New models for tracer diffusion coefficients of hard sphere and real systems: Application to gases, liquids and supercritical fluids

New models for tracer diffusion coefficients of hard sphere and real systems: Application to gases, liquids and supercritical fluids

J. of Supercritical Fluids 55 (2011) 898–923 Contents lists available at ScienceDirect The Journal of Supercritical Fluids journal homepage: www.els...

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J. of Supercritical Fluids 55 (2011) 898–923

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids journal homepage: www.elsevier.com/locate/supflu

New models for tracer diffusion coefficients of hard sphere and real systems: Application to gases, liquids and supercritical fluids Ana L. Magalhães, Francisco A. Da Silva, Carlos M. Silva ∗ Department of Chemistry, CICECO, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal

a r t i c l e

i n f o

Article history: Received 22 May 2010 Received in revised form 17 September 2010 Accepted 20 September 2010 Keywords: Tracer diffusion coefficient Rice and Gray Hard sphere Lennard–Jones Effective hard sphere diameter Self-diffusion coefficient

a b s t r a c t In this work very accurate expressions for the tracer diffusion coefficient of hard sphere (HS) and real systems are proposed. The new HS model depends explicitly on the reduced density of solvent, and on the ratios of the diameters and masses of solute and solvent. It provides a very good representation of molecular dynamics data taken from literature: average absolute relative deviation, AARD = 4.44%. With respect to real fluids, the proposed model was developed according to Rice and Gray approach, and is based on the previous HS equation. The model involves only one parameter and requires temperature, solvent density, and solute and solvent molecular weight and LJ force constants (these are estimated as function of the critical temperature and molar volume). Results calculated for 309 systems and 5341 data points gave rise to AARD = 4.26%, and shows the model interprets equally well the diffusive phenomena of gases, liquids and SCFs. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Infinite dilution diffusion coefficients (D12 ) are key transport properties in project and design of challenging chemical reactors and separation processes. For instance, unit operations based on supercritical fluid extraction frequently requires diffusivities difficult to grasp from experimental data, or to update to specific temperature and pressure conditions. Expressions for their calculation with limited available data constitute an ambitious objective. With such purpose, several models have appeared targeting specific temperature and pressure ranges, and fluid physical states [1–3]. More recently, wide range application models were proposed to predict tracer diffusion coefficients, namely those by Liu et al. [4], Liu and Ruckenstein [5], and Zhu et al. [6], whose validation involved between 74 and 120 binary systems (1033–1443 data points) with average absolute relative deviations (AARDs) between 8.42 and 17.31%. In this work we aim to develop new models for the tracer diffusivities of hard sphere (HS) and real fluids. Accordingly, an empirical correlation for D12,HS is firstly devised on the basis of molecular dynamics (MD) data available in the literature for the HS fluid. Then, a new equation for LJ tracer diffusivities, D12,LJ , is obtained by taking into account the Rice and Gray’s approach [5,7–9]. Finally, an adjustable interaction binary parameter is embodied in this LJ

∗ Corresponding author. Tel.: +351 234 401549; fax: +351 234 370084. E-mail address: [email protected] (C.M. Silva). 0896-8446/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.supflu.2010.09.031

model, in order to extend its application to real systems. The validation of our model for real molecules has been accomplished with the largest database ever compiled, which comprehends 309 systems and 5341 data points. 2. Theoretical approach The theoretical path adopted in this essay to develop the desired tracer diffusion models is shown in Fig. 1. Our approach considers four stages: it starts with ideal gases and passes to hard spheres by correcting Enskog theory; by introducing an attractive contribution and adopting effective hard sphere diameters, a LJ model arises from the HS model; finally, LJ model is extended to real systems. 2.1. Ideal gas The diffusion coefficient of dilute gases (i.e. in the limit of zero 0 , may be estimated from the rigorous kinetic theory by density), D12 [2,9]: 0 10 D12 =

3 2 812

 k T 1/2 B 2m12

(1)

where superscript “0” stands for dilute gas, subscripts “1” and “2” denote solvent and solute, respectively,  is number density, kB is Boltzmann constant, T is absolute temperature, m12 is the reduced mass of the system, and  12 is the distance between the centers of the molecules at collision. The values of  12 and m12 are obtained

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Nomenclature AARD B D F g(␴) HS kB k12 LJ m M MD NDP NS Na P P T V VD

average absolute relative deviation, AARD = NDP calc exp exp (100/NDP) i=1 |(D12,Real − D12,Real )/D12,Real |i parameter in Eq. (B.15) tracer diffusion coefficient, cm2 /s correction factor of HS system radial distribution function at contact hard sphere Boltzmann constant, 1.380658 × 10−16 g cm2 /s2 K binary interaction parameter Lennard–Jones mass of a molecule, g molecular weight, g/mol Molecular dynamics number of data points number of systems Avogadro constant, 6.0221367 × 1023 mol−1 pressure, bar Parachors temperature, K molar volume, cm3 /mol Parameter in Eq. (B.15), cm3 /mol

Fig. 1. Steps for the development of the new tracer diffusion coefficient model for real systems.

2.2. Enskog fluid A kinetic theory for transport coefficients of a dense HS system has been developed by Enskog [10], who considered that the molecular diameters are no longer negligible compared with interparticle distance, and modified the collision frequency in the fluid by the unlike pair radial distribution function at contact, g( 12 ). The Enskog equation for the tracer diffusion coefficient is: 1 D12,E 0 10 D12

=

1 g(12 )

(4)

In this work g( 12 ) is calculated as proposed by Mansoori et al. [11]:

Greek letters  viscosity, cP  friction coefficient ϕ1 HS packing fraction of solvent  number density, Na /V, cm−3 Lennard–Jones energy parameter, K ε/kB  Molecular diameter, cm

g(12 ) =



1 (1 − ϕ1 )

3

1 − ϕ1 +

2ϕ1 1 + 1 /2



1 − ϕ1 +

ϕ1 1 + 1 /2



(5)

where ϕ1 is the HS packing fraction of the solvent, which for N1 spheres occupying a volume V is given by:

Subscripts bp boiling point BAH Ben-Amotz and Herschbach c critical property eff effective hard sphere diameter (EHSD) E Enskog HS hard sphere fluid LJ Lennard–Jones fluid r reduced property R repulsive contribution Real real system S soft attractive contribution 1, 11 solvent 2 solute 12 binary property

ϕ1 =

N1 13 6V

=

  1 13 = 1∗ 6 6

(6)

1∗ ≡ 1 13 is the well known reduced number density. 2.3. Hard sphere fluid The Enskog theory is based on the molecular chaos approximation, and therefore it is not applicable over large density range. The errors introduced neglecting correlated motions between core collisions, which gives rise to both backscattering and vortex flow effects, have been studied in computer simulations for systems of a single test particle in a solvent, for selected size and mass ratios, to assess the necessary correction [2,9]. The HS tracer diffusivity is usually written as a modification of the Enskog theory, by introducing a correction factor hereafter denoted by F12 :



Superscripts 0 ideal gas * reduced quantity

D12,HS =

D12,HS D12,E



D12,E = F12 D12,E

(7)

This correction factor depends on the reduced density of the solvent, and on the size and mass ratios, so one may write: from the individual molecular diameters and masses by: 12,LJ = m12 =

1,LJ + 2,LJ 2

m1 m2 M1 M2 = m1 + m2 Na (M1 + M2 )

F12 = F12 (1∗ , 1 /2 , m1 /m2 ) (2) (3)

Eq. (1) is valid for monatomic gases and has to be modified if molecules have internal structure. Also, it is not applicable to dense gases and liquids since it is based upon the Boltzmann equation for the distribution function.

(8)

As shown by Alder et al. [12], the dependence of F12 is complex: at low densities, it increases with increasing  2 / 1 when m2 /m1 = 1.00, while diminishes with increasing  2 / 1 for m2 /m1 ≤ 0.10; at high densities, F12 always decreases as  1 / 2 increases. Additionally, it must satisfy the following four restrictions: (i) For equal diameters and masses, F12 reduces to the selfdiffusion correction factor, F11 , for which several good

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correlations are available [8,13–15]: lim

1 /2 →1 m1 /m2 →1

F12 =

The implied reduced temperatures are:

  D11,HS = F11 1∗ D11,E

(9)

∗ ≡ T12

(ii) At low density the Enskog theory has to be recovered: lim F12 (1∗ , 1 /2 , m1 /m2 ) = 1

(10)

∗ →0 1

(iii) At low density, the self-diffusion correction factor derived from Eq. (9), F11 , must satisfy Enskog theory:

 

lim F11 1∗ = 1

(11)

∗ →0

Ti∗ ≡

 

lim F11 1∗ = 0.

(12)

∗ →s∗ 1

A new F12 explicit and very accurate expression is proposed in the next section of this paper. Results will be compared with those provided by the well known models of Sung and Stell [16], Eaton and Akgerman [17], Easteal and Woolf [18], and Sun and Chen [19], whose equations are compiled in Appendix A. 2.4. Lennard–Jones fluid The LJ tracer diffusion coefficient is obtained by introducing an attractive contribution and an effective diameter ( eff ) on the HS model. In this essay, the first effect is taken into account in the framework of Rice and Gray [5,7–9] approach, which states that: D12,LJ

kB T = 12,R + 12,S

(13)

where ␨12,R is a repulsive friction coefficient, and ␨12,S is a soft attraction friction coefficient. From Eqs. (4) and (7) it is clear that:

g(12,eff ) 8 2 2m12 kB T 1 12,eff F12 3

12,R =

(14)

With respect to ␨12,S , an expression due to Ruckenstein and Liu [8] is chosen: 12,S =

8 0.4 2 1 12,eff 2m12 kB T ∗1.5 3 T

(15)

12

When Eqs. (14) and (15) are substituted in (13), the following explicit D12,LJ model is found: D12,LJ =

k T

1/2 8  2 k T 3 1 12,eff (2m12 B )

B



∗1.5 (g(12,eff )/F12 ) + (0.4/T12 )

(16)

(19)

kB T ε12,LJ

and the binary LJ diameter and energy are evaluated by the classical Lorentz–Berthelot combining rules:

12,LJ



ε2,LJ ε1,LJ × kB kB 1,LJ + 2,LJ = 2

ε12,LJ = kB

1

(iv) The self-diffusion correction factor derived from Eq. (9), F11 , should vanish at an intrinsic high density s∗ :

kB T εi,LJ

(20)

The individual LJ parameters are calculated as functions of the critical temperature and molar volume of each component [8,9] by, respectively: εi,LJ

Tc,i = 1.2593 kB 1/3 i,LJ = 0.7889 × 10−8 Vc,i

(21)

At this moment it is important to emphasize that the calculation of D12,LJ only depends on the existence of a good model for F12 . 2.5. Real fluids In this work, the model for tracer diffusivities of real systems is grafted onto former Eq. (16), by just introducing an adjustable binary parameter k12 in the diameter combining rule, akin to a thermodynamic binary correction. This model will be henceforth identified by LJ-1. Accordingly,  12 is now given by: 12,LJ = (1 − k12 )

1,LJ + 2,LJ 2

(22)

This tiny modification will allow us to reach excellent results for gaseous, supercritical and liquid systems without further complexity, as it consists on the introduction of only one fitted parameter for all density and temperature conditions of any system. Such fact reveals that the k12 goes through the real interpretation of the diffusive phenomenon, similarly to thermodynamic binary corrections. It is well known that mixtures of hard spheres, whose components interact by a hard-core potential diameter given by Eq. (2), are possibly deprived of typical phenomena of fluid mixtures, such as demixing or partial mixing of the components. It is worth noting that the k12 parameter was introduced in the diameter combining rule (Eq. (22)), although comparable results could be obtained by correcting Eq. (20) instead, as has been demonstrated elsewhere [4,9]. Moreover, in this way the effect of k12 is limited 2 to the 12,eff term on the denominator of Eq. (16), which simplifies enormously the required optimizations to determine their values from experimental data. 3. New hard sphere correction factor, F12

The effective diameter of the system, ␴12,eff , is computed by the expression of Ben-Amotz and Herschbach (BAH) according to Boltzmann criterion [9,20,21]:





∗ 12,eff ≡ 12,BAH = 1.153212,LJ 1 + 1.8975T12

1/2 −1/6

(17)

Moreover, it is worth noting that g( 12,eff ) imply the calculation of the both effective diameters,  1,eff and  2,eff , and also ϕ1,eff = 3 , as Eqs. (5) and (6) clearly show. In this case, the (/6)1 1,eff following single component analogue is adopted:





 −1/6 ∗ 1/2

i,eff ≡ i,BAH = 1.1532i,LJ 1 + 1.8975Ti

(18)

In this section it will be presented the new expression for F12 proposed in this work, whose final objective is to embody Eq. (16) for the LJ tracer diffusivities, later extended to real fluids. Our model is a simple empirical and explicit expression that satisfies all theoretical restrictions imposed by Eqs. (9)–(12):



F12 =











F11 + 1∗1.7 a ln 2 /1 + b ln2 2 /1 + c ln m2 /m1





1 + 1∗3.0 d ln 2 /1



 2

(23) This model arrived after an extensive search carried out with statistical TableCurve 3D® software followed by a modification. Using

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Fig. 2. F12 correction factor proposed in this work for the calculation of HS tracer diffusion coefficients, given by Eq. (23) against size and mass ratios, and for: (a) 1∗ = 0.4714 and (b) 1∗ = 0.8839; (䊉) MD data from Herman and Alder [22] and Alder et al. [12].

