# What exactly is entropy?

mishrashubham
I've been trying to understand this term ever since I first heard it from my friend. I have tried reading books, googled it and asking people but all I ever get are explanations about degree of disorder etc etc.

"Entropy is a thermodynamic property that is a measure of the energy not available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. "
That's what Wikipedia says about entropy and mentions about "useful work". I might be ignorant of scientific terminology but I assume useful work is work that is useful to humans. (No one ever seems to define 'useful work' though everyone keeps using it).

So my question is, the second law of thermodynamics concerns itself with entropy which I have gathered is defined with something based on human perception or is subjective to humans (useful work, disorder and the like). But isn't it supposed to be a law that must remain true regardless of whether humans existed or not. (I don't claim to understand this law as for that I must first understand what entropy is).

Why should a law be based upon something that is dependent on people (what might disorder for you might not be so for me).

Please help me in clearing this thing. Again I say that I'm not trying to criticize any law or scientist. The above are just mental arguments which I certainly look forward to be countered.

Thank You

all I ever get are explanations about degree of disorder etc etc. [...]
Why should a law be based upon something that is dependent on people (what might disorder for you might not be so for me).

Entropy is a fairly abstract concept, defined by formulas rather than ambiguous words. To call it a measure of disorder is just a way to visualize it with a particular situation in which the entropy concept applies. This shouldn't be taken too seriously, but it explains what happens if you represent a gas as a collection of hard spheres in motion.

What is your math background? (Some is needed to understand a better explanation than what you got so far.)

newflyer
Depending on what level of physics education you have, I found the statistical mechanics definition of entropy to much more tangible than the classical thermodynamics definition. Stat mech says that entropy is basically a measure of the number of ways to arrange a given system.

dsbeach
Machines based on thermodynamic principles run based upon energy or heat flow from high temperature to low temperature. Energy flows in at high temperature, work (in the physics sense) is done by the engine, and unused energy flows out at lower temperature. The "useful work" is simply the actual kinetic energy accomplished by the engine or machine as the engine runs. In a steam engine, for example, High temperature, high energy content steam is injected into a cylinder where it can expand and this expansion can be used to turn a crankshaft, etc. That is the work done by the engine. Low energy steam is then exhausted from the engine. The energy not used by the engine to do its work is kinetic energy of motion in the molecules of the exhausted gases from the engine. The efficiency of the engine is the work energy performed divided by the heat energy flowing into the engine at high temperature. The greater kinetic energy possessed by the atoms and molecules of the ejected gases from the engine represent matter in a more disorganized, less ordered state (KE for atoms or molecules means greater amplitude vibrations within the material, for example) and a higher state of entropy for that material than if it were at lower temperature.

Homework Helper
That's what Wikipedia says about entropy and mentions about "useful work". I might be ignorant of scientific terminology but I assume useful work is work that is useful to humans. (No one ever seems to define 'useful work' though everyone keeps using it).
Useful work is work done on macroscopic bodies (eg. turning wheels) rather than on molecules ("heat loss")

Heat flow from one body to another is a transfer of energy. Energy is the ability to do work. So when heat flow from a hotter body to a cooler body occurs, the molecules in the cooler body gain kinetic energy and those in the hotter body lose kinetic energy. So work is done on the molecules of the cooler body.

But just heating up the molecules will not turn car wheels, drive a generator etc. That requires useful work. That is what a heat engine can provide. But a heat engine can't convert all of the molecular kinetic energy into that kind of work. It can only convert a portion of it. The larger the temperature difference, the more useful work that can be produced. A Carnot engine provides the maximum amount of useful work that a heat engine operating between two given temperatures can provide.

Why should a law be based upon something that is dependent on people (what might disorder for you might not be so for me).
It isn't. That is why disorder is not a very good way to explain entropy. It is highly misleading.

Entropy is not a physical quantity any more than energy is. Energy is the ability to do work - to apply a force through a distance. Entropy is a measure of the ability of heat flow energy to do useful work (don't confuse work with power - maximizing useful work may require a very long time to do that work). Heat flow will always be in the direction of lower temperature - ie toward thermal equilibrium. The amount of useful work that you can get out of that heat flow depends on the temperature difference.

