Our values are:
a) For isothermal reversible expansion, We have that the change in the Internal Energy is 0, that is,
We can solve for the Work made in the system.
With work made it, this same quantity is necessary to add to the system through Heat. Then,
For Surrounding we apply the same relation,
Therefore,
b) For an isothermal irreversible expansion
We consider the external presion as 0, by this way, the work done is also 0.
Since isothermal irreversible expansion change in Internal Energy is 0, so
, since isothermal
Therefore
How the condition are the same, for the System the entropy is the same
But
c) And an adiabatic reversible expansion
Since the process is Adiabatic,
Therefore,