What is Entropy? The Great Unknown for the vast majority of technicians and engineers, let alone the public. Even those, who say they know what it is, only have a very vague perception without any deeper insight - it's not taught in schools and universities to a larger extend. Actually, there is no general agreement among scientists either and that may just be the cause why teachers not want to go too deep into the subject - they are unsure themselves.

Some say entropy is a measure for chaos and others that it is a measure for the dispersion of energy. The present definition of entropy is identical with the formulation of the Second Law of Thermo, basically saying that energy always tends to disperse and when it does, the total entropy of system AND environment increases. This again has to do with reversible and irreversible processes, environments and systems and the confusion becomes total. Anyway, entropy S is given by the relationship between energy Q and temperature T, as
S = Q/T

There are several ways to formulate the Second Law and though very different from each other, they are all considered to be equivalent - if one is wrong, all the others are wrong also. One of these (popular) formulations says that heat cannot flow spontaneously from a colder to a warmer region.

However, when you are in the tropics, where the air temperature can become above body temperature, your sweating skin cools your body, by which heat flows spontaneously from your cooler body to the warmer environment. In 'technology' this effect has been known and practiced since thousands of years, by keeping water cool in jars of porous material. Some of the water exudes (sweats) through the pores of that material and gives off its heat to the warmer surrounding air. The details of the process can be ignored, whether it is evaporation or not, because the result of it is given as an indisputable fact:
heat can flow spontaneously from a colder to a warmer region.

However, Clausius' statement did not include the term 'spontaneously'. His formulation was:
A process whose only final result is to transfer thermal energy from a cooler object to a warmer one, is impossible.

Now, the jar loses water and if not replenished, it will become empty and thus the transfer of heat from the jar to the surrounding air is not the only result of the process. Likewise, if you don't drink water, your sweating body will dry out and die in the end. Thus also here the transfer of heat is not the only result - Clausius was right! Nevertheless, as long as the process lasted, heat indeed did flow spontaneously from a colder to a warmer region, just that Clausius in exact wordings never denied that ...or did he?

So what is entropy? One thing we can all agree upon is that energy disperses, if it is not hindered to do so. This can also be seen as increasing disorder, because, as energy disperses, the molecules involved, move in  more chaotic patterns. However, it is true that shuffled cards, or a broken glass on the floor, are rather more chaotic conditions than a dispersion of energy. The confusing point is that one has to do work to restore the original order, not so much work to order the shuffled cards, but basically infinite work to restore the broken glass (without using new materials) to its original condition. This work disperses in the environment and decays to heat at that environment's temperature.

We are thus talking about closed loop, cycle processes here and if these processes are irreversible (they must be driven by an external source), the applied energy will disperse in the surroundings. The sweating jar and human body constitute irreversible processes, because the evaporated water will not by itself return as liquid to the jar or body - it's an open process. To make it a cycle process, work has to be done and then energy disperses again. Hence, if one sees entropy as a measure for disorder, one actually refers to the work done to restore the original order in a cycle process. If such restoration is not done, the shuffled cards and the broken glass on the floor indeed have nothing to do with entropy. But then, you can do things the easy, or the difficult way and thus the effort needed to restore the original condition, is not a given quantity. Therefore entropy cannot be a measure for disorder.

Is it a measure for the dispersion of energy? If so, well then, how much energy has dispersed with free expansion of and ideal gas? We only know the change of volume DV and the internal energy of the gas U, that remained constant during the process. If entropy would be the dispersion of energy, than it should be given by U/DV, because there is no change of temperature either. If we consider the equivalent isotherm process, we have a totally different procedure. In that one, energy is exchanged with the environment and the amount of that energy can be calculated, but it stands in no relation to the unchanged internal energy of the gas and it doesn't disperse from one given volume to another given volume either. If thus the volume-of-dispersion has no meaning, what then at all is meant with "dispersion"? On what grounds can we claim that the isotherm work, exchanged as heat Q with the environment, at a constant temperature T of the gas, is the change of entropy of the free expanded gas, equal to Q/T ?  Oh yes, the end state points of the two processes are the same and that is all the reason. Well, are the end state points really the same? Indeed for an ideal gas they are, but ideal gases do not exist - now we're getting somewhere!

