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Author Topic:   Entropy in Layman's Terms
Rrhain
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(1)
Message 1 of 51 (544059)
01-23-2010 3:33 AM


Dr Adequate says that the entropy cannot be expressed in layman's terms. As the author of a primer that I've posted here that derives the Second Law from first principles, I thought I'd take a crack at it.

There are many pithy sayings regarding the Second Law and entropy, but the problem with them is that they wind up being used in places where they make no sense. The reductions are accurate, but can only be understood within the concept of thermodynamics, not other areas. So if we're going to explain it in layman's terms, we have to start with making the point that entropy is a statement about energy. It isn't about "information" or "order" or "disorder."

In short, entropy is the energy in the system that cannot be used for work. Thermodynamics works by moving energy around from one place to another. But because this transfer is a physical process, this means that this energy will go into the physical transfer mechanism as well as to the delivery place. It's a bit like trying to carry water from one place to another: Some of it is always going to stick to the vessel. If I take a cup of water and pour it out, there will always be a few drops sticking to the cup. I can shake the cup and bang it, but there is always going to be something left over. That water that wets down the vessel is like entropy: It isn't being transferred from one place to another but is lost as you transfer all the rest of the water.

As you move the same energy around and around the system, you eventually come to a point where there isn't any energy left that can be used: It's all been bled away as entropy. But that assumes a completely closed system. If we had more water entering the system, we can afford to lose some of it sticking to the sides of the vessel.

This is why concepts of "information" or "disorder" make no sense in their regular senses: It isn't a question of "ordering" the energy. It's that it is no longer physically available to do any work. When we reduce the concept of the Second Law and entropy to the phrase, "The universe tends towards disorder," what we mean is that the energy distribution of the universe is tending toward a uniformity. You see, energy transfer depends upon variations in energy states. If everything is the same, then there can be no reaction.

Back to the water concept: If you have a cube of ice and a pot of boiling water, there will be energy flow between the two if we put them into a system: However, that energy flow only happens because of the disparity in their energy states. The heat flows from the boiling water into the ice and eventually, the water hits equilibrium. When everything hits the same temperature, no more energy flow takes place. All the heat is now unavailable to do any work because there isn't anywhere for it go. That doesn't mean there isn't any energy: It's just that there is no disparity between any two places in the system and thus, there can be no transfer of any energy of any kind.

Eventually, all the energy in the universe will be equally dispersed throughout the universe and no reactions of any significance can take place. That's what we mean when we say "disorder": There is no pattern to anything because everything is dispersed.

The better metaphors for entropy are the ones that I put through in my primer:

There are no perfect engines.

OR

There are no perfect refrigerators.

In the first sense, this has to do with the water example I gave above: Some of the water always sticks to the vessel. You cannot convert heat entirely into work. Some of the energy gets used on the engine itself, rather than being used to drive the engine. In the case of heat, that means some of the heat is used to heat the engine rather than being used to make work the engine facilitates. You can make the engine more efficient such that more energy is converted into work, but there will always be some heat lost to the engine itself and thus, there is no way to make it perfect.

In the second sense, this is a more practical idea: A refrigerator takes heat from a low-temperature area and moves it to a high-temperature area. But the only way it can do this is by doing some work to move it. Heat simply doesn't flow from low to high. When you put a piece of ice into boiling water, the ice doesn't become colder and dump the heat into the boiling water to make it hotter. Instead, the heat flow from hot to cold. And yet, a refrigerator is possible because of the work involved. You can make the refrigerator more efficient such that not as much work is required to move the heat, but there will always be some work required and thus, there is no way to make it perfect.

Entropy is simply energy unavailable to do work.


Rrhain

Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.
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Admin
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Message 2 of 51 (544080)
01-23-2010 8:15 AM


Thread Copied from Proposed New Topics Forum
Thread copied here from the Entropy in Layman's Terms thread in the Proposed New Topics forum.
    
Admin
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Message 3 of 51 (544082)
01-23-2010 8:19 AM


A Suggestion
I really like the idea of crafting a layperson-level explanation of thermodynamics that actually works for laypeople (even better if it also works for creationists, who represent a special class of laypeople), so I suggest using this thread not only to discuss Rrhain's attempt, but also to craft your own, whether original or just modifications to other attempts.

Edited by Admin, : Typo.


