Reversible and Irreversible Process 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.
 A: Answers to 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)?
There are an infinite number of reversible paths between the two different states.  They don't need to be isothermal and adibatic (or combinations of these), but it is easy to visualize a sequence of isothermal and adibatic steps to get from the intial  end state to the final end state.
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)?
As I said, the reversible steps do not have to be adiabatic or isothermal.  For example, consider polytropic steps.
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.
This is correct if you leave out the part about "corresponding irreversible paths."  You can't associate a particular irreversible path with any particular reversible path, or vice versa.  All the paths between the same two end states (both reversible and irreversible) will have the same $\Delta U$ and $\Delta S$
A: 1) The constant temperature curves of Ideal gas are not closed, so this implies that given two points with same temperature they must be joined by a unique constant temperature path, the only reversible process at constant temperature. But if you want any reversible process joining the two points, then any curve on the surface V=NkT/P that joins them will do! (think the surface as embedded in R^3). It makes no sense to talk about irreversible path, since there is no actual path, I mean no curve in the surface V=NkT/P. A curve/path is by definition a reversible process.
2) May be I am missing your point, but again, as long as you can draw a curve on the surface defined by the state equation, the reversible process exists.
3) I do not see a simple answer to this. I will think about it.
