Hello Uriel,

Yes, an electron, for example, can carry heat, i.e., it has a definite heat capacity. See the following article for a historical discussion of how Lord Kelvin (William Thomson) proved this fact back in the 1800s.

Bernard S. Finn; Thomson's Dilemma; Physics Today; Vol. 20; No. 9; September 1967; pp. 54-59.

But remember, the electron Thomson was considering were the conduction electrons in a metal. One can certainly average the heat capacity over all these electrons to obtain the heat capacity for a single electron, but the electron has to be part of an aggregate.

As far as entropy is concerned, I do not know whether a single particle can have entropy associated with itself. Entropy (or more precisely the change in entropy, dQ/T) is an extensive variable, i.e., it depends on the size of the system due the dQ in the numerator. Temperature and pressure, for example, are intensive variables since they do not depend on the size of the system. Volume, mass and heat are other examples of extensive variables. If you have a single electron, it has a finite dQ as I mentioned in the first paragraph, but the question is does it have a defined temperature by itself? Another way to ask this question is as follows. Does an electron obey the Zeroth law of thermodynamics? The Zeroth law is how temperatures are defined and operationally measured. When it is part of an aggregate, such as the conduction electrons in a metal, they certainly obey the Zeroth law, but does that make sense in the case of a single electron?

Can you tell us why you are asking this question, i.e., what is the context of the question?

Regards,

Tom Cuff

Dear Franklin,

If the single particle evolves stochastically (consider, for instance, a colloidal particle in water), one can associate a stochastic entropy with the state of the particle. This association is essential for stochastic thermodynamics. Such association is also meaningful for systems with discrete states, like a single spin coupled to a thermal bath, for instance. I highly recommend you to read the paper in the link below. It may help not only to answer your question, but also to digg deeper into stochastic thermodynamics.

As for heat, I'm not sure to understand your question. What a particle carries is energy. Heat is related to transfer of energy. Beyond this remark, in all the cases mentioned above the particle can indeed exchange heat with the thermal reservior and such heat can be defined even at the stochastic level. Hope it helps.

Sincerely,

Reinaldo.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.040602

Dear sir

Particle can carry energy in several figures one of them is heat. Energy devided to two sections usable and non-usable so particle can have entropy and exergy

Dear Mehdi Aliehyaei ,

There's the danger of quickly getting out of the context of the original question, but I feel compelled to answer anyway. Again, what a particle carries is energy. Heat is just a demonination for one on the forms in which that energy can be transferred. Heat is not a state function, so you cannot say "this macroscopic body, or this system or this particle, has this amount of heat".

Best,

Reinaldo.

Hi Franklin,

As I said before, I believe that many answers to your questions are in stochastic thermodynamics. Did you have a look to the paper I attached? It's is for stochastic systems, but stochastic thermodynamics has also been extended to quantum systems. Note that this is a field with already 20 years of activity. So, some of these ideas are actually not new, but somehow established.

Best wishes,

Reinaldo.

By the way, being precise, I didn't say heat is a form of energy. I'm saying that heat is a form of energy transfer. Subtle difference, but important. Energy is a state function. Heat is intrinsically related to a process.

Thanks Reinaldo !

Regards !! :)

Hello Reinaldo,

Can you tell us where you read that heat is only a form of energy transfer; that heat is not a form of energy; and that you cannot say that a macroscopic body has this or that amount of heat? I am curious about your source(s) for these parochial statements.

The first law of thermodynamics is usually quoted as saying that energy cannot be created or destroyed, it can only be only transformed. But you say that heat can only be transferred, and that it is not a form of energy, hence, it cannot be transformed, simply tranferred. I assume you would also say the same thing about work since it too is not a state variable, but is a path variable like heat.