--Percy
EvC Forum Director

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Iblis
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Posts: 663
Joined: 11-17-2005


Message 4 of 51 (544086)
01-23-2010 9:16 AM
Reply to: Message 3 by Admin
01-23-2010 8:19 AM


grimoire thermal
The Second Law of Thermodynamics says that 100% efficiency isn't achievable, because energy gets lost due to the way it spreads out, and getting it back itself is a use of energy, which costs more than it pays.

A good example is Maxwell's Demon, a pin-dancing "thought experiment" from the good old days when people knew what thermo- meant. A stove loses heat, can't be helped, so you have to fuel it. If it didn't lose heat, reduce in caloric content over time as it were, then it ought to stay hot forever and you wouldn't need to fuel it. What use it would be, I don't know, we use them to heat things, but let that pass for a moment. Can it really not be helped?

Let's postulate a demon, there in the stove, who chases the little hot air molecules for us, and grabs them, and drags them back into the stove, so the heat doesn't get lost. In a case like this, do we have 100% efficiency? No, because the demon is doing work, work requires energy, and energy is what he is supposedly saving for us, we have to fuel him somehow. So now instead of wood we have to burn sinners or something. Still fuel, energy still lost.

Only applies to information, particularly genetic information, in the sense that "a lot of work is done." Even here, all it means is that there has to be a fuel source. There is, the sun, both directly and through secondary manifestations like hot lava, radioactivity, chemical reactions, and the endless spinning of Rudolf Clausius in his grave. Nothing to do with emergent systems like evolution and intelligence being unable to develop on their own, they result from the inevitable increase in complexity among large stochastic systems, and represent increasing entropy overall, not decreasing, not ever.

* note that Maxwell actually used this fiend for a somewhat more complex purpose, as a gatekeeper with a magic flashlight; he still fails to prevent entropy however, no matter how he does the work


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Son Goku
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Posts: 1120
From: Ireland
Joined: 07-16-2005


Message 5 of 51 (544462)
01-26-2010 3:28 PM
Reply to: Message 1 by Rrhain
01-23-2010 3:33 AM


Entropy - The statistical mechanics viewpoint.
Rather than explain entropy from a thermodynamic point of view, I'll try to explain it from a statistical mechanical perspective. What this means is that instead of talking about it as a consequence of the movement of heat and energy (thermodynamics), I will discuss how it relates to the basically random (statistical) motions (mechanics) of billions of atoms.

Take a pile of hydrogen gas, it's composed of billions and billions of atom. Let's be specific and say we have 1 kilogram of hydrogen. That's
60,000,000,000,000,000,000,000,000 atoms. If we want to describe the hydrogen gas we need six numbers for each atom. Three to indicate where the atom is and three to indicate its momentum in each direction. So that's:
360,000,000,000,000,000,000,000,000 numbers over all.

However obviously nobody describes a gas this way. You use one number for its weight, one number for its volume and maybe, at most, a few numbers to describe its shape. This is vastly smaller.

So to describe the state of the gas on a microscopic scale, you need a huge amount of information and on a macroscopic scale you need far less. This leads to the idea of a microstate and macrostate. The microstate is the information on the current state of the atoms from an atomic point of view. The macrostate is the information needed to describe the gas from a larger/macro point of view.

The macrostate is obviously a much cruder way of describing the gas, so several states of the gas which have different microstates, will have identical macrostates.

As an example, say you were looking at the gas. If somebody displaced one single atom 5 nanometers to the right, then the microstate would be different, because the atom is in a different position. However the macrostate would be unchanged, because there is no visible difference at all on our scale.

Entropy is defined at the macroscale, it is essentially the quantity that "bridges" the two scales. The entropy of a macrostate is the number of microstates which produce that macrostate.*

You might ask, why does entropy increase and how is it related to disorder and work?

The answer to the first question is that entropy doesn't actually always increase, it's just massively unlikely not to.

Let's take two macrostates of our gas of hydrogen. In one macrostate all the hydrogen is spread across the room evenly. In the other macrostate it's packed into one corner. The first macrostate has a massive number of microstates associated with it. You can move loads of atoms and nothing will change on our scale. For the other, move enough atoms out of the corner and things will start to look different.

