Structure and Property Prediction of Sub- and Super-Critical Water

Transcription

1 Structure and Property Predcton of Sub- and Super-Crtcal Water Hassan Touba and G.Al Mansoor Department of Chemcal Engneerng, Unversty of Illnos at Chcago, (M/C 63), Chcago, Illnos , U.S.A. Paper presented at: The Thrteenth Symposum on Thermophyscal Propertes, June -7, 997, Boulder, Colorado, U.S.A. Emals: (). ; (). gal.mansoor@gmal.com ()

2 ABSTRACT In ths report two approaches are presented to predct the structure and PVT behavor of assocatng fluds wth emphass on water. One approach s the development of equatons of state based on the analytc chan assocaton theory (ACAT). An assocatng flud s assumed to be a mxture of monomer, dmer, trmer, etc., for whch the composton dstrbuton are obtaned. The resultng equatons are smple enough to be used for PVT calculatons. The second approach s the development of an analytcal expresson for the frst shell of the radal dstrbuton functon (RDF). Ths expresson satsfes the general functonalty of the RDF wth respect to ntermolecular potental, temperature and densty as well as all the lmtng values of RDF at hgh temperature and dlute gas, and nfnte separaton. Ths model has ntally been appled to smple potental energy functons, such as the Lennard-Jones and Khara functons. The results have been reported to be n good agreement wth the avalable computer smulaton data of RDF for these flud models and the expermental data for argon. Fnally, an effectve Khara par potental s derved for water whch ncorporates the hydrogen bondng usng the ACAT. The potental parameters of ths model are determned to predct the frst shell of the RDF data for water at varous subcrtcal temperatures and denstes. The predcted results for water at near crtcal and supercrtcal condtons are shown to be n agreement wth the data obtaned by neutron dffracton experments and wth the smulaton data. KEY WORDS: equaton of state; radal dstrbuton functon; potental energy functon; assocaton. ()

3 . INTRODUCTION H. Touba and G.A. Mansoor The key feature of water molecules s the hydrogen bondng whch refers to the formaton of chemcal aggregates or polymers. Nemethy and Scherga [], ndcated through ther studes on water structure that hydrogen bondng plays an mportant role n formng aggregates that can reach szes of up to one hundred H O molecules at room temperature. We have represented the assocaton of water molecules due to hydrogen bondng wth the followng chan reacton []. K (T) (H O) + (H O) (H O) + =,,..., () The equlbrum constant of the above reacton can be expressed as K = [x + /(x x )] [γ + /( γ γ )] = [x + /(x x )] Γ =,,..., () where x and γ are the mole fracton and actvty coeffcent of (H O), respectvely, and Γ s the rato of actvty coeffcents. For smplcty, we assume that all K 's and all Γ 's are the same, (.e. K=K =K =...= K =... and Γ=Γ =Γ =...=Γ =...). Let us defne κ as the rato of K / Γ, κ K / Γ = x + /(x x ) =,,..., (3) We may extend Eq. (3) to dfferent speces as follows: x = κ x x 3 = κ x x = κ x 3 : (4) x + = κ x x = κ x + : Snce the summaton of all mole fractons s unty, then κ x + = (5) = Consderng κ x <, we may show that the seres on the left sde of Eq. (5) converge to x /(- κ x ) = (6) (3)

4 Then the followng relaton wll result for the composton of the monomer: x =/(- κ) (7) Havng a large number of assocated components, compostons may be replaced wth a composton dstrbuton functon χ(i) where I s the number of assocated monomers. In ths case, the summaton n Eq. (5) can be replaced wth an ntegral. χ(i) di = (8) where χ(i) s defned as follows: χ(i) = χ κ I x I+ (9) χ s the normalzng factor and can be calculated by usng Eq. (8). χ = - (ln κ x )/ x () If we substtute χ and x nto Eq. (9), we get the followng result. χ(i) = - [κ /(+ κ)] I ln [κ /(+ κ)] () In what follows, we present a method for the development of equatons of state of assocatng fluds applyng the above composton dstrbuton whch s related to equlbrum mxtures of assocated speces. Moreover, ths composton dstrbuton wll be used n obtanng the molecular potental functon parameters. Havng the potental energy functon, we may be able to obtan the RDF applyng a general analytcal expresson for the frst shell of the RDF whch was ntally proposed by the authors [3] for smple potental energy functons, such as the Lennard-Jones and Khara functons.. APPLICATION OF COMPOSITION DISTRIBUTION TO EQUATIONS OF STATE The authors have used the composton dstrbuton already developed n order to extend the equatons of state parameters to assocatng fluds []. As an example ths extenson s frst (4)