Entropy is defined in thermodynamics as path integral of dQ/T over a reversible path between two thermodynamic states. The quantiy is defined this way because it was observed that this path integral is never negative in any process. It is either 0 (where the actual process between the two states was a reversible one) or greater than 0 (where an irreversible process has occurred).

So entropy is just a useful mathematical tool for analysing thermodynamic processes. Just like potential energy is a useful tool for analysying gravitational processes. Neither "exist" in the physical sense.

AM

Entropy is not a physical quantity [...]
So entropy is just a useful mathematical tool for analysing thermodynamic processes. Just like potential energy is a useful tool for analysying gravitational processes. Neither "exist" in the physical sense.

This is highly misleading. Both potential energy and entropy are measurable quantities, hence exist and are physical in every meaningful sense of the word.

Indeed, entropy is a very important thermodynamic state variable, and appears in thermodynamics on equal footing with energy, mass, and volume (the four basic extensive quantities in thermodynamics).

Feldoh
In statistical mechanics it's simply defined as a way to count the number of quantum states a system can exhibit.

newflyer
This is highly misleading. Both potential energy and entropy are measurable quantities, hence exist and are physical in every meaningful sense of the word. I'd like to see a gravitational potential energy meter. I'd like to see a gravitational potential energy meter.

Take the scales in your bath room.

newflyer
Take the scales in your bath room.

That measures force.

Studiot

"Exactly what is.......?"

and seek a simple answer in five words or less, no work longer than five letters.

Science is not like that, you have to work at it.

Here is a simple non rigorous answer.

Energy as shown by indicator diagrams.

It was realised that the area under a pressure - volume (PV) graph represented work or energy.

Clausius introduced Entropy to pair with temperature to be able to draw a similar graph for heat energy. Again the area under the entropy - temperature (TS) graph gives energy.

The hatched areas in the attachment both represent energy of some sort.

Similar indicator diagrams can be drawn for other pairs of quantities

Voltage - Charge
Force - Distance
Surface Tension - Area
Magnetic Field - Magnetic Moment

etc

#### Attachments

• energypairs.gif
3.1 KB · Views: 1,010
Homework Helper
This is highly misleading. Both potential energy and entropy are measurable quantities, hence exist and are physical in every meaningful sense of the word.
Well, let me ask you this: does vis viva really exist? It can be measured ie: it is$= mv^2$.

It was discovered that $mv^2$ is a conserved quantity in some kinds of collisions (elastic ones). Does that mean it physically exists any more than $\frac{1}{2}mv^2$? And, furthermore, as a quantity, it is relative to the frame of reference in which it is measured. Does something physically disappear or appear when I measure it in a different frame of reference?

Indeed, entropy is a very important thermodynamic state variable, and appears in thermodynamics on equal footing with energy, mass, and volume (the four basic extensive quantities in thermodynamics).
I agree that entropy is very important and useful. So is Gibbs free energy, enthalpy, Helmholtz free energy. They are all very useful. Does their physical existence depend on them being useful? I could make up a thermodynamic relationship that would not be very useful: let's call it the "Mason energy" defined as $Mh = TS/PV + U$. There, I just made it up. Does it physically exist?

AM

mishrashubham
Thanks to everyone who replied. This made things clearer. Also thanks Andrew to make me understand what useful work is. Alas as as far as mathematics go I know next to nothing (I am still in grade 10). So I can't understand calculus for now (though they'll be teaching us that next year).

So it turns out that entropy is not a concrete physical quantity but a mathematical tool created so that we can study such things better. I have been worrying about such things before this as well (the field concept for example).

So let me get this clear once again... if a heated body can transfer its heat energy to move objects around (thus doing work), it has entropy. Is this right?

Stat mech says that entropy is basically a measure of the number of ways to arrange a given system.

Can give any examples?

