger than the reversible isothermal expansion.

Here are my questions

1) How many different reversible paths can be drawn in between two different states at the same temperature (there can be infinite number of irreversible paths)?

2) How can heating an ideal gas at constant pressure and cooling at constant volume be a reversible process (these are not adiabatic or isothermal)?

3) Is it possible to say that : There can be many different reversible paths between two specified states, all of which will give larger work than corresponding irreversible paths but also vary in between themselves so that it is not possible to state which reversible path will give the highest work before specifying the path itself.

Thank you all in advance for your sincere help and answers.

UPDATE: From MIT thermodynamics course notes: the reversible one produces the maximum work of all possible processes between two states.

If so there should not be more than one reversible process between two states. Then how can the process path given in the question be reversible as we can already perform the same change with isothermal expansion.

  • $\begingroup$ I'd keep reading some first-year Statistical Mechanics books, and/or work through the line integral formalism which proves that statement about the reversible process being the one which produces max work. $\endgroup$ Commented Feb 24, 2016 at 13:40

1 Answer 1


1) One if you don't change temperature, infinite if you do.

2) It can't. Adding or removing heat can only be reversible when the temperature is not changing. This necessitates that both pressure and volume are changing.

3) Possible to say, however it would be incorrect. There are indeed an infinite number of reversible paths one could take between two states. (One can alternate between isothermal expansion/compression, and isentropic expansion/compression following any path that those allow). These paths can indeed produce different amounts of work. However, there is no guarantee that these paths will produce more work than an irreversible path.

What you're missing is that a reversible path can produce the most and least work of all paths constrained by a highest and lowest temperature. Usually what people care about is the work produced (or absorbed) by a cycle. This will be the difference in work between two paths. As such reversible paths offer the highest difference, specifically the reversible path that goes to the highest temperature, then isothermally expands/compresses will have the greatest magnitude work of all paths that stay below that maximum temperature, while the reversible path that cools to the lowest temperature and then isothermally expands/compresses will have the least magnitude of work of all paths that stay above that minimum temperature.


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