5 performed on the van der Waals equaton of state (VDW EOS). Snce an assocatng flud s a multcomponent mxture of dfferent polymers, equaton of state parameters a and b can be expressed by the followng mxng rules. a = x x a = x x (a a ) / = ( x a / ) () b = x x b = x x (b +b ) / = x b (3) H. Touba and G.A. Mansoor The above equatons can be replaced wth the followng expressons a = ( χ(i) [a(i)] / di ) (4) b = χ(i) b(i) di (5) For smplcty, the parameter a(i) s consdered to be a lnear functon of dstrbuton ndex "I" n the followng form, and snce "b" s proportonal to molecular volume, b /3 wll be lnearly proportonal to molecular length of assocatng speces, thus a(i) / = a / + a / I (6) b(i) /3 = b /3 I (7) where b s the equaton of state parameter for the monomer. Substtutng Eqs. (), (6) and (7) nto Eqs. (4) and (5), respectvely, and ntegratng, and then wrtng the resultng equatons wth respect to the crtcal propertes, we get: a = a c F(ξ) (8) b = b c ξ 3 (9) where F(ξ) = [(C + ξ)/(c +)] () C - (a / a ) / ln [κ c /(+κ c )] () (5)

6 ξ = ln κ c / ln κ = (ln K c - ln Γ c )/(ln K- ln Γ) () The assocaton constant appearng n the above equaton s expressed as ln K = (T S - H )/(RT) (3) where the reference change of enthalpy H and entropy S of assocaton are ndependent of temperature. The actvty coeffcent rato, Γ, s a functon of temperature, pressure and mole fracton. For smplcty, we may assume Γ to be also ndependent of pressure and mole fracton, and have the followng smple expresson. ln Γ = α / (RT) (4) where " α " s a constant. Usng Eqs. (3) and (4), the parameter "ξ" n Eq. () can be rewrtten as ξ = T r (+ ξ )/(T r + ξ ) (5) where ξ s defned as ξ - ( H + α )/( S T c ) (6) Snce " ξ" depends only on temperature, parameters a and b wll also be temperature dependent only. Therefore, the VDW EOS for assocatng fluds can be wrtten as P = RT/(V-b c ξ 3 ) -a c F(ξ) /V (7) The term V appearng n ths equaton s the true molar volume, and t s based on true number of moles. To calculate molar volume from Eq. (7) we need to have pressure and temperature of the system lke any equaton of state. Snce at the crtcal pont, the condtons ( P/ V) T,cr = ( P/ V ) T,cr = have to be satsfed, then for the VDW EOS, we wll have: (6)

7 a c =(7 R T c )/(64P c ) (8) b c = RT c /(8P c ) (9) whch are the same as the orgnal VDW EOS. The theory proposed here can be extended to other equatons of state by a smlar method as t s reported above for the VDW EOS. For example, consderng the Redlch-Kwong equaton of state (RK EOS), t can be shown that t takes the followng form for assocatng fluds: P = RT/(V-b c ξ 3 ) - a c F(ξ) /[T.5 V(V+b c ξ 3 )] (3) where "a c " and "b c " n ths equaton are the same as n the orgnal RK EOS. In the calculatons reported below, both the van der Waals and Redlch-Kwong equatons of state were used for water. Fve dfferent expermental sotherms wth reduced temperatures of.5,.7,.,.5 and. [4] were chosen for specfc volume calculatons of water whch cover both vapor and lqud phases equally. Also 35 saturaton data ponts (from trple pont to crtcal pont) were used for vapor pressure calculatons. Table shows the absolute average devatons (AAD%) of vapor pressure and specfc volume calculatons of water based on the orgnal VDW EOS and RK EOS and based on the mproved equatons applyng the values of assocatng parameters C and ξ. Accordng to ths table, C = ± whch mples that only the parameter ξ needs to be consdered for water n the assocatng VDW and RK EOS's. Another case of nterest may be the case where the unlke nteracton parameter a s represented by a = (a a ) / (-k ) where k s the couplng parameter of assocatng speces. Ths case has been studed n detal n Ref. []. Calculatons have also been reported for varous assocatng fluds based on these two cases. It has been demonstrated that ncorporatng the proposed theory wth equatons of state wll mprove the propertes calculatons n all cases. In the followng secton, another applcaton of the ACAT and the correspondng composton dstrbuton wll be studed n order to obtan molecular potental functon parameters for assocatng fluds. (7)