If you cannot say that a macroscopic body has an amount of heat, how do you explain the parameter known as heat capacity, either at constant volume or constant pressure? It would seem that you are implying that a macroscopic body or system can transfer something that it cannot contain. Consider the textbook by Walter Moore; Physical Chemistry, 3rd Ed.; Prentice Hall; 1962. On p. 42 it has a section, Section 8, titled: THE HEAT CONTENT OR ENTHALPY . In this section, Moore points out that the state variable E+PV (where E is is internal energy, P is pressure and V is volume) represents, in the case of a constant pressure process, the heat absorbed by the system, i.e., it represents the heat content of the system. In a footnote at the bottom of p. 42, Moore points out that heat content is a different function from heat capacity, and that because of this the term 'enthaphy' is preferred to the term 'heat content' to avoid confusion.

Can a macroscopic body posses or contain a state variable such as pressure, temperature or volume? It would seem to me that, like the path variables heat and work, they also can only be assessed at the boundary of the macroscopic body or system. For instance, to assess pressure I could enclose the gas (the system) inside a piston cylinder, and use a weight on the piston to determine the pressure: Force/Area (of the piston), assuming the piston is weightless and rides without friction in the piston cylinder bore. To determine temperature, I could invoke the zeroth law of thermodynamics and place the macroscopic body or system in thermal contact, at its boundaries, with other systems. And, of course, the volume of a macroscopic body is its boundaries.

Regards,

Tom Cuff

Hello Thomas Cuff ,

As far as understand, any reasonable book on thermodynamics would support my claim. My statement is not parochial, thermodynamics, and in particular, stochastic thermodynamics, are my main fields of research.

Heat is not a state function, it cannot characterize a system. State functions are energy and entropy in the microcanonical ensemble, and all the thermodynamic potentials associated to other ensembles (which can be accessed via Legendre transforms) e.g., Helmholtz and Gibbs free energies, enthalpy, and so on.

I believe, if you allow me to say it with all respect, that you have probably missinterpreted two key concepts of thermodynamics. It's of course a common thing, thermodynamics is much more subtle than what is appears.

The first point is the first law of thermodynamics. Written (in differential form) as dU =\delta W-\delta Q, it literally means that there're two ways of changing the energy of a system. One is by performing work on it (work corresponds to the energy change due to a modification of the energy surface), and by exchanging heat (heat is energy change just due to moving from one point in the energy surface to another point with different energy, i.e., it's associated to currents in phase space). To avoid controversy, it's worth clarifying that in the way I wrote the first law, heat is positive when it's released to the environment. One can, of course, change the sign of the heat, keeping in mind the corresponding prescription. I also clarify that the formula I wrote corresponds to a closed system (not to confuse with an isolated system). It can be generalized for other ensembles (in an open system, for instance, chemical potentials also enter in this expression).

The first law refers to variations of the energy of the system, not to the energy itself. This is clear, neither heat nor work are exact differentials (they depend on the process) but they add up to the variation of a state function, which is process independent (only initial and final states are important).

The second point is heat capacity. Yo ask "If you cannot say that a macroscopic body has an amount of heat, how do you explain the parameter known as heat capacity, either at constant volume or constant pressure? " By reading carefuly this question, it seems to me that you believe that heat capacity is about how much heat can a body "carry" and it's not. Heat capacity is, by definition, the amount of energy that a body has to absorbe or release in the form of heat in the given conditions, to vary its absolute temperature in one Kelvin. So, I don't understand how this lead you to support that a body can "carry" heat, or that heat can describe the body.

One last comment, regarding your reference and example. You seem to miss the point here. Heat, of course, in certain conditions may be an exact differential and characterize the states, as well as energy. Consider, for instance, the constant pressure process you mention. In that case, the work is an exact differential (note that PdV= d(PV) because P=const.), and given that energy is always an exact differential, then the heat will also be it, by virtue of the first law. This means that along those lines of isobaric processes, you can use heat as a state function. But this is not the general scenario. The generic thing is that heat is NOT a state function, since it's not an exact differential.

Best wishes,

Reinaldo.

By the way, Thomas, along those isobaric processes, heat (which in that particular case is a state function) and enthalpy, are the same thing, as you correctly stated. The only issue is that heat being a state function is not the general scenario.

Regards,

Reinaldo.

Hello Tom , Hello Reynaldo,

thank you both for adding to this conversation.

I am just in my third year of PhD, so I don't have much experience in physics and thermodynamics as you guys have.