To give you an idea, say the room was about one cubic meter in size. If we compare the case where the gas was concentrated in a corner one tenth of a cubic meter in size and the case where the gas is spread evenly around the room, the latter has a: (get ready for a massive number)
Google google google google google google google google google google google google,
times more microstates than the former.

That means that there are far, far more ways to arrange the atoms in the latter case to have things look identical, than there are in the former. You could say the latter macrostate is more generic and the former more special. Entropy is essentially a measure of how generic a macrostate is. The more generic a state is, the bigger the entropy.

This is why entropy will almost certainly increase. As time moves on the atoms are more likely to randomly wander into a high entropy state, because a high entropy states are literally more generic/common state, the odds are in their favour.

Finally, what has this to do with useable work?

Well if you think about it, in order for something to do work, it has to be arranged in a fashion where it can give up its energy to something else. You have to put (or find) it in a work capable state. To take my hydrogen gas example, you could get it to do work by arranging it to be tightly compressed, so that its pressure does work. Obviously there are macrostates more capable of doing work than others. The point is that a state that can do a lot of work is rarer than one that can do little, simply because one requires a specific arrangement and hence is special/rare and the other doesn't need any particular arranging and so is common.

Over time, just as I said, you're more likely to move into a common state (because they're common) and so you'll probably end up in a low work capable state.

Hence entropy will probably increase and useable work will probably decrease.

*More accurately it is the log of the number of microstates.


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Parasomnium
Member (Idle past 774 days)
Posts: 2191
Joined: 07-15-2003


(1)
Message 6 of 51 (544596)
01-27-2010 12:29 PM
Reply to: Message 3 by Admin
01-23-2010 8:19 AM


Here's my attempt
I posted the text below in a Message 646 somewhere else, but it doesn't really belong there. This thread seems a good opportunity to set things right.

---

On the Second Law of Thermodynamics
by Means of Mammals and Fleas

or

The Preservation of Good Ideas in the Struggle for Sanity

Imagine a sealed room with a flea infested panda sitting next to a flealess tiger. Imagine further that the fleas jump from one to the other randomly. Obviously, in the beginning there will be more fleas jumping from the panda to the tiger than vice versa. But as more and more fleas jump to the tiger, some fleas will be jumping back to the panda. Eventually, a situation will be arrived at with equally as many fleas on the panda as there are on the tiger. The fleas keep jumping to and fro, but the situation overall will be in equilibrium. The system (comprised of the panda, the tiger and the fleas) has gone from minimal entropy (i.e. minimal disorder: all fleas on the panda) to maximal entropy (i.e. maximal disorder: fleas equally distributed between both animals).

Will there ever again arise a situation in which there are more fleas on the panda than there are on the tiger, or the other way around? If the situation stays exactly as described above then the answer is no, it will never happen. But what if we add some energy to the mixture, what will happen then? Well, let's try. We'll open up the sealed room and set fire to the tiger. The fleas on the tiger don't relish being burnt and jump to the panda more readily. The fleas on the panda no longer smell a fresh tiger nearby, but a smouldering one, and they forego the jump. The result is that the fleas end up, en masse, on the panda. It's amazing what a little heat can do.

There is grandeur in this view of science, with its several laws, having been originally thought of by Some Scientists in some form or other; and that, whilst this planet has gone cycling on according to the laws of Another Scientist, from so simple a beginning, endless science most beautiful and most wonderful has been, and is being thought of.

---


"Ignorance more frequently begets confidence than does knowledge: it is those who know little, not those who know much, who so positively assert that this or that problem will never be solved by science." - Charles Darwin.
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Son
Member (Idle past 1908 days)
Posts: 346
From: France,Paris
Joined: 03-11-2009


Message 7 of 51 (544610)
01-27-2010 1:50 PM
Reply to: Message 6 by Parasomnium
01-27-2010 12:29 PM


Re: Here's my attempt
I think that this thread needs some kind of feedback to see if the explanations worsk well with the layman due to the particuliar nature of this thread. I don't have an explanation in layman's term but I can provide some feedback.

I think that Rrhain's explanation works well to explain how entropy works and is easily understandable for most people. The water analogy explains well why entropy says perpetual motion machine can't exist. I also liked how he explained the difference between energy and usable energy.

I thought that Son Goku explained well why entropy worked the way it does but I'm less sure that it could reach the masses because lots of people seem to dislike maths (but I may be mistaken).

