| What is Entropy? Some say entropy is a measure for | | | | change of entropy of an expanding ideal gas from |
| chaos (disorder) and others that it is a measure for the | | | | volume V1 to V2 is: Delta S = n.R.Ln (V2/V1) ( n = |
| dispersion of energy. It was firstly coined by Rudolph | | | | mole, R = gas constant ) expressed in terms of |
| Clausius in the mid 1800's, who also firstly formulated | | | | molecular units: Delta S = N.k.Ln (V2/V1). As P2/P1 = |
| the First and Second Law of Thermodynamics in its | | | | (V2/V1) N and thus ln(P2/P1) = N.ln (V2/V1) , we get: |
| still valid form. The Second Law basically says that | | | | Delta S = k.Ln(P2/P1) Then we can write this as: Delta |
| energy always tends to disperse and when it does, | | | | S = k.Ln(P2) - k.Ln(P1) From this it seems logic to |
| the total entropy of system AND environment | | | | define the entropy of every condition as: S = k.Ln(P) |
| increases. Entropy S is given by the ratio of energy Q | | | | (end intermezzo) |
| and temperature T , as S = Q/T. There are several | | | | Indeed, here Boltzmann's formula follows from letting |
| ways to formulate the Second Law and though very | | | | the free expansion of an ideal gas occur between the |
| different from each other, they are all considered to be | | | | same end state points in an isotherm expansion |
| equivalent - if one is wrong, all the others are wrong | | | | process, but because the resulting formula does not |
| also. | | | | contain temperature, it is applicable on all other |
| One popular, but wrong formulation says that heat | | | | situations, including JTE and thus even includes, must |
| cannot flow spontaneously from a colder to a warmer | | | | include ideal gases. Also mind that Boltzmann's formula |
| region. However, when you are in the tropics, where | | | | is valid under the condition of spontaneity only, in |
| the air temperature can become above body | | | | contrast to Clausius' formulation, where spontaneity is |
| temperature, your sweating skin cools your body, by | | | | not a condition for the change of entropy (his luck, |
| which heat flows spontaneously from your cooler | | | | otherwise the sweating body and jar would invalidate |
| body to the warmer environment. In 'technology' this | | | | the, "his" Second Law). There are no spontaneous |
| effect has been known and practiced since thousands | | | | reversible processes in the real world (an indefinitely |
| of years, by keeping water cool in jars of porous | | | | oscillating spring, is an ideal one). |
| material. Some of the water exudes (sweats) through | | | | Open systems always involve irreversible processes, |
| the pores of that material and gives off its heat to the | | | | by which energy disperses and that only means that |
| warmer surrounding air. Heat can flow spontaneously | | | | the quality of this energy (its density) becomes less. |
| from a colder to a warmer region. | | | | The lowest possible quality is when this energy cannot |
| This is not in conflict with the Second Law, because | | | | be recovered any more, as is when thermal energy |
| Clausius' statement did not include the term | | | | decays to heat at ambient temperature. All energy |
| 'spontaneously'. His formulation was: A process whose | | | | processes are about lowering the quality of a source's |
| only final result is to transfer thermal energy from a | | | | energy to that of the energy contained by the |
| cooler object to a warmer one, is impossible. | | | | environment. It happens spontaneously in nature all the |
| Now, the jar loses water and if not replenished, it will | | | | time and with our technology, we let it happen under |
| become empty and thus the transfer of heat from the | | | | controlled conditions. If the source quality is low, we |
| jar to the surrounding air is not the only result of the | | | | can't lower it much more and thus we won't be able to |
| process. Likewise, if you don't drink water, your | | | | make much use of it. The larger the difference |
| sweating body will dry out and die in the end. Thus also | | | | between source quality and that of the drain, or sink |
| here the transfer of heat is not the only result. | | | | (the surroundings, or environment) is, the more useful |
| Nevertheless, as long as the process lasted, heat | | | | work we can get out of it, meaning a higher efficiency |
| indeed did flow spontaneously from a colder to a | | | | of the process of consideration. |
| warmer region. This I have translated into a physical | | | | With entropy as a measure of probability, we can now |
| process, that possibly may revolutionize future energy | | | | introduce the notions of high and low entropy sources. |
| technology. | | | | High entropy sources give a low probability for efficient |
| So what is entropy? One thing we can all agree upon | | | | usage and low entropy sources give a high probability |
| is that energy disperses, if it is not hindered to do so. | | | | for it. In this context we can see entropy as a |
| This can also be seen as increasing disorder, because, | | | | measure for the quality of energy, though it is NOT a |
