The characteristic curves of water

Transcription

1 he characteristic curves of water Arnold Neumaier 1 Deartment of Mathematics, University of ienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria Ulrich K. Deiters Institute of Physical Chemistry, University of Cologne, Luxemburger Str. 116, D Köln, Germany 1 corresonding author, arnold.neumaier@univie.ac.at, el , fax

2 Abstract In 1960 E. H. Brown defined a set of characteristic curves (also known as ideal curves) of ure fluids, along which some thermodynamic roerties match those of an ideal gas. hese curves are used for testing the extraolation behaviour of equations of state. his work is revisited, and an elegant reresentation of the first-order characteristic curves as level curves of a master function is roosed. It is shown that Brown s ostulate that these curves are unique and dome-shaed in a double-logarithmic, reresentation may fail for fluids exhibiting a density anomaly. A careful study of the Amagat curve (Joule inversion curve) generated from the IAPWS-95 reference equation of state for water reveals the existence of an additional branch. Keywords: Brown s characteristic curve; ideal curve; Joule inversion; Joule homson inversion; water, IAPWS-95 equation of state 1 Introduction In 2002, Wagner and Pruß ublished a reference equation of state which became known as the IAPWS-95 (International Association for the Proerties of Water and Steam) equation [1]. his equation is a comlicated multi-arameter function which is able to describe all exerimental data for the thermodynamic roerties of water u to 600 MPa and 1000 K within their uncertainties. Later this equation of state became a art of the GERG (Groue Euroéen de Recherches Gazières) equation of state for mixtures [2]. As water is used as a working fluid in most of the world s thermal ower lants, lays a major role in geology and meteorology, and is an imortant solvent or reactant in chemical industry, the imortance of the IAPWS-95 equation cannot be underrated. One of the many tests that the IAPWS-95 had to ass was the calculation of Brown s characteristic curves (a.k.a. ideal curves). hese are curves in the ressure temerature lane along which one roerty of a real gas has the same value as that of an ideal gas. Brown defined a number of such curves [3] and roosed some rules about their shaes and arrangements. hese rules were in art based on thermodynamic laws and hysical insight, but to some extent also on exerience. Of course in comarison to the resent situation Brown was working with a rather limited set of exerimental data in But his rules were found to be good for nonolar fluids, and therefore the calculation of Brown s characteristic curves is recommended for all new equations of state [4, 5]. his immediately leads to the questions whether Brown s rules are alicable to a strongly olar and hydrogen-bonding fluid like water and in articular whether the IAPWS-95 reference equation of state obeys Brown s rules. 2 heory 2.1 he characteristic curves he comression factor Z of an ideal gas is 1 for all temeratures and molar volumes m, Z m R = 1. (1) 1

3 Moreover, for an ideal gas, the configurational internal energy U is zero for all volumes and temeratures. herefore all derivatives of Z or U with resect to temerature or ressure are zero, too. For a real gas, Z usually deviates from 1, and the derivatives ofz usually have nonzero values. In his work On the thermodynamic roerties of fluids, however, Brown [3] oints out that, for each thermodynamic roerty, there are secial states where it has the same value as in an ideal gas. For ure fluids, such states can be reresented by curves in the, lane. Brown studied derivatives of the comression factor and defined a hierarchy of such curves, which are nowadays called Brown s ideal curves or characteristic curves. In this work, the so-called first-order curves are of articular interest, i.e., curves along which a first-order derivative of Z vanishes 2. here are three such curves: 1. the Amagat curve, also called Joule inversion curve. Its mathematical condition is any one of the following: = 0, = 0, π ( ) U = 0, ( ) =. (2) π, the so-called internal ressure, is usually negative, i.e., an isothermal comression lowers the internal energy. At very high ressures or temeratures, however, the molecules are driven into the reulsive branches of their interaction otentials, and then a comression increases the internal energy. At the Joule inversion oint, the configurational internal energy is indeendent of density. he Amagat curve starts at high temeratures and zero ressure at the temerature A for which the second virial coefficientb 2 has its maximum, hence B 2( A ) = 0, (3) where the rime indicates a differentiation with resect to temerature. he terminal sloe at this endoint is 3 d = B 2 ( A)R A. (4) d A B 3 ( A ) From there it asses through a ressure maximum to lower temeratures, until it ends (in a, rojection) on the vaour ressure curve. For most substances, however, this endoint is not accessible because crystallization sets in before the endoint can be attained. he maximum of the Amagat curve lies at very high ressures, tyically times the critical ressure. For water, the maximum is exected around 2.6 GPa. his is a rather high value, beyond the scoe of resent technical alications, but within range of exeriments, and certainly of geological interest. 2. the Boyle curve, which is defined by one of = 0, = 0, ( ) =. (5) 2 Brown furthermore defined a number of second-order characteristic curves, but we shall not deal with them here. 3 he derivations of the endoint conditions and the terminal sloes are given in Aendix B. 2

