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How to introduce the concept of entropy

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Am I to understand that entropy is introduced as part of the first law of thermodynamics. And the properties of entropy are determined without a second law. And the second law establishes the condition for the growth of entropy. In other words. Is it possible to say that entropy is introduced as a convenient parameter for determining the inversion of the heat transfer sign, for example?

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19 minutes ago, Sheldunov said:

Am I to understand that entropy is introduced as part of the first law of thermodynamics. And the properties of entropy are determined without a second law. And the second law establishes the condition for the growth of entropy. In other words. Is it possible to say that entropy is introduced as a convenient parameter for determining the inversion of the heat transfer sign, for example?

 

AS far as I know Entropy is not commonly introduced in this way, although in my opinion it is a better way than the common stumbling explanation offered concerning Carnot cycles.
I think Entropy is best introduced in relation to indicator diagrams as a natural progression from PV work rather than Carnot cycles, which are best delayed until after the Second Law is broached.

Note since you have posted in Classical Physics I asssume you mean classical Thermodynamics?

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50 minutes ago, studiot said:

AS far as I know Entropy is not commonly introduced in this way, although in my opinion it is a better way than the common stumbling explanation offered concerning Carnot cycles.

Of course we can introduce entropy in many ways, but the question is can we introduce entropy precisely this way. So you think we can, aren't you?

2 hours ago, studiot said:

I think Entropy is best introduced in relation to indicator diagrams as a natural progression from PV work rather than Carnot cycles, which are best delayed until after the Second Law is broached.

Сould you please explain a little more details. I am not sure whether I understood you correctly.

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You can also introduce entropy axiomatically from statistical mechanics. From that approach temperature is more of a derived concept and entropy has a very intuitive meaning.

4 hours ago, studiot said:

I think Entropy is best introduced in relation to indicator diagrams as a natural progression from PV work rather than Carnot cycles, which are best delayed until after the Second Law is broached.

I always hated the Carnot approach, @studiot. I'm very interested in the method that you suggest.

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10 hours ago, joigus said:

You can also introduce entropy axiomatically from statistical mechanics.

Yes, but this is not about choosing between statistical or thermodynamical introducing. The question is about specifically THERMODYNAMICAL entropy definition and about using only the first law of thermodynamics in it.

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1 hour ago, Sheldunov said:

Yes, but this is not about choosing between statistical or thermodynamical introducing. The question is about specifically THERMODYNAMICAL entropy definition and about using only the first law of thermodynamics in it.

Sorry. I thought it was about how to introduce the concept of entropy.

Quote

How to introduce the concept of entropy

By Sheldunov, 16 hours ago in Classical Physics

My mistake.

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12 hours ago, Sheldunov said:

Yes, but this is not about choosing between statistical or thermodynamical introducing. The question is about specifically THERMODYNAMICAL entropy definition and about using only the first law of thermodynamics in it.

 

23 hours ago, joigus said:

I always hated the Carnot approach, @studiot. I'm very interested in the method that you suggest.

 

OK so studying Thermodynamics is like watching a good play, film or reading a good book.

The actors are introduced and a lttle bit of information is given about them.
Enough to know who they are some of their relationship to the other actors and why the audience should be interested in them.
As the play unfolds more information is revealed.

In the same way, in Thermodynamics we first learn about the main quantities, their relationship to the other main quantities and why we might be interested in them.

 

Now sonmeone who is just starting Thermodynamics is at the beginning of the play.

He has learned about

Boyle's Law P1V1 = P2 V2  (1662)

Charles Law V1T2 =V2T1  (1780)

Avogadros Gas Law PV = nRT (1812)

Something about energy (Young 1802)

Something about 'work' as the force times distance

Something about 'heat' being mass times specific heat

So he is in the same position of scientists in the first half of the 19th century that is 50 years before the first version of the First Law of Thermodynamics.

So to get to the First Law some study needs to be carried out filling in the gaps and strengthening the definitions of these variables.

Additionally he will need to learn some of the structure of Thermodynamics.
 

In particular what is a system and what is a process and what is a state and a state variable.

Once these ideas have been absorbed (they are all important) additional details can be studied and understood.

Such the meaning of isolated, open, closed and flow systems.

The difference between reversible and irreversible processes

The difference between intensive and extensive variables

and so on.

 

So our student learns about different types of energy, including something called internal energy (symbol here U) and arrives at the First Law

This connects the State Variable: Internal Energy to two variables heat and work.
These are not system properties but variables that connect the system to its surroundings.

U2  -  U1 = q + w

But he notices that whilst q and w can easily be measured or directly calculated from measurable variables , U cannot be directly measured.

Furthermore the other state variables already mentioned P, V and T can also be easily measured and the equations stated allow calculation of w.

So Internal Energy is the first quantity he has come across that he cannot directly measure.
However he realises that it is a very important quantity since it is like a bank balance which = money in - money out.

This was so important to early engineers in the development of steam engines that they invented a special diagram called an indicator diagram to show the work part  - w.

This work is the area under a P - V graph or plot

And they even invented a mechanism to fix to steam engines called an 'indicator', which is where the name came from.

 

Now you will have noticed by now that there are three measurable connected variables, P, V and T and that we have only used one of them.

So guess what?
Someone said, "Wouldn't it be nice if we had another variable we could plot against temperature (the unused variable) to calculate q in the same way ?"

Bingo there you have it Entropy as Clausius named it.

