# Entropy in a reversible process

According to 2nd law of thermodynamics entropy never decreases, it's either zero or bigger.

The problem with the definition is that it doesn't specify WHICH entropy never decreases? Of the system that we are observing, of the environment or both at the same time.

If a system goes from state A to state B in a reversible manner , there is an entropy change, because the system goes from a macrostate A with a certain multiplicity to another one with another multiplicity value, which means entropy changes. Now if we reverse the process and the system goes from B to A, since entropy is a state variable,when the system goes back to state A, entropy will have the value that it had in the beginning, and that can only happen if the entropy decreases (assuming that the value of entropy is bigger when the system is in state B compared to when it is in state A). Which means that the entropy in a cyclic reversible process increases and decreases.

Am I missing something here?

• The entropy of the universe = system + environment
– Rol
Nov 23 '21 at 22:43
• Both at the same time. Nov 24 '21 at 0:48

## 2 Answers

The second law entropy statement refers to the entropy of an isolated system.

If a system can move reversibly between states of different entropy, then it is not an isolated system. But the combination of system with its environment might together be isolated. So then any entropy moving out of the system goes into the environment, and any entropy moving out of the environment goes into the system. Since entropy is conserved in reversible processes it is just like the flow of any other conserved quantity.

The problem with the definition is that it doesn't specify WHICH entropy never decreases? Of the system that we are observing, of the environment or both at the same time.

As already indicated, the definition applies to an isolated system, which is one that exchanges neither mass nor energy with its surroundings, or equivalently it applies to the change in entropy of the system plus the change in entropy of the surroundings, which together constitute the universe.

If a system goes from state A to state B in a reversible manner , there is an entropy change...

There is not always an entropy change of the system when it undergoes a reversible process. The entropy change for a reversible adiabatic process is zero.

Now if we reverse the process and the system goes from B to A, since entropy is a state variable, when the system goes back to state A, entropy will have the value that it had in the beginning...

That is correct. But is applies whether the process is reversible or irreversible. If the process is reversible, then the change in entropy of both the system and surroundings will be zero when returning the system to its original state. If the process is irreversible, the entropy of the system will return to its original state but the change of the surroundings will be greater than zero.

...and that can only happen if the entropy decreases (assuming that the value of entropy is bigger when the system is in state B compared to when it is in state A).

But the entropy at state B does not have to be greater than state A. An example is the change in entropy for a reversible isothermal compression from state A to state B of an ideal gas, which is $$\Delta S_{sys}=-Q/T$$ where heat $$Q$$ leaves the system and $$T$$ is the constant system equilibrium temperature.

Which means that the entropy in a cyclic reversible process increases and decreases

Again, not necessarily. A reversible adiabatic expansion from state A to state B followed by a reversible adiabatic compression from state B to state A involves no entropy change in either direction.

Hope this helps.