4 his curve originates on the temeratures axis at the Boyle temerature B, i.e., at the temerature for which B 2 ( B ) = 0. (6) Its terminal sloe is 3 d d = B 2 ( B)R B B 2B 3 ( B ). (7) From there the curve asses through a ressure maximum, and ends on the vaour ressure curve 4 near to the critical oint. 3. the Charles curve, also known as Joule homson inversion curve. It is defined by any one of ( ) ( ) ( ) ( ) Z Z H = 0, = 0, = 0, = ( ), = 0. (8) H he Charles curve starts on the temerature axis at the temerature at which the sloe of the second virial coefficient matches that of the secant, B 2( C ) = B 2( C ) C ; (9) the terminal sloe is 3 d d = C B 2 ( C)R C B 3 ( C) 2 C B 3 ( C ). (10) Like the Amagat and Boyle curves, it runs through a ressure maximum to lower temeratures and ends on the vaour ressure curve. he Charles curve marks the transition from cooling to heating uon isenthalic throttling [6]. In his article, Brown formulates some ostulates about the behaviour of the first-order characteristic curves: here is exactly one Amagat, Boyle, and Charles curve. he Amagat, Boyle, and Charles curves must not cross, but surround each other (Boyle inside Charles inside Amagat), as can be seen in Fig. 1, which shows these curves for the IAPWS-95 equation of state for water. In a double-logarithmic diagram, the Amagat, Boyle, and Charles curves have negative curvatures everywhere (i.e., they are dome-shaed, with a single maximum and no inflection oints), and their sloes tend to infinity for low ressures. 4 if metastable states are excluded; otherwise, the endoint is on the liquid sinodal. 3

5 2.2 Resonse functions Of considerable ractical interest are various thermodynamic resonse functions, which describe how a roerty changes when some other roerty is varied. Basic second-order thermodynamic resonse functions are the isochoric and isobaric heat caacities, which for a ure system are defined by C ( ) U = ( ) S and C ( ) H = ( ) S, (11) the isobaric thermal exansivity, α 1 ( ), (12) and the isothermal comressibility, κ 1 ( ). (13) For the study of Brown s characteristic curves it is advantageous to define dimensionless resonse functions (n: amount of substance): Because of the well-known thermodynamic relation c C nr, c C nr, c α. (14) C = C + α2 κ, the isothermal comressibility can be exressed by κ = Z c κ with c κ = c c c 2, (15) wherec κ denotes another dimensionless resonse function. 2.3 Stability rivial restrictions on ossible thermodynamic states are > 0, > 0, and, for stable fluid states, > 0. In addition, there are restrictions due to thermodynamic stability: In all states where a haseφis thermodynamically stable or metastable, the Gibbs energyg φ (,,n) must be concave in and ; for a ure substance, this is also sufficient for thermodynamic stability. he necessary and sufficient condition for the thermodynamic stability of a ure substance at fixed ressure and temerature is that the negative symmetric Hessian matrix G of the molar Gibbs energyg m (,) as a function ofand is ositive semidefinite. Because of ( ) Gm = S m and ( ) Gm = m, 4

