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Maxwell Relations for Substances with Negative Thermal Expansion and Negative Compressibility

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Maxwell Relations for Substances with Negative Thermal Expansion and Negative Compressibility Physical Science International Journal 14(4): 1-6, 2017; Article no.PSIJ.33660 ISSN: 2348-0130 Maxwell Relations for Substances with Negative Thermal Expansion and Negative Compressibility Igor A. Stepanov 1* 1Institute of Science and Innovative Technologies, Liepaja University, Liela 14, Liepaja, LV −3401, Latvia. Author’s contribution The sole author designed, analyzed and interpreted and prepared the manuscript. Article Information DOI: 10.9734/PSIJ/2017/33660 Editor(s): (1) Alexander Tolstoguzov, Department of Physics, Universidade Nova de Lisboa, Portugal. (2) Christian Brosseau, Distinguished Professor, Department of Physics, Université de Bretagne Occidentale, France. Reviewers: (1) Edisson Sávio de Góes Maciel, Instituto Tecnológico de Aeronáutica, Brazil. (2) Imdat Taymaz, Sakarya University, Turkey. (3) Mahmoud Huleihil, Beit-berl academic college, Israel. Complete Peer review History: http://www.sciencedomain.org/review-history/19154 Received 25 th April 2017 Accepted 19 th May 2017 Short Research Article nd Published 22 May 2017 ABSTRACT It is shown that taking into account the negative compressibility of substances changes Maxwell relations. The earlier results of the author indicating that these relations differ for substances with negative thermal expansion have received additional confirmation. Universal Maxwell relations have been derived. The results obtained have been confirmed experimentally by a number of authors. Keywords: Maxwell relations; thermodynamics; negative thermal expansion; negative compressibility; heat exchange; compression. 1. INTRODUCTION for such substances differ from normal relations. Moreover, additional evidences are found that Recently, negative compressibility materials were Maxwell relations differ for substances with discovered [1 −9]. For them, isothermal a negative thermal expansion coefficient compressibility β =−1 V( ∂ V ∂ P ) is negative. In α =1 V() ∂ V ∂ T . General Maxwell relations T P this paper, it is shown that the Maxwell relations have been derived which take into account the _____________________________________________________________________________________________________ *Corresponding author: E-mail: [email protected]; Stepanov; PSIJ, 14(4): 1-6, 2017; Article no.PSIJ.33660 sign of compressibility and thermal expansion. This is the derivative for the heat exchange Experiments show that the first and third general process. However, for mechanical compression Maxwell relations are correct. this derivative must be obtained from Eq. (4), and it is: 2. THEORY ∂U = − P. (7) The first Maxwell relation is: ∂ V P ∂T ∂ P = − (1) Therefore, the thermodynamics of compression ∂ ∂ V S S V differs from the thermodynamics of the heat exchange. where S is entropy. This relation stems from the first law of thermodynamics. This law for heat From Eq. (3) it follows that the first general exchange can be written as: Maxwell relation will be as follows: δQ= TSd = d U + PV d (2) ∂T ∂ P = − sign(α ) . (8) ∂ ∂ VS S V δ where Q is the heat introduced into the system and U is internal energy. One introduces a In the Appendix it is shown that this has been quantity of heat into the system and it turns into confirmed by many experiments and that Eq. (1) the change in internal energy and work produced contradicts them. by the system. On the left-hand side, the motive force of the process is written, and its effect The second Maxwell relation is: is written on the right-hand side. In [10 −16] the general form of the first law of ∂T ∂ V thermodynamics for the heat exchange was = . (9) ∂ ∂ obtained: P S S P δ = = + α QTSd d U sign()d. PV (3) Its traditional derivation is the following [19]. One introduces d U from Eq. (2) into the differential of However, for the heating of substance by enthalpy: compression , the first law of thermodynamics cannot be derived from Eq. (2) and must be dH= d U + PV d + VP d (10) derived independently [10,11,13,17]: and obtains: Pd V= − d U (4) = + dH TS d VP d. (11) (without the heat losses). Again, the motive force of the process is written on the left-hand side, From this equation, Eq. (9) results. and its effect is written on the right-hand side. One can prove this result very easily. Equation One can notice a mistake in this derivation. Let (2) cannot describe the compression of the us prove that Eq. (2) is valid only for a constant substances with negative compressibility. The pressure. Let us assume that the pressure is not equation for that must be the following one: constant in it. One can notice that δQ = TdS = dHP