8 3. APPLICATION OF COMPOSITION DISTRIBUTION TO POTENTIAL PARAMETERS The composton dstrbuton prevously developed, may also be ncorporated n the potental energy functon. For smplcty, we assume that the assocatng speces form an deal soluton,.e. Γ= and accordng to Eq. (3), κ =K. Therefore, Eq. () has the followng form. χ(i) = -[K /(+K)] I ln [K /(+K)] (3) The assocaton constant appearng n the above equaton may be determned from Eq. (3). The most relable values for H and S are those obtaned usng spectroscopc methods, such as Raman spectroscopy as reported by Walrafen et al. [5]: H = - kj/mole, S = -5 J/mole K Applyng the conformal soluton theory, whch assumes that there exsts a pure hypothetcal flud wth the same propertes as those of the mxture at the same densty and temperature, the par potental φ can be represented as follows [6]: φ = ε φ oo (r /σ ) (3) where subscrpt (oo) denotes the reference flud, r s the ntermolecular dstance and the parameters σ and ε represent the molecular dameter and energy parameter, respectvely. Among the statstcal mechancal conformal soluton theores of mxtures, one-flud van der Waals theory s smple to use and accurate enough wth the followng form: σ 3 3 = x x σ ε σ 3 3 = x x ε σ δ 3 3 = x x δ (33) (34) (35) (8)

9 Applyng the combnng rules σ 3 = (σ 3 + σ 3 ) /, δ 3 = (δ 3 +δ 3 ) / and ε = (ε ε ) / for unlke-nteracton potental parameters, we get the followng expressons: σ 3 = x x [(σ 3 + σ 3 3 )/ ] = x σ δ 3 = x x [(δ 3 + δ 3 3 )/ ] = x δ ε σ 3 = x x (ε ε ) / [(σ 3 + σ 3 )/ ] = (x ε / σ 3 ) / x ε (36) (37) (38) Consderng the number of assocatng speces to be very large, we can replace the above summatons wth ntegrals and the compostons wth the composton dstrbuton functon. σ 3 = δ 3 = χ(i) [σ(i)] 3 di (39) χ(i) [δ(i)] 3 di (4) ε σ 3 = χ(i) [ε(i)] / [σ(i)] 3 di χ(i) [ε(i)] / di (4) We relate parameters [σ(i)] 3, [δ(i)] 3 and [ε(i)] / to be functons of dstrbuton ndex "I". [σ(i)] 3 = σ 3 I ξ (4) [δ(i)] 3 = δ 3 I ξ (43) [ε(i)] / = ε / I ξ 3 [+ ξ ( /I -)+ ξ 3 4 (-I)] (44) where σ, δ and ε are the potental energy functon parameters for the monomer and ξ, ξ, ξ 3 and ξ 4 are constants. wll have Substtutng Eqs. (3) and (39)-(4) nto Eqs. (4)-(44), respectvely, and ntegratng, we (9)

10 σ = σ [Γ (+ ξ ) θ ξ ]/3 (45) δ = δ [Γ (+ ξ ) θ ξ ]/3 (46) ε = ε θ ξ 3 [Γ (+ ξ + ξ 3 ) Γ (+ ξ 3 )/ Γ (+ξ )] {+ ξ 3 /[(ξ + ξ 3 ) θ]- ξ 3 + ξ 4 [-(+ξ + ξ 3 ) θ]}. {+/θ- ξ 3 + ξ 4 [-(+ ξ 3 ) θ]} (47) where Γ s the gamma functon and θ -/ln [K /(+K)]. At ths pont, we are gong to use these potental parameters n an analytcal expresson for the frst shell of the RDF proposed by the authors [3] for smple potental energy functons. 4. AN ANALYTIC EXPRESSION FOR THE FIRST SHELL The radal dstrbuton functon of water s the most nformatve feature of ts molecular structure. In general, two lmtng condtons that the radal dstrbuton functon has to satsfy are the case of dlute gases n whch the densty approaches zero and the case of hard spheres where the temperature approaches nfnty. The RDF of dlute gases can be derved from the statstcal thermodynamcs as g dg (y)=exp [-β φ(y)] (48) where g dg (y) s the dlute gas RDF, y r/σ, β /(k T) and k s the Boltzmann's constant. The authors [3] proposed the followng functonal form for the frst shell of radal dstrbuton functon whch satsfes these two lmtng cases. g(y)= m g hs () exp [-m β φ(x)]+ (-m ) exp [-m β φ(y) -c (y-d * )] for y d * (49) g(y)=m g hs (x)+ (-m ) exp [-m β φ(y) -c (y-d * )] for d * y < y m (5) where x r/d, d * d/σ, d s the locaton of maxmum of RDF whch n the case of hard sphere RDF corresponds to the hard core dameter, y m s the mnmum of the RDF after the frst peak, ()

