Fundamental Research 1 (2021) 6–9 Contents lists available at ScienceDirect Fundamental Research journal homepage: http://www.keaipublishing.com/en/journals/fundamental-research/ Article The uniqueness of the integration factor associated with the exchanged heat in thermodynamics Yu-Han Ma a,b, Hui Dong b, Hai-Tao Quan c, Chang-Pu Sun a,b,∗ a Beijing Computational Science Research Center , Beijing 100193 , China b Graduate School of China Academy of Engineering Physics , No. 10 Xibeiwang East Road, Haidian District , Beijing 100193 , China c School of Physics , Peking University , Beijing 100871 , China a r t i c l e i n f o a b s t r a c t Keywords: State functions play important roles in thermodynamics. Different from the process function, such as the ex- Integration factor changed heat and the applied work , the change of the state function can be expressed as an exact differ- Process function ential. We prove here that, for a generic thermodynamic system, only the inverse of the temperature, namely 1∕ , Exchanged heat can serve as the integration factor for the exchanged heat . The uniqueness of the integration factor invalidates Thermodynamic entropy any attempt to define other state functions associated with the exchanged heat, and in turn, reveals the incorrect- Uniqueness theorem 2 ness of defining the entransy ℎ = ∕2 as a state function by treating T as an integration factor. We further show the errors in the derivation of entransy by treating the heat capacity as a temperature-independent constant. 1. Introduction prove that only one factor, namely 1∕ , can serve as the integration fac- tor to convert the exchanged heat into the state function. Such unique- State functions, e.g., the internal energy and the entropy, in ther- ness invalidates any attempt to find new state functions associated with modynamics characterize important features of the system in a thermal the exchanged heat . equilibrium state [1] . Physically, some quantities are process-dependent It is worth mentioning that the introduction of the so-called “state and thus cannot be treated as state functions, e.g., the exchanged heat function ”, the entransy defined via ℎ = [5] , in the realm of the and the applied work . An integration factor can be utilized heat transfer is an example of such attempt. The entransy is claimed to be to convert the process function, e.g., the exchanged heat, into an exact a “state function ”of a system characterizing its potential of heat transfer differential (the change of a state function). Mathematically, the require- [5] . Since its appearance, it has triggered a lot of debates [6-10] over ment of the state function can be expressed as follows: the change of the its validity as state function [6] as well as its usefulness [7-10] in the state function remains unchanged with any topological variation of the practical application in the field of heat transfer. In this paper, from the integration path on the parameter space [2] . In addition, the number of fundamental principles of thermodynamics [11-13] and with the help the integration factors is usually limited for any system with more than of statistical mechanics, we reveal the improperness and incorrectness two thermodynamic variables. Such number of the integration factor is of such a concept. further reduced in order to define a universal state function without the The rest of the paper is organized as follow. In Sec. 2, we prove the dependence on the system characteristics [2] . It is a common sense that uniqueness of the integration factor 1∕ and the resultant entropy as the inverse of the temperature, i.e., 1∕ , serves as the integration factor the state function, then we further discuss the errors in the definition of for the exchanged heat , and thus a state function, the entropy can the entransy. The conclusions are given in Sec. 3. be defined. A relevant question arises here: is there any other universal integration factor associated with the exchanged heat for an arbi- trary system? Several attempts on this issue have been made for both 2. Proof of the uniqueness of the integration factor ∕ specific systems [3] and generic systems [4] . However, the uniqueness of the integration factor of interest for a generic thermodynamic system In this section, we will prove the uniqueness of the integration fac- remains unexplored. In our current paper, from the first principle, we tor associated with the exchanged heat . As an illustration, we first demonstrate the result in the ideal gas with the internal energy , the temperature , the volume , and the pressure . ∗ Corresponding author. E-mail address: [email protected] (C.