I have drawn the indicator diagram for Temperature v Entropy side by side with the one for Pressure v Volume so you it can be seen that they have the same format.

indicator1.jpg.7adf546eb1272a09efb183844df11a75.jpg

Entropy is not some mystical property.

It is simply the thing we need to multiply temperature by to get a particular energy in this case the heat exchanged across the system boundary.

Again we cannot directly measure entropy, but simple equations exist to calculate it from easily measurable ones.

 

Now that we are not frightened of it we can study more details and proceed to the next level of the development of Thermodynamics.

Also if there are any details in the foregoing that are unclear please ask.

 

 

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11 hours ago, studiot said:

So guess what?
Someone said, "Wouldn't it be nice if we had another variable we could plot against temperature (the unused variable) to calculate q in the same way ?"

That was a great summary of how scientists and engineers arrived at entropy from the formalism of thermodynamics. +1

11 hours ago, studiot said:

Also if there are any details in the foregoing that are unclear please ask.

It was pretty clear to me.

11 hours ago, studiot said:

Now that we are not frightened of it we can study more details and proceed to the next level of the development of Thermodynamics.

Perhaps the next step will be fixing of the zero of entropy against the zero of temperature...? But, please, carry on. :-)

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16 hours ago, studiot said:

OK so studying Thermodynamics is like watching a good play, film or reading a good book.

Thank you very much for your explanation. Your concept of teaching thermodynamics is quite good and comprehensive and mainly matches ours. Of course we can continue the story of thermodynamics as a play, but my question was about one little precise act of this play. This act is about introducing entropy without the second law. 

So this little act is about one specific question that matter the most for me in our conversation. The question is can I suggest that we can use only first law to introduce entropy as state variable? And second law is used to determining cardinal property of entropy (growth in closed system in irreversible process).

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3 minutes ago, Sheldunov said:

Thank you very much for your explanation. Your concept of teaching thermodynamics is quite good and comprehensive and mainly matches ours. Of course we can continue the story of thermodynamics as a play, but my question was about one little precise act of this play. This act is about introducing entropy without the second law. 

So this little act is about one specific question that matter the most for me in our conversation. The question is can I suggest that we can use only first law to introduce entropy as state variable? And second law is used to determining cardinal property of entropy (growth in closed system in irreversible process).

I posted the 'play' because I don't know if you are asking because you know what entropy is and want to teach it or because you are studying it yourself?

So the 'play' was a bit of a middle way.

I guess from what you are saying that you are teaching it.

Then the main point from the play was that you don't tell them everything at once.

Just some basic but important part and let them get used to it before telling them more.

In particular I don't know the level of maths available. It is easy to show that entropy is a state variable if they can do cyclic integrals.
 

Otherwise you can just say "This is a state variable and we will come to how and why later"

or you could discuss it in the context of the difference between reversible and irreversible changes.

You could also note there are many more derived quantities such as Free energy, Chemical Potential and so on depending upon which subject you are teaching it in.

 

You do not need the Second Law  - although that came first for historical reasons.
If you come to the second law before meeting entropy then the law can be nothing more than a definition for you. Not a Law at all.

Here is an Engineering introduction along these lines.

entropy1.thumb.jpg.2a11fae8f756dbfe4f8ff6e710ba53ba.jpg

entropy2.thumb.jpg.cebcce5d099fb29f58b4c20c002fa27e.jpg

entropy3.thumb.jpg.d1137636fa5112f7767326aa9545cb80.jpg

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19 hours ago, studiot said:

You do not need the Second Law  - although that came first for historical reasons.
If you come to the second law before meeting entropy then the law can be nothing more than a definition for you. Not a Law at all.

I am very grateful to you for your answer! Thank you very much!

19 hours ago, studiot said:

In particular I don't know the level of maths available. It is easy to show that entropy is a state variable if they can do cyclic integrals.

Could you please explain, how we can prove that entropy is a state variable in general? I know how to do it with ideal gas, but I would be very pleased if you tell me how to do it in general.

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Posted (edited)
On 6/5/2020 at 6:36 PM, Sheldunov said:

Am I to understand that entropy is introduced as part of the first law of thermodynamics. And the properties of entropy are determined without a second law. And the second law establishes the condition for the growth of entropy. In other words. Is it possible to say that entropy is introduced as a convenient parameter for determining the inversion of the heat transfer sign, for example?

Yes and no. It's easy enough to set up a theoretical process for an ideal gas that runs up and down the P, V relation PV^(Cp/Cv) = constant using just the 1st Law and the concept of reversible heating (ie no departure from thermal equilibrium). This relationship is the classic textbook case for obtaining the maximum shaft work output from gas expansion, (or minimum shaftwork for gas compression in the reverse sense).

Explaining why this conserved quantity represents a brick wall for power generation cannot be explained by the First Law alone - for that you need a concept that looks a little deeper into this reversible heating idea and recognises that 1 Joule at 1000 K is a hell of a lot more useful than 1 Joule at 100 K. Hence we end up with the idea of entropy S, defined by its derivative dq = TdS and find that PV^(Cp/Cv) = constant is synonymous with S = constant. This conserved quantity is quite distinct and independent of the 1st Law conserved quantity PV/T = constant. For the general case of course dS>0 due to dissipative factors like friction, ergo the 2nd Law.

The directional signing of Q and W is purely conventional reflecting industrial processes where heat is typically input to a gas and shaft work is output - Q and W are both defined as positive in this scenario.  

 

 

Edited by sethoflagos

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