How can entropy of universe or surroundings be a state function?

If a system goes from state A to B in a reversible process, then $(\Delta S)\_{\text{universe}}=0$. However, my confusion starts if we take an irreversible route. For the system, even if we take the irreversible route, we can always calculate it using the reversible route from A to B, so the system entropy is a state function. However, the surroundings entropy change is not negative of the system's entropy change in an irreversible process, meaning that $(\Delta S)\_{\text{universe}}\neq 0$, despite the process still being between states of the system A and B. This suggests to me that $(\Delta S)\_{\text{universe}}$ is not a state function but depends on the path (reversible vs irreversible). I was under the impression that entropy is a state function.


Is this reasoning correct?




**TL;DR:** entropy, for reversible and irreversible processes, for system, surroundings and universe, is always a state function. You are not accurately considering the initial and final states of surroundings (and universe) for reversible and irreversible processes.


### The details


In system going from state $A$ to $B$ reversibly, say the state of the surroundings changes from $C$ to $D$. In system going from state $A$ to $B$ irreversibly, considering the same initial state of the surroundings, $C$, the final state will differ from $D$ (say $D'$)


[![Entropy change of system and surroundings for reversible and irreversible processes](https://i.stack.imgur.com/H4wWX.png)](https://i.stack.imgur.com/H4wWX.png)


For the reversible process,


$$
\begin{align}
(\Delta S)\_\text{uni} &= (\Delta S)\_\text{sys} + (\Delta S)\_\text{surr}\\
&= S\_B-S\_A + S\_D-S\_C \tag{1}
\end{align}
$$


For the irreversible process,


$$
\begin{align}
(\Delta S)\_\text{uni} &= (\Delta S)\_\text{sys} + (\Delta S)\_\text{surr}\\
&= S\_B-S\_A + S\_{D'}-S\_C \tag{2}
\end{align}
$$


From equations (1) and (2), the difference between the entropy change of the universe when the system undergoes an irreversible and a reversible process is given by $S\_{D'}-S\_D > 0$, where the inequality is a consequence of the second law of thermodynamics.


 4

There is no reversible route from A to B if those are the states of the universe.


If A and B are system states, and you take different paths, the surrounding state will be distinct for distinct paths. So in one case the entropy of the universe might increase by a certain amount, and in the other case it increases by a different amount.


However, if the entire universe is changed on a reversible path, there is nothing outside of the universe that can drive that. So either the universe is at equilibrium, with no more increase in entropy, or things are still happening and entropy is increasing.


Bottom line: there is no way to go reversible from a state A to a state B of the entire universe.


 1

A State Function has a value for a given state of a system. This means that if something undergoes a process and eventually returns to the exact same conditions as it started, its state functions U, G, H, S, PV will be exactly the same. The energy terms U, H, G are conserved so these will remain constant in the universe. S is not conserved and must increase for anything that happens. Since S has not changed in our system it must have increased in the universe. The trick is to expand our immediate universe to determine where and how the entropy increase happens.


This can be difficult but if reduced to extremes is very simple: Everything on Earth depends on Energy from the Sun, Directed energy of higher energy photons of LOW entropy. To maintain an energy steady state High entropy energy must be emitted. This is low energy infrared photons with higher entropy.


An example: A large Oak tree is measured to determine the energy etc. of everything down to the tilt of its leaves in the Sun and the rotational states of all its molecules. The tree is cut down, all its roots removed and used for lumber furniture and fuel. One of its acorns is planted in the exact same spot and with much TLC Grows to an exact clone of the tree down to the rotational states of the molecules. Its state functions are the same as the original tree! but the entropy increase of the universe needed to accomplish this is not only beyond all the supercomputers but beyond imagination. In the end one is content to simply have a new oak tree.


 -2