Reversibility and isothermal processes

Question

I would like to know why any reversible process has to be isothermal, according to my sources.

Why can´t we consider adiabatic processes to be reversible?

EDITED

https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter5.html (In "Mechanical and thermal reversibility" section)

For summing up, this text says that any reversible process has to be constantly in thermodynamical equilibrium and frictionless (No problem until this point). But later on, it claims that has to be isothermal too, what is a more restrictive condition than to be in thermal equilibrium (if isothermal, the process is in thermal equilibrium, but the converse is not true)

Why this another condition, is there any intuitive explanation?

Answer 1

This is what it says on the linked page:

...thermal reversibility requires that all heat transfer is isothermal.

This statement is only true in the context of that web-page, where it is being assumed that the system is in contact with a thermal reservoir (i.e. an object whose thermal mass is so large that it's temperature doesn't change while exchanging energy via heat with the system).

We know that a process in which there is energy transfer via heat between two objects at different temperatures is irreversible, and so in order for a process to be reversible in this context, the system and the environment must be at the same temperature at all times during the heat transfer. Since the environment's temperature doesn't change, neither can the system's.

Thus:

A reversible process in which energy is exchanged via heat between a finite system and an infinite reservoir must be isothermal.

Now, it is possible to design a reversible process in which the system's temperature changes and the system exchanges energy via heat with its environment. If you bring a system into contact with a sequence of thermal reservoirs, each at an infinitesimally larger temperature than the last, and you allow the system and reservoir to equilibrate at each step, then the system gains energy via heat, and its temperature goes up. Crucially, this heat exchange occurs when the system is in contact with a reservoir at the same temperature, so the entire process is reversible.

Answer 2

Only condition is all the changes must be infinitesmal, this is the only way to keep intensive variables (most importantly, temperature and pressure) homogeneous during every point of time during process like you said, and I think you misinterpreted isothermal, by isothermal, it should mean that difference between system and surrounding temperature must be infinitesmal, it doesn't have anything to do with the kind of process we are performing, to set up adiabetic reversible,just put surroundings at exact same temperature as system {at every point of time during the process} , since no temperature difference, no heat will flow and the condition of process is fulfilled or simply using an isolated system would be more than enough, with other systems also like a piston having infinitesmal pressure difference only for the reasons mentioned above.