6 we have 2 G m 2 G m G = 2 2 G m 2 G m 2 = mκ m α C,m m α = = ( ) m ( ) Sm m c Rc κ Rc c m ( ) m ( ) Sm. (16) G is ositive semidefinite only if all its diagonal elements and its determinant are nonnegative. In view of Eq. (15) this is equivalent to the thermodynamic stability conditionsc c 0. In articular, thermodynamic stability imlies κ 0 andc κ 0. he condition c κ 0, c while c κ c remains finite defines the so-called sinodal, the boundary to thermodynamic instability. Not a stability condition, but generally assumed in the literature as a consistency criterion is the entroy condition lim ( ) S = 0. (17) his giveslim c = 0 and, using the thermodynamic stability criteria,lim c = 0, hence lim c κc 2 = 0. (18) 2.4 Derivatives of the comression factor o comute the derivatives of Z aearing in the definitions of the Amagat, Boyle, and Charles curves, Eqs. (2 8), we define some dimensionless logarithmic sloes, namely the dimensionless isochoric sloe q A the dimensionless bulk modulus q B ( ) ln ln ( ) ln ln = and the dimensionless thermal suscetibility ( ) ln q C = ln = ( ) ( ) = c κc Z, (19) = c κ Z, (20) ( ) = c = q A. (21) q B 5

7 hen thez derivatives become A 1 : + A 2 : B 1 : B 2 : + C 1 : + C 2 : + = c κ c Z = Z(q A 1), ( ) Z = Z 1 = Z 1 q A, c κ c q B ( ) Z = Z 1 = Z 1 q B, c κ q B = Z c κ = Z(1 q B ), Details of the derivations are given in Aendix A. = Z(c 1) = Z q B q A = Z(1 q C ), q B ( = Z 1 1 ) = Z q ( A q B = Z 1 1 ) c q A q C In the following, we consider the comression factor as a function Z(,) of temerature and ressure, continuous excet along the coexistence curve, where one gets different limitsz l andz g when one aroaches it from the liquid or the vaour side, resectively. he limiting behaviour of thermodynamic quantities as temerature or ressure tend to zero or infinity is known from statistical mechanics and summarized in able 1: At low densities the behaviour of a fluid is described by the virial equation of state,. (22) Z(ρ,) = 1+B 2 ()ρ+o ( ρ 2), (23) where ρ = 1 m denotes the molar density. B 2(), the second virial coefficient, can be comuted from the intermolecular air otential. In articular, for realistic air otentials, i.e. otential functions not having a rigid core, the limit lim B 2 () = 0 is aroached from above. Substitution of Eq. (23) into Eqs. (19 21) immediately yields the low-density limits of theq X : lim q A = 1+ d(b 2) ρ, ρ 0 d lim q B = 1+B 2 ρ, ρ 0 lim q C = 1+ db 2 ρ 0 d ρ. As a consequence,z,q A, q B, and q C all aroach 1 for ρ 0. he high-density behaviour of matter is governed by formulas derived from the so-called quantum-statistical model, an extension of the homas Fermi theory 5. his model [7] was used to derive the entries for as well as 0 in able 1. he low-ressure, high-density behaviour is derived from the inet equation [8], which has been reorted to work very well for solids [7]. (24) 5 Equations of state matching the homas Fermi asymtotics aear to be valid for materials at extremely high ressures as found in fusion lasmas and neutron stars [7]. Some researchers aly it to all states of aggregation, whereas others argue that this might be inaroriate for substances under terrestrial conditions where lim q B > 5 in non-lasma matter. 3 6