Real gases do not expand freely at constant temperature, which is known as the Joule-Thomson effect (JTE). Most real gases expand at a decreasing temperature, but that depends on the initial temperature of the process. If that temperature is below the so called inversion temperature, free expansion occurs at decreasing temperatures. If the initial temperature is above the inversion temperature, free expansion occurs at increasing temperatures. Now somebody tells me how to calculate the change of entropy on this? It is not in my physics books and I have found it nowhere on the web. What I did find was a TS diagram on this page:
http://www.mmr.com/PDFs/JTCooling_TSB014fm.pdf ,   shown here to the left.

Anyway, in the same test set-up, with the same initial temperature, the result of the expansion becomes different, depending whether the expanding gas is an ideal, or a real one. Because the dispersion of energy is the same in both cases, but the end temperatures are not, the change of entropy cannot be the same and this rules out entropy to be a measure for the dispersion of energy!

This website, mmr.com, never answered my question how to calculate the entropy change, so I assume they used a standard TS-diagram and plotted the measured expansion temperatures as curves of constant internal energy, between 235 and 435 J/g. Indeed, as it is free expansion, the internal energy doesn't change and so the temperature curves collapse with those for internal energy (horizontally for ideal gases)

As thus entropy for JTE cannot be calculated, I don't give a hoot for the entropy values shown in this TS-diagram. Because ideal gases do not exist, entropy calculations based on them have no relevance in the real world either and that is for me the end of entropy as a thermodynamic property of matter AND as a measure of the dispersion of energy, let alone to see entropy as an absolute physical dimension of whatever (third "law" of thermo) - what options are left?

 

Around 100 years ago, one of the greatest scientific genius ever, Ludwig Boltzmann, gave the answer. In short, entropy is the probability for a given number of micro-states to occur spontaneously, written as S = k•Ln(W) and DS = k•Ln(W1/W2) where W stands for probability (Wahrscheinlichkeit in German language) and k is Boltzmann's constant. My more simple formulation is: the probability to predict a certain object to be at a certain time in a certain place. In fact this means DISORDER again, but now we can accept it, because entropy is about probability and not about a physical property of matter. In Boltzmann's formula the terms of energy and temperature do not occur, but mind my argumentation above - it takes work to restore order, just that this work is not of a given amount - do it the easy, or the difficult way (f.ex. filling a bottle with water, using, or not using a funnel). Boltzmann's formula can be derived as follows:

Assume an inert gas in a closed reservoir V. Further assuming that the motions of the gas molecules are at random and independent from each other, we can calculate the probability that all the molecules at a certain arbitrary moment are situated in a smaller volume V1 within V . The prop ability that one molecule in the chosen moment is situated in V1 is equal to the ratio: V1/V. If there are N molecules, the probability that all of them are in the smaller volume V1 and at the same moment in time, is: (V1/V)^N
If we denote this probability with P1 , then we can write:  P1 = (V1/V)^N
In the same analogue manner it is valid, that the probability that all the molecules at a certain moment are within a volume V2 is equal to: P2 = (V2/V)^N
In order to establish a connection between these two probabilities and entropy, we should remind that the change of entropy of an expanding gas from volume V1 to V2 is: DS = n.R•Ln (V2/V1)     ( n = mole,   R = gas constant )  

Indeed, here Boltzmann's formula follows from letting the free expansion of an ideal gas occur between the same end state points as an isotherm expansion, but as the resulting formula is independent from temperature, it is applicable on all other situations, including JTE and thus even includes, must include ideal gases. Also mind that Boltzmann's formula is valid under the condition of spontaneousity only, in contrast to Clausius' formulation, where spontaneousity is not a condition for entropy (his luck, otherwise the sweating body and jar would invalidate the, "his" Second Law). There are no spontaneous reversible processes in the real world (an indefinitely oscillating spring, is an ideal one). However, I may have found a way around it, though with great help of James Watt -see here

Open systems always involve irreversible processes, by which energy disperses and that only means that the quality of this energy (its density) becomes less. The lowest possible quality is when this energy cannot be recovered any more, as is when thermal energy decays to heat at ambient temperature. All energy processes are about lowering the quality of a source's energy to that of the energy contained by the environment. It happens spontaneously in nature all the time and with our technology, we let it happen under controlled conditions. If the source quality is low, we can't lower it much more and thus we won't be able to make much use of it. The larger the difference between source quality and that of the drain, or sink (the surroundings, or environment) is, the more useful work we can get out of it, meaning a higher efficiency of the process of consideration.