Overall, those were my 2 favorites explanation, Rrhain explaining how entropy works whereas Son Goku explaining why it works this way.
(If this post has no place in this thread, tell me, I will delete the content)


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gragbarder
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Posts: 30
Joined: 03-19-2010


Message 8 of 51 (551129)
03-21-2010 8:49 AM


For lay people ...

The second law of thermodynamics states that the TOTAL entropy of an ISOLATED system cannot decrease.

Entropy is a measure of disorder, and an isolated system is one for which neither matter nor energy enter or exit. Open systems can exist within an isolated system, and in such local regions entropy can decrease, as long as that decrease is offset by a larger increase in the open system's surroundings, such that the TOTAL entropy of the ISOLATED system does not decrease.


    
Fiver
Junior Member (Idle past 3042 days)
Posts: 26
From: Provo, UT
Joined: 04-17-2010


Message 9 of 51 (556142)
04-17-2010 3:38 PM
Reply to: Message 1 by Rrhain
01-23-2010 3:33 AM


Yes, you're right... altogether a very good description. I'm a bit of a physics nerd, but I'm more interested in Computer Science, especially Information Theory. 'Entropy' is a word that is thrown around in both fields and is (I find) generally difficult to get grasp on. The creationist ideas that I.T. 'Entropy' is the same thing as Thermodynamic 'Entropy', and that both equate to the general concept of 'disorder' is particularly misleading.

For example, if we consider a bag of marbles, some glowing hot, and some freezing cold, with all varieties of temperatures represented in the same bag, the common laymen would say that this bag shows far less 'order' than an identical bag in which all of the marbles are the same luke-warm temperatur (that second bag seems more neat and orderly, wouldn't you say?) And yet, from my understanding of thermodynamics, the second bag, with uniform temperature, is the one which demonstrates the greater amount of entropy.

Can someone correct or confirm my thinking here?


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Son Goku
Member (Idle past 42 days)
Posts: 1120
From: Ireland
Joined: 07-16-2005


Message 10 of 51 (557151)
04-23-2010 4:37 AM
Reply to: Message 9 by Fiver
04-17-2010 3:38 PM


Entropy and Disorder
Yes, you are correct. It is an example that entropy is not really related to what humans call order/disorder. There are less arrangements of the marbles which result in the bag with a speration of cold and hot marbles, hence it has less entropy.

Of course since all of this is related to statistics, you need a large amount of marbles for the statement to be meaningful.


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purpledawn
Member (Idle past 1535 days)
Posts: 4453
From: Indiana
Joined: 04-25-2004


Message 11 of 51 (557157)
04-23-2010 7:32 AM
Reply to: Message 1 by Rrhain
01-23-2010 3:33 AM


Bucket, Fridge, Engine
As a Layperson, I enjoy learning, but do need more visuals to understand some of the scientific concepts and jargon. How long I retain this information is another matter since I don't think of energy every day.

quote:
As you move the same energy around and around the system, you eventually come to a point where there isn't any energy left that can be used: It's all been bled away as entropy. But that assumes a completely closed system. If we had more water entering the system, we can afford to lose some of it sticking to the sides of the vessel.
I understand the water in the bucket analogy. Thank you

quote:
This is why concepts of "information" or "disorder" make no sense in their regular senses: It isn't a question of "ordering" the energy. It's that it is no longer physically available to do any work. When we reduce the concept of the Second Law and entropy to the phrase, "The universe tends towards disorder," what we mean is that the energy distribution of the universe is tending toward a uniformity. You see, energy transfer depends upon variations in energy states. If everything is the same, then there can be no reaction.
Yes, jargon tends to complicate the issue; especially when the common meaning of the word isn't used. That's why I stay out of the science arena. Religion has the same problem. Catch phrases or jargon are spoken that don't carry the common meaning.