11 g hs (x) s the hard sphere RDF for whch the Werthem s analytcal soluton [7] of Percus-Yevck equaton for the frst shell of hard sphere radal dstrbuton functon has been utlzed. Both Eqs. (49) and (5) converge to the the followng smple equaton at the maxmum of the frst peak of the RDF (at y=d * ). H. Touba and G.A. Mansoor g(d * )=m g hs ()+(-m ) exp [-m β φ(d * )] (5) The parameters c and c appearng n Eqs. (49) and (5) must be determned from the fact that the g(y) has to be maxmum at dstance d *,.e. [ g(y)/ y] y=d* =. c = -m m β φ () g hs () / {d * (-m ) exp [-m β φ(d * )]} -m β φ (d * ) (5) c = m g hs () /{d * (-m ) exp [-m β φ(d * )]} -m β φ (d * ) (53) The parameters m, m and d * n the RDF equaton are expressed as functons of the dmensonless temperature T * k T/ε, and dmensonless densty ρ * ρσ 3 such that they satsfy the lmtng condtons of RDF. m = exp [-4.93/( ρ * T * )] (54) m = exp [.68 ρ * (-/ T * )] (55) d * =R m exp (-.483 ρ * T *.5 ) (56) where R m s the locaton of dlute gas RDF peak whch can be calculated by solvng the equaton: [ φ(y)/ y] y = R = φ (y =R m ) = (57) m Accordng to Eq. (54), as the temperature approaches nfnty, m wll approach unty. Therefore, we conclude that Eq. (5) approaches the lmtng case of hard sphere RDF, g hs for y d * (or x ). For the case where the densty s very low, accordng to Eqs. (5)-(56), m =c =c = and m =. Hence Eqs. (49) and (5) reduce to Eq. (48), the dlute gas RDF. ()

12 The present model was tested versus the frst shell RDF data for the Lennard-Jones and Khara fluds and versus the expermental results for the argon. A good agreement was reported to be between the calculated RDF and the smulated and expermental data [3]. In the followng secton, the above expressons for the frst shell of RDF are oned wth the Khara potental energy functon whose parameters, n the case of water, has been found by usng the ACAT and the conformal soluton theory. 5. APPLICATION TO KIHARA POTENTIAL FUNCTION The Khara potental n whch each molecule s assumed to have an mpenetrable hard core of dameter δ accounts for the ntermolecular forces of water provded that ts parameters nclude the hydrogen bondng effects. In order to apply the precedng frst-shell RDF model to Khara potental, we derve the followng expressons for φ(y), φ(x), φ(d * ), φ (d * ), φ (), and R m. φ(y)= 4ε {[(-δ * )/(y- δ * )] - [(- δ * )/(y- δ * )] 6 } (58) φ(x)= 4ε {[(- δ * /d * )/(x- δ * /d * )] - [(- δ * /d * )/(x- δ * /d * )] 6 } (59) φ(d * )= 4ε {[(- δ * )/(d * - δ * )] - [(- δ * )/(d * - δ * )] 6 } (6) φ (d * )= [-4ε/(d * - δ * )] { [(- δ * )/(d * - δ * )] - [(- δ * )/(d * - δ * )] 6 } (6) φ ()= -4ε/(- δ * /d * ) (6) R m = δ * + /6 (- δ * ) (63) where δ * = δ/σ. Fg. shows the varatons of calculated values of Khara parameters usng the proposed model for the frst shell of RDF and the expermental data reported by Narten and Levy [8]. It can be nferred from ths fgure that σ does not practcally change wth temperature ()

