This is an obvious conflict but was ignored for a long time.
By Clausius’ inequality we have \oint {dQ/T}≤0, and by the theory of exact differential we have \oint (dS)=0. Hence the conclusion that dS=dQ/T is an exact differential cannot be proven in mathematics.
As the state function change must be independent of the path taken, the definition of the state function must be path-independent; this is a fundamental principle of exact differential. However, as an only exception, Clausius’ definition dS=dQ/T does not conform this fundamental principle because the restriction to reversible processes must be attached.
In thermodynamics, the very fact is that the exact differential dS itself never need to attach with the restriction to reversible processes, except only for Clausius’ definition dS=dQ/T. This shows that an additional restriction to reversible processes is not an essential requirement for the definition of entropy, and thermodynamics itself has already contained such a conclusion.
From non- equilibrium thermodynamics, we have dS=deS+diS, and in this opinion, dQ/T is really the entropy flux deS, it means that Clausius’ definition does not contain diS. Assuming an additional restriction to reversible processes is just in order for that diS=0. In this case, dS=dQ/T is actually the partial differential deS=dQ/T. By the inequality dS(= deS+diS)≥dQ/T (=deS), when dS >dQ/T, then what is dS? The total differential dS had not been established.
In my opinion, dQ/T is only the entropy of dQ but not the definition the state function S. A perfect definition of the state function S should and must conform the fundamental principle of exact differential; it must be a total differential! In Clausius’ approach, the total differentials of entropy and the second law (or the total entropy production) both are not established.
Thus, I think, Clausius’ definition should be revised; I understand that many will question this idea because it sounds incredible.
Before you argue with me, I recommend that you consider such questions:
1) Clausius’ definition cannot be proven in mathematics but can only be established basing on imaginary heat engine cycles. Why ?
2) Why Clausius’ definition be an only exception?
3) Thermodynamics itself has already contained some implicit corrections.
Reference please see
https://www.researchgate.net/publication/236983852_arXiv1201.4284v4
Data Internal Heat Energy, Entropy and The Second Law: A New Pers...
In truth $dS \equiv \frac{dq_{rev}}{T}$ --> the path is defined!
Thus ends any further need for discussion.
Dear Victor, thanks for your comment, I like elegant expound.
I agree. Gibbs entropy formula is an elegant approach, and “…expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, perpetual motion machines and the like.” In Gibbs approach (and other statistical approaches), the entropy change dS is independent of the path followed.
My interest is this theoretical conflict itself that involves some important topics, and I agree your opinion that Clausius' entropy formula may tell us “a way to calculate the entropy change of a given process”. But I believe, there should be a thermodynamic approach, in which, the definition of entropy can also be “…expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, perpetual motion machines and the like.”
I don’t believe that a state function, which represents the “state properties of the individual equilibrium state”, can only be defined basing on imaginary engine cycles, and must attach with the restriction to reversible path. In my opinion, dQ/T is only the entropy of dQ but unsuitable as the definition of the state function S.
@T Suye: "In my opinion, dQ/T is only the entropy of dQ but unsuitable as the definition of the state function S."
Your confusion here then is that you are making a false connection to start. The definition $dS \equiv dq_{rev}/T$ is absolutely not a definition of $S$. The definition of $S$ as a state function (which it is) and the definition of $dS$ as an exact differential (which it is) do not conflict. The former is a physical statement about the value of a measurable quantity in a system at any instant in time. The latter is a mathematical statement about the final result after integration over any path to obtain the change in a value from an initial state to a final state.
The widely carried misconception also is to call $dS$ a state function in the sense that $dS$ represents a measurable value at any instant in time. You would be wrong in that same way to call $dU$ a state function. No infinitesimal term is measurable until after it is evaluated through a process. Therefore, no $dX$ term is truly a state function in this regard.
Otherwise, no one would ever get anywhere in their travels because of Zeno's paradox.
From my point of viewer, to solve Clausius problem is similar to solve irreversibility problem. The problem of irreversibility is solved by construction mechanics structured particles (SP-system from potential interaction material points (MP)).To be brief. Let us change MP on the SP. In this case the energy of the external field is not only to change the velocity of the SP as for MT, but also changes internal energy of the SP. Therefore the dynamics of the SP is determined by changes two types of energy: internal energy and the motion energy of the body while maintaining their sum. Therefore, the acceleration of the SP is not uniquely associated with the flow of external energy, as it takes place in the case for MP but also will go to increasing of the internal energy of SP. In this case entropy determines the part energy which goes to increasing the internal energy
For more details, see (Somsikov V. M. Nonequilibrium systems and mechanics of the structured particles. Elsever. Chaos and Complex system. 2013, XV, 581 p. 31-40 ).