quote:
Back to the water concept: If you have a cube of ice and a pot of boiling water, there will be energy flow between the two if we put them into a system: However, that energy flow only happens because of the disparity in their energy states. The heat flows from the boiling water into the ice and eventually, the water hits equilibrium. When everything hits the same temperature, no more energy flow takes place. All the heat is now unavailable to do any work because there isn't anywhere for it go. That doesn't mean there isn't any energy: It's just that there is no disparity between any two places in the system and thus, there can be no transfer of any energy of any kind.
I understand that the water eventually reaches the same temperature. I'm a little fuzzy on the energy transfer. Are you saying that energy transfer is like a balancing act? (I'm visualizing Parasomnium's Panda and Tiger) Energy is expended to melt the ice and once done no more energy is needed for that purpose. Kinda like when I'm relaxing in my chair and then grandson spills his drink on the floor. I now have to leave my relaxed position and expend energy to clean up the spill. Once done, I will go back to my relaxed position. The spill is the disparity. Once cleaned up, balance is back and I don't have to expend that energy.

quote:
Entropy is simply energy unavailable to do work.
IOW, "wasted" energy. Not used for the work at hand. Makes me think of my daughter as a teenager cleaning out the dishwasher. Take out a few dishes and then dance around the floor (headphones). The dancing around the floor is wasted energy.

quote:
There are no perfect engines.
Because there is always going to be wasted energy, right?

quote:
There are no perfect refrigerators.
I didn't quite understand where the wasted energy is in this one unless it is still the heat wasted in the moving process.

Overall I feel I understand what you're saying. Unfortunately, if we got into a science discussion, as soon as one goes back to the jargon the visual is lost. Son Goku lost me.

Thanks for the lesson. Learn something new everyday.


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Percy
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Posts: 18309
From: New Hampshire
Joined: 12-23-2000
Member Rating: 2.7


Message 12 of 51 (557159)
04-23-2010 8:28 AM
Reply to: Message 11 by purpledawn
04-23-2010 7:32 AM


Re: Bucket, Fridge, Engine
Hi Purpledawn,

I'm coming in in the middle and may not be able to help, but let me try.

Concerning your example of your grandson spilling his drink, cleaning it up, then going back to your chair, that's not really equivalent to what is being explained. Let me try a different visual.

Imagine a water wheel such as would be used in the 1800's to run a grinding mill for grain. There are many historic mills still around, and here's a picture of one (click to enlarge):

Now imagine that all the water we have to drive the wheel is in a water tank, and the tank has a valve that is currently closed. If we open the valve then water will flow from the water tank down the sluice and onto the wheel, thereby making it turn. The water in the tank has potential energy, the potential to do work by driving the wheel.

So let's imagine that we open the valve and let the water drive the wheel. The water flows down the sluice and onto the wheel and the wheel begins to turn. The wheel turns for a while, then the tank runs out of water and the wheel stops turning. All the potential energy in the water that was in the tank has been transformed into kinetic energy when it made the wheel turn (and presumably turned a millstone that ground some grain), and now that water is sitting in the pool below, unable to perform any more work for us. The water has no more potential energy.

In thermodynamic terms we would say that when the water was in the tank that it's potential energy to do work represented a lower entropy level, and that as the water flowed down the sluice and turned the wheel then went into the pool of water that its ability to do work was used up and its entropy increased. The water can no longer be used to drive the wheel because it is below the wheel now instead of above it.

One way this water could do further work is if there is another water wheel further down stream at a lower level. Another way is if we do work by filling pails with the water and lugging the water back up to the tank and pouring it in. After we've expended all this work on the water it will again have a great deal of potential energy in the tank, and we can again open the valve and let the water do work by turning the wheel again.

Hope this helps.

--Percy


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Taq
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Posts: 7673
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Message 13 of 51 (557182)
04-23-2010 12:51 PM
Reply to: Message 11 by purpledawn
04-23-2010 7:32 AM


Re: Bucket, Fridge, Engine
Kinda like when I'm relaxing in my chair and then grandson spills his drink on the floor. I now have to leave my relaxed position and expend energy to clean up the spill. Once done, I will go back to my relaxed position. The spill is the disparity. Once cleaned up, balance is back and I don't have to expend that energy.

The ATP (the main energy molecule in your cells) you use to stand up is forever gone. Your body must use ingested food to replace it, and the food will contain less energy than it did to start with. The mere act of sitting down does not replace that ATP. There is no machine or system you could use during the act of sitting down to replace all of the energy used in standing up (i.e. no perpetual energy machine).

IOW, "wasted" energy. Not used for the work at hand.

For the most part. For example, an internal combustion engine is fairly ineffecient. It pumps out hot air that is not used and it creates a lot of sound waves that are not used.