13 whch mples that ξ n Eq. (45) s almost zero. The values of δ and ξ are found to be.35 Å and.. Consequently, Eqs. (45)-(47) reduce to σ σ =.68 Å (64) δ =.35 θ /3 (65) ε = ε {θ ξ 3 Γ (+ ξ 3 ) {+/θ - ξ 3 + ξ 4 [-(+ ξ 3 ) θ]} } (66) where ε /k =3 K, ξ 3 =.4 and ξ 4 =.36. Fg. represents the calculated frst shell of the RDF of water at varous temperatures usng the effectve Khara potental functon for assocatng fluds. The curves have been compared wth the expermental data for RDF at the same condtons determned by Narten and Levy [8]. In order to verfy the valdty of the proposed model for near crtcal and supercrtcal condtons, we have calculated the frst shell of RDF for the condtons T=3 C, ρ=.7 g/cm 3 and T=4 C, ρ =.66 g/cm 3 for whch the expermental data were reported by Postorno et al. [9] and the molecular smulaton data based on ST model were reported by Chalvo and Cummngs []. Accordng to Fg. 3, there s a good general agreement between the proposed theory and the expermental and smulated data as far as the locaton and heght of the peak are concerned. However, the bump on the left hand sde of the frst peak n the expermental data could not be reproduced wth the proposed model. Chalvo and Cummngs [] beleved that ths bump was a false feature of the RDF of water ntroduced by the dffracton data analyss. Ther molecular smulaton model also faled to predct ths bump. One of the mportant features of the expermental RDF data of water was reported to be the fact that as the temperature s rased from ambent temperature to about 7 C, the frst peak dmnshes a lttle n heght, whle by further ncreasng the temperature to supercrtcal regon, there s a rse n the frst peak agan [9]. Ths pecularty of water whch can be consdered n all expermental and smulated data [8-] s also predcted by the proposed model. (3)

14 Fg. 4 represents the varatons of the frst peak of RDF of water at reduced temperatures (T r =T/T c =T * /T c * ) of.5,. and.5 and reduced denstes (ρ r =ρ/ρ c =ρ * /ρ c * ) of.5,. and., respectvely. In order to compare the results obtaned for water wth the RDF of Lennard- Jones fluds, the correspondng RDF curves for Lennard-Jones fluds at the smlar condtons have been plotted on the same fgure. We have chosen the crtcal temperature and densty of Lennard-Jones fluds [,] T c * =.3 and ρ c * =.3. The frst peaks of Lennard-Jones fluds RDF are hgher at subcrtcal and lower at supercrtcal condtons than those of water. In order to nvestgate the suffcency of the nformaton for the frst shell of RDF, the concept of the radus of nfluence has been utlzed whch was proposed by Mansoor and Ely [3] for the theory of local compostons. Ths concept s appled to calculate the dstance at whch the RDF could be truncated for determnng the sothermal compressblty. In all cases studed, t s shown that the radus of truncaton s located nsde the frst shell of RDF. The sothermal compressblty, κ T, can be calculated by applyng the equaton: κ * T = /(ρ * T * )+ (4π/T * ) [g(y) -] y dy (67) where κ T * κ T ε/ σ 3. The ntegral n Eq. (67) can be wrtten n the followng form. [g(y) -] y dy = R κ [g(y) -] y dy+ Rκ [g(y) -] y dy (68) The radus of truncaton of RDF, R κ, s chosen such that the second ntegral dsappears,.e. [g(y) -] y dy= (69) R κ In general, Eq. (69) has several roots for R κ. However, we mpose the constrant that R κ has to be wthn the frst shell of RDF. Therefore κ * T = /(ρ * T * )+ (4π/T * R κ ) [g(y) -] y dy (7) (4)