"But I believe, there should be a thermodynamic approach, in which, the definition of entropy can also be '…expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, perpetual motion machines and the like.' "
You can write the entropy as a function of the partition sum. No circles or machines involved.
Dear Tang,
for irreversible processes \delta Q / T is not exact differential anymore and dS is not equal to it. You need to take into account the entropy production due to the processes like diffusion, chemical reactions etc.,
The questions you recommend to consider at the end of your note could be reformulated as a single one: why the reversible processes are of such distinguishing importance when considering the entropy. The answer is that performing the reversible process (like Carno cycle) on can immediately recognize the existence of a state function (entropy) and to calculate its change.
At last, your line of arguing appeared to be similar to the line, which led to the extension of equilibrium thermodynamics to the thermodynamics of open inhomogenious systems rather long time ago.
So, there is no necessity to revise Clausius equality, which defines the change of the entropy of closed homogenious system at the reversible process and is pretty good with that.
Dear Alexander,
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 ( Arnold Sommerfeld).
The fourth time I “go through it”, I think, there have been some problems that are not clear in thermodynamics itself.
We can only recognize the existence of a partial differential dQ/T in a heat exchange process and calculate its change, but cannot immediately recognize the existence of a state function (entropy) and to calculate the total differential of which by the aid of a reversible process (like Carnot cycle), in that dS=dQ/T is not a full image of the function S, it is only the entropy of dQ.
As you’ve said, “for irreversible processes \delta Q / T is not exact differential anymore and dS is not equal to it,” and then, in this case, what is dS equal to? We need a perfect definition of entropy S, by which, dS is an exact / total differential, for arbitrary reversible or irreversible processes, and for arbitrary system, the entropy changes can all be calculated by dS=….
dS=dQ/T is not a full description for the state properties of the individual equilibrium state. It looks like pretty good, the reason is that the flaws had been implicitly revised by the fundamental equation of classical thermodynamics, but the revision for the definition of entropy itself has not been completed.
The definition dS must be a total differential of the function S!
But dS=dQ/T is not.
So, there is necessity and possibility to revise Clausius definition, and it is not an isolated revision because that relates some important themes.
My point was that using the partition sum one obtains a definition of S depending only on the state of the system. Therefore S as a state function and dS as total differential. Isn't that exactly what you are looking for? This approach incorporates statistical information of the system that is not available from the thermodynamic point of view. It is therefore not surprising that a comprehenisve understanding of entropy for non-reversible processes can hardly be derived from pure thermodynamics.
Dear Stefan,
From statistical point, you are right.
But statistical models are established basing on some postulates, its valid range is less than thermodynamics because those postulates do not need to be considered for the latter. Another is the fact that the canonical variables /coordinates p_i and q_i cannot correspond to the independent state variables T, V, x and N_j by one-to-one. For a generalized thermodynamic system, some thermodynamic properties can hardly be described by statistical approach.
So I am looking for a pure thermodynamic approach. Indeed, it was a little difficult. But I think I had found it.
Dear Vyacheslav, I think that your approach contains a good idea (the structured particles), but the two questions should be considered.
1) The internal energy U may be divided into the two parts according to their contribution to entropy. One part makes contribution to entropy, another makes no contribution to entropy. In a dynamic approach, these two parts should be divided with a similar meaning.
By the fundamental equation of classical thermodynamics we have
dS=dU/T-Ydx/T-\sum(μdN)/T+pdV/T.
Ydx and \sum(μdN) are the partial differentials of dU, so we know that Ydx and both \sum(μdN) make no contribution to entropy.
How to divide these two parts in a dynamic approach is still a problem, it’s also the problem in statistical approach.
2) There are three basic sources of the total entropy production or irreversibility. The “friction force” only corresponds to one, i.e., Ydx→dq (Free energy was converted into heat energy）.
Dear Tang Suye. Our aim was to find an explanation for the problem of irreversibility in the frame of the deterministic laws of Newton. We find this explanation on the basis of a simple model SP. First of all we replace in classical mechanics the material points (MP) on structural particles (SP). The equation of motion of SP obtained from its energy. But the energy of the SP should be presented in the form of the motion energy of SP and its internal energy. Notice here that the internal energy is the energy of interaction between the MP and the energy of the relative motion (this is not exactly the same thermodynamic internal energy). Both types of energy are independent. This means that when the SP in the non homogeneity field of forces, the work of this field going to the change motion energy of SP and its internal energy. As you can see, there is not yet chem. potential; there is the energy of the valence bonds, etc. We consider a very simple case. But even this simple case allows us to introduce the concept of entropy in classical mechanics, and then go to the justification of thermodynamics. We proposed a simple definition of entropy for SP in the framework of classical mechanics. But in general cases of equilibrium open systems, the notion of entropy, as the concept of energy are very complicated. There's a huge layer of problems and work.(More details you can easily find in my works which published in . arXiv )
Dear Vyacheslav, Sorry for the late response.