TableCurve 3D® , equation no 1204 (of rational type with logarithmic variable dependence) gave rise to the best results from the huge set of empirical equations of its database. As described by TableCurve 3D® user manual it «fits 310 polynomials, 300 rationals, scans 453.696.714 selective subset equations fitting up to 36.582 of them». Afterwards, the equation thus obtained was modified to refine the solvent density influence upon F12 . Accordingly, coefficients a, b, c and d were made linearly dependent on 1∗ :

⎧ a = −1.6763821∗ + 1.638561 ⎪ ⎪ ⎪ ⎨ ∗

ture, namely Sung and Stell [16], Eaton and Akgerman [17], Easteal and Woolf [18], and Sun and Chen [19]. A concise description of these models is given in Appendix A. From Table 1 it may be concluded that our model performs undoubtedly better, since the remaining four deviations are much higher: AARDthis work = 4.44 % against AARDSung−Stell = 63.89 %, AARDEaton−Akgerman = 67.78 %, AARDEasteal−Woolf = 18.20 %, and AARDSun−Chen = 24.46 %. In Fig. 3 these five models are plotted together  with  MD dataof Alder et al. [12] in the following way: F12 = F12 1∗ , F12 = F12 m2 /m1 , and



b = −8.5168301 + 8.631536

(24)

⎪ c = −1.3203471∗ + 1.351067 ⎪ ⎪ ⎩ ∗

d = −5.0625461 + 5.409662

The embodied self-diffusion factor, F11 , is calculated by the simple and very accurate expression due to Ruckenstein and Liu [8]: F11 = 1 + 0.946051∗1.5 + 1.40221∗3 − 5.68981∗5 + 2.66261∗7 (25) The constants of the linear relationships in Eq. (24) were optimized using 43 MD data points of Herman and Alder [22] and Alder et al. [12] for the HS system, in the following ranges: 1∗ = 0.4714 − 0.9428, m2 /m1 = 0.01 − 4.00, and  2 / 1 = 0.25 − 1.00. Such optimization was accomplished using fminsearch function of Matlab 2006b® ; the objective function adopted was the average abso  NDP cacl lute relative deviation, AARD (%) = 100/NDP × |(F12 − i=1 MD )/F MD | . F12 12 i The representation accomplished by Eqs. (23)–(25) is shown graphically in Fig. 2 for two different reduced densities, 0.4714 (middle density) and 0.8839 (high density). It is possible to observe the good fitting provided by the new F12 expression, even in the steepest descent region. It is notorious the strong dependence of F12 with  2 / 1 ratio when density increases. The average absolute relative deviation found confirms such finding since it reaches only AARD = 4.44% (see Table 1). The performance of the proposed F12 model has been compared with that achieved by four well known models taken from litera-



F12 = F12 2 /1 . Taking into account Fig. 3a one concludes that the expressions of Sun and Chen [19], and Eaton and Akgerman [17] break the zero density constraint and provide raw quantitative agreement with MD data in the middle density range. The wrong behaviour offered by these two models is also reinforced in Fig. 3b, where m2 /m1 is the independent variable, and 1∗ = 0.4714 and  2 / 1 = 1.00. From Fig. 3c, where F12 is graphed against diameter ratio for 1∗ = 0.9428 and m2 /m1 = 0.001, it is possible to conclude that the theory of Sung and Stell [16] does present a noticeable wrong trend, since F12 increases while data points decrease. Additionally, the calculated results for Easteal and Woolf [18] are poor in the low and middle density regions, what is also illustrated in Fig. 3a. In the whole, all models from literature exhibit outstanding deficiencies, whereas the results achieved by our model are very accurate for all representations and ranges shown. Such good performance certainly due to the impositions stressed on Eq. (23) and also to the number of parameters involved in Eq. (24) which is superior to the remaining models (see Appendix A). 4. New tracer diffusion coefficient model for real systems As has been mentioned above, the new model proposed in this work for the limiting mutual diffusion coefficients of real substances (LJ-1) is based on Eq. (16) for the LJ fluid, by introducing an interaction parameter k12 in the diameter combining rule, Eq. (22). Accordingly, it is possible to fit k12 to experimental data available in the literature for any system.

Table 1 Calculated deviations for the F12 models studied in this work. Model

This work (Eqs. (23)–(25))

Sung and Stell (Eqs. (A.1)–(A.5))

Eaton and Akgerman (Eqs. (A.6)–(A.9))

Easteal and Woolf (Eqs. (A.10) and (A.11))

Sun and Chen (Eq. (A.12))

AARD (%)

4.44

63.89

67.78

18.20

24.46

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Fig. 3. F12 correction factors of the HS tracer diffusion coefficient plotted against: (a) solvent reduced density ( 2 / 1 = 0.75, m2 /m1 = 1.00), (b) molecular weight ratio (1∗ = 0.4714,  2 / 1 = 1.00), (c) molecular size ratio (1∗ = 0.9428, m2 /m1 = 0.01). Legend: () MD data of Alder et al. [12]; (1) this work, Eqs. (23)–(25); (2) Easteal and Woolf [18]; (3) Sung and Stell [16]; (4) Sun and Chen [19]; and (5) Eaton and Akgerman [17].

4.1. Database In this work it has been compiled the largest database ever used: 309 systems performing 5341 points, covering gas (37 sytems/422 points), liquid (101 systems/641 points) and supercritical (171 systems/4278 points) mixtures. Database contains diffusivities collected from open literature. It is evident the increasing number of publications for supercritical systems in comparison to gases and liquids, especially during last decade, which confirms the growing interest on green separation and reaction techniques. Table 2 contains the systems studied, the reduced ranges of temperature, pressure, and solvent density for each system (reduction performed with critical constants), number of data points (NDP), and data sources. As much as possible, all published data were used. However, systems with data available only in graphical form have been rejected. 4.2. Data for the calculations In Table 3 the name, molecular formula, CAS number, molecular weight, critical constants (Tc , Pc and Vc ), normal boiling point (Tbp ), and molar volume at normal boiling point (Vbp ) are listed for all molecules involved in calculations. All data sources are identified. 5. Results and discussion Table 4 shows a compilation of the detailed results obtained with our model and the equations adopted for comparison, namely:

the hydrodynamic expressions of Wilke–Chang (WC) [1,3,129], Tyn–Calus (TC) [1,130], Scheibel (Sch) [1,131] and Reddy–Doraiswy (RD) [1,132], with zero parameters; the predictive He–Yu–Su model (HYS) [133] (specific for supercritical systems); the predictive equation of Zhu et al. (Zhu) [6]; and Dymond correlation (DHB) [2,9,134] (2 parameters). The full expressions of these models may be found in Appendix B. Global results were compiled in Table 5. With regard to the calculation procedure and results, one may detach the following comments:

(1) The proposed LJ-1 model involves only one parameter (k12 ). The required input data are temperature, solvent density, and, for both components, the molecular weight and LJ force constants. (The solute and solvent LJ parameters are estimated as function of Tc and Vc by Eq. (21); whenever unknown, they may be estimated by appropriated models, as those identified in the notes of Table 3). Hence, the prediction of D12,Real is accomplished explicitly: D12,Real = D12,Real (T, 1 ; M1 , M2 ,  LJ,1 ,  LJ,2 , ε1,LJ , ε2,LJ ,  s for 309 systems in Table 4, but for k12 ). This work provides k12 distinct systems it is possible to firstly optimize k12 whenever a few data points are available. It is worth noting that the optimization is very simple to carry out, since Eq. (16) is linear in −2 (1 − k12 ) . (2) The calculation results in Table 5 show that the LJ-1 equation behaves equally well for gases (AARD = 2.63%), liquids (AARD = 6.22%), and SCFs (AARD = 4.12%), giving rise to a grand AARD of only 4.26%. Such results prove that one binary

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Table 2 Systems studied and data sources. System

Tr,1

Solvent (1)

Solute (2)

2,3-Dimethylbutane

Benzene Naphthalene Phenanthrene Toluene 1,1,1,5,5,5-Hexafluoroacetylacetone 1,1 -Dimethylferrocene 1,2-Dichlorobenzene 1,2-Diethylbenzene 1,3,5-Trimethylbenzene 1,3-Divinylbenzene 1,4-Diethylbenzene 15-Crown-5 1-Naphthol 1-Phenyldodecane 1-Phenylethanol 1-Phenylhexane 1-Phenyloctane 1-Propanol 2,2,4,4-Tetramethyl-3-pentanone 2,3-Dimethylaniline 2,3-Dimethylnaphthalene 2,4-Dimethyl-3-pentanone 2,4-Dimethylphenol 2,6-Dimethylaniline 2,6-Dimethylnaphthalene 2,7-Dimethylnaphthalene 2-Bromoanisole 2-Butanone 2-Ethyltoluene 2-Fluoroanisole 2-Heptanone 2-Methylanisole 2-Naphthol 2-Nitroanisole 2-Nonanone 2-Pentanone 2-Phenyl-1-propanol 2-Phenylethanol 2-Phenylethyl acetate 2-Propanol 3-Ethyltoluene 3-Nitrotoluene 3-Pentanone 3-Phenyl-1-propanol 3-Phenylpropyl acetate 4-Ethyltoluene 4-Heptanone 4-Methylanisole 5-Nonanone 5-tert-Butyl-m-xylene 6-Undecanone Acetone Acridine Adamantanone ␣-Linolenic acid Allylbenzene Aniline Anisole Anthracene ␣-Pinene Arachidonic acid (AA) AA ethyl ester ␣-Tocopherol ␤-Carotene Behenic acid ethyl ester Benzene Benzoic acid Benzyl acetate Benzylacetone Biphenyl ␤-Pinene Bromobenzene

Pr,1

r,1

NDP

Data sources

Supercritical systems

Carbon dioxide

1.046–1.096 1.046–1.096 1.046–1.096 1.046–1.096 1.013–1.046 1.013–1.063 1.029–1.095 1.030–1.096 0.997–1.096 1.030–1.096 1.030–1.096 1.013–1.030 1.013–1.046 1.029–1.095 1.030–1.096 1.029–1.095 1.029–1.095 1.030 1.031 1.029–1.095 1.013 1.033 1.029–1.095 1.029–1.095 1.013 1.013 1.030–1.096 1.013–1.079 1.029–1.095 1.030–1.096 1.034 1.029–1.095 1.013–1.046 1.029–1.095 1.034 1.013–1.034 1.030–1.096 1.030–1.096 1.030–1.096 1.030 1.029–1.095 1.029–1.095 1.013–1.079 1.030–1.096 1.030–1.096 1.029–1.095 1.031 1.029–1.095 1.034 1.013–1.309 1.034 0.997–1.096 1.013–1.079 1.031 1.013–1.128 1.030–1.096 1.030–1.096 1.029–1.095 1.029–1.095 1.030–1.096 1.013–1.128 1.013–1.112 1.013–1.096 1.013–1.096 1.013–1.046 0.997–1.096 0.964–1.048 1.030–1.096 1.030–1.096 0.964–1.063 1.030–1.096 1.029–1.095

1.710–5.080 1.710–5.080 1.710–5.080 2.005–5.080 1.411–3.008 1.114–5.436 2.033–4.743 2.033–4.743 1.287–4.743 2.033–4.743 2.033–4.743 1.188–4.070 1.436–2.195 2.033–4.743 2.033–4.743 2.033–4.743 2.033–4.743 1.287–2.168 1.355–2.168 2.033–4.743 1.341–2.629 1.355–2.439 2.033–4.743 2.033–4.743 1.233–2.642 1.450–2.710 2.033–4.743 1.129–4.679 2.033–4.743 2.033–4.743 1.423–2.439 2.033–4.743 1.341–2.060 2.033–4.743 1.355–2.033 1.203–3.963 2.033–4.743 2.033–4.743 2.033–4.743 1.287–2.304 2.033–4.743 2.033–4.743 1.172–4.684 2.033–4.743 2.033–4.743 2.033–4.743 1.355–2.168 2.033–4.743 1.355–2.439 2.033–4.743 1.355–2.439 1.076–5.435 2.337–3.734 1.355–2.033 1.152–4.084 2.033–4.743 2.033–4.743 2.033–4.743 4.065–47.425 1.626–2.710 1.287–4.131 1.141–4.058 1.153–4.107 1.236–4.111 1.310–2.852 1.084–4.743 0.962–4.065 2.033–4.743 2.033–4.743 0.962–2.317 1.626–2.710 2.033–4.743

