How exactly is heat supplied in an isothermal process

How exactly is heat supplied in an reversible isothermal process. If the temperature of system and surroundings always remains the same, then how come surrounding is supplying heat?


I studied that T of system and surrounding is the same while studying entropy.




Reversibility has two distinct meanings.


1. Bidirectionality. You can melt ice to water and freeze it back to ice.
2. Being in equilibrium along the whole process path. This can be just approximated in real processes by being near equilibrium.


The second meaning requires that reversible process is returned to its initial state by returning the neighbourhood to its initial state.


If the above cannot be done, the process is not reversible(2), even if it is reversible(1).




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The heat is in your case provided as usually, but infinitely slowly for infinitely long time, with infinitely small temperature difference between the system and its surrounding.


A reversible isothermal process is an idealised process lasting infinite (long enough) time, so heat can be provided at zero (small enough to neglect) temperature difference.


$$Q \propto \Delta T \cdot \Delta t$$


$$\lim\_{\Delta t \to \infty}{\Delta T} = 0$$


E.g., you need 1 hour to transfer heat Q at temperature difference 1 K.  

You need 2 hours for 0.5 K difference.  

4 hours for 0.25 K
etc.
For time going to infinity and temperature difference going to zero, systems converge to being reversible.


With infinite time, zero temperature difference is needed to provide finite heat.


In reality, scientists choose "good enough" approach with small enough differences and long enough time.


Truly reversible processes do not exist in reality, as they are assumed to be in equilibrium all the time, having infinite time to achieve anything.


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Ice fusion at $0°$C in "hotter" bath (just hotter than $0°$C) is a good example of a nearly reversible isothermal process. Heat can be introduced in such a system by having a mixture of ice + water in a metallic container, and dipping it into a bigger container full of water at $+1$°C.


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