Can give any examples?

entropy, S, is defined to be: S = k * ln(W)

k is the boltzmann constant, approximately 1.38*10^-23 J/K

W is the number of ways to arrange the system

//===================================================================
//===================================================================

in complicated examples W is difficult to calculate (there is an alternative formula). but consider the simple case of arranging 4 particles in a room: there is only 1 way of arranging all the particles on the left side of the room. hence W = 1. the entropy can be calculated:

S = k * ln(W)

= (1.38*10^-23) * ln(1)

= 0

however, there are 6 ways where two particles are on the left and two particles are on the right (think of arranging 4 people into pairs). hence W = 6. the entropy of this system is:

S = k * ln(W)

= (1.38*10^-23) * ln(6)

= 2.47*10^-23

//===================================================================
//===================================================================

there are more ways of arranging the particles so half are on one side, than there are ways of arranging all four of the particles to one side. the most probable state as it turns out, is one in which we have half the particles on one side, and this is called the equilibrium state. entropy is always a maximum at equilibrium, and thermodynamic systems tend to equilibrium. this explains why particles fill up the entire room rather than just go to one side.

Homework Helper
So let me get this clear once again... if a heated body can transfer its heat energy to move objects around (thus doing work), it has entropy. Is this right?
Not quite.

Think of a reservoir at a temperature Th relative to a reservoir at Tc (ie. temperature difference of Th-Tc). At this temperature difference, a given quantity of heat flow (Qh) from the hot to cold reservoir (using some system called a heat engine) has the potential to do a certain amount of work. The maximum amount of (useful) mechanical work (we'll just call it work) that can be obtained from Qh is Wm where Wm = Qh(1-Tc/Th) (where Tc and Th are the absolute temperatures in Kelvins).

In practice Qh will do less work than Wm, call it Wr. The amount by which Wr differs from Wm is of interest. Someone wanted to measure that difference and invented entropy (a guy named Clausius).

AM

That measures force.

it measures the gravitational force, which is the derivative of the gravitational potential.
For heights of the order of a bathroom, the potential is linear in the distance from the
center of the earth. Knowing all that, you can change the labels on your bathroom scales and get a measuring apparatus for the gravitational potential.

"Mason energy" defined as $Mh = TS/PV + U$. There, I just made it up. Does it physically exist?

It depends on your definition of existence, which seems to be very different from mine.

So it turns out that entropy is not a concrete physical quantity but a mathematical tool created so that we can study such things better.

In these terms, all concepts of theoretical physics are not a concrete physical quantity but a mathematical tool created so that we can study such things better. What has been said of entropy can be said equally well of temperature, force, mass, speed, position, time. You can't measure anything reliably without mathematical tools.

They get their physical existence only through mathematics and their usefulness.

mishrashubham
in complicated examples W is difficult to calculate (there is an alternative formula). but consider the simple case of arranging 4 particles in a room: View attachment 31512

there is only 1 way of arranging all the particles on the left side of the room. hence W = 1. the entropy can be calculated:

S = k * ln(W)

= (1.38*10^-23) * ln(1)

= 0

But doesn't that depend on where I want to put the division? Let us suppose I divide the box into 3 parts, left, right and middle. Then one of the four particles lying on the left half of your box may be situated in the middle of my box. If I go on increasing the number of divisions I will increase the number of ways to arrange the system thus reaching infinity. (Please I am not trying to simply contradict. Sometimes it is expected of me to understand certain certain obvious things but usually I don't. So please bear with me)

In these terms, all concepts of theoretical physics are not a concrete physical quantity but a mathematical tool created so that we can study such things better. What has been said of entropy can be said equally well of temperature, force, mass, speed, position, time. You can't measure anything reliably without mathematical tools.

They get their physical existence only through mathematics and their usefulness.

There is a difference between being a mathematical tool and being measured by a mathematical tool. Mass as you said measures the amount of matter. In that sense mass is a mathematical tool while matter physically exists. That is what I meant.

Studiot
Hello , Mish you are obviously a thoughtful sort.

What did you make of my comment?

mishrashubham
Hello , Mish you are obviously a thoughtful sort.

What did you make of my comment?

Haha Don't worry Studiot what i said was not because of your comment. Your post has helped a lot. The reason why I tell people to bear with me is because I have had school teachers erupting at me just because I made a contradictory question to what she was teaching. I just wanted to a better explanation and not disprove someone. But some people have inflated egos.

