The uniqueness of the integration factor associated with the exchanged heat in thermodynamics
The uniqueness of the integration factor associated with the exchanged heat in thermodynamics
Yu-Han Ma , Hui Dong , Hai-Tao Quan and Chang-Pu Sun * Address:
Beijing Computational Science Research Center, Beijing 100193, China
Address:
Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
Address:
School of Physics, Peking University, Beijing, 100871, China * Correspondence: [email protected]
Abstract
State functions play important roles in thermodynamics. Different from the process function, such as the exchanged heat ๐ฟ๐ and the applied work ๐ฟ๐ , the change of the state function can be expressed as an exact differential. We prove here that, for a generic thermodynamic system, only the inverse of the temperature, namely , can serve as the integration factor for the exchanged heat ๐ฟ๐ . The uniqueness of the integration factor invalidates any attempt to define other state functions associated with the exchanged heat, and in turn, reveals the incorrectness of defining the entransy ๐ธ ๐ฃโ = ๐ถ ๐ ๐ /2 as a state function by treating ๐ as an integration factor. We further show the errors in the derivation of entransy by treating the heat capacity ๐ถ ๐ as a temperature-independent constant. I. Introduction
State functions, e.g., the internal energy and the entropy, in thermodynamics characterize important features of the system in a thermal equilibrium state [1]. Physically, some quantities are process-dependent and thus cannot be treated as state functions, e.g., the exchanged heat ๐ฟ๐ and the applied work ๐ฟ๐ . An integration factor ๐ can be utilized to convert the process function, e.g., the exchanged heat, into an exact differential (the change of a state function). Mathematically, the requirement of the state function can be expressed as follows: the change of the state function remains unchanged with any topological variation of the integration path on the parameter space [2]. In addition, the number of the integration factors is usually limited for any system with more than two thermodynamic variables. Such number of the integration factor is further reduced in order to define a universal state function without the dependence on the system characteristics [2]. It is a common sense that the inverse of the temperature, i.e., serves as the integration factor for the exchanged heat ๐ฟ๐ , and thus a state function, the entropy can be defined. A relevant question arises here: is there any other universal integration factor associated with the exchanged heat ๐ฟ๐ for an arbitrary system? Several attempts on this issue have been made for both specific systems [3] and generic systems [4]. However, the uniqueness of the integration factor of interest for a generic thermodynamic system remains unexplored. In our current paper, from the first principle, we prove that only one factor, namely , can serve as the integration factor to convert the exchanged heat into the state function. Such uniqueness invalidates any attempt to find new state functions associated with the exchanged heat ๐ฟ๐ . It is worth mentioning that the introduction of the so-called โstate functionโ, the entransy defined via ๐ฟ๐ธ ๐ฃโ = ๐๐ฟ๐ [5], in the realm of the heat transfer is an example of such attempt. The entransy is claimed to be a โstate functionโ of a system characterizing its potential of heat transfer [5]. Since its appearance, it has triggered a lot of debates [6-10] over its validity as state