in Eq. (2) is a full differential, where d HP is = the enthalpy change at a constant pressure. Pd V d U . (5) Therefore, the derivatives (∂1 ∂ V ) and U One can adduce another good argument. (∂P ∂ U ) must be equal. However, for the According to tables of thermodynamic derivatives V [18], ideal gas, they equal 0 and 2 , respectively. In 3V ∂U C reality, these derivatives must also be taken at a = −P + P . (6) constant pressure, thus both are equal to zero. ∂V α V P The differential d U from Eq. (2) does not equal 2 Stepanov; PSIJ, 14(4): 1-6, 2017; Article no.PSIJ.33660 dU from Eq. (10) because in Eq. (2) it is for Consider its traditional derivation [19]. One constant pressure, but in Eq. (10) it is for varying introduces d U from Eq. (2) into the differential of pressure. (In Eq. (4), of course, the pressure can Helmholtz energy: vary.) The correct derivation must be the following. For heat exchange with varying dF= d U − TS d − ST d (21) pressure and volume [17,20]: which results in: δQ= TSdd = U + PV d + VP d. (12) dF= − PV d − ST d. (22) One can see that: From this, Eq. (20) is obtained. However, this derivation is non-strict: an equation which = + TSd TS d1 TS d 2 (13) describes the heat exchange at a constant pressure with varying volume is introduced into where the equation which describes a process with a constant volume and varying pressure. A more strict derivation should be one such as: δQ= TSd = d U + PV d (14) 1 1 1 = dF d F1 + d F 2 (23) is the heat exchange at a constant pressure, and Where δ = = + Q2 TSd 2 d U 2 VP d (15) ddF= U − TS d − ST d =− ST d (24) 11 1 1 1 is the heat exchange at a constant volume [17,20,21]. From Eq. (15), omitting the corresponds to the quantity of heat introduced subscripts, the second Maxwell relation can be into the system at a constant volume (we derived: introduce into Eq. (24) d U from Eq. (2) with d V = 0), and ∂T ∂ V = − . (16) ddF= U − TS d − ST d =− PVST dd. − (25) ∂ ∂ 2222 2 PS S P corresponds to the expansion/contraction of the This equation has a different sign compared with system due to the heat exchange at a constant Eq. (9). pressure (into Eq. (25) we introduce d U from Eq. (2)). Summing up Eqs. (24) and (25), one gets From the well-known thermodynamic identity Eq. (22). [22], it follows that: Let us take the thermal expansion coefficient into ∂P ∂ V ∂ P α account. Introducing Eq. (3) into Eq. (25), and dP= d T =− dd. TT = (17) summing Eqs. (24) and (25), one obtains: ∂ ∂ ∂ β TV T P V T dF= − sign()dα PVST − d (26) This means that Eq. (15) will look like: and the third general Maxwell relation becomes: = + αβ TSUd d sign( )d. VP (18) ∂P ∂ S From this equation the second general Maxwell sign()α = . (27) ∂T ∂ V relation follows: V T ∂T ∂ V In the Appendix it is shown that this equation has = − sign(αβ ) . (19) ∂ ∂ been confirmed by many experiments and that PS S P Eq. (20) contradicts them. The third Maxwell relation is: The fourth Maxwell relation is: ∂P ∂ S ∂V ∂ S = . (20) = − . (28) ∂ ∂ ∂ ∂ T V V T T P P T 3 Stepanov; PSIJ, 14(4): 1-6, 2017; Article no.PSIJ.33660 Consider its traditional derivation [19]. One STd= sign(αβ )d VP (38) 2 introduces d U from Eq. (2) into the differential of Gibbs energy: and that the fourth general Maxwell relation is: ddG=+ U PV d + VPTS ddd − − ST (29) ∂V ∂ S sign()αβ = − . (39) ∂T ∂ P P T and obtains: 3. CONCLUSION = − dG VP d ST d. (30) It has been shown that the negative From this, Eq. (28) results. compressibility of substances effects the Maxwell relations. The earlier results of the author One can see that this derivation is non-strict: d U indicating that negative thermal expansion also from Eq. (2) is at a constant pressure and does effects these relations have been strongly not equal d U from Eq. (29), which is at a varying confirmed. General Maxwell relations have been pressure. Let us try to derive Eq. (28) more obtained which take into account the sign of strictly. One introduces a quantity of heat ( TdS) compressibility and thermal expansion: Eqs. (8), into the system at varying volumes and (19), (27), and (39). Inaccuracies in the previous pressures and the Gibbs energy of the system derivation of the Maxwell relations are shown changes: and corrected. Previously, the dependence of the second, third, and fourth Maxwell relations on the = + dG d G1 d G 2 (31) sign of thermal expansion was published [11,14,15,20]. In the present paper, the dependence of all Maxwell relations on the sign where d G1 is the change in the Gibbs energy at a constant pressure: of thermal expansion and compressibility is given. The first and third general Maxwell relations have been supported experimentally. It ddG= U + PVTS d − d − ST d (32) 11 11 is shown that their previous versions fail to describe the experiments of a number of authors.
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