Studiot
All my comments were meant to be encouraging, sorry if they were a bit abrupt. The thoughtful bit was said because I thought you answered others rather well.

But I really mean my comment about the nature of entropy - your question.

It is hard to do entropy without getting mathematical, but that is what I tried to do.

mishrashubham
All my comments were meant to be encouraging, sorry if they were a bit abrupt. The thoughtful bit was said because I thought you answered others rather well.

But I really mean my comment about the nature of entropy - your question.

It is hard to do entropy without getting mathematical, but that is what I tried to do.

I understand, I never took your comments as discouraging. I understood what you tried to say about areas under graphs.

Homework Helper
it measures the gravitational force, which is the derivative of the gravitational potential.
For heights of the order of a bathroom, the potential is linear in the distance from the
center of the earth. Knowing all that, you can change the labels on your bathroom scales and get a measuring apparatus for the gravitational potential.
It measures weight. Gravitational potential energy is relative. You can calculate the gravitational potential energy of the object being weighed (in relation to some other point) only if you know the distance of that object from that point. The scale does not tell you that. From the measured force and knowing the relationship between gravitational potential energy and force AND knowing the vertical distance from the scale from the point you have chosen, you could determine the gravitational potential energy of the object whose weight is being measured in relation to that point.

AM

It measures weight. Gravitational potential energy is relative. You can calculate the gravitational potential energy of the object being weighed (in relation to some other point) only if you know the distance of that object from that point. The scale does not tell you that. From the measured force and knowing the relationship between gravitational potential energy and force AND knowing the vertical distance from the scale from the point you have chosen, you could determine the gravitational potential energy of the object whose weight is being measured in relation to that point.

Of course it is relative, just as energy is and temperature was before Nernst's law was discussed. That only differences are measurable doesn't make it unphysical.
For mass and length , only quotients are measurable (since one needs to have a reference mass and reference length to define the units). So according to you criteria, mass and length would be unphysical.

Physical is whatever physicists can measure, after having specified the necessary reference objects.

The floor of the bathroom (or,if you prefer, the mean sea level of the Mediterranean sea) is therefore as adequate a reference point for defining the gravitational potential as the Urkilogram is for defining mass.

There is a difference between being a mathematical tool and being measured by a mathematical tool. Mass as you said measures the amount of matter. In that sense mass is a mathematical tool while matter physically exists. That is what I meant.

Well, mass is not a mathematical concept - you cant find it defined in terms of mathematical axioms, it is not a concept of algebra, geometry, number theory, or calculus, say. Thus it is an intrinsically physical tool.

Moreover, matter is an abstraction from reality just as mass. How do you know that matter exists? You know that apples exist, and you postulate that it consists of matter,
like other objects, too. (But what about air? This hasn't been matter for a long time, but today it is.) You throw an apple and it smashes a window, and you postulate that there is energy. So matter and energy exist because they are useful unifying concepts. The same is the case with mass, temperature, and any of the concepts of physics. Take away their existence and physics is empty.

What exists, is Nature around us, classified by humans into different kinds of objects with different properties. But to talk objectively about Nature we need to ascribe reality to these objects....

Physics began when people found that certain parts of Nature can be described using mathematics in such a way that it was easier to control. A long history of research lead to concepts that we now take for granted and think of physically existing.

But of course, since existence is not a physical concepts, different people have different concepts of what it means to exist. Again, the question is which of these concepts is most useful. The dominant view in physics is, i believe, that things exist if they are measurable in principle, or (on the quantum level) if their properties in space and time can be described in terms of mathematical theories that provide a uniform description of large parts of nature.

Mueiz
I've been trying to understand this term ever since I first heard it from my friend. I have tried reading books, googled it and asking people but all I ever get are explanations about degree of disorder etc etc.

"Entropy is a thermodynamic property that is a measure of the energy not available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. "
That's what Wikipedia says about entropy and mentions about "useful work". I might be ignorant of scientific terminology but I assume useful work is work that is useful to humans. (No one ever seems to define 'useful work' though everyone keeps using it).