function [6] as well as its usefulness [7-10] in the practical application in the field of heat transfer. In this paper, from the fundamental principles of thermodynamics [11-13] and with the help of statistical mechanics, we reveal the improperness and incorrectness of such a concept. The rest of the paper is organized as follow. In Sec. II, we prove the uniqueness of the integration factor and the resultant entropy as the state function. In Sec. III, we further discuss the errors in the definition of the entransy. The conclusions are given in Sec. IV II. Proof of the uniqueness of the integration factor
In this section, we will prove the uniqueness of the integration factor associated with the exchanged heat ๐ฟ๐ . As an illustration, we first demonstrate the result in the ideal gas with the internal energy ๐ , the temperature ๐ , the volume ๐ , and the pressure ๐ . According to the first law of thermodynamics, the changed heat of the gas reads ๐ฟ๐ = ๐๐ โ๐ฟ๐ , which can be further written as (with the work done on the ideal gas ๐ฟ๐ = โ๐๐๐ ) ๐ฟ๐ = ๐๐ + ๐๐๐. (1) An infinitesimal change of a generic thermodynamic function ฮ can be defined as ๐ฟฮ =๐(๐)๐ฟ๐ , namely, ๐ฟฮ = ๐(๐)(๐๐ + ๐๐๐). (2) Figure 1 . The evolution path of the ideal gas in the parameter space
๐ โ ๐
As illustrated in Fig. 1, the state (๐ , ๐ ) at point A can be connected by two paths ( ๐ ๐๐๐ ๐ ) to the state (๐ ๐ , ๐ ๐ ) at point B in the ๐ โ ๐ space. If ฮ is a state function, it is required that โซ ๐ฟฮ ๐ AโB = โซ ๐ฟฮ ๐ AโB [2]. Accordingly, the loop integral of ๐ฟฮ in the ๐ โ ๐ space is strictly zero, namely, โฎ ๐ฟฮ = โซ ๐ฟฮ ๐ AโB + โซ ๐ฟฮ ๐ BโA = โซ ๐ฟฮ ๐ AโB โ โซ ๐ฟฮ ๐ AโB = 0. (3)
With Eq. (2), the above equation can be specifically written as
ฮฮ = โฎ ๐ฟฮ = โฎ ๐(๐) (๐ถ ๐ ๐๐ + ๐๐ ๐๐ ๐๐) = 0, (4) where we have used ๐๐ = ๐ถ ๐ ๐๐ and the equation of state of the ideal gas ๐๐ = ๐๐ ๐ . Here, ๐ถ ๐ is the heat capacity at a constant volume, ๐ is the number of moles of the gas, and ๐ is the gas constant. With Greenโs theorem, the loop integral in Eq. (3) can be rewritten in the form of the surface integration as ฮฮ = โฌ { ๐๐๐ [๐(๐)๐ถ ๐ ] โ ๐๐๐ [๐(๐) ๐๐ ๐๐ ]} ๐๐๐๐ = 0. (5) This equation is valid irrespective of the integral intervals in the
๐ โ ๐ space. Hence, we have ๐(๐) ๐๐ถ ๐ ๐๐ โ ๐๐๐ [๐(๐) ๐๐ ๐๐ ] = 0. (6) The fact ๐๐ถ ๐ /๐๐ = 0 for the ideal gas further simplifies the above equations to ๐๐๐ [๐(๐) ๐๐ ๐๐ ] = 0, (7) whose solution can be uniquely determined as ๐(๐) = ๐ผ๐ , (8) with ๐ผ being an arbitrary constant independent of the temperature ๐ . Without losing generality, we choose ๐ผ = 1 , i.e., ๐ฟฮ = ๐ฟ๐/๐. Thus, ๐ฟฮ is nothing but the change of the thermodynamic entropy ๐๐ . In summary, we prove that for the classical ideal gas only one state function associated with the exchanged heat, namely the entropy, can be defined. A similar proof for such an ideal gas system was proposed by Weiss [3]. Having shown the uniqueness of the integration factor associated with the exchanged heat for the classical ideal gas, in the following, we will prove a theorem on the uniqueness of the integration factor associated with ๐ฟ๐ for a generic thermodynamic system. Theorem:
For a generic thermodynamic system, the universal thermodynamic state function ฮ can be defined as ๐ฮ = ๐(๐, ๐)๐ฟ๐ , if and only if ๐(๐, ๐) = ฮฑ/๐ with ฮฑ, a system-independent constant. Proof:
For a generic thermodynamic system with the internal energy ๐ , the first law of thermodynamic law reads ๐ฟ๐ = ๐๐ โ ๐ฟ๐ = ๐๐ โ ๐๐๐, (9) where ๐ and ๐ are the generalized force and the generalized displacement, respectively. Similar to the discussions for the ideal gas system, the condition for ๐ฟฮ = ๐(๐, ๐)๐ฟ๐ to be an exact differential ( ฮ to be a state function) is: For an arbitrary loop in the ๐ โ ๐ space, we always have
ฮฮ = โฎ ๐(๐, ๐)๐ฟ๐ = โฎ ๐(๐, ๐) (๐๐ โ ๐๐๐) = 0. (10)
Noticing the relation for the internal energy ๐๐ = ๐๐๐๐ ๐๐ + ๐๐๐๐ ๐๐, (11)
Eq. (10) can be rewritten in the form of the surface integration with Greenโs theorem as
ฮฮ = โฌ { ๐๐๐ [๐(๐, ๐) ๐๐๐๐] โ ๐๐๐ [๐(๐, ๐) (๐๐๐๐ โ ๐)]} ๐๐๐๐ = 0. (12)
The above equation is valid irrespective of the integration intervals. Hence, we have ๐๐๐ [๐(๐, ๐) ๐๐๐๐] โ ๐๐๐ [๐(๐, ๐) (๐๐๐๐ โ ๐)] = 0, (13) which can be further simplified as ๐๐๐๐ ๐๐(๐, ๐)๐๐ + ๐(๐, ๐) ๐๐๐๐ + (๐ โ ๐๐๐๐ ) ๐๐(๐, ๐)๐๐ = 0. (14)
In statistical mechanics, the internal energy and the generalized force can be written as [1]
๐ = โ ๐ ๐ ๐ธ ๐๐=๐๐=1 , (15) and ๐ = (โ ๐ ๐ ๐๐ธ ๐ ๐๐ ๐=๐๐=1 ) , (16) where ๐ธ ๐ = ๐ธ ๐ (๐) (๐ = 1,2 โฏ ๐) is the ๐ -th energy level of the system [ For simplicity, we consider a system with discrete energy levels. But it is straightforward to extend our discussions to systems with a continuous energy spectrum. ] , and it is ๐ -dependent. ๐ ๐ = ๐ โ๐ฝ๐ธ ๐ (๐) โ ๐ โ๐ฝ๐ธ ๐ (๐)๐=๐๐=1 โก ๐ โ๐ฝ๐ธ ๐ ๐ (17) is the corresponding thermal equilibrium distribution of the system on the ๐ -th energy level with ๐ฝ = 1/(๐ ๐ต ๐) as the inverse temperature and ๐ ๐ต as the Boltzmann constant. Combining Eqs. (15-17), we find ๐๐๐๐ = ๐๐๐ (โ ๐ โ๐ฝ๐ธ ๐ ๐ ๐ธ ๐๐=๐๐=1 ) = โฉ๐ธ ๐2 โช โ ๐ ๐ ๐ต ๐ โก ๐ถ ๐ , (18) ๐๐๐๐ = ๐๐๐ (โ ๐ โ๐ฝ๐ธ ๐ ๐ ๐๐ธ ๐ ๐๐ ๐=๐๐=1 ) = ๐ฝ๐ (โ ๐ ๐ ๐ธ ๐ ๐๐ธ ๐ ๐๐ ๐=๐๐=1 โ ๐๐) , (19) and ๐๐๐๐ = ๐๐๐ (โ ๐ โ๐ฝ๐ธ ๐ ๐ ๐ธ ๐๐=๐๐=1 ) = ๐ + ๐ฝ๐๐ โ ๐ฝ โ ๐ ๐ ๐ธ ๐ ๐๐ธ ๐ ๐๐ ๐=๐๐=1 . (20) Substituting the above three relations into Eq. (14), we obtain ๐ถ ๐ ๐๐(๐, ๐)๐๐ + ฮ (๐๐(๐, ๐)๐๐ + ๐(๐, ๐)๐ ) = 0, (21) where ฮ โก ๐ฝ (โ ๐ ๐ ๐ธ ๐ ๐๐ธ ๐ ๐๐ ๐=๐๐=1 โ ๐๐) = ๐ ๐๐๐๐ (22) is determined by the equation of state ๐ = ๐(๐, ๐) of the system. With the assumption of the factorized structure ๐(๐, ๐) = ๐(๐)โ(๐) , Eq. (21) becomes ๐ถ ๐ ๐lnโ(๐)๐๐ + ฮ (๐ln๐(๐)๐๐ + 1๐) = 0. (23) The solution to Eq. (23) follows: (๐ln๐(๐)๐๐ + 1๐) = ๐ = โ ๐ถ ๐ ฮ ๐lnโ(๐)๐๐ (24) where ๐ = ๐(๐) is a function of ๐ independent of ๐ . The solution to Eq. (24) is โ(๐) = โ ๐ โ๐ โซ ฮ๐ถ ๐ ๐๐ , ๐(๐) = ๐ ๐ ๐ โซ ๐๐๐ . (25)
And the integration factor ๐(๐, ๐) can be explicitly written as ๐(๐, ๐) = ๐ผ๐ ๐ โ๐ โซ ฮ๐ถ ๐ ๐๐+โซ ๐๐๐ . (26) Here, ๐ , โ are integral constants independent of ๐ and ๐ , and ๐ผ = ๐ โ . In order to be consistent with the factorized structure assumption for ๐(๐, ๐) , ฮ/๐ถ ๐ also needs to have a factorized structure of ๐ and ๐ . In the equation above, the form of ๐(๐, ๐) is not unique due to the multiple choice of the function ๐(๐) . The dependence of ๐(๐, ๐) on ฮ/๐ถ ๐ with the specific thermodynamic heat capacity and the equation of state prohibit the definition of the universal quantity. Therefore, ๐ can only be set to be ๐ = 0 to make ฮ a universal state function ๐(๐, ๐) = ๐ผ๐ . (27) We have proven the uniqueness theorem of the integration factor associated with the exchanged heat. Without loss of generality we set ๐ผ = 1 , and we obtain ๐ฟฮ = ๐ฟ๐๐ = ๐๐, (28) which indicates that, associated with the exchanged heat ๐ฟ๐ , the entropy defined via ๐๐ =๐ฟ๐/๐ is the only universal state function. The existence of the integration factor for the exchanged heat is known in thermodynamics. We have shown no other integration factors exist for the