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Pressure-Volume Work for Metastable Liquid and Solid at Zero Pressure

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Pressure-Volume Work for Metastable Liquid and Solid at Zero Pressure entropy Article Pressure-Volume Work for Metastable Liquid and Solid at Zero Pressure Attila R. Imre 1,2,* ID , Krzysztof W. Wojciechowski 3,4 ID ,Gábor Györke 2, Axel Groniewsky 2 and Jakub. W. Narojczyk 3 ID 1 Thermohydraulics Department, MTA Centre for Energy Research, P.O. Box 49, 1525 Budapest, Hungary 2 Budapest University of Technology and Economics, Department of Energy Engineering, Muegyetem rkp. 3, H-1111 Budapest, Hungary; [email protected] (G.G.); [email protected] (A.G.) 3 Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 17, PL-60-179 Poznan, Poland; [email protected] (K.W.W.); [email protected] (J.W.N.) 4 President Stanisław Wojciechowski State University of Applied Sciences, Nowy Swiat 4, 62-800 Kalisz, Poland * Correspondence: [email protected] or [email protected] Received: 15 February 2018; Accepted: 20 April 2018; Published: 3 May 2018 Abstract: Unlike with gases, for liquids and solids the pressure of a system can be not only positive, but also negative, or even zero. Upon isobaric heat exchange (heating or cooling) at p = 0, the volume work (p-V) should be zero, assuming the general validity of traditional dW = dWp = −pdV equality. This means that at zero pressure, a special process can be realized; a macroscopic change of volume achieved by isobaric heating/cooling without any work done by the system on its surroundings or by the surroundings on the system. A neologism is proposed for these dWp = 0 (and in general, also for non-trivial dW = 0 and W = 0) processes: “aergiatic” (from Greek: , “inactivity”). In this way, two phenomenologically similar processes—adiabatic without any heat exchange, and aergiatic without any work—would have matching, but well-distinguishable terms. Keywords: adiabatic; aergiatic; isobaric; heat exchange; metastate; metastability; negative pressure; spinodal 1. Introduction Isobaric heating/cooling is one of the basic theoretical processes of thermodynamics, as well as of energy engineering. Some of the basic thermodynamic cycles—like Rankine or Joule-Brayton—comprise one or more steps, when heating/cooling happens at constant pressure. A schematic diagram of the Joule-Brayton cycle can be seen in Figure1. This cycle is frequently used in gas turbines to convert heat to movement, and then to electricity. This has an isentropic (reverse and adiabatic) compression (1-2), then an isobaric heat intake (2-3), an isentropic expansion (3-4) and an isobaric heat rejection (cooling) (4-1), before starting the cycle again. During the isobaric heat exchange, temperature, as well as volume, change simultaneously, i.e., part of the heat can be converted into kinetic energy. Although in most cases Joule-Brayton cycles are open to a given environment (using an open internal combustion chamber), it is also possible for heating and cooling to apply heat exchangers, making a thermodynamically closed system. In a similar manner, isobaric heating for condensed matter (liquids as well as solids) is also a common process. In the case of isobaric heating (common heating of any object on atmospheric pressure, for example), the volume change can be described by the isobaric heat expansion. Although the volume effect on heating is smaller for gases that that of liquids and solids, the description of the process is the same. Entropy 2018, 20, 338; doi:10.3390/e20050338 www.mdpi.com/journal/entropy EntropyEntropy 20182018, 20, 20, x, 338FOR PEER REVIEW 2 of2 of12 12 FigureFigure 1. 1. SchematicSchematic representation representation of of a a Joule Joule-Brayton-Brayton thermodynamic cyclecycle onon temperature-entropytemperature-entropy (a ) (aand) and pressure-volume pressure-volume space space (b), ( consistingb), consisting an isentropic an isentropic (reversible (reversible adiabatic) adiabatic) compression, compression, an isobaric an isobaricheating heating an isentropic an isentropic expansion expansion and an and isobaric an isobaric cooling. cooling. InIn this this paper, paper, we wouldwould likelike to to address address the the problem problem that that occurs occurs in condensed in condensed matters matters where, where under, undersome some circumstances, circumstances the absolute, the absolute scalar scalar pressure pressure of the of system the system can be can non-positive, be non-positive, i.e., p i.e.≤ 0, p[1 ≤– 30]); [1a– special3]); a special process process can exist can in exist an isobaric in an isobaric heat exchange heat exchange (heating or(heating cooling) or at cooling)p = 0. During at p = this0. During process, thisa finite process, volume a finite change volume can occur,change while can theoccur pressure-volume, while the pressure work-volume remains work at zero. remains Although at zero.p = 0 Altstateshough are p considered = 0 states are as “common”considered statesas “common” for condensed states mattersfor condensed (unlike formatters gases, (unlike where forp = gases, 0 is the whereabsolute p = lower0 is the limit absolute for the lower existence limit of for any the gaseous existence state), of any the existencegaseous state), of these the processes existence shows of these that processesin some sense,shows thesethat in states some are sense, special. these states are special. 