What I can add to the discusion is what I remind from the years in college where the Thermodynamics' and Heat Transfer professors use to teach, or at least these statements are what from those years, I always kept these concepts in my mind, may be oversimplified or may be addressed to just a particular set of cases :

" Heat is energy, and its units are Joules, it has to be represented with the letter Q (or q) but not Q_dot, and Heat transfer refers to a Heat Flux (or a Heat Rate) which is an energy transfer and it has to be represented by the transcription Q_dot (or q_dot) and it is measured in J/s "

I don't know wheteher this is a simplified explanation, or is just and approximation of the more general definition of the difference between both.

Maybe my professors were n't so good after all.

Best Regards to both !,

Thank you for your words, is so nice to read and learn from such interesting replys.

:)

Hello Franklin,

Yes, other thing is dissipation rate (or heat transfer rate as you say), which the rate at which heat is released (or absorbed) per unit time. Heat, itself, relates to a change in the energy of the system, and is associated to transitions between states (or equivalently, to currents in phase space, looking at it from a statistical mechanics perspective).

On the other side of your original question, as anti-intuitive as it sounds, we can now associate entropy to a single particle. The average of that entropy (which is a fluctuating quantity in nature) coincides with Shannon entropy.

In any case, thanks to you for the interesting question!!!

Hello Reinaldo,

I do not know what you mean when you say a careful reading of my question about the term 'heat capacity'. The word 'capacity' means the ability to receive, contain, or absorb so it is perfectly logical to think of a substance containing thermal energy in the form of heat. Walter Moore's textbook used the term 'heat content' as a synonym for the term 'enthalpy', and the word 'content' means that which a thing contains. On pp. 70-71, Moore's book has diagrams of the Carnot heat engine, in which there is a hot and cold "heat reservoir". In this case, a reservoir is a receptacle for something that can be contained, and, in this case, the something is heat.

I suppose Moore could have called the term 'heat capacity' something like 'energy capacity' or 'thermal energy capacity', but he did not. Now you can say that the term 'heat capacity' is a misnomer brought about by misguided, inappropriate, or misleading use of language, but these terms ('heat capacity', 'heat content', 'heat reservoir', etc.) are common currency amongst those studying, teaching, or applying thermodynamics. And, of course, the language of science and technology is filled with misnomers such as the term 'semiconductor' - from the German compound noun 'Halbleiter' (= half + conductor), when the only semiconductors known were minerals such as galena, PbS; where half the compound is a conductor (metal), in this case lead (Pb) - or the term 'thermionic' - when early investigations revealed that many different charged particles were emitted in early vacuum tubes with the poor vacuums and uncertain processing of materials, though, today an electron is not considered an ion.

Saying repeatedly that thermodynamics is "subtle" seems to me simply a dodge for not saying what you mean, and meaning what you say; where the 'you' I am speaking of are all those people interpreting the science of thermodynamics. The subject of thermodynamics is not subtle, but it is obscured by language. Ludwig Wittengstein said something to the effect that the limits of my language are the limits of my science.

Regards,

Tom Cuff

Hi Tom,

Just a quick answer (I'm travelling). It's unfortunate, but yes, there're some terms in thermodynamics which do have (I believe due to merely historical reasons) misleading names. I didn't made up the definition of heat capacity. It's the actual definition (said with my own words) so the discussion about the use of the word "capacity" is a mere issue of language which I don't believe worth discussing further. As regards the heat reservoir, it's a system that gives an takes energy ONLY in the form of heat (i.e. without doing work). Thus, a heat reservoir can also be characterized by heat itself (refer to the first law again to convince yourself that in any process occurring in a heat reservoir, heat is an exact diffefential). But again, you're only bringing the exceptions or special cases to the rule that heat is not, in general, a state function. Please, give it a thought, I've not made mutually contradicting statements so far.

I do believe that thermodynamics is subtle. Sommerfeld once said "Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you any more". It's a funny quote, but I do agree with it saving some distances.

Best,

Reinaldo.