1.4320–1.9083 1.4320–1.9083 1.4320–1.9083 1.4320–1.9083 1.2102–1.8699 0.8278–2.0767 1.2955–1.9971 1.2953–1.9973 1.2482–1.9104 1.2953–1.9973 1.2953–1.9973 0.8997–1.9430 1.1308–1.7453 1.2953–1.9973 1.2955–1.9979 1.2953–1.9973 1.2953–1.9973 1.2360–1.6971 1.3304–1.6937 1.2955–1.9971 1.5133–1.8374 1.2971–1.7377 1.2955–1.9971 1.2955–1.9971 1.4272–1.8392 1.5707–1.8482 1.2953–1.9973 1.2427–2.0287 1.2959–1.9104 1.2953–1.9973 1.3637–1.7339 1.2955–1.9971 0.7020–1.7176 1.2955–1.9971 1.2820–1.6464 1.1841–1.9341 1.2955–1.9979 1.2955–1.9979 1.2953–1.9973 1.2360–1.7247 1.2959–1.9104 1.2955–1.9971 1.2610–2.0286 1.2955–1.9979 1.2953–1.9973 1.2959–1.9104 1.3304–1.6937 1.2955–1.9971 1.2820–1.7339 1.2953–2.0331 1.2820–1.7339 0.7722–2.0762 1.5149–1.9564 1.3304–1.6621 1.1629–1.9845 1.2953–1.9973 1.2959–1.9104 1.2955–1.9971 0.7681–1.9949 0.9530–1.7986 1.1880–1.9890 1.0685–1.8157 1.3113–1.9871 1.3346–1.9875 1.2802–1.8136 0.5974–1.9973 1.1462–1.9424 1.2953–1.9973 1.2953–1.9973 1.1479–1.9458 0.9530–1.7986 1.2959–1.9104

11 9 11 10 15 68 15 15 24 15 15 29 11 15 15 15 15 17 9 15 3 8 15 15 6 6 15 38 15 15 11 15 16 15 10 23 15 15 15 18 15 15 39 15 15 15 9 15 12 31 13 178 6 8 56 15 15 15 22 15 75 48 82 90 17 222 29 15 15 24 15 15

[23] [23] [23] [23] [24] [25] [26] [27] [28–30] [31] [27] [32] [138] [33] [34] [33] [33] [35] [36] [37] [38] [36] [39] [37] [38,40] [38,40] [31] [41] [42] [31] [36] [37] [138] [26] [36] [41] [34] [34] [43] [35] [42] [37] [41] [34] [43] [42] [36] [37] [36] [27] [36] [28,41,44,45] [46] [36] [47] [31] [30] [39] [138] [48] [49] [50] [47,51,52] [47,51,52] [53] [29,54–59] [35,60,61] [43] [62] [61] [48] [63]

904

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Table 2 (Continued) System Solvent (1)

Tr,1

Pr,1

r,1

1.310–2.852 1.088–1.943 1.310–2.852 1.310–2.852 2.033–4.743 2.168–3.591 1.626–2.710 1.314–5.420 1.466–2.243 1.355–2.439 1.355–2.439 1.355–2.439 2.034–4.743 2.033–4.743 1.098–2.197 1.098–2.197 1.355–3.389 1.626–2.710 1.256–4.083 1.141–4.058 1.310–2.852 1.176–4.085 1.141–4.058 1.310–2.852 1.287–3.388 1.024–1.938 2.033–4.743 2.033–4.743 2.033–4.743 1.087–5.466 2.033–4.743 1.176–4.133 1.138–2.169 1.099–4.553 1.310–3.352 2.033–4.743 2.033–4.743 1.626–2.710 1.152–4.106 1.897–4.553 1.287–2.846 1.356–3.388 1.310–2.852 1.247–4.065 1.084–1.897 1.626–2.168 0.911–13.550 2.033–4.743 1.220–1.423 1.220–1.423 1.220–1.423 1.220–1.423 2.033–4.743 1.220–1.423 1.220–1.423 1.220–1.423 2.033–4.743 1.152–4.743 1.220–1.423 1.220–1.423 1.282–4.079 1.165–1.491 1.084–1.491 1.152–5.420 1.310–2.852 1.252–2.317 1.308–3.591 1.089–4.103 1.310–2.852 2.033–4.743 2.033–4.743 14.499–47.425 1.762–2.439

1.2802–1.8136 0.9110–1.7197 1.2802–1.8136 1.2802–1.8136 1.2959–1.9104 1.7069–1.8776 0.9530–1.7986 1.2883–2.0413 1.2844–1.7589 1.2971–1.7377 1.2971–1.7377 1.2971–1.7377 1.6664–2.0237 1.2953–1.9973 0.4182–1.7041 0.4182–1.7041 1.3414–1.8777 0.9530–1.7986 1.2190–1.9849 1.0685–1.8157 1.2802–1.8136 1.1571–1.9514 1.0685–1.8157 1.2802–1.8136 1.2360–1.8776 0.4545–1.7197 1.2953–1.9973 1.2953–1.9973 1.2953–1.9973 0.6002–2.0771 1.2959–1.9104 0.9758–1.9482 0.7165–1.6974 0.6222–1.9821 0.8698–1.9223 1.2959–1.9104 1.2953–1.9973 0.9530–1.7986 1.1880–1.9870 1.5767–1.9821 1.2360–1.8121 1.3430–1.8776 1.2802–1.8136 0.9770–1.8776 0.4807–1.2360 1.5271–1.6937 0.6480–2.3739 1.2953–1.9973 1.5592–1.7450 1.5592–1.7450 1.5592–1.7450 1.5592–1.7450 1.2955–1.9971 1.5592–1.7450 1.5592–1.7450 1.5592–1.7450 1.2955–2.0343 0.7638–1.9973 1.5592–1.7450 1.5592–1.7450 1.2246–1.9436 0.8052–1.4594 0.5939–1.4594 1.2016–2.0757 1.2802–1.8136 1.1479–1.8669 1.0821–1.9242 0.7574–1.9868 1.2802–1.8136 1.2953–1.9973 1.2955–1.9979 0.8321–1.9885 1.5621–1.7339

NDP

Data sources

Solute (2) Butyric acid ethyl ester Caffeine Capric acid ethyl ester Caprylic acid ethyl ester Chlorobenzene Chrysene Citral Cobalt(III) acetylacetonate Copper(II) trifluoroacetylacetonate Cycloheptanone Cyclononanone Cyclopentanone Dibenzo-24-crown-8 Dibenzyl ether Diethyl ether Diisopropyl ether Diolein d-limonene Docosahexaenoic acid (DHA) DHA ethyl ester DHA methyl ester Eicosapentaenoic acid (EPA) EPA ethyl ester EPA methyl ester Ethanol Ethyl acetate Ethyl benzoate Ethylbenzene Eugenol Ferrocene Fluorobenzene ␥-Linolenic acid ␥-Linolenic acid ethyl ester ␥-Linolenic acid methyl ester Hexachlorobenzene Iodobenzene i-Propylbenzene Linalool Linoleic acid Linoleic acid methyl ester Methanol Monoolein Myristic acid ethyl ester Myristoleic acid Myristoleic acid methyl ester N-(4-methoxybenzy-lidene)-4-n-butylaniline Naphthalene n-Butylbenzene n-Decane n-Dodecane n-Heptane n-Hexane Nitrobenzene n-Nonane n-Octane n-Pentane n-Pentylbenzene n-Propylbenzene n-Tetradecane n-Undecane Oleic acid Oleic acid ethyl ester Oleic acid methyl ester Palladium(II) acetylacetonate Palmitic acid ethyl ester p-Dichlorobenzene Phenanthrene Phenol Phenylacetic acid Phenylacetylene Phenylmethanol Pyrene Squalene

1.013–1.046 1.013–1.079 1.013–1.046 1.013–1.046 1.029–1.095 0.997–1.096 1.030–1.096 1.030–1.096 1.013–1.046 1.033 1.033 1.033 1.013–1.030 1.030–1.096 1.030–1.096 1.030–1.096 1.030 1.030–1.096 1.013–1.128 1.013–1.112 1.013–1.046 1.013–1.128 1.013–1.112 1.013–1.046 1.030 1.013–1.079 1.030–1.096 1.030–1.096 1.030–1.096 1.013–1.063 1.029–1.095 1.013–1.128 1.030–1.128 1.030–1.128 1.013–1.079 1.029–1.095 1.030–1.096 1.030–1.096 1.013–1.128 1.013–1.079 1.030 1.030 1.013–1.046 1.030–1.128 1.030–1.128 1.031 0.948–1.096 1.029–1.095 0.984–1.013 0.984–1.013 0.984–1.013 0.984–1.013 1.029–1.095 0.984–1.013 0.984–1.013 0.984–1.013 1.013–1.309 1.013–1.096 0.984–1.013 0.984–1.013 1.030 1.030 1.030 1.013–1.128 1.013–1.046 0.980–1.046 0.997–1.096 1.013–1.079 1.013–1.046 1.030–1.096 1.030–1.096 1.029–1.095 1.034

16 21 16 16 15 4 15 38 12 8 8 8 28 15 15 15 9 15 63 65 17 55 48 17 24 15 15 15 15 98 15 142 41 52 14 15 15 15 71 21 10 11 16 42 79 5 83 15 5 5 5 5 15 5 5 5 31 34 5 5 19 5 19 125 17 13 19 109 16 15 15 18 5

[64,65] [66] [64,65] [64,65] [63] [28] [67] [68] [24] [69] [69] [69] [32] [43] [70] [70] [71] [67] [72] [50,53] [53] [72] [50] [53] [35] [66] [62] [55] [62] [25] [63] [73] [73] [73,74] [75] [63] [55] [54] [49] [74,76] [35] [71] [64,65] [77] [77] [36] [28,38,78,79] [33] [80] [80] [80] [80] [39] [80] [80] [80] [33] [29,55] [80] [80] [71] [71] [71] [68] [81] [61] [28,75] [35,45,51,52,66] [82] [27] [34] [138] [36]

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

905

Table 2 (Continued) System Solvent (1)

Chlorotrifluoromethane Ethane Sulfur hexafluoride

Tr,1

Pr,1

r,1

NDP

Data sources

1.310–2.852 2.033–4.743 2.033–4.743 1.098–2.197 1.430–3.037 1.089–4.743 1.348–4.098 1.119–4.073 1.449–2.924 1.220–4.072 1.237–1.900 1.153–4.095 1.396–2.852 1.355–4.065 1.214–4.068 1.150–2.093 1.220–2.819 1.449–2.295 1.414–2.295 1.117–3.816 0.931–3.816 1.729–3.191 1.117–3.816 1.729–3.191 0.798–3.989 0.931–3.816

1.2802–1.8136 1.2959–1.9104 1.2955–1.9971 0.4182–1.7041 1.2102–1.8870 0.9687–1.9973 1.3330–1.9452 1.2041–1.9840 1.2375–1.8731 1.2680–1.9841 1.1011–1.6305 1.3113–1.9844 1.2802–1.8136 1.3418–1.9425 1.0177–1.9432 0.6873–1.6510 0.6908–1.7270 1.5210–1.9683 1.5190–1.9925 0.6806–1.9056 0.4083–1.9056 1.2485–1.9065 0.6806–1.9056 1.5089–2.0278 0.4083–2.2459 0.4083–1.9056

17 15 15 15 15 35 27 101 15 38 10 80 15 16 20 10 8 6 9 10 9 6 6 5 52 11

[53] [30] [26] [70] [24] [55,59,66] [83] [83] [24] [83] [71] [52,84] [82] [32] [52,76] [85] [85] [86] [86] [85] [85] [81] [85] [81] [85] [85]

2.7566–2.8675 2.7566–2.8675 2.7566–2.8675 2.7566–2.8675 2.7566–2.8675 2.7566–2.8675 2.3861–2.7813 2.3861–2.7813 2.5947–2.9057 2.5947–2.9057 2.3861–2.7813 2.3861–2.7813 1.7566–2.8325 1.7566–2.8325 2.3861–2.8325 2.3861–2.8325 2.3861–2.8325 2.3861–2.8325 1.7566–2.8618 2.3861–2.8325 2.8291–3.0824 2.8291–3.0824 2.8291–3.0824 2.6077–3.0831 2.8298–3.0831 2.8291–3.0824 2.8291–3.0824 2.6077–3.0831 2.6077–3.0831 2.8291–3.0824 2.6077–3.0831 2.6077–3.0831 2.6077–3.0831 2.6077–3.0831 2.6077–3.0831 3.0682–3.1519 3.0389–3.1519 3.0682–3.1519 2.1292–3.1028 2.1292–3.1028 2.1292–3.1028 3.0682–3.1519 3.0682–3.1519 3.0389–3.1519 2.1896–3.1067 2.1896–3.1067

4 4 4 4 4 4 6 6 5 5 6 6 8 8 7 7 7 6 12 6 4 4 4 3 3 4 4 3 3 4 4 3 4 4 4 4 5 4 9 9 9 4 4 5 5 5

[87] [87] [87] [87] [87] [87] [88] [88] [89] [89] [88] [88] [19] [19] [88] [88] [88] [88] [19,90] [88] [91] [91] [91] [92] [92] [91] [91] [92] [92] [91] [92] [92] [92] [92] [92] [90] [90] [90] [93] [93] [93] [90] [90] [90] [94] [94]

Solute (2) Stearic acid ethyl ester Styrene tert-Butylbenzene Tetrahydrofuran Thenoyltrifluoroacetone Toluene Triarachidonin Trierucin Trifluoroacetylacetone Trinervonin Triolein Ubiquinone CoQ10 Vanillin Vitamin K1 Vitamin K3 Acetone p-Xylene 1-Octene 1-Tetradecene 1,3,5-Trimethylbenzene Benzene Benzoic acid Carbon tetrachloride Naphthalene p-Xylene Toluene