8 able 1: Limiting values of the comression factor and related quantities. conditions quantity a Z q A q B q C ρ id /R 0,ρ < ρ c 1+B 2 ρ id 1+(B 2 ) ρ id 1+B 2 ρ id 1+B 2 ρid 0,ρ > ρ c 0 O( 1 ) a 2 R 2/5 O( 2/5 5 ) 3 0,ρ > ρ c O( 1 ) O( 2 ) > 5 3 = c, = c Z c < 1 q Ac > 1 0 sinodal finite finite 0 0 DM b finite 0 finite Amagat finite 1 Boyle finite 1 Charles finite 1 a A rime denotes differentiation w.r.t.. b DM = temerature density maximum Fig. 2 reresents, in nonlinearly transformed q A,q B coordinates, the IAWPS-95 reference equation of 1995 for water [9]. hat the 270 K isotherm is curved to the left is a consequence of the density anomaly of water. 2.5 First-order characteristic curves A comarison of the definitions of Brown s first-order characteristic curves, Eqs. (2 8), with Eqs. (19 21) immediately shows that these curves can be characterized by q A = 1 Amagat curve, q B = 1 Boyle curve, q A = q B Charles curve. (25) herefore these three characteristic curves are level curves of the thermodynamic variable q R 1 q B q A +2q B 1 = 2 in the homas Fermi limit, 1 2 along the Amagat curve, 1 3 along the Charles curve, ±0 along the Boyle curve. (26) he value q R = 2 comes from the homas Fermi theory, which gives q A 0 and q B 5 3 in the high density limit; this limit is not attainable. he level curves q R = const therefore interolate continuously between the characteristic curves, which exlains their onion ringlike arrangement. A double-logarithmic lot of q R for water, based on he IAPWS reference equation of state, as a function of and is given in Fig. 3, exhibiting the tyical shae of the characteristic curves. (he bottom left art is in the metastable domain, where the IAWPS-95 equation is not reliable.). 7

9 For water, the minimal value of q R attained (for = 2000 MPa, the maximal value of the ressure tried) was found to be aroximately ; the maximal value attained (for = 0 and slightly below the critical temerature) is about ; the value at the critical oint is about Since excet at critical oints q A andq B deend continuously on and, the existence of all three characteristic curves follows from the inequalities 6 or equivalently q A <1 < q B if is large or is small, q A >1 > q B close to the critical oint, c < Z c κ < 1 if is large or is small, c > Z c κ > 1 close to the critical oint. (27) By the discussion in Section 2.4, this is satisfied for real fluids; cf. able 1. he characteristic curves are believed to be unique, nonintersecting, 7 and to have a characteristic shae. Because of Eq. (27), it is easy to see that, at the critical temerature, all three curves must lie above the critical oint and surround it. hese roerties are robably shared by all level curves q R = const 0. It is suggested in [5] that, in a double-logarithmic lot, the characteristic curves should have a unique maximum and no inflection oints. he only argument for this (given in the aendix of [4]) seems to be based on the corresonding-states rincile and hence has little force. Under Brown s assumtions, c > 0 (28) must hold in a fluid hase, for then and only then the criteria A 1 and A 2 as well as C 1 and C 2 in Eq. (22) are equivalent. 2.6 he Amagat curve of water at low temeratures c 0, however, might cause a second Amagat and a second Charles curve, for a change of sign ofc causes a change of sign of A 2 and C 2, as will be discussed in a moment. Moreover, a negativec has imlications for the behavior of ressure isotherms. Indeed, two isotherms, with temeratures 1 and 2 cross if there is a densityρsuch that(ρ, 1 ) = (ρ, 2 ). Because of the mean value theorem, this imlies the existence of an intermediate temerature for which d(ρ,) d = ( ) = Rc c κ m m 6 he suorting arguments of Brown [3] are not fully justified. He assumes, beyond the three laws of thermodynamics and the entroy condition Eq. (17), the additional condition lim Z/ = v min()/r > 0 at constant, which is inaroriate in view of homas Fermi theory. 7 At a oint where two of the curves intersect, we must have c = Z/c κ = 1, hence all three curves intersect. Brown argues that this imlies that Z attains a global minimum there and that this is imossible, but his argument is not rigorous. 8