So my question is, the second law of thermodynamics concerns itself with entropy which I have gathered is defined with something based on human perception or is subjective to humans (useful work, disorder and the like). But isn't it supposed to be a law that must remain true regardless of whether humans existed or not. (I don't claim to understand this law as for that I must first understand what entropy is).

Why should a law be based upon something that is dependent on people (what might disorder for you might not be so for me).

Please help me in clearing this thing. Again I say that I'm not trying to criticize any law or scientist. The above are just mental arguments which I certainly look forward to be countered.

Thank You

Entropy is defined as path integral of dQ/T
This is the precise definition of entropy
But there is something which is not less important than definition if you want to know the meaning of a physical concept.
It is the significance .
The significance of entropy is different in different branches of science:
Theromdynamics: entropy is a property of the system that increase in all isolated processes.
Statistical Physics:entropy is a measure to disorder.
Mechanical Enginearing:entropy is a measure for energy not available to useful work.
Philosophy :entropy is an arrow of time.
Biology: entropy is a measure for ageing problems
(In PF : entropy is a confusing concept about which many questions are asked and no satisfactory answer is found )

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Homework Helper
Statistical mechanics definition of entropy ... thermodynamics definition.
It never helps when scientists decide to use the same term to describe different things.

This is highly misleading. Both potential energy and entropy are measurable quantities.
Potential energy is normally calculable, but not measurable. To "measure" gravitational potential energy, you'd have to have a device that measured weight and distance as it moved an object from one height to another.

Kevin_Axion
Here is something I posted a while back about entropy:

Entropy is the following, (this analogy derives for Brian Greene's book The Fabric of the Cosmos) say you have the book War and Peace by Leo Tolstoy the book is 1475 pages long. You have the ordered state, which is sequential pages, 1 through 1475, you can call this state $$\varphi$$. Now you take the state $$\varphi$$ or the book of pages and throw it in the air. Now it is clear that $$\varphi$$ becomes disordered, so lets call these disordered states $$\zeta$$ . Now if you throw it unendingly it becomes clear that the amount of disordered states $$\zeta$$ is far greater than the one ordered/sequential state $$\varphi$$. Therefore we can say $$\varphi$$ < $$\zeta$$. This is true because there is one ordered state which is sequential and approximately $$10^{500}$$ disordered states, I got this number by just approximating the amount Brian Greene said. Disorderd states $$\zeta$$ can just have 2 pages out of place or 500. Therefore the book began in an ordered state and as it evolved which is the throwing, it will become disordered. Now chaos (or the butterfly effect) has the analogy that say you have a butterfly in Africa. This butterfly flaps its wings and at first it has a very small effect on the air pressure, but over time as the weather system evolves this small pressure system the butterfly created increases exponentially and hence the slight change in initial conditions can cause a hurricane to be formed across the Atlantic. Or at the beginning of the Universe if the expansion rate was slower by a billionth it would have expanded far to quickly for gravity to create a stable super clusters and structure formation. Or if the electromagnetic force was weaker molecules wouldn't form.

Kevin

yuiop
I've been trying to understand this term ever since I first heard it from my friend. I have tried reading books, googled it and asking people but all I ever get are explanations about degree of disorder etc etc.

"Entropy is a thermodynamic property that is a measure of the energy not available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. "
That's what Wikipedia says about entropy and mentions about "useful work". I might be ignorant of scientific terminology but I assume useful work is work that is useful to humans. (No one ever seems to define 'useful work' though everyone keeps using it).

So my question is, the second law of thermodynamics concerns itself with entropy which I have gathered is defined with something based on human perception or is subjective to humans (useful work, disorder and the like). But isn't it supposed to be a law that must remain true regardless of whether humans existed or not. (I don't claim to understand this law as for that I must first understand what entropy is).