1.013–1.046 1.030–1.096 1.029–1.095 1.030–1.096 1.013–1.046 1.007–1.096 1.030 1.013–1.063 1.013–1.046 1.013–1.063 1.030 1.013–1.096 1.013–1.046 1.030 1.030 1.037 1.053 0.970–1.055 0.960–1.055 1.029 1.029 1.030–1.061 1.029 0.998–1.030 0.889–1.061 1.029

1,3,5-Trimethylbenzene Benzene Ethylbenzene o-Xylene p-Xylene Toluene Argon Carbon tetrachloride Ethane Ethylene Krypton Methane Phenanthrene p-Xylene Tetrabutyltin Tetraethyltin Tetramethyltin Tetrapropyltin Toluene Xenon 12-Crown-4 15-Crown-5 18-Crown-6 Argon Carbon tetrachloride Dicyclohexano-18-crown-6 Dicyclohexano-24-crown-8 Krypton Methane s-Trioxane Tetrabutyltin Tetraethyltin Tetramethyltin Tetrapropyltin Xenon 1,3,5-Trimethylbenzene Acetone Benzene Carbon dioxide Carbon monoxide Hydrogen Linoleic acid methyl ester m-Xylene Naphthalene n-Decane n-Hexadecane

0.557–0.612 0.557–0.612 0.557–0.612 0.557–0.612 0.557–0.612 0.557–0.612 0.566–0.751 0.566–0.751 0.507–0.656 0.507–0.656 0.566–0.751 0.566–0.751 0.539–0.945 0.539–0.945 0.539–0.751 0.539–0.751 0.539–0.751 0.539–0.751 0.539–0.945 0.539–0.751 0.483–0.604 0.483–0.604 0.483–0.604 0.482–0.701 0.482–0.604 0.483–0.604 0.483–0.604 0.482–0.701 0.482–0.701 0.483–0.604 0.482–0.701 0.482–0.701 0.482–0.701 0.482–0.701 0.482–0.701 0.461–0.506 0.461–0.521 0.461–0.506 0.462–0.862 0.462–0.862 0.462–0.862 0.461–0.506 0.461–0.506 0.461–0.521 0.462–0.860 0.462–0.860

Liquid systems 2,2,4-Trimethylpentane

Cyclohexane

n-Decane

n-Dodecane

sat.p.a sat.p. 3.931; sat.p. 8.791 8.791 8.791 0.765–1.898 0.765–1.898 0.765–1.898 8.791 8.791 8.791 0.776–0.796 0.776–0.796

906

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Table 2 (Continued) System Solvent (1)

n-Eicosane

n-heptane

n-Hexadecane

n-hexane

n-octacosane

n-octane

Propane

Tr,1

Pr,1

r,1

NDP

Data sources

0.776–1.890 0.776–0.796 8.791 1.226 1.226 1.226 1.226 1.226 1.226 0.037–1.268 0.037–1.296 0.037–1.296 0.037–1.268 0.037–1.296 0.991–2.454 0.991–2.454 0.991–2.454 1.004–1.021 1.004–1.021 1.004–2.486 1.004–1.021

2.1896–3.1067 2.1896–3.1067 3.0682–3.1519 2.6453–3.1064 2.6453–3.1064 2.6453–3.1064 2.6449–3.1064 2.6449–3.1064 2.6449–3.1064 2.2353–2.9290 2.2353–2.9437 2.1638–2.9437 2.2353–2.9290 2.2353–2.9437 2.3098–3.0928 2.3098–3.0928 2.3098–3.0928 2.3606–3.0917 2.3606–3.0917 2.3606–3.0950 2.3606–3.0917 1.3481–2.8637 2.7521–2.8637 2.8122–3.4674 1.3481–3.4493 2.8122–3.4665 2.7907–2.8637 2.7521–2.8637 1.3481–2.8637 1.3481–2.6705 1.3481–2.8637 1.3481–3.4592 2.6435–3.0842 2.6435–3.0842 2.6435–3.0842 2.6435–3.0789 2.6435–3.0789 2.6435–3.0789 2.8882–2.9974 2.5858–3.0098 2.8882–2.9974 2.7361–3.0098 2.8882–2.9974 2.5858–3.0098 2.5858–3.0098 2.8882–2.9974 2.8882–2.9974 2.4799–3.0159 2.4799–3.0159 2.4799–3.0159 2.4799–3.0159 2.8882–2.9974 2.5858–3.0098 2.0441–2.3664 2.1270–2.3848

9 5 4 5 5 5 5 5 5 5 5 8 4 5 10 10 10 5 5 10 5 20 5 7 36 10 4 5 20 15 17 28 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 5 4 4 4 4 8 8

[94] [94] [90] [95] [95] [95] [95] [95] [95] [94] [94] [94] [94] [94] [93] [93] [93] [96] [96] [96] [96] [90,97] [90] [98] [89,90,97,98] [98] [76,90] [90] [90,97,98] [97] [90,97] [90,97] [99] [99] [99] [99] [99] [99] [87] [92] [87] [92] [87] [92] [92] [87] [87] [92] [92] [92] [92] [87] [92] [86] [86]

0.0014–0.0030 0.0031–0.0077 0.0014–0.0030 0.0014–0.0030 0.0014–0.0030 0.0007–0.0077 0.0013–0.0030 0.0365–1.6855 0.0033–0.0036 0.0033–0.0036 0.0033–0.0036 0.0025–0.0064 0.0027–0.0058 0.0514–1.5935 0.0035–0.0049 0.0008–0.0059

9 5 8 9 8 25 9 48 7 7 7 5 14 49 7 17

[100] [101] [100] [100] [100] [101–103] [100] [104] [105] [105] [105] [101] [105,106] [104] [105] [101,107]

Solute (2) n-Octane n-Tetradecane Toluene Carbon dioxide Carbon monoxide Hydrogen n-Dodecane n-Hexadecane n-Octane n-Decane n-Dodecane n-Hexadecane n-Octane n-Tetradecane Carbon dioxide Carbon monoxide Hydrogen n-Decane n-Dodecane n-Octane n-Tetradecane 1,3,5-Trimethylbenzene Acetone Acetonitrile Benzene Carbon disulphide Indole m-Xylene Naphthalene Phenanthrene p-Xylene Toluene Carbon dioxide Carbon monoxide Hydrogen n-Dodecane n-Hexadecane n-Octane 1,3,5-Trimethylbenzene Argon Benzene Carbon tetrachloride Ethyl benzene Krypton Methane o-Xylene p-Xylene Tetrabutyltin Tetraethyltin Tetramethyltin Tetrapropyltin Toluene Xenon 1-Octene 1-Tetradecene

0.462–0.860 0.462–0.860 0.461–0.506 0.488–0.696 0.488–0.696 0.488–0.696 0.489–0.696 0.489–0.696 0.489–0.696 0.553–0.883 0.553–0.883 0.553–0.883 0.553–0.883 0.553–0.883 0.448–0.781 0.448–0.781 0.448–0.781 0.448–0.781 0.448–0.781 0.448–0.781 0.448–0.781 0.597–1.070 0.597–0.657 0.588 0.588–1.070 0.588 0.617 0.597–0.657 0.597–1.070 0.657–1.070 0.617–1.070 0.588–1.070 0.429–0.618 0.429–0.618 0.429–0.618 0.432–0.618 0.432–0.618 0.432–0.618 0.533–0.586 0.524–0.709 0.533–0.586 0.524–0.656 0.533–0.586 0.524–0.709 0.524–0.709 0.533–0.586 0.533–0.586 0.524–0.761 0.524–0.761 0.524–0.761 0.524–0.761 0.533–0.586 0.524–0.709 0.802–0.913 0.791–0.912

b

5.316 0.034–128.11 c

0.034–127.575 5.316–8.306 5.316 b

sat.p.; Pr,1 > 1 b d

2.108 2.108 2.108 2.108 2.108 2.108 1.308–2.198 2.092–2.165 Gas systems

Argon

Carbon dioxide Carbon monoxide Deuterium Ethane Ethylene Helium

Ethane Hydrogen i-Butane Methane n-Butane Neon Propane Ethylene Hydrogen Helium Hydrogen Hydrogen Nitrogen Carbon dioxide Nitrogen Hydrogen

2.035–4.464 0.782–1.963 2.026–4.464 2.043–4.464 2.032–4.464 0.782–9.041 2.027–4.503 0.981–1.145 1.037–1.131 2.372–2.588 2.372–2.588 2.995–7.682 1.022–2.198 1.056–1.233 1.143–1.593 22.736–176.686

0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.146–2.752 0.014 0.029 0.029 0.061 0.021 0.217–3.724 0.020 0.446

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

907

Table 2 (Continued) System

Tr,1

Solvent (1)

Solute (2)

Krypton

Argon Helium Neon Xenon Carbon dioxide Tetrachloroethene Deuterium Helium Hydrogen Xenon Helium Hydrogen Methane n-Butane Propane Helium Hydrogen Cyclohexane Methylcyclohexane 1,1,1-Trichloroethane Tetrachloroethene

Methane Neon

Nitrogen

Oxygen Sulfur hexafluoride Tetrafluoromethane

1.619–5.708 1.710–5.703 1.304–5.838 1.713–5.313 1.502–1.932 1.487–1.802 2.590–6.644 1.725–8.874 2.590–6.644 6.149–8.874 2.363–3.947 2.402–3.591 2.486–5.319 2.484–5.319 2.508–5.319 1.929–3.222 1.961–2.931 0.888–1.077 0.888–1.077 1.244–1.508 1.244–1.508

Pr,1

r,1

NDP

0.018 0.018 0.018 0.018 0.022 0.022 0.037 0.037 0.037 0.037 0.030 0.030 0.030 0.030 0.030 0.020 0.020 0.027 0.027 0.027 0.027

0.0009–0.0033 0.0009–0.0031 0.0009–0.0041 0.0010–0.0031 0.0033–0.0042 0.0035–0.0043 0.0017–0.0044 0.0013–0.0066 0.0017–0.0044 0.0013–0.0019 0.0022–0.0037 0.0024–0.0036 0.0016–0.0035 0.0016–0.0035 0.0016–0.0035 0.0018–0.0030 0.0020–0.0030 0.0071–0.0086 0.0071–0.0086 0.0050–0.0060 0.0050–0.0060

6 6 17 8 10 5 5 24 5 6 8 29 7 5 6 8 13 5 5 5 5

Note: An hyphen means not available. a Saturation pressure. b Pr,1 = 0.498 and 0.532, at saturation pressure and other points at Pr,1 > 1. c Pr,1 from 0.034 to 122.69 and at saturation pressure. d Pr,1 from 0.034 to 124.58 and at saturation pressure.

Fig. 4. Comparison between calculated and experimental tracer diffusivities for supercritical, liquid and gas systems.

Data sources

[108] [108] [103,108] [108] [109] [110] [101] [101–103] [101] [103] [111] [112] [106] [106] [106] [111] [112] [113] [113] [110] [110]

908

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Table 3 Data for pure substances. Substance

Formula

CAS number

M (g/mol)

1,1,1,5,5,5-Hexafluoroacetylacetone 1,1,1-Trichloroethane 1,1 -Dimethylferrocene 1,2-Dichlorobenzene 1,2-Diethylbenzene 1,3,5-Trimethylbenzene 1,3-Divinylbenzene 1,4-Diethylbenzene 12-Crown-4 15-Crown-5 18-Crown-6 1-Naphthol 1-Octene 1-Phenyldodecane 1-Phenylethanol 1-Phenylhexane 1-Phenyloctane 1-Propanol 1-Tetradecene 2,2,4,4-Tetramethyl-3-pentanone 2,2,4-Trimethylpentane 2,3-Dimethylaniline 2,3-Dimethylbutane 2,3-Dimethylnaphthalene 2,4-Dimethyl-3-pentanone 2,4-Dimethylphenol 2,6-Dimethylaniline 2,6-Dimethylnaphthalene 2,7-Dimethylnaphthalene 2-Bromoanisole 2-Butanone 2-Ethyltoluene 2-Fluoroanisole 2-Heptanone 2-Methylanisole 2-Naphthol 2-Nitroanisole 2-Nonanone 2-Pentanone 2-Phenyl-1-propanol 2-Phenylethanol 2-Phenylethyl acetate 2-Propanol 3-Ethyltoluene 3-Nitrotoluene 3-Pentanone 3-Phenyl-1-propanol 3-Phenylpropyl acetate 4-Ethyltoluene 4-Heptanone 4-Methylanisole 5-Nonanone 5-tert-Butyl-m-xylene 6-Undecanone Acetone Acetonitrile Acridine Adamantanone ␣-Linolenic acid Allylbenzene Aniline Anthracene Anisole ␣-Pinene Arachidonic acid (AA) AA ethyl ester Argon ␣-Tocopherol ␤-Carotene Behenic acid ethyl ester Benzene Benzoic acid Benzyl acetate

