A Hypothetical Situation:
Two thermally isolated identical systems have heat capacities which vary as:
$$Cv =βT^4$$($β$ is constant)
Initially one system is at $300K$ and the other at $400K$. The systems are then brought into thermal contact and the combined system is allowed to reach thermal equilibrium.
Note: I am skipping all the rigorous mathematical calculation because of typing issue.
Now, if I want to calculate (skipping Mathematical Calculation) the final Temperature of the combined system, I can do the calculation in two ways:
- Through a reversible process.
- Through an irreversible process.
As the whole system(both the systems) is thermally isolated no energy will be lost. But for the two different ways the final temperatures of the combined system will be different. I can understand it Mathematically but I am wondering what is actually happening physically!
For the irreversible case the change in entropy is greater than zero; So the heat exchange between the two systems will be same i.e.,
$$∫Cv dT1 = ∫Cv dT2$$(integration limit is according to the question and final temperature)
But, for the reversible case the change in entropy is equal to zero($∆S1+∆S2=0$). So, though the total energy is conserved, the heat exchange between the two systems is not same! I am wondering how is this possible?
I mean, if one system is releasing $q$ amount of heat and the other system is gaining the same amount of heat but the final temperature of the combined system is not in accordance with the consumed heat. At first, I thought that actually some heat is only used for gaining entropy but not for the temperature. But I don't know how it is possible to consume heat without rising temperature?(This is also not the phase transition stage so how is it possible?)