10 vanishes. Hence crossing isotherms aear in regions where c and hence the thermal exansivity α = c / change their sign. Because of Inequality (28), this would be imossible in a fluid hase under Brown s assumtions. he exerimentally observed density anomaly of water, however, results in a negative thermal exansivity for < C at atmosheric ressure. As a result, the Amagat line of water, which enters the metastable 8 region at low temeratures and a ressure of about 1100 MPa, becomes stable again at about 390 MPa and then remains stable down to zero ressure, as shown in Fig. 4. A closer study of the low-temerature, high-ressure region reveals two eculiarities: Close to the endoint, the Amagat curve has a negative sloe in, coordinates. In the doublelogarithmic reresentation, this curve must therefore have an inflection oint. his is at variance with Brown s ostulates. he other eculiariaty is the existence of a second Amagat curve (Fig. 4), which is metastable with resect to crystallization. One might be temted to write it off as an artifact of the IAPWS- 95 equation, but the matter is more comlicated: o exlain this henomenon we consider Fig. 5: In this diagram, the Amagat, Boyle, and Charles conditions corresond to a vertical, horizontal, and diagonal line, resectively. he crossing oint of these lines is a hyothetical state of infinite temerature and low ressure. he high-temerature endoints of the characteristic curves lie in its vicinity; the low-temerature endoints lie outward, at high q A or q B values. he indicated ath reresents an isothermal exansion starting at a very high ressure. his ath necessary crosses the Amagat, Charles, and Boyle lines in this sequence and exactly once, as ostulated by Brown. For a fluid exhibiting a density anomaly, however, the initial state may lie at q A < 0 (state 1 in the diagram). From here, an alternative exansion ath is ossible that intersects the Charles and then the Amagat line, and thus gives rise to a second Charles and Amagat curve in a, diagram, resectively. As a check, we looked at the equation of state of Holten et al. [10], which describes the lowtemerature and suercooled regions, articularly the solid liquid equilibria of water. his equation redicts a second Amagat curve, too. But here the arrangement of the curve branches is different. A comarison of the Amagat curves obtained for the IAPWS-95 equation and that of Holten and al. suggests that, in a continuous deformation connecting the two thermodynamic descritions, a bifurcation occurs between these two equations at which the Amagat curves exchange branches. It is conceivable, of course, that both the IAPWS-95 equation and that of Holten et al. suffer from artifacts. But if this is the case, they do so because there is sensitive sot in the, lane, and there may be a hysical exlanation for this sensitivity. his is corroborated by the observation that the ortion of the IAPWS-95 Amagat curve running from the high-temerature endoint to the temerature minimum at about K and 255 MPa is a locus of minima of U m ( m ), whereas the ortion from there to the endoint at about 277 K is a locus of maxima. Evidently, there is a curve of inflection oints of U m ( m ) lying between the two Amagat curve branches. he shaded regions in Figs. 4a and 4b indicate the region of the density anomaly (α < 0). Where its border is close to an Amagat curve, this curve is a locus of maxima. hus the secondary Amagat curve that the IAPWS-95 equation redicts is also a locus of maxima. Below 660 MPa, the border is a locus of oints for whichα = 0 holds; above that ressure, the border is a locus of oles. he transition oint aears to be the origin of the secondary Amagat curve. 8 with resect to crystallization; the melting ressure curve in Fig. 4 consists of several segments, as ice undergoes several hase changes with increasing ressure (ice I III I II) [1]. 9