C5 H2 F6 O2 C2 H3 Cl3 C12 H14 Fe C6 H4 Cl2 C10 H14 C9 H12 C10 H10 C10 H14 C8 H16 O4 C10 H20 O5 C12 H24 O6 C12 H8 O C8 H16 C18 H30 C8 H10 O C12 H18 C14 H22 C3 H8 O C14 H28 C9 H18 O C8 H18 C8 H11 N C6 H14 C12 H12 C7 H14 O C8 H10 O C8 H11 N C12 H12 C12 H12 C7 H7 BrO C4 H8 O C9 H12 C7 H7 FO C7 H14 O C8 H10 O C10 H8 O C7 H7 NO3 C9 H18 O C5 H10 O C9 H12 O C8 H10 O C10 H12 O2 C3 H8 O C9 H12 C7 H7 NO2 C5 H10 O C9 H12 O C11 H14 O2 C9 H12 C7 H14 O C8 H10 O C9 H18 O C12 H18 C11 H22 O C3 H6 O C2 H3 N C13 H9 N C10 H14 O C18 H30 O2 C9 H10 C6 H7 N C14 H10 C7 H8 O C10 H16 C20 H32 O2 C22 H36 O2 Ar C29 H50 O2 C40 H56 C24 H48 O2 C6 H6 C7 H6 O2 C9 H10 O2

1552-22-1 71-55-6 1291-47-0 95-50-1 135-01-3 108-67-8 108-57-6 105-05-5 294-93-9 33100-27-5 17455-13-9 90-15-3 111-66-0 123-01-3 98-85-1 1077-16-3 2189-60-8 71-23-8 1120-36-1 815-24-7 540-84-1 87-59-2 79-29-8 581-40-8 565-80-0 105-67-9 87-62-7 581-42-0 582-16-1 578-57-4 78-93-3 611-14-3 321-28-8 110-43-0 578-58-5 135-19-3 91-23-6 821-55-6 107-87-9 1123-85-9 60-12-8 103-45-5 67-63-0 620-14-4 99-08-1 96-22-0 122-97-4 122-72-5 622-96-8 123-19-3 104-93-8 502-56-7 98-19-1 927-49-1 666-52-4 75-05-8 260-94-6 700-58-3 463-40-1 300-57-2 62-53-3 120-12-7 100-66-3 80-56-8 506-32-1 1808-26-0 7440-37-1 59-02-9 7235-40-7 5908-87-2 71-43-2 65-85-0 140-11-4

208.06 133.41 214.09 147.00 134.22 120.20 130.19 134.22 176.21 220.27 264.32 144.17 112.22 246.44 122.17 162.28 190.33 60.10 196.38 142.24 114.23 121.18 86.18 156.23 114.19 122.17 121.18 156.23 156.23 187.04 72.11 120.20 126.13 114.19 122.17 144.17 153.14 142.24 86.13 136.20 122.17 164.10 60.10 120.20 137.14 86.13 136.20 178.30 120.20 114.19 122.17 142.24 162.28 170.30 58.08 41.05 179.22 150.22 278.44 118.18 93.13 178.23 108.14 136.24 304.47 332.53 39.95 430.71 536.88 368.64 78.11 122.12 150.18

Tc (K)

Pc (bar)

Vc (cm3 /mol)

569.07a 545.00b 514.45c 729.00b 668.00d 637.30b 692.00d 657.96d 780.66e 876.80e 970.51e 802.00d 566.70b 774.26d 675.30f 698.00d 729.00d 536.80b 689.00a 627.18c 544.00b 717.00f 500.00b 777.78d 597.13c 707.60b 722.00g 777.00d 778.00d 737.58f 536.80b 651.00b 644.81f 611.50b 648.79f 811.40i 782.00d 644.29d 561.10b 662.02f 684.00d 712.23f 508.30b 637.00b 734.00d 561.00b 702.30f 718.70f 640.00b 595.31d 655.36f 640.00g 684.85f 678.50h 508.10b 545.50b 905.00h 759.15c 780.00h 639.86f 699.00b 873.00d 641.65d 632.00d 1013.42e 960.63a 150.80b 964.30h 1450.76e 984.94a 562.20b 752.00b 699.00d

27.17a 43.00b 27.41c 41.00b 28.80d 31.30b 31.20d 28.03d 33.59e 28.72e 24.95e 47.37d 26.20b 15.79d 40.60f 23.80d 20.20d 51.70b 15.60b 30.29c 25.70b 36.30f 31.30b 30.06d 35.22c 44.00d 42.00g 31.70d 31.70d 40.04f 42.10b 30.40b 38.11f 34.40b 35.60f 47.40i 37.60d 24.53d 36.90b 36.90f 39.20d 30.12f 47.60b 28.40b 38.00d 37.30b 36.40f 27.23f 29.40b 29.96d 35.60f 23.20h 23.90f 20.52h 47.00b 48.30b 36.40h 31.55c 14.40h 33.50f 53.10b 29.00d 41.75d 27.60d 12.74e 11.31a 48.70b 10.80h 6.90e 9.15a 48.90b 45.60b 31.80d

406.05a 281.00d 400.64c 360.00b 502.00d 433.00d 440.00d 497.00d 444.75e 548.75e 652.75e 375.50d 464.00b 1000.00d 392.15f 618.00d 703.00d 219.00b 817.00d 407.72c 468.00b 400.38f 358.00b 521.50d 324.85c 390.00d 400.38f 520.00d 520.00d 378.05f 267.00b 460.00b 328.87f 421.00d 371.70f 375.50i 422.00d 545.50d 301.00b 443.23f 387.00d 524.15f 220.00b 490.00b 441.00d 336.00b 455.45f 580.37f 470.00b 433.50d 371.70f 560.00h 591.75f 692.00h 209.00b 173.00b 543.00h 368.22c 1070.00h 419.80f 274.00b 554.00d 337.00d 504.00d 1093.20e 1195.26a 74.90b 1720.00h 1934.95e 1394.66a 259.00b 341.00b 449.00d

Tbp (K) 410.70j 347.20b 353.55k 452.00b 456.61d 437.90b 472.65d 456.94d 540.08j 625.60j 711.12j 561.15d 394.40b 600.76d 478.16l 499.26d 537.55d 370.30b 524.30b 425.35k 372.40b 494.66l 331.10b 541.16d 400.85k 484.10b 491.05g 535.15d 536.15d 489.16l 352.70b 438.30b 427.66l 424.20b 444.16l 558.65k 546.15d 346.95g 375.40b 476.16l 492.05d 505.16l 355.40b 434.50b 505.00d 375.10b 508.16l 518.16l 435.20b 416.67g 448.66l 461.60g 480.16l 500.55h 329.20b 354.80b 619.15h 519.85k 632.00h 429.16l 457.60b 615.18d 426.73d 429.29d 819.15j 777.62j 87.30b 787.80h 1209.38j 806.74j 353.20b 523.00b 486.65d

Vbp m (cm3 /mol)

 LJ p (Å)

154.40 104.98 152.24 136.10 192.83 165.15 167.95 190.82 169.85 211.69 253.92 142.24 177.56 397.05 148.86 239.77 274.44 80.84 321.26 155.06 179.17 152.14 135.31 200.69 122.20 148.01 152.14 200.09 200.09 143.26 99.50 175.96 123.79 160.36 140.74 142.24 160.76 210.38 112.82 169.24 146.81 201.76 81.23 188.01 168.35 126.60 174.14 224.50 179.97 165.35 140.74 216.24 229.11 269.95 76.98 63.14 209.37 139.36 426.23 159.88 102.24 213.82d 127.00 193.64 435.92 478.66 26.26 700.94 793.00 562.66 96.38 128.58 171.55

5.84182 451.89 5.16723 432.78 5.81576 408.52 5.61207 578.89 6.26984 530.45 5.96831 506.07 6.00030 549.51 6.24896 522.48 6.02181 619.92 6.45873 696.26 6.84338 770.67 5.69148 636.86 6.10747 450.01 7.88900 614.83 5.77439 536.25 6.71972 554.28 7.01467 578.89 4.75521 426.27 7.37501 547.13 5.84982 498.04 6.12497 431.99 5.81450 569.36 5.60165 397.05 6.35000 617.63 5.42312 474.18 5.76382 561.90 5.81450 573.33 6.34390 617.01 6.34390 617.80 5.70434 585.71 5.07995 426.27 6.08987 516.95 5.44540 512.04 5.91266 485.59 5.67222 515.20 5.69148 644.33 5.91733 620.98 6.44595 511.63 5.28702 445.56 6.01495 525.70 5.74900 543.16 6.36073 565.58 4.76244 403.64 6.21948 505.84 6.00484 582.86 5.48447 445.49 6.06972 557.69 6.58047 570.71 6.13368 508.22 5.97060 472.73 5.67222 520.42 6.50256 508.22 6.62320 543.83 6.97789 538.79 4.68171 403.48 4.39579 433.18 6.43609 718.65 5.65446 602.83 8.06894 619.39 5.90703 508.11 5.12396 555.07 6.47926 693.24 5.48991 509.53 6.27816 501.87 8.12684 804.75 8.37226 762.83 3.32544 119.75 9.45217 765.74 9.83057 1152.04 8.81410 782.13 5.02869 446.44 5.51155 597.16 6.04093 555.07

εLJ /kB p (K)

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

909

Table 3 (Continued) Pc (bar) Vc (cm3 /mol) Tbp (K)

Vbp m (cm3 /mol)

722.51f 789.00b 643.00d 670.00b 579.00i 855.60i 699.30i 655.70i 304.10b 552.00b 132.90b 556.40b 632.40b 302.00b 979.00d 692.70e 573.48c 412.85c 671.19c 553.50b 702.10c 626.00d 38.40b 1396.77e 777.00d 1177.47e 1357.66e 466.70b 500.30b 1025.00h 660.00d 1075.45a 1023.28a 999.34a 1020.90a 968.16a 890.55a 305.40b 513.90b 523.20b 668.70b 617.20b 282.40b 735.31f 786.27c 560.10b 958.98a 937.01c 882.79a 5.19b 825.00d 33.00b 408.20b 721.00b 631.10b 209.40b 645.80e 775.00h 870.78a 190.40b 512.60b 572.20b 885.00h 617.10b 789.35a 854.23e 777.79e 962.06c

31.20f 38.50b 27.60d 45.20b 31.40i 41.50i 17.88i 21.18i 73.80b 79.00b 35.00b 45.60b 45.20b 38.70b 23.90d 23.15e 2.52c 20.63c 36.86c 40.70b 31.47c 58.50d 16.50b 15.80e 25.60d 16.24e 13.48e 36.40b 28.80b 7.92h 27.50d 12.41a 10.84a 11.41a 13.47a 11.67a 11.90a 48.80b 61.40b 38.30b 23.20b 36.00b 50.40b 33.52f 32.07c 45.50b 14.17a 17.56c 12.92a 2.27b 28.50d 12.90b 36.50b 45.20b 32.10b 55.00b 25.95e 14.10h 12.54a 46.00b 80.90b 34.70b 12.40h 35.40b 13.89a 16.97e 15.26e 21.33c

500.50f 502.00b 506.00d 324.00b 400.00i 488.00i 733.50i 621.50i 93.90b 160.00b 93.20b 275.90b 308.00b 180.40b 690.00d 591.00e 640.95c 441.13c 297.87c 308.00b 380.74c 258.00d 60.30b 1174.35e 608.00d 1002.75e 1210.75e 280.00b 386.00b 2150.00h 524.00d 1148.05a 1262.06a 1206.56a 1059.15a 1173.16a 1187.03a 148.30b 167.10b 286.00b 489.00d 374.00b 130.40b 447.23f 317.77c 269.00b 992.35a 797.37c 1050.86a 57.40b 526.00d 64.30b 263.00b 351.00b 427.70d 91.20b 558.00e 990.00h 1070.95a 99.20b 118.00b 368.00b 1210.00h 376.00b 950.66a 819.90e 876.45e 592.93c

506.66l 529.30b 439.19d 429.20b 393.15m 451.15m 518.15m 480.15m 194.70d 319.00b 81.70b 349.90b 404.90b 193.20b 714.15d 502.20n 411.55o 299.15k 453.15k 353.80b 478.25k 403.80d 23.60b 1111.44j 561.45d 906.84j 1077.88j 307.60b 341.70b 920.00h 449.65d 873.23j 831.70j 808.82j 823.31j 781.78j 758.90j 184.60b 351.40b 350.30b 485.90b 409.30b 169.30b 526.36l 522.15o 357.90b 769.23j 663.73k 704.82j 4.25b 582.55d 20.30b 261.40b 461.60b 425.60b 119.90b 472.00n 628.00h 700.66j 111.60b 337.70b 374.10b 714.00h 412.30b 623.70j 669.39j 604.98j 677.05k

192.23 192.83 194.44 121.87 151.99 187.20 286.94 241.20 33.28 58.18 33.02 102.98 115.57 65.98 269.13 228.81 249.11 168.40 111.59 115.57 144.33 95.99 20.92 469.89 235.71 398.19 485.16 104.58 146.42 885.61 201.70 458.86 506.73 483.40 421.70 469.39 475.20 53.73 60.89 106.93 187.60 141.65 46.95 170.84 119.42 100.28 393.87 313.17 418.24 19.87 202.51 22.38 97.94 132.53 163.03 32.28 215.44 392.89 426.62 35.25 42.28 139.27 484.85 142.44 376.54 322.45 345.80 229.59

748.40b 425.20b 660.50b 617.70b 658.20b 767.00b

40.50b 38.00b 28.90b 21.20b 18.20b 11.10b

413.00b 255.00b 497.00b 603.00b 713.00b 1190.00d

491.10b 272.70b 456.50b 447.30b 489.50b 617.00b

157.17 94.82 190.82 233.68 278.54 476.45

Substance

Formula

CAS number M (g/mol) Tc (K)