-P. Sun). https://doi.org/10.1016/j.fmre.2020.11.003 Available online 10 December 2020 2667-3258/© 2021 The Authors. Publishing Services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) Y.-H. Ma, H. Dong, H.-T. Quan et al. Fundamental Research 1 (2021) 6–9 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 for a generic thermodynamic system. Theorem. For a generic thermodynamic system, the universal thermo- dynamic state function Λ can be defined as Λ= ( , ) , if and only if ( , ) = α∕ with , a system-independent constant. Proof. For a generic thermodynamic system with the internal energy , the first law of thermodynamic law reads = − = − , (9) where and are the generalized force and the generalized displace- ment, respectively. Similar to the discussions for the ideal gas system, the condition for Λ= ( , ) to be an exact differential ( Λ to be a Fig. 1. The evolution path of the ideal gas in the parameter space − . state function) is: For an arbitrary loop in the − space, we always have According to the first law of thermodynamics, the changed heat of ΔΛ = ( , ) = ( , )( − ) = 0 . (10) the gas reads = − , which can be further written as (with the ∮ ∮ work done on the ideal gas = − ) Noticing the relation for the internal energy . = + (1) = + , (11) An infinitesimal change of a generic thermodynamic function Λ can be defined as Λ= ( ) , namely, Eq. (10) can be rewritten in the form of the surface integration with Green’s theorem as Λ= ( )( + ) . (2) { [ ] [ ( )]} , , . , ΔΛ = ( ) − ( ) − = 0 (12) As illustrated in Fig. 1, the state ( 0 0 ) at point A can be connected ∫∫ by two paths ( ) to the state ( , ) at point B in the − 1 2 The above equation is valid irrespective of the integration intervals. ∫ 1 ∫ 2 space. If Λ is a state function, it is required that A →B Λ= A →B Λ [2]. Hence, we have Accordingly, the loop integral of Λ in the − space is strictly zero, [ ] [ ( )] namely, ( , ) − ( , ) − = 0 , (13) 1 2 1 2 Λ= ∫ Λ+ ∫ Λ= ∫ Λ− ∫ Λ= 0 . (3) which can be further simplified as ∮ → → → → ( ) A B B A A B A B ( , ) ( , ) + ( , ) + − = 0 . (14) With Eq. (2) , the above equation can be specifically written as ( ) In statistical mechanics, the internal energy and the generalized force ΔΛ = Λ= ( ) + = 0 , (4) ∮ ∮ can be written as [1] where we have used = and the equation of state of the ideal ∑= , gas = . Here, is the heat capacity at a constant volume, is = (15) the number of moles of the gas, and is the gas constant. With Green’s =1 theorem, the loop integral in Eq. (3) can be rewritten in the form of the and ( ) surface integration as = ∑ { [ ]} , [ ] = (16) ΔΛ = ( ) − ( ) = 0 . (5) =1 ∫∫ , 1 This equation is valid irrespective of the integral intervals in the − where = ( ) ( = 1 2 … ) is the th energy level of the system , space. Hence, we have and it is -dependent. [ ] − ( ) − ≡ ( ) − ( ) = 0 . (6) = ∑ (17) = − ( ) =1 The fact ∕ = 0 for the ideal gas further simplifies the above is the corresponding thermal equilibrium distribution of the system on equations to [ ] the -th energy level with = 1∕( ) as the inverse temperature and ( ) = 0 , (7) as the Boltzmann constant. Combining Eqs. (15)–(17), we find ( ) = 2 2 ∑ − ⟨ ⟩ − whose solution can be uniquely determined as = = ≡ , (18) 2 ( ) = , (8) =1 ( ) ( ) = = ∑ − ∑ with being an arbitrary constant independent of the temperature . = = − , (19) Without losing generality, we choose = 1 , i.e., Λ= ∕ . Thus, Λ is =1 =1 nothing but the change of the thermodynamic entropy . In summary, we prove that for the classical ideal gas only one state function associ- 1 For simplicity, we consider a system with discrete energy levels. But it is ated with the exchanged heat, namely the entropy, can be defined. A straightforward to extend our discussions to systems with continuous energy similar proof for such an ideal gas system was proposed by Weiss [3] . spectrum. 7 Y.-H. Ma, H. Dong, H.-T. Quan et al. Fundamental Research 1 (2021) 6–9 2 and ∕2 , with the analogy to the definition of the energy of the elec- ( ) = = tronic capacity. To assure the equivalence of the two definitions above, ∑ − ∑ = = + − . (20) the capacity at a constant volume is assumed as a constant to simplify =1 =1 ∫ 2 . the integration ℎ = 0 ( ) = ∕2 However, such assump- Substituting the above three relations into Eq. (14) , we obtain tion is only valid for some systems, such as classical ideal gas, in the ( ) high temperature regime, but not valid for any real solid-state materi- ( , ) ( , ) ( , ) + Θ + = 0 , (21) als in an arbitrary circumstance. One typical heat capacity as the function of temperature is known as the Debye’s law, which shows 3 where that, in the low temperature regime of ≪ Θ , ∝ rather than a ( ) = ∑ constant [14]. Here Θ is the Debye temperature. Such oversimplified Θ ≡ − = (22) assumption prevents the practical application.