exchanged heat ๐ฟ๐ . The entransy ๐ฟ๐ธ ๐ฃโ = ๐๐ฟ๐ is introduced as a โstate functionโ for the purpose of optimizing the heat transfer [5]. The supporters of the entransy claim it as a new state function by regarding a new integration factor ๐ , which contradicts the theorem. The theorem directly excludes the entransy as a state function. Clearly, the entransy is essentially different from the well-defined thermodynamic state quantities such as the internal energy, the free energy, and the entropy. Thus, we conclude that the entransy introduced in Ref. [5] cannot be regarded as a fundamental thermodynamic quantity. And the entransy is introduced for the system with a fixed volume [5]. Such assumption leaves the temperature ๐ as the only variable. It is meaningless to talk about the state function of a single-valued function since in thermodynamics a state function is exclusively a function of two or more variables. For the case of the single-valued function, the entransy is written explicitly by the integration ๐ธ ๐ฃโ = โซ ๐ถ ๐ (๐)๐๐๐ ๐0 . Alternatively, the definition of the entransy [5] via the internal energy is given as E ๐ฃโ = ๐๐/2 = ๐ถ ๐ ๐ /2, with the analogy to the definition of the energy of the electronic capacity. To assure the equivalence of the two definitions above, the capacity at a constant volume is assumed as a constant to simplify the integration ๐ธ ๐ฃโ = โซ ๐ถ ๐ (๐)๐๐๐ ๐0 =๐ถ ๐ ๐ /2. However, such assumption is only valid for some systems, such as classical ideal gas, in the high temperature regime, but not valid for any real solid-state materials in an arbitrary circumstance. One typical heat capacity ๐ถ ๐ as the function of temperature ๐ is known as the Debyeโs law, which shows that, in the low temperature regime of ๐ โช ฮ ๐ท , ๐ถ ๐ โ ๐ rather than a constant [14]. Here ฮ ๐ท is the Debye temperature. Such oversimplified assumption prevents the practical application. III. Conclusions
In summary, it is a common sense that is an integration factor associated with the exchanged heat ๐ฟ๐ . The uniqueness of this integration factor has been proven for some specific systems with given equations of state [3]. In this paper, from the perspective of statistical mechanics, we prove the uniqueness of as the integration factor associated with the exchanged heat ๐ฟ๐ for a generic thermodynamic system without referring to its equation of state. Such a theorem prevents the possibility of defining any new state function associated with the exchanged heat other than the entropy. With this theorem, we clearly exclude the possibility of the entransy as a state function in thermodynamics. In addition, we have also shown errors in the derivation of the entransy with the false assumption of a temperature-independent heat capacity. We conclude that the entransy cannot be a state function in thermodynamics and the false assumption in the derivation prevents its practical applications in heat transfer in any real materials. Acknowledgments
Y. H. Ma is grateful to R. X. Zhai and Z. Y. Fei for helpful discussions. This work was supported by the National Natural Science Foundation of China (NSFC) (Grants No. 11534002, No. 12088101, No. 11875049, No. U1730449, No. U1930402, No. U1930403, No. 11775001, No. 11534002, and No. 11825001), and the National Basic Research Program of China (Grants No. 2016YFA0301201).
Declaration of Competing Interest
The authors declared that they have no conflicts of interest to this work.
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