Benzylacetone Biphenyl ␤-Pinene Bromobenzene Butyric acid ethyl ester Caffeine Capric acid ethyl ester Caprylic acid ethyl ester Carbon dioxide Carbon disulphide Carbon monoxide Carbon tetrachloride Chlorobenzene Chlorotrifluoromethane Chrysene Citral Cobalt (III) acetylacetonate Copper(II) trifluoroacetylacetonate Cycloheptanone Cyclohexane Cyclononanone Cyclopentanone Deuterium Dibenzo-24-crown-8 Dibenzyl ether Dicyclohexano-18-crown-6 Dicyclohexano-24-crown-8 Diethyl ether Diisopropyl ether Diolein d-limonene Docosahexaenoic acid (DHA) DHA ethyl ester DHA methyl ester Eicosapentaenoic acid (EPA) EPA ethyl ester EPA methyl ester Ethane Ethanol Ethyl acetate Ethyl benzoate Ethylbenzene Ethylene Eugenol Ferrocene Fluorobenzene ␥-Linolenic acid ␥-Linolenic acid ethyl ester ␥-Linolenic acid methyl ester Helium Hexachlorobenzene Hydrogen i-Butane Iodobenzene i-Propylbenzene Krypton Linalool Linoleic acid Linoleic acid methyl ester Methane Methanol Methylcyclohexane Monoolein m-Xylene Myristic acid ethyl ester Myristoleic acid Myristoleic acid methyl ester N-(4-methoxybenzylidene)-4-nbutylaniline Naphthalene n-Butane n-Butylbenzene n-Decane n-Dodecane n-Eicosane

C10 H12 O C12 H10 C10 H16 C6 H5 Br C6 H12 O2 C8 H10 N4 O2 C12 H24 O2 C10 H20 O2 CO2 CS2 CO CCl4 C6 H5 Cl CClF3 C18 H12 C10 H16 O C15 H21 CoO6 C10 H8 CuF6 O4 C7 H12 O C6 H12 C9 H16 O C5 H8 O D2 C24 H32 O8 C14 H14 O C20 H36 O6 C24 H44 O8 C4 H10 O C6 H14 O C39 H72 O5 C10 H16 C22 H32 O2 C24 H36 O2 C23 H34 O2 C20 H30 O2 C22 H34 O2 C21 H32 O2 C2 H6 C2 H6 O C4 H8 O2 C9 H10 O2 C8 H10 C2 H4 C10 H12 O2 C10 H10 Fe C6 H5 F C18 H30 O2 C20 H34 O2 C19 H32 O2 He C6 Cl6 H2 C4 H10 C6 H5 I C9 H12 Kr C10 H18 O C18 H32 O2 C19 H34 O2 CH4 CH4 O C7 H14 C21 H40 O4 C8 H10 C16 H32 O2 C14 H26 O2 C15 H28 O2 C18 H21 NO

2550-26-7 92-52-4 127-91-3 108-86-1 105-54-4 58-08-2 110-38-3 106-32-1 124-38-9 75-15-0 630-08-0 56-23-5 108-90-7 75-72-9 218-01-9 5392-40-5 21679-46-9 14324-82-4 502-42-1 110-82-7 3350-30-9 120-92-3 7782-39-0 14174-09-5 103-50-4 16069-36-6 17455-23-1 60-29-7 108-20-3 2465-32-9 138-86-3 6217-54-5 84494-72-4 28061-46-3 10417-94-4 84494-70-2 2734-47-6 74-84-0 64-17-5 141-78-6 93-89-0 100-41-4 74-85-1 97-53-0 102-54-5 462-06-6 506-26-3 1191-41-9 301-00-8 7440-59-7 118-74-1 1333-74-0 75-28-5 591-50-4 98-82-8 7439-90-9 78-70-6 60-33-3 112-63-0 74-82-8 67-56-1 108-87-2 111-03-5 108-38-3 124-06-1 544-64-9 56219-06-8 26227-73-6

148.20 154.21 136.24 157.01 116.20 194.20 200.00 172.30 44.01 76.13 28.01 153.82 112.56 104.46 228.29 152.24 356.26 369.70 112.17 84.16 140.22 84.12 4.03 448.51 198.27 372.50 460.61 74.12 102.18 621.99 136.24 328.49 356.55 342.52 302.46 330.51 316.48 30.07 46.07 88.11 150.18 106.17 28.05 164.20 186.04 96.10 278.44 306.48 292.46 4.00 284.78 2.02 58.12 204.01 120.20 83.80 154.25 280.45 294.48 16.04 32.04 98.19 356.55 106.17 256.43 226.36 240.39 267.37

C10 H8 C4 H10 C10 H14 C10 H22 C12 H26 C20 H42

91-20-3 106-97-8 104-51-8 124-18-5 112-40-3 112-95-8

128.17 58.12 134.22 142.29 170.34 282.56

 LJ p (Å)

εLJ /kB p (K)

6.26359 573.74 6.26984 626.54 6.28645 510.60 5.41839 532.04 5.81266 459.78 6.21101 679.43 7.11468 555.31 6.73239 520.69 3.58573 241.48 4.28281 438.34 3.57679 105.53 5.13577 441.83 5.32769 502.18 4.45760 239.82 6.97116 777.42 6.62040 550.07 6.80190 455.40 6.00543 327.84 5.26863 532.99 5.32769 439.53 5.71783 557.53 5.02221 497.10 3.09358 30.49 8.32315 1109.16 6.68328 617.01 7.89622 935.02 8.40827 1078.11 5.16109 370.60 5.74404 397.28 10.18204 813.94 6.36013 524.10 8.26055 854.01 8.52541 812.58 8.39856 793.57 8.04157 810.69 8.32034 768.81 8.35300 707.18 4.17576 242.52 4.34524 408.08 5.19769 415.47 6.21524 531.01 5.68389 490.11 4.00050 224.25 6.03299 583.90 5.38344 624.37 5.09260 444.77 7.86883 761.52 7.31546 744.07 8.02054 701.02 3.04317 4.12 6.36821 655.13 3.16052 26.21 5.05445 324.15 5.56490 572.54 5.94386 501.15 3.55102 166.28 6.49481 512.82 7.86262 615.42 8.07133 691.48 3.65196 151.20 3.86945 407.05 5.65333 454.38 8.40654 702.77 5.69401 490.03 7.75706 626.82 7.38373 678.34 7.54972 617.64 6.62760 763.96 5.87497 5.00267 6.24896 6.66491 7.04777 8.35996

594.30 337.65 524.50 490.51 522.67 609.07

910

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Table 3 (Continued) Substance

Formula

CAS number

Neon n-Heptane n-Hexadecane n-Hexane Nitrobenzene Nitrogen n-Nonane n-Octacosane n-Octane n-Pentane n-Pentylbenzene n-Propylbenzene n-Tetradecane n-Undecane Oleic acid Oleic acid ethyl ester Oleic acid methyl ester Oxygen o-Xylene Palladium(II) acetylacetonate Palmitic acid ethyl ester p-Dichlorobenzene Phenanthrene Phenol Phenylacetic acid Phenylacetylene Phenylmethanol Propane p-Xylene Pyrene Squalene Stearic acid ethyl ester s-Trioxane Styrene Sulfur hexafluoride tert-Butylbenzene Tetrabutyltin Tetrachloroethene Tetraethyltin Tetrafluoromethane Tetrahydrofuran Tetramethyltin Tetrapropyltin Thenoyltrifluoroacetone Toluene Triarachidonin Trierucin Trifluoroacetylacetone Trinervonin Triolein Ubiquinone CoQ10 Vanillin Vitamin K1 Vitamin K3 Xenon

Ne C7 H16 C16 H34 C6 H14 C6 H5 NO2 N2 C9 H20 C28 H58 C8 H18 C5 H12 C11 H16 C9 H12 C14 H30 C11 H24 C18 H34 O2 C20 H38 O2 C19 H36 O2 O2 C8 H10 C10 H14 O4 Pd C18 H36 O2 C6 H4 Cl2 C14 H10 C6 H6 O C8 H8 O2 C8 H6 C7 H8 O C3 H8 C8 H10 C16 H10 C30 H50 C20 H40 O2 C3 H6 O3 C8 H8 SF6 C10 H14 C16 H36 Sn C2 Cl4 C8 H20 Sn CF4 C4 H8 O C4 H12 Sn C12 H28 Sn C8 H5 F3 O2 S C7 H8 C63 H98 O6 C69 H128 O6 C5 H5 F3 O2 C75 H140 O6 C57 H104 O6 C59 H90 O4 C8 H8 O3 C31 H48 O2 C11 H8 O2 Xe

7727-37-9 20.18 142-82-5 100.21 544-76-3 226.45 110-54-3 86.18 98-95-3 123.11 7727-37-9 28.01 111-84-2 128.26 630-02-4 394.77 111-65-9 114.23 109-66-0 72.15 538-68-1 148.25 103-65-1 120.20 629-59-4 198.39 1120-21-4 156.31 112-80-1 282.47 111-62-6 310.52 112-62-9 296.49 7782-44-7 32.00 95-47-6 106.17 14024-61-4 304.64 628-97-7 284.48 106-46-7 147.00 85-01-8 178.23 108-95-2 94.11 103-82-2 136.15 536-74-3 102.14 100-51-6 108.14 74-98-6 44.09 106-42-3 106.17 129-00-0 202.26 111-02-4 410.72 111-61-5 312.54 110-88-3 90.08 100-42-5 104.15 2551-62-4 146.05 98-06-6 134.22 1461-25-2 347.17 127-18-4 165.83 597-64-8 234.95 75-73-0 88.01 109-99-9 72.11 594-27-4 178.85 2176-98-9 291.06 326-91-0 222.18 108-88-3 92.14 23314-57-0 951.45 2752-99-0 1053.75 367-57-7 154.09 81913-24-8 1137.91 122-32-7 885.43 303-98-0 863.34 121-33-5 152.15 84-80-0 452.71 58-27-5 172.18 7440-63-3 131.30

a b c d e f g h i j k l m n o p

M (g/mol)

Tc (K)

Pc (bar)

Vc (cm3 /mol)

44.40b 540.30b 722.00b 507.50b 719.00d 126.20b 594.60b 864.27d 568.80b 469.70b 679.90d 638.20b 693.00b 638.80b 781.00h 891.97e 868.65e 154.60b 630.30b 651.12c 835.62a 684.75d 873.00b 694.20b 783.55e 655.43d 720.20b 369.80b 616.20b 936.00d 974.94c 883.39a 604.00d 647.00b 318.70b 660.00b 767.97c 620.20b 655.92c 227.60b 540.10b 511.77c 759.88c 838.69c 591.80b 1499.66c 1549.28c 594.02a 1601.10c 1448.04c 1522.50c 777.00d 1329.54e 893.85e 289.70b

27.60b 27.40b 14.10b 30.10b 44.00d 33.90b 22.90b 6.55d 24.90b 33.70b 26.04d 32.00b 14.40b 19.70b 13.90h 11.38e 12.01e 50.40b 37.30b 4.13c 12.36a 40.70d 29.00d 61.30b 38.50e 44.03d 44.00b 42.50b 35.10b 26.10d 13.23c 11.09a 58.20d 39.90b 37.60b 29.60b 17.25c 47.60b 25.75c 37.40b 51.90b 34.18c 20.66c 26.32c 41.00b 6.51c 5.62c 32.89a 5.20c 6.70c 7.09c 40.10d 8.58e 31.96e 58.40b

41.60b 432.00b 930.00d 370.00b 349.00d 89.80b 548.00b 1603.50d 492.00b 304.00b 550.00d 440.00b 830.00b 660.00b 1000.00h 1154.20e 1098.65e 73.40b 369.00b 435.41c 1061.66a 351.00d 554.00b 229.00b 422.60e 337.50d 335.00d 203.00b 379.00b 630.00d 1128.14c 1172.66a 206.00d 352.00d 198.80b 492.00d 760.75c 289.60b 429.28c 139.60b 224.00b 263.54c 595.01c 428.15c 316.00b 2341.53c 2832.93c 365.58a 3081.54c 2335.72c 2146.17c 415.00d 1620.20e 537.20e 118.40b

Average of the values by the Joback [1,114,115] and Somayajulu [116] methods. Taken from Reid et al. [1]. Estimated by the Klincewicz [1,117] method. Taken from Yaws [118]. Average of the values by the Joback [1,114] and Ambrose [1,119,120] methods. Average of the values by the Joback [1,114,115] and Wen–Qiang [121] methods. Taken from Korea Thermophysical Properties Data Bank (KDB) [122]. Taken from ASPEN database [123]. Taken from Table 4 of Liu and Ruckenstein [5]. Estimated by the Joback [1,114,115] method. Taken from ChemSpider [124]. Taken from Lide [125]. Estimated by the Tyn–Calus [1,126] expression. Taken from Perry and Green [127]. Taken from LookChem [128]. Estimated by the Eq. (21).