2.2. Isobaric Isobaric Heat Heat Exchange Exchange and and the the First First Law Law of of Thermodynamics Thermodynamics TheThe total total change change of of internal internal energy energy for for a aclosed closed system system can can be be written written as: as: dUdU == dδQQ + δWdW (1)(1) byby the the First First Law Law of of Thermodynamics Thermodynamics (in (in the the form form recommended recommended by by the the IUPAC IUPAC [4,5]), [4,5]), where where δQdQ andand δWdW areare the the heat heat and and work work of of the the infinitesimal infinitesimal process. process. During During a athermodynamic thermodynamic cycle cycle—where—where the the initialinitial and and end end states states are are the the same same—the—the total total change change of of the the internal internal energy energy will will be be zero, zero, but but some some of of thethe heat heat can can be be converted converted into into work work or or vice vice versa. versa. AtAt various various steps steps of of the the cycle, cycle, for for example example during during the the step of isobaric heating, thethe work—whichwork—which is iscalled called volume, volume, or orp-V p-Vwork work inin thisthis case—cancase—can bebe calculatedcalculated as:as: V Z2 W = − p(V)dV = −p ∗ V = − () = − ∗ ∆D (2)(2) V1 where V1 and V2 are the volumes of the initial and final state respectively. The second equality holds where V1 and V2 are the volumes of the initial and final state respectively. The second equality holds only for isobaric processes, where p is the constant (volume-independent) pressure of the process, only for isobaric processes, where p is the constant (volume-independent) pressure of the process, and DV is the volume difference between the initial and final states. The negative sign is due to the and ΔV is the volume difference between the initial and final states. The negative sign is due to the fact that during compression (negative DV), we transfer energy to the system; therefore, the work fact that during compression (negative ΔV), we transfer energy to the system; therefore, the work done on the system should be positive. In general, p-V work is the work done by the system on the done on the system should be positive. In general, p-V work is the work done by the system on the environment during expansion, or that done by the environment on the system upon compression. environment during expansion, or that done by the environment on the system upon compression. One should be aware that the system and the environment are in equilibrium, i.e., the internal and One should be aware that the system and the environment are in equilibrium, i.e., the internal and external pressures are equal. In this way, the volume change is not the result of a pressure change. external pressures are equal. In this way, the volume change is not the result of a pressure change. When there is an isobaric heat exchange, it is caused by thermal expansion or contraction by isobaric When there is an isobaric heat exchange, it is caused by thermal expansion or contraction by isobaric heating or cooling. heating or cooling. Although work—just like heat—is not a state function, during isobaric processes (as well as Although work—just like heat—is not a state function, during isobaric processes (as well as during any other simple process with a fixed path in the field of state variables), the infinitesimal during any other simple process with a fixed path in the field of state variables), the infinitesimal change of work can be written as dWp, instead of the path-dependent dW: change of work can be written as dWp, instead of the path-dependent δW: | ≡ (3) EntropyEntropy 20182018, ,2020, ,x 338 FOR PEER REVIEW 33 of of 12 12 Hereafter this nomenclature will be used, although in Appendix A, generalized (not necessarily j ≡ isobaric) δW also will be used. dW p dWp (3) InHereafter gases, where this nomenclature only p > 0 states will exist be and used, the although system always in Appendix expandsA, generalizedupon heating (not and necessarily contracts upisobaric)on cooling,dW also the previous will be used. statement (that compression corresponds to a positive work, done on the system)In gases,seems whereto be appropriate. only p > 0 states For condensed exist and the matters system (liquids, always solids expands and upon glasses) heating however, and contracts where pupon ≤ 0 states cooling, [1– the3], or previous materials statement with negative (that compression thermal expansion corresponds (like to water a positive below work, 4 °C) done exist, on the the determinationsystem) seems toof bethe appropriate.
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