Entropy, path to point
There was a little arm-waving in the earlier post about entropy, as it did not reveal the details of how we got from a cyclic reversible process to a general property of substances undergoing irreversible processes. Most of this was developed during the middle of the 19th century, and is a marvel of intellectual prowess. It also had a big help from calculus, as it facilitates analyzing change, and this topic is about change!
The mathematical detail that should be understood is substance has properties, that define its state. It’s not relevant how it got to that state, just that a unique set of properties define its state. So, it could be a certain combination of temperature and pressure makes water ice; another combination liquid, yet another a vapor. Or two states of water could be as a liquid, but one hot, the other cold. In other words, the path it took to get there doesn’t matter. One could step up the pressure first, and then change the temperature next and arrive at a state. Likewise, one could change the temperature first, and then step up the pressure, and it could arrive at exactly the same state. So the properties are independent of the way or path it took to get there.
On the other hand, the way energy is changed in a process can depend on the path (as an example of two different paths, a less efficient heat engine will produce less power for the same heat input)[1]. So this discussion involves two types of concepts: work and heat (energy), vs. state properties. State properties are conservative, and independent of the process path; work and heat are not conservative, so can vary with the path. Another way to view it is the endpoints from beginning to end of a process will always be the same for properties, no matter what the path, while heat and work input and output could vary for those same property endpoints, depending on the path. One conceptual path is the reversible path, already mentioned, which depends on zero friction and very small temperature differences for heat transfer to take place.
The “bad” refrigerator was used in the first post on entropy. This was used to show all the heat contributions at variable surrounding temperatures that took place during a cycle had to be less than or equal to zero. With another internally reversible[2] device (heat engine or refrigerator) run in both modes, the sum of the first will be more than or equal to zero, and the other sum less than or equal to zero. The only conclusion is all the heat contributions at variable temperatures that took place in the cycle either way has to be zero when added up! This coincides exactly with the mathematical definition of a conservative property (end points the same, regardless of the path), so the ratio of heat to temperature must be a property!
That must’ve been an exciting discovery when it was made, and so it was named entropy. However, this all involved reversible cycles, nothing that exists in the real world. It has to be further shown that this property does also apply to real irreversible cycles, and furthermore, to change of state processes that aren’t necessarily in a cycle. One way to do that is go from an originating state to a final state with a reversible process, and then return the originating state with an irreversible process, thus making a cycle with two different processes. Chained together this is an irreversible cyclic process, and it was shown earlier (see previous post) that the sum of the heat transferred at temperatures T must be less than zero. Reverse the internally reversible process, and algebra will show that the conclusion is the sum of the heat transferred at temperatures T for the internally reversible process is greater than the sum of the heat transferred at temperatures T for the irreversible.[3] Or, in other words, the change in the property of entropy for the irreversible process is greater than the sum of the heats at temperatures T.
A bit of a contradiction at first pass, but resolved considering that entropy is a property, only defined by end states, and independent of the path. So, the irreversible and reversible processes described have the same change in entropy from beginning to end, but the sum of the heats at temperatures T is greater than the change in entropy for the same change of state. This leads to a way to calculate the change in entropy: use a reversible process to calculate the sum of the heats at temperatures T, and use that change in entropy to describe the change in entropy for the irreversible processes of the real world.
There’s a couple of more considerations, a general one of the fate of entropy in the universe, and the special one of the way entropy relates to other state properties. And some notes on why any of this matters. That will be found in the third and final write-up of this subject. See the following link.
[1] Two reversible cycle devices, of completely different operation, will have the same work output and the same efficiency, if their heat source and sinks are the same respective temperatures. Their output is independent of their respective paths, in other words. On the other hand, an irreversible device will have a lower efficiency and lower work output at the same source and sink temperatures, and a different irreversible device will have yet a different efficiency and work output than the first irreversible one. These devices are path dependent for their operation.
[2] To help understand the difference, a reversible cycle is one that in theory could be reversed and regain exactly the same state it started at. An internally reversible one is a sub-category of that, being a cycle that is practically reversible, such as a heat pump reversed and operated as refrigerator or cooler.
[3] Algebraically, you could let Xtot stand for the sum of the small heats at given temperatures of this cycle being equal to the sums of the two parts of this cycle, one internally reversible Xint_rev, the other the irreversible leg, Xirr. So Xtot = Xirr + Xint_rev. The internally reversible one, Xint_rev, can be reversed, so Xtot = Xirr - Xint_rev. Recalling the “bad” refrigerator argument made in the prior discussion, Xtot is also less than zero, so Xirr - Xint_rev < 0, and algebra simplifies that to Xint_rev > Xirr. The change in the property of entropy was earlier defined as what we’re calling Xint_rev, so this reduces to the conclusion that the change in entropy is greater than Xirr, the irreversible process.