Tbp (K) 27.10b 371.60b 560.00b 341.90b 483.95d 77.40b 424.00b 704.75h 398.80b 309.20b 478.61d 432.40b 526.70b 469.10b 633.00h 719.38j 696.50j 90.20b 417.60b 460.75o 669.46j 447.21d 613.00b 455.00b 554.63j 418.36d 478.60b 231.10b 411.50b 667.95d 702.45k 715.22j 387.65d 418.30b 209.60b 442.30b 548.45k 394.40b 456.25k 145.10b 338.00b 347.65k 536.35k 584.42k 383.80b 1135.95k 1182.75k 416.12j 1229.05k 1091.85k 1142.15k 558.00d 1099.02j 638.20j 165.00b

Vbp m (cm3 /mol) 14.18 164.75 367.97 140.06 131.74 31.76 211.39 651.26 188.81 114.00 212.20 167.95 326.62 256.88 397.05 461.44 438.19 25.71 139.67 166.11 422.74 132.53 213.82 84.71 161.00 127.20 126.21 74.66 143.63 244.65 450.53 469.18 75.82 132.93 73.04 188.81 298.12 108.35 163.66 50.43 82.78 98.15 230.44 163.21 118.72 968.46 1182.46 138.31 1291.44 965.94 883.95 157.96 658.37 207.03 42.43

 LJ p (Å) 2.73350 5.96371 7.70045 5.66356 5.55431 3.53276 6.45578 9.23375 6.22793 5.30452 6.46363 6.00030 7.41392 6.86863 7.88900 8.27527 8.14032 3.30309 5.65845 5.97936 8.04792 5.56490 6.47926 4.82652 5.92014 5.49262 5.47903 4.63647 5.70911 6.76294 8.21252 8.31916 4.65920 5.57018 4.60427 6.22793 7.20172 5.21941 5.95117 4.09245 4.79113 5.05791 6.63534 5.94594 5.37342 10.47583 11.16265 5.64091 11.48007 10.46716 10.17599 5.88443 9.26570 6.41309 3.87382

εLJ /kB p (K) 35.26 429.05 573.33 403.00 570.95 100.21 472.17 686.31 451.68 372.98 539.90 506.79 550.31 507.27 620.19 708.31 689.79 122.77 500.52 517.05 663.56 543.75 693.24 551.26 622.21 520.47 571.91 293.66 489.32 743.27 774.19 701.49 479.63 513.78 253.08 524.10 609.84 492.50 520.86 180.74 428.89 406.39 603.41 666.00 469.94 1190.87 1230.27 471.71 1271.42 1149.88 1209.01 617.01 1055.78 709.80 230.05

Table 4 Calculated Results. System

LJ-1 (this work: Eqs. (16)–(25))

DHB (Eq. (B.15))

VD (cm3 /mol) B × 107 (mol/cm s K0.5 )

Zhu (Eqs. (B.8)–(B.14))

HYS (Eqs. (B.6) and (B.7))

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

Solvent (1)

Solute (2)

NDP

k12

2,3-Dimethylbutane

Benzene Naphthalene Phenanthrene Toluene 1,1,1,5,5,5Hexafluoroacetylacetone 1,1 -Dimethylferrocene 1,2-Dichlorobenzene 1,2-Diethylbenzene 1,3,5Trimethylbenzene 1,3-Divinylbenzene 1,4-Diethylbenzene 15-Crown-5 1-Naphthol 1-Phenyldodecane 1-Phenylethanol 1-Phenylhexane 1-Phenyloctane 1-Propanol 2,2,4,4-Tetramethyl-3pentanone 2,3-Dimethylaniline 2,3Dimethylnaphthalene 2,4-Dimethyl-3pentanone 2,4-Dimethylphenol 2,6-Dimethylaniline 2,6Dimethylnaphthalene 2,7Dimethylnaphthalene 2-Bromoanisole 2-Butanone 2-Ethyltoluene 2-Fluoroanisole 2-Heptanone 2-Methylanisole 2-Naphthol 2-Nitroanisole 2-Nonanone 2-Pentanone 2-Phenyl-1-propanol 2-Phenylethanol 2-Phenylethyl acetate 2-Propanol 3-Ethyltoluene 3-Nitrotoluene

11 9 11 10 15

−0.03812 −0.01904 −0.02096 −0.03062 −0.01094

1.25 1.17 1.30 1.32 4.31

0.9661 0.7527 0.5871 0.9458 1.1460

70.08 66.74 54.23 77.91 11.05

1.56 1.75 1.53 1.79 4.36

7.93 2.41 5.23 4.82 23.39

4.46 2.76 2.49 4.75 10.10

75.95 65.82 61.35 72.49 18.95

81.84 85.96 90.49 84.58 27.98

156.21 119.78 104.02 140.62 33.14

56.59 68.14 77.61 62.29 91.59

68 15 15 24

0.01041 0.06039 0.08839 0.04073

2.44 2.12 2.17 4.97

1.2835 1.5427 1.3663 1.2531

11.85 16.61 15.06 11.01

3.67 2.07 2.61 4.29

29.12 10.51 11.78 13.33

17.13 9.25 3.30 10.57

12.16 6.89 6.19 7.27

20.35 11.69 6.01 14.10

25.53 19.05 6.09 16.95

79.91 65.11 61.99 71.51

15 15 29 11 15 15 15 15 17 9

0.08070 0.08194 0.11868 0.14034 0.08194 0.03349 0.09175 0.09793 0.03343 0.21054

1.57 2.41 5.30 1.82 2.33 1.84 2.10 2.62 5.04 2.52

1.5589 1.3865 0.9986 2.1460 0.9485 1.3428 1.2534 1.1894 1.3448 3.0352

18.35 15.72 1.19 24.53 17.21 14.22 16.08 16.53 −10.04 29.08

1.39 4.06 5.98 0.88 3.14 3.15 2.71 3.65 3.00 0.76

12.10 11.17 21.17 9.22 26.54 11.15 14.59 20.79 8.71 24.62

2.34 3.56 13.43 18.59 16.09 7.66 4.98 3.77 10.36 33.00

3.74 5.65 7.85 5.77 6.96 10.31 7.52 8.65 15.43 27.01

7.74 6.72 12.64 0.39 18.09 17.97 7.80 8.49 11.60 21.41

10.19 6.97 11.20 5.67 7.99 23.56 5.06 3.91 33.81 18.31

62.36 63.10 74.02 48.49 94.23 75.96 68.63 72.00 56.46 17.70

15 3

0.00436 0.09695

2.26 1.02

1.2259 1.5361

13.77 21.43

2.38 1.08

12.08 10.35

16.97 1.25

16.04 3.66

24.54 8.29

29.92 7.70

86.17 66.40

8

−0.04379

3.06

1.7583

25.46

2.33

27.67

15.38

29.97

34.48

46.54

96.69

15 15 6

0.03564 0.03267 0.08942

3.34 3.36 4.52

1.1649 1.1330 1.1736

8.33 8.66 10.78

3.63 3.34 4.24

11.95 11.78 15.38

8.03 12.35 5.28

9.42 11.47 7.15

16.90 19.64 9.26

22.58 24.80 8.88

74.26 78.84 67.76

6

0.09355

4.35

1.5069

19.63

4.50

11.91

4.68

6.91

6.00

5.64

60.69

15 38 15 15 11 15 16 15 10 23 15 15 15 18 15 15

0.02773 0.07918 0.07968 0.00637 0.21051 0.02917 0.14634 0.03639 0.24779 0.07783 0.02865 0.02805 0.03280 0.05193 0.08838 0.07620

2.14 1.62 3.41 2.12 3.88 2.25 2.84 1.86 3.53 2.05 2.00 1.91 2.68 4.07 3.69 2.25

1.2561 2.0672 1.4172 1.3944 3.4792 1.3055 1.9948 1.2984 2.8331 1.7185 1.2825 1.3652 1.1080 1.5735 1.4175 1.3849

12.79 16.91 14.72 14.19 32.29 12.12 21.27 15.75 29.00 12.62 15.24 15.18 13.22 −1.37 14.91 14.90

3.66 2.60 3.76 2.58 1.83 2.79 1.73 2.39 2.35 2.52 2.64 3.03 3.06 2.21 4.00 3.96

10.23 9.83 9.30 9.93 22.02 9.69 6.52 10.88 25.05 7.78 15.83 10.31 13.89 7.42 9.82 9.58

11.27 3.98 5.62 3.35 26.86 5.18 23.33 3.48 29.36 4.35 9.53 8.49 9.82 6.43 6.07 2.86

16.52 5.38 8.95 18.48 30.11 9.67 7.84 11.47 35.76 4.45 9.72 12.10 8.63 9.57 11.68 4.00

23.81 5.39 4.63 22.85 24.33 16.19 2.46 20.75 27.22 4.27 19.88 19.59 22.22 6.96 5.28 8.67

30.64 10.33 4.63 33.51 21.82 23.02 5.02 24.69 28.14 13.04 22.66 25.60 21.45 26.96 5.07 11.30

83.97 38.58 53.63 79.91 13.71 72.34 45.23 81.49 12.37 47.94 81.11 78.14 87.91 48.70 50.98 64.09

AARD

AARD

Supercritical systems

Carbon dioxide

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923 911

912

Table 4 (Continued) System

Solvent (1)

Solute (2)

DHB (Eq. (B.15))

NDP

B × 107 (mol/cm s K0.5 )

k12

AARD

VD (cm3 /mol)

AARD

Zhu (Eqs. (B.8)–(B.14))

HYS (Eqs. (B.6) and (B.7))

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

39 15 15 15 9 15 12 31 13 75 48 178 6 8 56 15 15 15 22 15 82 90 17 222 29 15 15 24 15 15 16 21 16 16

0.10192 0.04329 0.03716 0.06296 0.24990 −0.00839 0.22800 0.08648 0.25178 0.11788 0.13880 0.06228 0.08914 0.19714 0.11547 0.07426 −0.05610 0.04049 0.12949 0.09612 0.21519 0.17275 0.17179 0.10391 0.07014 0.04221 0.05000 0.12074 0.12276 0.03365 0.08239 0.03195 0.11303 0.10235

1.45 1.66 3.39 3.10 1.67 1.84 4.07 2.10 3.76 4.71 0.97 3.76 2.87 2.53 3.94 3.49 3.28 2.17 2.33 3.06 2.85 2.68 1.53 7.54 5.57 2.21 2.71 4.43 3.65 4.66 2.80 8.16 2.43 1.82

2.0419 1.5158 1.0565 1.4780 3.1752 1.3473 3.0575 1.6033 2.8101 0.8591 1.1233 2.1316 1.2634 1.4379 0.9745 1.3060 1.1791 1.4822 1.6433 1.7055 0.9419 0.6748 1.2263 1.4921 1.8170 1.3600 1.0795 1.3557 1.3716 1.4373 2.2177 0.7164 1.7729 1.7619

19.27 20.64 12.96 17.40 27.31 16.11 32.15 22.88 31.76 13.44 23.26 13.74 16.55 −2.78 15.74 10.81 8.61 13.90 22.35 21.73 18.16 16.79 27.32 −1.03 21.69 16.99 9.68 13.85 9.22 12.25 27.36 −17.53 29.09 26.83

2.03 2.28 3.37 3.07 0.47 3.21 1.10 3.02 2.53 2.51 1.17 5.05 2.85 2.59 2.79 3.34 2.46 2.97 1.75 3.83 2.21 2.26 0.86 7.59 6.34 3.02 3.76 3.39 4.97 4.36 1.89 4.87 1.44 1.67

9.04 9.50 14.86 8.89 29.62 12.16 25.83 13.60 27.76 49.72 37.41 11.71 15.47 29.47 26.31 10.60 21.45 7.70 20.69 9.38 64.17 144.58 65.88 11.77 8.54 9.82 12.63 15.21 10.14 9.08 3.91 37.00 10.56 7.39

4.22 7.82 9.79 8.12 32.43 12.63 26.31 3.34 28.03 7.61 1.53 4.15 3.50 31.54 5.15 4.39 25.07 3.35 10.72 4.11 9.43 8.79 2.73 10.21 6.05 3.24 9.19 8.45 6.66 14.21 3.54 6.83 2.88 2.45

9.48 6.21 7.12 7.46 36.53 17.52 34.31 8.45 38.53 9.70 15.16 5.64 4.93 18.17 14.24 5.36 33.34 7.33 10.38 7.00 31.55 14.88 21.34 9.14 9.27 7.79 6.19 10.13 11.72 7.81 4.31 22.62 13.46 10.23

6.20 16.54 21.63 4.10 30.93 24.50 25.24 5.81 27.40 16.13 10.26 5.48 12.06 13.45 9.32 8.42 33.91 11.77 2.86 4.66 5.63 20.36 5.03 9.23 13.02 17.75 17.90 6.94 4.07 10.55 2.96 36.24 3.77 4.14

3.73 18.65 18.93 4.86 29.03 31.83 26.49 3.65 30.77 7.12 0.94 20.26 10.72 9.10 4.65 11.96 51.80 20.82 1.72 4.68 16.48 5.82 6.43 13.21 21.26 20.20 18.07 6.96 4.08 20.44 7.13 37.03 2.46 1.73

38.28 76.57 89.00 56.37 4.11 84.67 15.74 64.73 14.91 92.80 84.76 38.16 72.93 28.25 81.08 62.71 92.40 64.09 57.43 59.39 65.05 112.10 78.86 40.08 65.32 78.14 80.40 56.82 51.21 61.51 53.47 107.91 64.43 62.85

