# What exactly is entropy?

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

A. Neumaier
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.)

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.

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.

Andrew Mason
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

A. Neumaier
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).

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

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.

A. Neumaier I'd like to see a gravitational potential energy meter.

Take the scales in your bath room.

Take the scales in your bath room.

That measures force.

"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
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Andrew Mason
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

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.

Andrew Mason
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

A. Neumaier
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.

A. Neumaier
"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.

A. Neumaier
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.

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.

Hello , Mish you are obviously a thoughtful sort.

What did you make of my comment?

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.

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.

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.

Andrew Mason
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

A. Neumaier