Refrigerators are another good example. They actually produce more heat than they do "cold". A refridgerator works by insulating the cold area from the heat it produces. However, overall there is an increase in heat due to the activity of the refrigerator. Hypothetically, if you opened all of the refrigerators in the world for a month it would actually heat the atmosphere, not cool it.


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purpledawn
Member (Idle past 1535 days)
Posts: 4453
From: Indiana
Joined: 04-25-2004


Message 14 of 51 (557190)
04-23-2010 1:57 PM
Reply to: Message 12 by Percy
04-23-2010 8:28 AM


Re: Bucket, Fridge, Engine
quote:
The water has no more potential energy.

In thermodynamic terms we would say that when the water was in the tank that it's potential energy to do work represented a lower entropy level, and that as the water flowed down the sluice and turned the wheel then went into the pool of water that its ability to do work was used up and its entropy increased. The water can no longer be used to drive the wheel because it is below the wheel now instead of above it.


If I'm trying to explain something to someone outside the field, I would avoid using the jargon. I would put the jargon in parenthesis. Right now I have to go back to the opening post and refresh my memory on entropy.

Entropy is simply energy unavailable to do work.

I understand the water wheel. It's easy. The water in the tank is available or in a position to drive the wheel (work). The water in the pool is not available or in a position to drive the wheel (work).

So lower entropy means more energy available for work and higher entropy means less energy available for work.

quote:
Concerning your example of your grandson spilling his drink, cleaning it up, then going back to your chair, that's not really equivalent to what is being explained. Let me try a different visual.
Your example doesn't have the idea of equilibrium as the water and ice example.

One thing to remember with laypeople is not to get to literal with our analogies. When we try to create an analogy or visual that helps us remember what something means, it may not be scientifically correct.

I am the heat, the spill is the ice cube. The hot water is balanced (me resting in a chair) until an ice cube is thrown in (the spill occurs). Rrhain said: If you have a cube of ice and a pot of boiling water, there will be energy flow between the two if we put them into a system: However, that energy flow only happens because of the disparity in their energy states. The heat flows from the boiling water into the ice and eventually, the water hits equilibrium.

In the hot water example, the hot water still contains energy, but there is no "work" to do. When the ice cube is dropped in there is "work" to do. Hence the spill thought.

I understand what you're saying concerning the water wheel, but it doesn't give me the idea of balance or equilibrium. That's the one I'm trying to make sure I understand.

Since entropy isn't something I say or think of every day; I would always have to go back and read the definition.

Unfortunately, entropy sounds like a disease.


This message is a reply to:
 Message 12 by Percy, posted 04-23-2010 8:28 AM Percy has responded

Replies to this message:
 Message 16 by Percy, posted 04-24-2010 7:11 AM purpledawn has responded

  
purpledawn
Member (Idle past 1535 days)
Posts: 4453
From: Indiana
Joined: 04-25-2004


Message 15 of 51 (557192)
04-23-2010 2:11 PM
Reply to: Message 13 by Taq
04-23-2010 12:51 PM


Re: Bucket, Fridge, Engine
quote:
The ATP (the main energy molecule in your cells) you use to stand up is forever gone. Your body must use ingested food to replace it, and the food will contain less energy than it did to start with. The mere act of sitting down does not replace that ATP. There is no machine or system you could use during the act of sitting down to replace all of the energy used in standing up (i.e. no perpetual energy machine).
As I was explaining to Percy, don't take lay analogies so literally. I was afraid someone would do what you just did. I'm the heat, the spill is the ice cube.

quote:
Refrigerators are another good example. They actually produce more heat than they do "cold". A refridgerator works by insulating the cold area from the heat it produces. However, overall there is an increase in heat due to the activity of the refrigerator. Hypothetically, if you opened all of the refrigerators in the world for a month it would actually heat the atmosphere, not cool it.
So it is the heat generated that is wasted. Heat that is not available to be used for anything else.

So entropy can be wasted energy or it can be unusable energy for a specific job.

The hot air is wasted energy. Not reused for anything. But in the case of Percy's waterwheel, the water at the bottom is unusable energy in its present position for that specific job.


This message is a reply to:
 Message 13 by Taq, posted 04-23-2010 12:51 PM Taq has responded

Replies to this message:
 Message 23 by Taq, posted 04-24-2010 11:52 AM purpledawn has not yet responded

  
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