With entropy as a measure of probability, we can now introduce the notions of high and low entropy sources. High entropy sources give a low probability for efficient usage and low entropy sources give a high probability for it. In this context we can see entropy as a measure for the quality of energy, though it is NOT a physical property of energy. Energy is simply energy, regardless its quality, just like a banana remains a banana, whether it is the only one on your table, or one of many on the tree. With entropy as a measure of the quality of energy, we have a useful tool to judge the viability of certain projects.

If we take wind energy for example, it has a low quality (low density), close to that of the environment and so the efficiency of converting it to high density energy (electrical power) becomes very low - it is a high entropy source. Likewise with solar energy that is widely spread in the environment and thus has a low quality. Likewise with energy from biomass, the source of which is widely spread vegetation. We have to do a lot of work to bring it in the location of usage and to prepare it into a useable form (increase its quality) and so the overall efficiency becomes very low. Fuels on the other hand, have a high energy density, stored in chemical, or nuclear form - they are low-entropy sources. This is why we can make high efficient use of them and that gives the economical viability.

The only natural low-entropy source that solar energy provides, is that of hydro-electric power. The solar energy collects water from wide areas (like the oceans) through rain into a high reservoir of limited size (increase of energy density), thus providing a low-entropy source, that we can make efficient use of. Does the water in that reservoir "have" a low entropy? One could say that, but it has nothing to do with the physical properties of the water. Once it has fallen down, passed throug the turbine(s) and flown away from there, one could in the same manner say that it has got a high entropy then, but it is still the same water. Boltzmann's formula simply says that the probability to find a certain water molecule in a certain place at  a certain moment was higher in the reservoir, than it is in the stream that flows out from the turbine(s). The same could be said from a tiny fish in that water.

Indeed, increased disorder, but that is not the essence of entropy, just an effect of it. Different it is with the potential energy that was converted to heat and mechanical energy in the turbine(s). This was solar energy, that evaporated the original water from mainly the oceans and let it rain into the reservoir. Had it not done that, it would have been just as spread out in Nature as it becomes in and after the turbine(s) - also the mechanical energy will finally decay to heat at ambient temperature. Thus is terms of dispersion of energy, the total change of entropy was zero - nothing has changed for planet Earth as a whole. In terms of probability, the entropy has increased, but also this has no meaning for planet Earth. It had meaning for us though, as we needed that mechanial energy to make electricity at a reasonable price.

We can clearly see this again in the TS-diagram. The temperature scale can be made to reflect internal energy, simply by multiplying it with the specific heat C_v of the gas concerned. As specific entropy has the same dimension as specific heat, the entropy scale then becomes dimensionless (divided with C_v).  It means that with the JTE, the specific heat of the expanding gas changes and that causes the change of temperature during free expansion at constant internal energy. The entropy scale in the TS-diagram above thus basically reflects a factor, by which the specific heat of the gas changes - this factor is dimensionless.

If we apply this on Boltzmann's formula, we have to divide k with C_v and mass m ( because entropy is an extensive notion), thus: S = (k/m•C_v)•Ln(P)  and it becomes dimensionless. Hence, entropy has no dimension and can be seen as the relationship between effort and result, which directly relates to probability. The lower the probability for a process to happen spontaneously, the larger the effort will be to make it happen. This means entropy is not valid in thermo only, but can also be applied in economics, marketing, psychology, etc, everywhere there is a relationship between effort and result (the base formula for that simply becomes S = C•Ln(P), where C is a dimensionless system constant) .
This is what entropy is all about, nothing more!

Finally, being a thermo-engineer myself, I can tell you that no engineer has to know a thing about entropy, to enable him/her to design a thermo machine. Fortunately, because most engineers have only a vague idea of what entropy is and so they can ignore the subject in their work, as James Watt did for example (think!).

See other views on the subject here:
www.entropysite.com/entropy_isnot_disorder.html
www.secondlaw.com