| as energy disperses, the molecules involved, move in | | | | physical property of energy. Energy is simply energy, |
| more chaotic patterns. However, it is true that shuffled | | | | regardless its quality, just like a banana remains a |
| cards, or a broken glass on the floor, are rather more | | | | banana, whether it is the only one on your table, or one |
| chaotic conditions than a dispersion of energy. The | | | | of many on the tree. With entropy as a measure of |
| confusing point is that one has to do work to restore | | | | the quality of energy, we have a useful tool to judge |
| the original order, not so much work to order the | | | | the viability of certain projects. |
| shuffled cards, but basically infinite work to restore the | | | | If we take wind energy for example, it has a low |
| broken glass (without using new materials) to its original | | | | quality (low density), close to that of the environment |
| condition. This work disperses in the environment and | | | | and so the efficiency of converting it to high density |
| decays to heat at that environment's temperature. We | | | | energy (electrical power) becomes very low - it is a |
| are thus talking about closed loop, cycle processes | | | | high entropy source. Likewise with solar energy that is |
| here and if these processes are irreversible (they | | | | widely spread in the environment and thus has a low |
| must be driven by an external source), the applied | | | | quality. Likewise with energy from biomass, the source |
| energy will disperse in the surroundings. | | | | of which is widely spread vegetation. We have to do |
| The sweating jar and human body constitute | | | | a lot of work to bring it in the location of usage and to |
| irreversible processes, because the evaporated water | | | | prepare it into a useable form (increase its quality) and |
| will not by itself return as liquid to the jar or body - it's | | | | so the overall efficiency becomes very low. Fuels on |
| an open process. To make it a cycle process, work | | | | the other hand, have a high energy density, stored in |
| has to be done and then energy disperses again. | | | | chemical, or nuclear form - they are low-entropy |
| Hence, if one sees entropy as a measure for disorder, | | | | sources. This is why we can make high efficient use |
| one actually refers to the work done to restore the | | | | of them and that gives the economical viability. |
| original order in a cycle process. If such restoration is | | | | The only natural low-entropy source that solar energy |
| not done, the shuffled cards and the broken glass on | | | | provides, is that of hydro-electric power. The forces of |
| the floor indeed have nothing to do with entropy. But | | | | Nature (driven by the Sun) collect rain water into high |
| then, you can do things the easy, or the difficult way | | | | situated reservoirs (increase of energy density), thus |
| and thus the effort needed to restore the original | | | | providing a low-entropy source, that we can make |
| condition, is not a given quantity. Therefore entropy | | | | efficient use of. Does the water in those reservoirs |
| cannot be a measure for disorder. | | | | "have" a low entropy? One could say that, but it has |
| Is it a measure for the dispersion of energy? If so, then | | | | nothing to do with the physical properties of the water. |
| the change of entropy should be independent from | | | | Once it has fallen down, passed through the turbine(s) |
| whether ideal, or real gases are concerned. On the | | | | and flown away from there, one could in the same |
| contrary, real gases behave differently from ideal | | | | manner say that it has got a high entropy then, but it is |
| gases. Unlike ideal gases, real gases do not expand | | | | still the same water. Boltzmann's formula simply says |
| freely at constant temperature, which is known as the | | | | that the probability to find a certain water molecule in a |
| Joule-Thomson effect (JTE). Most real gases expand | | | | certain place at a certain moment was higher in the |
| at decreasing temperatures, but some do at increasing | | | | reservoir, than it is in the stream that flows out from |
| temperatures. Now somebody tells me how to | | | | the turbine(s) and disperses in the surroundings. The |
| calculate the change of entropy on this? It is not in my | | | | same could be said from a tiny fish in that water. |
| physics books and I have found it nowhere on the | | | | Indeed, increased disorder, but that is not the essence |
| web. | | | | of entropy, just an effect of it. Different it is with the |
| Anyway, in the same test set-up, with the same initial | | | | potential energy that was converted to heat and |
| temperature (same internal energy), the result of the | | | | mechanical energy in the turbine(s). This was solar |
| expansion becomes different, depending whether the | | | | energy, that evaporated the original water from mainly |
| expanding gas is an ideal, or a real one. Because the | | | | the oceans and let it rain into the reservoir. Had it not |
| end temperatures are not the same, the change of | | | | done that, it would have been just as spread out in |