Hello Reinaldo,

Your statement that terms such as 'heat capacity', 'heat content', and 'heat reservoir' are "merely historical" shows that, to paraphrase Sommerfeld, you have gotten too used to your view of thermodynamics. Early practitioners of thermodynamics believed that substances contained heat, hence, terms such as 'heat capacity', which implied from the common English language meaning of the word 'capacity' that the body contained heat. Their belief was, what you might call, an article of faith. Today, you believe that heat is only a form of energy transfer; that heat is not a form of energy; and that you cannot say that a macroscopic body has this or that amount of heat. But you are worshiping at someone else's pew since I know that you yourself did not come up with these newer parochial (narrower) views on heat.

As far as I can tell, the ideas you espouse about heat were codified in a chapter in the following book:

J.A. Beattie, W.H. Stockmayer; An account of the thermodynamics of gases, principally in terms of the Beattie-Bridgman equation; in H.S. Taylor, S. Glasstone (Eds.); A Treatise on Physical Chemistry, Volume 2, 3rd Edition; Van Nostrand; 1951; pp. 187-290. [Note, this is a hard book to locate as there do not appear to be any copies available on the used book market, although Vol. 1 appears to be available; only the 1924 version is available as a viewable scanned digital copy at various web sites, the 1954 edition is scanned but not viewable due to copyright restrictions; and it is not held by many libraries.]

This citation I found in the book by Castellan, who swears fealty to the same ideas about heat as you do, although, on the plus side he, at least, knows the source of these ideas.

Gilbert W. Castellan; Physical Chemistry; Addison-Wesley Publishing Company; 1964; pp. 89-91 (text) & 695 (citation).

It is not clear to me what experiments, if any, persuaded Beattie and Stockmayer - and by proxy Castellan - to adopt their newer ideas about heat and discard the so-called older ideas.

As far as the idea that the change in internal energy is an exact differential or state variable, let me say that operationally this is what you would call a special case. Take, for example, the Joule expansion experiment or its more sensitive version, the Joule-Thomson expansion experiment. If you read their original papers, you will see that one of the precautions they had to take, when allowing the compressed gas in chamber A to expand into the vacuum of chamber B, was to prevent the generation of sound as this energy could not be accounted for easily.

Anyway, I have enjoyed our discussion, and I too wish to thank Uriel for his original question. I am still not sure I understand your explanation about single particle entropy as I found it terse and too couched in technical jargon. And, yes, I know I could just read the paper by Seifert you referenced, but since I assume you read it, I thought you could provide an English language paraphrasing of it for us. I will leave this discussion now, and let someone else continue it, if they wish, though no one else has stepped forward, so far.

Regards,

Tom Cuff

Well, Tom, it's sad to hear that you continue to see my points as parochial and qualify them as cult-related stuff. You have not, by the way, succeeded to find a flaw in my arguments. I think I've clearly stated the fact heat is not a state function except for few finely tuned cases where it's an exact differential and you can, defining a reference state, use it to label all states along the processes where it remains to be a state function. For me (probably too accostumed to it, true) this is one of the most basic (in the sense of fundamental) aspects of thermodynamics, and I'm genuinely surprised by this discussion.

Even though you keep relying on cases where heat is an exact differential, I believe you would agree (eventually) that in the space of all possible processes, those you so strongly rely on represent a set of zero measure. So, if you are not convinced about the mathematical fact that a generically path dependent quantity cannot be used to characterize states, then I have nothing else to add here.

I appreciate your culture (I can see you read a lot) which hepls to support the discussion, but when exchanging arguments, sometimes debunking (at least trying) the argument of the person you're arguing with is a helpful excercise, because it is during that process that either you end up agreeing with him or you find the flaw in his argument. I'm not trying to discuss on history, I'm discussing on mathematical facts and their associated physical consequences.

We can of course discuss the paper of Seifert. I don't know if Franklin would agree, it's up to him, he was originally interested on entropy for single particles. Otherwise we can create a different thread.

Best wishes,

Reinaldo.