11 his shows that the behaviour of the Amagat curve(s) at low temeratures is linked to the sign of α. Evidently, Brown s ostulates do not fully aly to fluids exhibiting a density anomaly. 2.7 he Amagat curve of water at high temeratures At the high-temerature endoint, the Amagat curve obtained with the IAPWS-95 equation exhibits a ositive sloe, again in contradiction to Brown s ostulates. he ositive sloe imlies that the third virial coefficient of water increases with temerature (cf.aendix B). For nonolar fluids,b 3 is usually negative at low temeratures, asses through a (ositive) maximum below the critical temerature, and then declines towards zero. For olar fluids the maximum is less ronounced [11]; for water, the maximum is very shallow, as can be seen in Fig. 6. Unfortunately, the exerimental data sets for water do not agree very well, reliable exerimental data are scarce beyond 800 K, and none aear to exist beyond 2000 K. But even so, exeriments and theoretical calculations based on olarizable interaction otentials all agree that the maximum occurs in the range K [12, 13]. Beyond this maximum, the sloe of B 3 is negative, and hence the terminal sloe of the Amagat curve must be negative, too. herefore the ositive terminal sloe derived from the IAPWS-95 equation is robably an artifact. It must be ointed out that this wrong behaviour of the IAPWS-95 equation is of little imortance, for this equation is valid u to 1000 K only. Moreover, at 5000 K, water would undergo decomosition to a significant extent. But as in ractical alications equations of state are often used beyond their limits of validity, it is imortant to know about this roblem of the IAPWS-95 equation. 3 Conclusion In this work a novel formulation of Brown s first-order characteristic curves is roosed, in which these curves are obtained as level functions of a master function q R = const. As already observed by Brown, the Amagat curve surrounds the Charles curve in a double-logarithmic, diagram, and the Charles curve surrounds the Boyle curve. Brown ostulated that these curves are unique and dome-shaed, with a single ressure maximum and no inflection oints. We show here, and we verify it at the examle of water, that for fluids exhibiting a density anomaly, the Amagat and Charles curves may have more than one branch. For such fluids it is ossible to have a vanishing thermal exansivity, α = 0, as well as (ρ) isotherm crossing. For water, the initial sloe of the Amagat curve (i.e., the sloe at low ressures and at low temeratures) is negative, which is in conflict with Brown s ostulates. he terminal sloe at high temeratures is ositive, again in conflict with Brown s ostulates, but here it can be shown that, most likely, the IAPWS-95 equation of state is at fault and cannot be extraolated 9 to 5000 K. We conclude that Brown s analysis of the characteristic curves, articularly of the Amagat curve, is qualitatively correct for fluids having α > 0 for all stable thermodynamic states. Caution is advised when the characteristic curves are comuted for fluids exhibiting a density anomaly. 9 he IAPWS-95 equation is officially valid u to 1000 K. 10

12 Symbols B i C C ith virial coefficient isobaric heat caacity isochoric heat caacity c (dimensionless) thermodynamic resonse function, X {,,, κ} (Eqs. (14 15)) G G H n Gibbs energy Hessian ofg m (,) configurational enthaly amount of substance ressure q X dimensionless logarithmic sloe,x {A,B,C} (Eqs. (19 21)) R S U Z α κ ρ universal gas constant configurational entroy temerature configurational internal energy volume comression factor, Z = m /(R) isobaric thermal exansivity isothermal comressibility molar density,ρ = 1 m Subscrits A B c C m Amagat (Joule inversion) curve Boyle curve critical roerty Charles (Joule homson inversion) curve molar roerty References [1] W. Wagner, A. Pruß, J. Phys. Chem. Ref. Data 31, (2002) 387 [2] O. Kunz, R. Klimeck, W. Wagner, M. Jaeschke, he GERG-2004 Wide-Range Reference Equation of State for Natural Gases, vol. 15 of GERG (Groue Euroéen de Recherches Gazières) echnical Monograhs, DI-erlag, Düsseldorf (2007) [3] E. H. Brown, Bull. Intnl. Inst. Refrig., Paris, Annexe , (1960)