Newton´s laws are the equations of the energy conservation, and correspond to the first law of thermodynamics. T-symmetry is the basic symmetry of Newton´s equations. Indeed, it is only the symmetry of the equations, we're not sure if it is also the basic symmetry of a dynamic phenomenon itself by these equations, but we can be sure these equations don’t involve the theme about dissipation. I call these equations “the first law physics”. If we want to find an explanation for the problem of irreversibility in a mechanical model, we must extend the theoretical structure of mechanics from “the first law physics” to “the second law physics.”
“Perhaps, the fundamental structure of theoretical physics might be divided into the two main lines according to different laws, one being the first law physics, the other one being the second law physics. We can do or not, that is the problem.”
I feel that the energy of the structured particles should correspond to Gibbs free energy.
Dear Tang Suye.
In reality in the nature there are only a systems. And the body can arise only because the elements of which they are composed are also systems. For the motion of any body we waste energy not only on their motion but also on the change of the internal energy. The internal energy of the body is not involved in the motion. If the body is defined by a system of material points (MP), the internal energy corresponds to the energy of the relative motion MP. My key statement is that the dynamics of body is defined by the space symmetry and symmetry of the body. Accordingly, the energy of the body is also divided into two parts. It is only one part of it is involved in the motion.
The fact that systems can be irreversible should not be surprising. The system but not MP can be heated. Quantity is transformed into quality. The motion of the system is determined by duality of the symmetries. The reason is the existence of the internal degrees of freedom in the system.
Gibbs considered the state of the systems. And you're right that his theory is acceptable for structured body. But as far as I know, the question of how to change the internal state of the system as it moves in a non-uniform force field, he did not affect.
Dear Vyacheslav, Sorry for the late response.
Your models may be useful for explaining Stefan’ questions, how thermodynamic energy corresponds to dynamic energy? (see my first topic) It involves the theoretical extension of dynamics, and we can draw some interesting results from this extension.
The energy of the structured particles involves Gibbs free energy, and the well known fact is that Gibbs free energy makes no contribution to entropy, and the degrees of freedom of the free energy also makes no contribution to entropy. Friction involves the free energy Yx to be converted into heat energy q; can you distinguish G and q in the models of the structured particles.
The state of thermodynamic entropy contains two items; it can be easily confirmed by all of the concrete examples or the general equation (in current theory and both my approach), such as an idea gas, a photon gas, or a simple solid. These two items correspond to different microscopic sources; you models only involve one item. How can you make sure that the expression of your entropy is a total differential?
Dear Tang Suye.
The motion equation for structured particles ( SP) has been obtained subject to the fulfillment the Newton’s laws for material Points (MP). Indeed, the energy of SP is the sum of the energies of MP. The energy of each MP is equal to the energy of motion and potential energy in the external field and the field of forces acted from another of MP. The motion equations an each MP are Newton’s motion equation and connected with the Newton’s second law for MP. But in a result of construction of the motion equation for SP basing on the Newton’s laws for MP it was found that the Newton’s second law is not fulfilled for SP. It is because the work of external forces going not only to the motion of SP but also changes its internal energy. It is the main difference between the dynamics of the SP and MP.
In contrast to the Newton’s equations for MP, the SP motion equation is irreversible. The breaking-symmetry of the time for SP takes place because the motion energy of SP transformed into the internal energy and the internal energy can’t go back to the SP motion energy. Therefor we can say that the internal energy is the energy of the chaos.
Dear Vyacheslav,
I partly agree with your point of view, the differences between us is that, from thermodynamics, 1) only some parts of the internal energy of the structured particles make contribution to entropy, not all of them. 2) thermodynamic entropy involves the two items, one is the distribution of the heat energy, the other one may be the distribution of the particles. Your equation of entropy only involves one item, so we can not make sure that you equation is a total differential. Please see the link in the top floor, Eqs.(7) (8) (9) and Eqs.(47) (48).