Dear Tom & Reinaldo:

Thank you for the valuable discussion, I don't want you guys to think I did n't care about it, I followed up all the answers and arguments, although I would like to end the discussion on whether or not Heat is a state function, whether is relevant to consider it as a state function just in some particular cases, or about the terminology related with the Heat Capacity term. I have learned a lot from both and I read more than once all of your arguments to try to follow up the discussion,

I cannot elaborate much on what does state function means or represents in particular cases, or in the general case of thermodynamics, as well as on discussing on the historical experiments, or particular examples on the field, since I find difficult to carry on with a discussion with this level of language and terminology. I haven't lived in an English speaking country so i'm not highly proficient in expressing my points in English, so you may understand why I haven't expressed them about your ideas so far,. So I don't want to take the risk of trying to discuss someone's argument and then find myself in the necessity to elaborate my language and terminology more than what I'm capable of.

So very briefly. What I undersand after have though several times on your ideas, and from what I have further read cause of this discussion:

Heat is not a State Function, I think we can find that in any Thermodynamic's book. So now I understand more clearly that we cannot characterize a system with the Heat, since is not a parameter which could help us to describe a physical state of the system, I understand (good enough) why Reinaldo call it a Transfer of energy, since it just describes transitions between equilibrium states (in the simplest scenario) which depends on the path. So we say: "Some amount of heat has been removed from the system" meaning thermal energy has been transferred from the system to the surroundings.

So, a Process dependent Function as Heat and Work cannot characterize the system since its values depend on the transitions between energy states distributed in it, and since the system never reach thermodynamic equilibrium, we cannot use these variables (heat and work) to characterize a particular state of it.

by the way, in University Heat Transfer and Thermodynamics professors does n't discuss this with students, but leave them to think that Heat is a form of Energy without explaining further whether it serves to characterize the system or not, and of what does this mean.

So I feel that an important part of the conclusion is an issue of language, but just after you have understood the relations between state functions, state variables, and the system, etc.

Same with the term Heat Capacity, I really think Tom has made very good points about the term Heat Capacity , and all the historical examples as Reinaldo said, to bring up the important historical proved examples always helps to a constructive discussion. I also think that is important to always recall the definition of the concept : "the amount of energy that a body has to absorbe or release in the form of heat in the given conditions, to vary its absolute temperature in one Kelvin" as in words of Reinaldo.

I learn a lot from these discussion, I hope I get something to cotribute.

Can we discuss on the second question a little more?

Reinaldo, I have gave a quick look to the paper but I its total understanding of it is out of my hands.

When he speaks about "Stochastic trajectory does he refers to a single particle, does n't he?

I don't know if trying to understand this work may be the more practical way for me.

Does the concept that a particle (elementary or quasielementary particle) can have entropy is straightforward for you ?

I think It still bugging me cuz I continue thinking about this in terms of, either the statistical's entropy formula S=k_b*ln(W) , or in terms of the classical thermodynamics' entropy formula: dS=dQ/T

Regards !, thank you for your time both,

The so-called "entropy " doesn't exist at all.

During the process of deriving the so-called entropy, in fact, ΔQ/T can not be turned into dQ/T. That is, the so-called "entropy " doesn't exist at all.

The so-called entropy was such a concept that was derived by mistake in history.

It is well known that calculus has a definition,

any theory should follow the same principle of calculus; thermodynamics, of course, is no exception, for there's no other calculus at all, this is common sense.

Based on the definition of calculus, we know:

to the definite integral ∫T f(T)dQ, only when Q=F(T), ∫T f(T)dQ=∫T f(T)dF(T) is meaningful.

As long as Q is not a single-valued function of T, namely, Q=F( T, X, …), then,

∫T f(T)dQ=∫T f(T)dF(T, X, …) is meaningless.

1) Now, on the one hand, we all know that Q is not a single-valued function of T, this alone is enough to determine that the definite integral ∫T f(T)dQ=∫T 1/TdQ is meaningless.

2) On the other hand, In fact, Q=f(P, V, T), then

∫T 1/TdQ = ∫T 1/Tdf(T, V, P)= ∫T dF(T, V, P) is certainly meaningless. ( in ∫T , T is subscript ).

We know that dQ/T is used for the definite integral ∫T 1/TdQ, while ∫T 1/TdQ is meaningless, so, ΔQ/T can not be turned into dQ/T at all.

that is, the so-called "entropy " doesn't exist at all.