13 [4] R. San, W. Wagner, Int. J. hermohys. 18, (1997) 1415 [5] U. K. Deiters, K. M. de Reuck, Pure Al. Chem. 69, (1997) 1237, doi: /ac [6] J. P. Joule, W. homson, Phil. Mag. (Series 4) 4, (1852) 481 [7] J. Hama, K. Suito, J. Phys.: Condens. Matter 8, (1996) 67 [8] P. inet, F. Ferrante, J. R. Smith, J. J. Rose, J. Phys. C: Solid State Phys. 19, (1986) L467 [9] W. Wagner, K. Brachthäuser, R. Kleinrahm, H. W. Lösch, Int. J. hermohys. 16, (1995) 399 [10]. Holten, J. Sengers, M. A. Anisimov, J. Phys. Chem. Ref. Data 43, (2014) :1 [11] J. S. Rowlinson, J. Chem. Phys. 19, (1951) 827 [12] K. M. Benjamin, A. J. Schultz, D. A. Kofke, J. Phys. Chem. C 111, (2007) [13] K. M. Benjamin, J. K. Singh, A. J. Schultz, D. A. Kofke, J. Phys. Chem. B 111, (2007) [14] A. Schaber, Zum thermischen erhalten fluider Stoffe Eine systematische Untersuchung der charakteristischen Kurven des homogenen Zustandsgebiets, Ph.D. thesis, echnische Hochschule Karlsruhe, Germany (1965) [15] M. P. ukalovich, M. S. rakhtengerts, G. A. Siridonov, eloenergetica 14, (1967) 65, russian language [16] G. S. Kell, G. E. McLaurin, E. Whalley, J. Chem. Phys. 48, (1965) 3805, doi: / [17] I. M. Abdulagatov, A. R. Bazaev, R. K. Gasanov, A. E. Ramazanova, J. Chem. hermodyn. 28, (1996) 1037 Aendix A hermodynamic derivatives All derivatives are comuted at constant amount of substancen. isochoric tension coefficient: β ( ) ( ) ( ) = P = α κ. (A.1) 12

14 derivatives aearing in Eq. (22): = ( ) nr nr 2 = Zα Z κ = c κc Z, = ( ) nr 2 + nr = Z ( 1 κ ) = Z ( 1 Z ) α c κ c = ( ) + nr nr = Z (1 κ ) = Z ) (1 Zcκ, = ( ) + nr nr = κ 1 nr = Z c κ, = ( ) nr nr 2 = Zα Z = Z (c 1), = ( ) nr 2 + nr = Z ( 1 1 ) = Z ( 1 1 ). α c, (A.2) Aendix B he terminal sloes of the characteristic curves at high temeratures Schaber resents exressions for the terminal sloes of Brown s characteristic curves in his thesis [14], but does not give the roofs. For the readers s convenience, these roofs are offered here. For small ressures and large molar volumes the fluid can be described with a 3-term virial equation (cf. Eq. (23)), Z = 1+ B 2() + B 3() m m 2. (B.1) he molar volume as a function of ressure is then, neglecting higher-order terms, m R +B 2()+ B 3(). (B.2) R Aendix B.1 he terminal sloe of the Amagat curve Alying the first one of the Amagat criteria in Eq. (2) to the virial equation yields = B 2 () + B 3 () m m m 2 = 0 or B 2()+ B 3 () m = 0. (B.3) 13

15 In the limit 0, m the second term can be neglected, and therefore B 2 () = 0 is the criterion for the endoint of the Amagat curve, which corresonds to a maximum of the second virial coefficient. Let A denote the temerature of this endoint, and = A and = the temerature and ressure deviations, resectively, from this oint. hen, in the vicinity of A, B 2 () can be aroximated by B 2 () B 2 ( A), where B 2 ( A) denotes the curvature of the second virial coefficient function at the Amagat endoint. Substituting this into Eq. (B.3) and switching from molar volume to ressure yields and, after some rearrangements, B 2( A ) + In the limit 0, A this reduces to d lim A d B 3 () R +B = 0 = B 2 ( A) B 3 () (R + B 2()+...). qr = 1 2 = B 2 ( A)R A B 3 ( A). (B.4) Aendix B.2 he terminal sloe of the Boyle curve Alying the criterion Eq. (5) to the virial equation gives = B 2() m m 2 2B 3() m 3 and hence B 2 ()+ = 0 2B 3 () R +B 2 ()+... = 0. (B.5) In the limit 0, this reduces to B 2 = 0: At the endoint of the Boyle curve, at the Boyle temerature B, the second virial coefficient vanishes. or Again using deviation variables, we can write the revious equation as B 2( B ) + 2B 3 () R + B 2 ()+... = 0. = B 2 ( B) 2B 3 () (R + B 2()+...). In the limit 0, B this reduces to d lim B d = qr =0 B 2 ( B)R B. (B.6) 2B 3 ( B ) 14