15 Fg. 5 represents schematcally the value of R κ whch gves rse to the equalty of dashed areas above and below the horzontal axs to the rght of R κ. Snce g(y) s a functon of temperature and densty, R κ s also a functon of temperature and densty. Fg. 8 llustrates the varatons of R κ wth the temperature for water at P sat,.5 and kbar. In order to obtan R κ, we have utlzed the expermental sothermal compressblty data reported by Helgeson and Krkham [4]. Accordng to Fg. 6, the radus of truncaton of RDF s wthn the frst shell for all the condtons reported. 6. CONCLUSION We have demonstrated that the analytc chan assocaton theory can be ncorporated nto equatons of state. When ths theory s appled to cubc equatons of state, wth certan assumptons the cubc nature of these equatons can be retaned. The resultng equatons are smple enough to be used for PVT calculatons. Numercal calculatons for densty and vapor pressure have been performed over wde ranges of temperature and pressure for water as one representatve assocatng flud. It can be observed that the results are greatly mproved when applyng the assocaton theory and the error s n the order of magntude of applyng the equaton to non-assocatng systems. We have also derved an effectve Khara par potental for water whch ncorporates the hydrogen bondng by usng the ACAT and the conformal soluton theory. The potental parameters have been obtaned based on the frst shell of RDF expermental data for water by usng an analytcal expresson for the frst shell of RDF prevously proposed by the authors whch satsfes the lmtng cases of hard sphere radal dstrbuton functon at hgh temperatures and the dlute gas radal dstrbuton functon at very low denstes. The calculated values of RDF compare well wth sub-crtcal expermental data and wth smulaton and expermental near-crtcal and supercrtcal data. (5)

16 REFERENCES [] G. Nemethy and G.H. Scheraga, J. Chem. Phys., 36 (96) [] H. Touba and G.A. Mansoor, Flud Phase Equlbra, 9 (996) [3] H. Touba and G.A. Mansoor, Int. J. Thermophys., (to appear 997). [4] L. Haar, J.S. Gallagher, and G.S. Kell, NBS/NRS steam tables. McGraw-Hll, New York 984. [5] G.E. Walrafen, M.R. Fsher, M.S. Hokmabad and W.H. Yang, J. Chem. Phys., 85 (986) [6] W.B. Brown, Phlosophcal transactons of the Royal Socety of London, seres A: Mathematcal and physcal scences, 5 (957) 75. [7] M.S. Werthem, Phys. Rev. Lett. (963) [8] A.H. Narten and H.A. Levy, J. Chem. Phys., 55 (97) [9] P. Postorno, R.H. Tromp, M.A. Rcc, A.K. Soper and G.W. Nelson, Nature, 366 (993) []A.A. Chalvo and P.T. Cummngs, J. Phys. Chem., (996) []J.K. Johnson, J.A. Zollweg and K.E. Gubbns, Mol. Phys., 78 (993) []A. Lotf, J. Vrabec and J. Fscher, Mol. Phys., 76 (99) 39. [3]G.A. Mansoor and J.F. Ely, Flud Phase Equlbra, (985) [4]H.C. Helgeson and D.H. Krkham, Amercan J. Sc., 74 (974) 89. Table. Absolute average devaton of vapor pressure and specfc volume of water. EOS ξ C AAD% n P sat AAD% n V VDW (orgnal) VDW (proposed theory) ± RK (orgnal) RK (proposed theory).9 ± (6)

17 3.8 σ, Å δ, Å ε/ k, Kelvn T, C Fg.. Varatons of the Khara parameters for water wth temperature usng the proposed model for the frst shell of RDF (7)

18 4 C C 5 C 5 C g(r) 75 C C 5 C C Fg r, Å Comparson of the proposed model for the frst shell of the water RDF and the expermental data at varous temperatures. (8)

19 3 C Expermental data Smulaton data Proposed model g(r) 4 C r, Å Fg. 3. Comparson of the predcted frst shell of the water RDF wth the expermental data and the molecular smulaton results at near crtcal and supercrtcal condtons. (9)

20 5 4 3 ρ r =.5 for water ρ r =. for water ρ r =. for water ρ r =.5 for LJ ρ r =. for LJ ρ r =. for LJ g(x) T r =.5 T r =. T r = x=r/d Fg.4. Varatons of the frst peak of the water RDF and a Lennard-Jones flud at dfferent condtons. ()

21 .5 [g(y)-] y.5 R Κ y=r/σ Fg. 5. Radus of truncaton R κ for the sothermal compressblty ntegral [g(y)-] y R Κ P=P sat P=.5 kbar P= kbar 3 4 T, C Fg. 6. Varatons of the radus of truncaton R κ wth temperature and densty for water. ()