Why should a law be based upon something that is dependent on people (what might disorder for you might not be so for me).
Hi Mish. "Useful work" does not have to be useful to humans. A bubble of hot lava under the Earth's crust has low entropy and potential to do work (force x distance) but when it erupts humans might not think its work is useful if the volcano decimates a populated city. A better way to define "useful work" is "coherent motion". In a hot gas under pressure (like the ignited fuel mixture in a car engine) the molecules in the gas have kinetic energy and are bouncing all over the place in random directions. If the piston is locked in place near maximum compression, this random motion of the molecules is not changing much. On a microscopic scale a lot is going on, but on a macroscopic scale the volume of the gas is not changing and the pressure of the gas is not changing. Now if the piston is unlocked, this random incoherent motion of the gas molecules is converted into the coherent directed motion of the piston molecules. The parallel motion of a large number of molecules in a given direction or "Coherent motion" rather than random motion, is our definition of "useful work". It does not have to useful from the point of view of humans and if the motion of the piston caused the car to move and run over a pedestrian, then the victim certainly would not consider it useful :tongue:.

Now let us consider a cylinder of gas with a double sided piston in the middle. Initially the pressure is equal on both sides of the piston and average number of molecules per unit volume is the same. Now we artificially move the piston to one end increasing the pressure one side and decreasing the pressure the other side. Technically by the rules of thermodynamics we should let the temperature equalise on both sides of the piston. Now we can calculate the entropy of the system using pressure and volume etc. Another way to calculate the entropy on a microscopic level is to calculate the number of ways to distribute the molecules. Imagine that the volume is divided up into lots of tiny compartments just big enough to contain a single molecule. In the low pressure side there are a lot of compartments that are empty and so there are lots of ways to rearrange the gas molecules. This relates to the "degrees of freedom" or "counting states" or "disorder" or "chaos" that you may read about. Anyway, we say the low pressure side has a greater entropy than the high pressure side. We can also calculate the total entropy of the system and let's say this works out to be x. Now we release the piston and it naturally moves back to the centre of the cylinder doing work (coherent motion). Now we calculate all the different ways of rearranging molecules (or use temperature and pressure and volume) to calculate the entropy of the system again and find that the total entropy is now greater than x. The total energy of the total system has not changed, but as a result of the increase of the entropy, the energy available to do useful (coherent) work has reduced. Does that help any?

Gold Member
Science has a lot of concepts in it. Some of them are nearer to our everyday experience than others and you can sometimes fool yourself that you 'understand' something because it seems to relate directly to some everyday phenomenon or sensation. Not everything is lioke that though.
SO, for instance, we have a feel for what Mass is, or what a Force is but Temperature is a bit harder to nail and Special Relativity is even harder still.
In the end, the only way to appreciate these things as fully as possible is to use Maths. It's the only way have a hope of avoiding pitfalls and misconceptions. Yes, its a bit elitist but so is professional sport and music; not everyone can or will get into it deep enough to be an 'expert'. That's life. You just can't always expect a two word answer to what may seem to be a simple question.

Studiot
Hello, yuiop.

I am a bit worried about your piston example, perhaps you could clarify a few points?

What is the system in this case?

For the following I assume you mean the system to comprise the two gas chambers and the piston?

Now when the piston is moved sideways work is input to the system by whatever pushes the piston. Does this raise the temperature of either chamber and then what happens to this temperature?

When you release the piston you say 'work is done' Done on what? Does not the work done by the expanding gas in the high pressure chamber equal the work done compressing the gas in the low pressure one? Since they are both part of the system, what do you mean by work is done?

Since entropy is a state variable and you have taken the system round a complete cycle, returning it to its starting point, how does the total entropy differ at the end from the beginning?

Galap
Entropy is the fact that, in a system, there tend to be many more states we call 'uninteresting' than states we call 'interesting'.

But doesn't that depend on where I want to put the division? Let us suppose I divide the box into 3 parts, left, right and middle. Then one of the four particles lying on the left half of your box may be situated in the middle of my box. If I go on increasing the number of divisions I will increase the number of ways to arrange the system thus reaching infinity.

the example was simplified; i assumed there were only two individual particle states ("on the left" or "on the right"). more generally, http://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)#Counting_of_microstates" is not so easy, although in certain situations you can make a good approximation.

there is an alternate formula for calculating entropy through heat and temperature (dS = dQ/T) which is equivalent to the number of ways method (using S = k * ln(W)). you can use either formula to calculate the entropy, and in the situations where the approximate to the number of ways is good, both the formulas agree.

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