16 Aendix B.3 he terminal sloe of the Charles curve For the derivation of this roerty it is advantageous to start from the ressure-based virial equation of state, Z = 1+ B 2 ()+ B 3 ()+ 2. (B.7) he ressure-based virial coefficients are related to the volume-based ones by B 2 () = B 2() R and B3 () = B 3() B 2 2 () (R) 2. (B.8) Alying the aroriate criterion in Eq. (8) gives = R or ( ) B 2 B() ( ) ( 2 () + B 3 R () 2B 3() 2B 2 ()B 2 ()+ 2B2 2 () ) = 0. B 2() B 2() + R ( B 3() 2B 3() 2B 2 ()B 2()+ 2B2 2 () ) = 0. (B.9) he endoint, at 0, is evidently characterized byb 2 () B 2()/ = 0. In order to obtain the terminal sloe, we exand this criterion around the endoint temerature C, B 2() B ( 2() B 2( C ) B 2 ( C) + B ) 2( C ) C C 2 = B 2( C ). hen Eq. (B.9) becomes B 2( C ) = ( B R 3() 2B [ 3() 2B 2 () B 2 () B ]) 2() = 0. In the limit 0, C, where the term in brackets vanishes, this yields the terminal sloe, d lim C d qr = 1 3 = B 2 ( C)R C B 3 ( C) 2 C B 3 ( C ). (B.10) 15

17 Aendix C Figures /MPa /K Fig. 1: Overview of Brown s first-order characteristic curves for the IAPWS-95 equation of state. : Amagat curve, Boyle curve, Charles curve, grey: vaour ressure curve, : critical oint. he melting ressure curves have been omitted for clarity. 16

18 1 A MPa 270 K 5 MPa 300 K 400 K C 10 MPa 0.6 f(q B ) 20 MPa MPa MPa 600 K 0 B f(q A ) Fig. 2: Isolines of ressure and temerature in theq A,q B lane, comuted for the IAPWS-95 equation of state [1]. : isobars MPa, : isotherms K isotherms, : connodes, red: boundary of the vaour liquid 2-hase region, blue: Amagat (A), Boyle (B), and Charles (C) curves. he nonlinearly transformed coordinates f(q X ) = q X /(10+ q X ),X = A,B were chosen to make details discernible. 17

19 1 0.5 q R log 2 ( / c ) log 2 ( / c ) 2 3 Fig. 3: q R as a function of ressure and temerature for water (IAPWS-95 equation of state). he level curves forq R = 1 2, 1 3, and 0 define the Amagat curve, Charles curve, and Boyle curve, resectively. 18

20 /MPa /K (a) Overview: double-logarithmic reresentation. /GPa /K /MPa /K (b) Detail: low-temerature range. (c) Detail: high-temerature range. Fig. 4: Amagat curves of water. : IAPWS-95 equation of state, : equation of state of Holten et al., : melting ressure curve, : trile oint; grey area: region of negativeα, grey curve: oles ofα, : inflection oints of U m ( m ) (the latter three calculated for the IAPWS-95 equation). 19

21 q B A C B q A Fig. 5: Schematic reresentation of an exansion from very high ressure towards the critical region. : Amagat condition (q A = 1), : Boyle condition (q B = 1), : Charles condition (q A = q B ), : critical oint, : homas Fermi limit ( 0,. he high-temerature endoints of the characteristic curves are in the vicinity of the intersection oint (q A = q B = 1), the low-temerature endoints at high q A or q B values : exansion ath of a normal fluid, 1 : ossible starting oint of a fluid exhibiting a density anomaly. 20

22 B 3 /(cm 6 mol 1 ) /K Fig. 6: he third virial coefficient B 3 of water as a function of recirocal temerature. : comuted from the GCPM water interaction otential [12], : sline interolation through these oints with the restriction that B 3 and B 3 must vanish for ; filled symbols: exerimental data: : ukalovich et al. [15], : Kell et al. [16], and : Abdulagatov et al. [17]. 21