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Entropy 2020, 22, 437 15 of 26 14 of 27 FFigiguurere111.1E.xpeErximpernitmalencotaelxisctoeenxciestdenencesitydemnseiatsyurmemeaesnutsretmakeenntsditraekcetlny fdroirmectTlayblfero1mof TGaiblglen1etoafl. [5G0i]l;gaetnTect athl.e[5e0x]p;eartimTecntthael edxepnesritiymegnatpalcdoeincsidtyesgawpitchoitnhceidliensewaritdhetnhseitylindeiafrfedrencseitybedtiwffeerenncthee subpetewrcereintictahlegsauspearncdritlicqauligdasphanasdelsiqausidepfhinaesdesbays dtheefinseigdnboyfththeesirgignidofittyhestraitgeidfiutyncstiaotne faunndctoiobntaainded frobmtaitnhedNfrIoSmTt2h0e1N7ITShTe2rm01o7pThhyesrimcaolpthaybsleicsal[4t]a;btlhese[p4]e;rtchoelapteiorcnolaotciion(dlaoschie(da)sdhecdr)edaesecreliansealrilnyeawrliyth dwenitshitydeinstihtye isnutphercsruiptiecraclrriatincaglerandgeaapnpdearptpoeaoritgoinoartigeiantateheatBtohyeleBoteymleptermatpuerea;tuthre;ltahregelaregde rdeadta data points are the LRD mean coexisting densities. points are the LRD mean coexisting densities. 3.3. Physical-Constant Equations-of-State From the modern literature of thermodynamic p-V-T data on atomic and molecular fluids, one couldIcnhorodseratnoydoemneoonfstrhaete2a00neflcueisdssitylisotfedthirneethsepNarIaStTetehqeuramtio-npsh-oyfs-isctalted,awtaebtaknek, oasr anyexoanmepolfe,thae hsuunpderrecdristiocafleixspotehreimrmenTta=lp35a0peKr,soirnTt/hTel=ite1r.1a5tu,roef.CNOum.TehriecaclrietiqcuaaltainondsB-ofy-sletattemfproemrattuhresliftoeraCtuOre, c22 haocwcoervdeirn,gshtouNldISTaldsoatabeanvkie[w4]eadrewTith=c3i0rc5uKmasnpdecTtion=,7e2s5peKc.iaUllsyinignththeesavmiceinmiteythofdtdhesccrriibteicdal cB tepmrepveiroautsulrye.foTrhaernguomn e[8ri]c,atlhreepcoresxeisnttiantgiodnesnosfitriaews aetxTperairmeefnotuanlldy tmoebaesuρre(dgadsa)t=a p7o.7i7n1tsm[4o]l/uLseadndin cc pρar(almiqe)t=er1iz3a.4t6iomnsola/rLe. Spurebdsetiteurtminignethdebseypahnyasipcarilocroinastsaunmt vpatilounesaitnthoeeoqutasteiotnosf (t5hetoex7)isftoernTce=of35a0vKan, c dwereWobatalisnstihneguglasriatnydatliaqucriditisctaltde ednesnitsyit,yabnoducnodnstiρnui(tgyaosf) t=he7.s3u0p5emrcorli/tLicalnedqρuati(olinq-)o=f-s1t2a.t0e2inmaoll/Lit,s PB PA daenridvathtieverisg.iTdhiteyswo(-3c5a0lleKd)u=n0iv.3e1r8skaTl e(x0p.9o2n5eknJt/smfolr).the description of the thermodynamic properties in theimImfwedeitartyetvoicfiitntithyeoNfITScThdaavteaafolsrothbeewnhionlceoirsroetchtleyrmemcopnlotiyneudo.uTshlyusfr,oemve0n,tjuhostuugphtnoooonnlye5h0asMePvae,r muesainsugraed6-atevrmanipsohliynngodmeinasliftoyrdeixffaemrepnlcee, wupe toobtTacinditrheectrleys,ualtl:the NIST thermodynamic state functions 65432 have mp i=st–a0k.e0n0l0y00p8rρesu+m0e.0d0a04c5o7nρtin–u0it.y00o9f3g7a3sρ an+d0l.i0q9u6i2d45toρ b–e 0th.5e9u63n1d7eρrly+in3g.4s5c9ie2n64ceρ–fr0o.m189th2e39owutisteht. 2 meaOnnsequraeraesorengrfeosrsitohneRina=d0e.q9u9a9c1y25o.f complex multiparameter equations-of-state with increased experiImnetnhtiasltyppreicaislicoonntiisntuhoautsthoveecroanlltipnoulyounsomfuianlctfiiot,ntahleflorwmesr-aoredefur ncodeaffimceinetnatlslyariencsocrierencttifiicnaltlhye vmicienaintyingolfesTsc.Iandconintrathste,bsyupuesrincrgitvicirailalmcoide-ffiracniegnetsbaentwdepehnysgicaaslapnrodpelirqtiueisdofpthaesemse.sTohpehamseesbopuhnadse,, cothnefiNneISdTwdiathtainfopretrhceolwathionleloracnigtheaotfbtohurenedrtehgeiognasscandbleiqruepidropdhuacsedswbyithiegshsern-toiradlleyr1d0i0s%copntriencuisiitoiens, cwanherenatdhielygabse, ildiqeunitdifaiend. mAesiom-sptaletensuhmaveeridcaiffl edriefnfetreqnutiattiions-of-NstIaStTe. eWqeuatsisounms-eoft-hsatat teh,efroerisexaat mleapslte, canthdierdm-onrdsteraptehtahse etrxainstseinticoeno, fi.eth.,easduipsceorcnrtiitnicuailtymiensothpehathseirdanderloivcatievtehoefpGhiabsbesbeonuenrgdys,aatlothneggaansy isaontdhelriqmu,ido-fsatantyebflouuidnd(ea.rgie.,sCinOa2cactoTrd/Twc=ith1.i2n5e)qfuoarliatnieyso(2fathned230).0fluidsintheNISTThermophysical PropeTrthye deaxtpaerbimaneknt[a4l].daTthaefsreombotuhnedNarIiSeTs tarbeulsamtionotshaerde roevperodbuycethdeineqFuigautiroen1s2-owf-sithatetheuseadmeto ppareacmiseitoenrizaes the oriigiinalledxaptearicmanenbtealodbattai.ned, i.e., accurate to 5 or 6 figures, as evidenced by the mean-squared regression (R2) in the trendline polynomial coefficients as given. In addition, shown in 3.2. Rigidity Symmetry at State Bounds Figure 12 are redefined equations to give the virial coefficients as shown on the plots. The coefficient a1 in the liquid-state expansion is taken to be equal to the value of w The literature p-V-T data show inequalities that distinguish gas from liquid in the supercritical in the mesophase to describe the third-order discontinuity between the mesophase and the liquid state. There is a symmetry between region, and bound the phases according to the respective percolation loci PB (bonded cluster of gas the rigidity of gas and liquid on either side of the mesophase. This observation is empirical though it molecules) and PA (available volume or voids in liquid state) which evidently coincide with discontinuities, probably appearing in the third derivatives of Gibbs energy with p or T. Rigidity (w) is the work required to isothermally increase the density of a fluid; with dimensions of a molar energy. 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