Consider that irreversibility should have the same sources both in thermodynamics and dynamics, if it is true, we must look for a connection between the two theories for mutual verification. The first step is that we need to confirm what state that entropy represented; we need a total differential, an explicit function.
However, in current thermodynamics, δQ/T is actually the entropy of δQ, and in the fundamental equation of thermodynamics [see my paper Eq.(6)], dS is an implicit function, both the two equations can hardly correspond directly to dynamics. You may find that my equation Eq.(9) may provide a new foundation for establishing such a connection.
Dear Tang Suye.
I agree with Stefan Schnabel that the modern concept of entropy cannot be determined within the framework of thermodynamics only one. So we cannot to answer your question within the framework of thermodynamics. For answer your question it is necessary to involved, for example, statistical physics. I think that the desire to understand the essence of entropy has led to the creation of the statistical physics by Boltzmann. But Boltzmann failed to achieve its goal - to find the link the thermodynamics with the mechanics basing on a molecular-kinetic model of a gas. From my point of view without solving this problem one can hardly expect a deeper physical understanding of physics, in particular, the entropy.
In an effort to complete the task set by Boltzmann, we obtained the equation of motion of a structured particles (SP) which is equilibrium system of the potentially interacting material points (MP) in a heterogeneous space. I emphasize that this equation you will not get in the formalisms of classical mechanics as these formalisms are constructed basing on the hypothesis of holonomic constraints. The equation of motion of the SP was obtained by us from the condition of duality symmetries of space and the system without using the holonomic hypothesis. This has led to a qualitative difference between mechanics SP and classical mechanics for MP (into the SP mechanics we have symmetry breaking of time).
According to the equation of motion SP the work of the external forces expended not only on the motion of the SP but also on the increase of the internal energy. Let us take a closed non-equilibrium system (NES) which consist from a set of SP in relative motion to each other. According to these equations the NES system will be go to the equilibration as a result of the transformation of the energy of motion of SP into their internal energy. This has allowed us to offer a deterministic definition of entropy which follows from the fact that the internal energy is not able to go into the energy of motion SP. Hence the entropy is a ratio of the part of the motion energy which transformed into internal energy of the SP, to the value of the internal energy. Physically, this determination is very clear. Of course, in the future remains to compare the deterministic definition of entropy with the definitions of Clausius , Boltzmann , Sinai , etc. It is necessary to establish the limitations of various definitions of entropy. So the mechanics of SP allows explaining the thermodynamic empirical laws basing on the strict laws of the classical mechanics.
Dear Vyacheslav Somsikov and Stefan Schnabel,
I agree with this opinion “Boltzmann failed to achieve its goal - to find the link the thermodynamics with the mechanics basing on a molecular-kinetic model”. But we must pay attention to such a fact: The valid scope of statistical physics is less that of thermodynamics because of the equal a priori probability postulate, as we known that there is no similar postulate in thermodynamics and both dynamics, so Boltzmann approach can not be a substitute for thermodynamics.
Boltzmann approach can only be applied to such case: the random thermal motion is stronger than the attraction between the particles, such as a gas, a liquid. In Boltzmann approach, we can not obtain structures and orders due to the equal a priori probability postulate can not be applied to some thermodynamics quantities in general, such as Yx and Gibbs free energy μN.
The internal energy of structured particles involves Gibbs free energy. We must pay attention to the fact that the equal a priori probability postulate is not universally applicable to thermodynamic system.
Dear Tang Suye.
In constructing the mechanics of structured particles (SP) is convenient to assume SP as equilibrium. This eliminates the need to consider the collective motions within the SP. Otherwise, we would be able to take SP as a set of equilibrium subsystems and again reduce the problem to the motion of equilibrium systems. When calculating the dynamics of the SP we do not use statistical laws. We are taking only Newton’s laws for the MP. So in our tasks it does not matter for what statistics the SP elements obey. Another thing is that if we want to get the solution of the problem, for example, to calculate the entropy. Here we can take, for example, Gibbs statistics, after a good enough justification. But these are the other tasks which refer to justification and the study of thermodynamics and statistical physics.
Of cause these are very important tasks and we hope to solve it.
Dear Vyacheslav Somsikov,
Indeed, looking for a connection between the two theories is an interesting subject; there are some important issues that we hope to solve.
Many have ignored the obvious fact that statistical theory can not establish a closed architecture, due to many thermodynamic phenomena absolutely can not be described by the equal a priori probability postulate. So that, statistical approach can not be an alternative to thermodynamics.
Dear Tang Suye.
I agree with you. Please, send me your paper in connection with thermodynamics if you can. My mail: [email protected]
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