| entropy cannot be the same and this rules out entropy | | | | Nature as it becomes in and after the turbine(s) - also |
| to be a measure for the dispersion of energy! | | | | the mechanical energy will finally decay to heat at |
| As thus entropy for JTE cannot be calculated and | | | | ambient temperature. Thus in terms of dispersion of |
| ideal gases do not exist, entropy calculations based on | | | | energy, the total change of entropy was zero - nothing |
| the latter have no relevance in the real world and that | | | | has changed for planet Earth as a whole. In terms of |
| is for me the end of entropy as a thermodynamic | | | | probability, the entropy has increased, but also this has |
| property of matter AND as a measure of the | | | | no meaning for planet Earth. It had meaning for us |
| dispersion of energy, let alone to see entropy as an | | | | though, as we needed that mechanical energy to |
| absolute physical dimension of whatever (third "law" of | | | | make electricity at a reasonable price. |
| thermo) - what options are left? | | | | We can clearly see this again in the TS-diagram |
| Around 100 years ago, one of the greatest scientific | | | | (Temperature-Entropy). The temperature scale can be |
| genius ever, Ludwig Boltzmann, gave the answer. | | | | made to reflect internal energy, simply by multiplying it |
| According to him, entropy is the probability for a given | | | | with the specific heat Cv of the gas concerned. As |
| number of micro-states to occur spontaneously , | | | | specific entropy has the same dimension as specific |
| written as S = k.Ln(W) and Delta S = k.Ln(W1/W2) | | | | heat, the entropy scale then becomes dimensionless |
| where W stands for probability ( Wahrscheinlichkeit in | | | | (divided with Cv ). It means that with the JTE, the |
| German language) and k is Boltzmann's constant. My | | | | specific heat of the expanding gas changes and that |
| more simple formulation is: the probability to predict a | | | | causes the change of temperature during free |
| certain object to be at a certain time in a certain place. | | | | expansion at constant internal energy. The entropy |
| In fact this means DISORDER again, but now we can | | | | scale in the TS-diagram thus basically reflects a factor, |
| accept it, because entropy is about the probability for it | | | | by which the specific heat of the gas changes - this |
| and not about a physical property of matter. In | | | | factor is dimensionless. If we apply this on Boltzmann's |
| Boltzmann's formula the terms of energy and | | | | formula, we have to divide k with Cv and mass m ( |
| temperature do not occur, but mind my argumentation | | | | because entropy is an extensive notion), thus: S = (k |
| above - it takes work to restore order, just that this | | | | (m.Cv) ).Ln(P) and it becomes dimensionless. This is for |
| work is not of a given amount - do it the easy, or the | | | | me the mathematical and physical prove that Entropy |
| difficult way (f.ex. filling a bottle with water, using a cup, | | | | is not a physical property of whatever - it is a |
| or a funnel). | | | | quantitative notion only. |
| (Intermezzo - you can jump over it) | | | | Hence, entropy has no dimension and can be seen as |
| Boltzmann's formula can be derived as follows: | | | | the relationship between effort and result, which |
| Assume an inert gas in a closed reservoir V . Further | | | | directly relates to probability. The lower the probability |
| assuming that the motions of the gas molecules are at | | | | for a process to happen spontaneously, the larger the |
| random and independent from each other, we can | | | | effort will be to make it happen. This means entropy is |
| calculate the probability that all the molecules at a | | | | not valid in thermo only, but can also be applied in |
| certain arbitrary moment are situated in a smaller | | | | economics, marketing, psychology, etc, everywhere |
| volume V1 within V . The probability that one molecule | | | | there is a relationship between effort and result (the |
| in the chosen moment is situated in V1 is equal to the | | | | base formula for that simply becomes S = CoLn(P), |
| ratio: V1/V . If there are N molecules, the probability that | | | | where C is a system constant) |
| all of them are in the smaller volume V1 and at the | | | | This is what entropy is all about, nothing more! Finally, |
| same moment in time, is: (V1/V) N If we denote this | | | | being a thermo-engineer myself, I can tell you that no |
| probability with P1 , then we can write: P1 = (V1/V) N In | | | | engineer has to know a thing about entropy, to enable |
| the same analogue manner it is valid, that the | | | | him/her to design a thermo machine. Fortunately, |
| probability that all the molecules at a certain moment | | | | because most engineers have only a vague idea of |
| are within a volume V2 is equal to: P2 = (V2/V) N In | | | | what entropy is and so they can ignore the subject in |
| order to establish a connection between these two | | | | their work, as James Watt did for example (think! |
| probabilities and entropy, we should remind that the | | | | |