15 4 15 38

0.03049 0.16203 0.09956 −0.02178

4.12 1.06 2.52 1.80

1.4476 1.8594 1.3860 1.0715

11.52 27.80 16.29 17.87

3.61 2.61 4.31 2.13

8.91 43.90 10.55 125.95

5.24 13.69 3.50 12.44

8.22 16.16 8.63 11.53

10.29 2.80 5.29 29.97

21.63 5.59 3.45 25.24

60.41 56.60 63.44 104.07

12

−0.13615

4.63

1.2764

22.66

5.04

72.60

17.87

37.09

49.65

53.26

125.99

8 8 8 28 15 15 15

−0.00071 0.01692 0.01838 0.16529 0.05662 −0.02679 0.02806

2.80 3.02 0.99 1.65 2.24 15.41 11.52

1.8569 1.7583 1.8382 1.0944 1.0702 1.0474 1.0782

23.71 25.46 18.40 24.24 14.04 −26.59 −16.29

1.85 2.33 1.03 1.93 3.22 4.98 7.87

8.58 10.68 13.18 99.02 15.81 32.29 29.92

5.33 4.12 7.64 9.04 5.56 6.98 8.47

24.01 17.62 20.31 12.73 5.32 11.80 7.14

26.38 25.13 19.56 13.08 20.72 12.38 6.46

40.45 31.85 37.55 2.71 17.22 24.74 9.41

83.17 86.08 70.71 89.14 88.51 59.29 52.65

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

3-Pentanone 3-Phenyl-1-propanol 3-Phenylpropyl acetate 4-Ethyltoluene 4-Heptanone 4-Methylanisole 5-Nonanone 5-tert-Butyl-m-xylene 6-Undecanone Arachidonic acid (AA) AA ethyl ester Acetone Acridine Adamantanone ␣-Linolenic acid Allylbenzene Aniline Anisole Anthracene ␣-Pinene ␣-Tocopherol ␤-Carotene Behenic acid ethyl ester Benzene Benzoic acid Benzyl acetate Benzylacetone Biphenyl ␤-Pinene Bromobenzene Butyric acid ethyl ester Caffeine Capric acid ethyl ester Caprylic acid ethyl ester Chlorobenzene Chrysene Citral Cobalt(III) acetylacetonate Copper(II) trifluoroacetylacetonate Cycloheptanone Cyclononanone Cyclopentanone Dibenzo-24-crown-8 Dibenzyl ether Diethyl ether Diisopropyl ether

LJ-1 (this work: Eqs. (16)–(25))

Table 4 (Continued) System

Solvent (1)

Solute (2)

NDP

B × 107 (mol/cm s K0.5 )

Diolein d-limonene Docosahexaenoic acid (DHA) DHA ethyl ester DHA methyl ester Eicosapentaenoic acid (EPA) EPA ethyl ester EPA methyl ester Ethanol Ethyl acetate Ethyl benzoate Ethylbenzene Eugenol Ferrocene Fluorobenzene ␥-Linolenic acid ␥-Linolenic acid ethyl ester ␥-Linolenic acid methyl ester Hexachlorobenzene Iodobenzene i-Propylbenzene Linalool Linoleic acid Linoleic acid methyl ester Methanol Monoolein Myristic acid ethyl ester Myristoleic acid Myristoleic acid methyl ester N-(4methoxybenzylidene) -4-n-Butylaniline Naphthalene n-Butylbenzene n-Decane n-Dodecane n-Heptane n-Hexane Nitrobenzene n-Nonane n-Octane n-Pentane n-Pentylbenzene n-Propylbenzene n-Tetradecane n-Undecane

9 15 63

0.18338 0.10598 0.11616

3.74 3.69 2.51

0.6769 1.3735 0.9162

14.02 12.04 18.60

1.61 4.06 1.63

77.97 10.49 58.53

4.84 4.03 9.19

23.69 9.32 7.28

9.91 4.27 19.86

5.27 4.20 8.29

95.83 57.32 100.02

65 17 55

0.15248 0.15558 0.11572

1.60 1.07 3.17

1.0708 1.2079 0.9184

22.09 25.70 16.15

1.45 0.92 1.79

48.46 52.38 46.01

1.67 2.34 7.35

16.73 16.76 7.79

9.27 8.37 17.55

2.00 2.47 7.30

84.14 81.78 94.51

48 17 24 15 15 15 15 98 15 142 41

0.13979 0.14362 0.05064 −0.00519 0.04491 0.09268 0.00224 0.04990 0.01155 0.10814 0.04630

1.18 1.55 2.97 18.11 3.66 1.54 2.68 3.06 3.57 5.00 6.69

1.1626 1.2964 1.9621 0.8254 1.8993 1.8248 1.3885 1.2424 1.7808 0.8364 0.8603

24.23 27.43 6.43 −49.61 27.23 18.85 20.62 6.91 17.76 9.15 6.15

1.06 0.49 3.21 6.71 2.94 2.28 3.58 6.35 4.22 2.15 5.10

36.48 37.76 12.41 36.32 12.20 9.47 10.54 21.24 10.40 33.37 43.72

1.55 1.65 4.48 8.62 4.95 4.72 7.33 19.48 4.34 7.90 9.79

14.98 17.37 11.29 12.75 3.88 7.44 17.29 17.32 11.04 7.79 6.92

10.14 7.27 5.39 13.77 12.77 3.41 28.35 20.92 11.16 16.90 27.87

1.02 3.34 32.94 26.10 13.39 4.76 31.11 32.42 26.57 8.37 19.79

84.21 79.62 39.86 62.20 72.14 46.77 94.09 76.45 59.41 92.13 105.41

52

0.11534

7.34

0.8588

7.53

7.58

47.11

6.85

13.41

10.20

4.68

82.00

14 15 15 15 71 21

0.11598 0.02343 0.09392 0.09694 0.09491 0.12807

8.71 3.37 2.45 3.44 5.68 2.17

0.8331 1.2680 1.6274 1.3508 0.8351 1.0645

−12.47 11.18 17.00 14.09 9.74 19.77

4.18 2.72 2.00 4.02 3.73 1.66

25.04 11.69 10.94 11.26 30.68 56.92

25.68 16.88 4.82 3.52 7.14 2.11

10.99 12.21 9.27 7.24 9.63 15.74

13.01 17.69 4.11 5.63 15.88 7.44

12.63 26.12 4.28 4.13 7.28 3.30

71.35 73.53 49.12 63.29 90.41 77.97

10 11 16

0.05609 0.09642 0.13321

3.93 3.33 2.36

2.1980 0.8199 1.4546

1.60 13.88 27.11

2.14 1.22 2.14

20.87 24.17 24.88

6.17 5.13 3.07

16.79 8.71 15.97

7.10 18.90 4.92

46.84 7.31 3.60

33.18 99.51 71.66

42 79

0.06120 0.01906

5.82 9.00

0.8465 0.7247

4.77 −15.38

2.66 10.07

25.97 68.29

13.91 11.93

5.68 10.41

22.90 19.80

14.76 13.01

98.00 94.13

5

0.19411

1.41

2.1419

29.00

0.33

42.04

30.23

17.85

5.56

7.95

47.09

83 15 5 5 5 5 15 5 5 5 31 34 5 5

0.09961 0.08608 0.24037 0.25131 0.18633 0.15090 0.05167 0.22570 0.20710 0.09826 0.08492 0.06420 0.25003 0.25308

8.78 1.90 3.04 4.81 2.80 2.58 2.42 3.06 3.16 2.87 2.20 10.64 6.96 3.79

1.4146 1.4109 3.8443 4.3507 3.9421 3.9838 1.2975 4.0604 4.1527 4.0586 1.6168 0.8647 4.3703 4.1553

11.79 15.95 36.48 39.93 35.67 35.55 9.79 37.02 37.05 35.82 21.86 −17.49 41.99 38.28

8.29 2.97 1.47 2.99 1.00 2.09 3.33 1.43 1.66 1.67 3.95 5.03 3.32 1.85

18.59 11.18 32.31 37.54 20.99 16.06 9.96 29.33 25.60 9.12 12.96 25.18 43.05 35.87

10.29 2.77 27.51 29.20 19.03 14.11 4.21 25.69 22.91 6.62 3.96 8.65 25.26 29.69

10.81 6.29 38.51 40.87 28.91 22.91 8.79 36.45 33.41 13.23 8.27 12.75 38.78 40.94

10.77 5.39 29.10 29.79 22.68 18.40 13.98 27.94 25.91 11.26 5.37 14.62 25.35 30.82

11.51 5.62 31.07 33.31 20.51 13.51 22.29 28.90 25.59 2.49 4.32 15.88 30.40 33.61

56.10 61.07 10.62 11.47 16.50 20.98 67.96 11.31 13.16 28.89 62.60 66.76 20.42 8.96

k12

AARD

VD (cm3 /mol)

AARD

Zhu (Eqs. (B.8)–(B.14))

HYS (Eqs. (B.6) and (B.7))

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

913

DHB (Eq. (B.15))

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

LJ-1 (this work: Eqs. (16)–(25))

914

Table 4 (Continued) System

Solvent (1)

Ethane Sulfur hexafluoride

Oleic acid Oleic acid ethyl ester Oleic acid methyl ester Palladium(II) acetylacetonate Palmitic acid ethyl ester p-Dichlorobenzene Phenanthrene Phenol Phenylacetic acid Phenylacetylene Phenylmethanol Pyrene Squalene Stearic acid ethyl ester Styrene tert-Butylbenzene Tetrahydrofuran Thenoyltrifluoroacetone Toluene Triarachidonin Trierucin Trifluoroacetylacetone Trinervonin Triolein Ubiquinone CoQ10 Vanillin Vitamin K1 Vitamin K3 Acetone p-Xylene 1-Octene 1-Tetradecene 1,3,5Trimethylbenzene Benzene Benzoic acid Carbon tetrachloride Naphthalene p-Xylene Toluene

DHB (Eq. (B.15))

NDP

VD (cm3 /mol) B × 107 (mol/cm s K0.5 )

k12

AARD

AARD

Zhu (Eqs. (B.8)–(B.14))

HYS (Eqs. (B.6) and (B.7))

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

19 5 19 125

0.09848 0.04134 0.00526 −0.02284

5.40 6.29 8.22 2.56

0.8088 0.5458 0.5031 1.2460

9.94 −23.03 −34.23 17.11

2.14 0.97 1.93 4.65

37.36 53.72 65.15 91.51

7.78 5.13 6.26 18.95

10.03 5.72 6.50 21.93

16.61 27.68 31.10 32.80

7.86 16.67 19.06 36.33

91.81 113.19 117.76 100.26

17

0.13180

1.38

1.3142

26.47

0.61

30.82

1.46

15.14

8.02

1.75

78.79

13 19 109 16 15 15 18 5 17 15 15 15 15 35 27 101 15 38 10 80 15 16 20 10 8 6 9 10

0.04635 0.15914 0.03366 0.07464 0.04583 0.03313 0.12463 0.14122 0.13988 0.03001 0.09222 0.01091 −0.02344 0.07407 0.18755 0.16378 0.05745 0.1806 0.16986 0.17148 0.04536 0.23026 0.13593 −0.11516 −0.15246 0.03142 0.11067 −0.06452

3.63 6.09 3.09 1.55 1.59 1.61 2.38 4.21 1.41 3.97 3.10 13.12 3.19 4.01 6.00 7.58 1.61 6.61 5.73 3.43 1.70 3.00 4.59 4.27 8.93 5.28 2.05 5.00

1.5067 1.3448 1.3641 1.7122 1.6377 1.5257 1.5497 1.9069 1.2474 1.6454 1.8018 1.1871 1.1938 1.6490 0.4912 0.3731 1.7229 0.3962 0.4442 0.6220 1.5345 0.8940 1.1807 0.9096 0.4851 1.2050 1.5925 0.5377

16.27 12.04 2.62 22.71 17.26 16.57 23.47 36.68 26.37 18.93 23.59 −22.18 18.09 11.46 6.85 −2.73 19.57 2.77 −1.58 16.09 21.13 16.80 8.84 19.58 −13.32 12.75 38.24 18.96

3.72 5.03 4.61 1.80 1.58 2.54 1.84 1.87 1.03 4.39 3.62 4.99 3.05 4.14 0.82 2.95 2.18 2.86 1.26 4.09 2.03 2.22 2.70 2.56 2.28 1.40 1.61 4.59

8.76 23.94 8.02 6.41 7.61 8.05 26.23 50.04 41.53 8.35 9.25 22.16 13.70 8.44 149.23 146.62 6.36 163.95 100.27 141.53 10.92 106.89 24.21 5.85 38.21 11.82 18.87 14.17

8.46 13.72 5.21 1.62 4.46 7.02 6.84 10.81 1.65 9.17 3.26 8.80 4.04 5.43 10.98 7.37 14.09 6.58 5.95 7.87 2.63 9.99 8.88 8.40 7.10 4.02 7.84 6.88

10.61 13.96 21.47 4.25 7.80 13.91 9.18 13.56 16.25 5.38 7.92 15.96 30.20 5.32 17.49 13.57 3.74 16.62 11.80 13.90 12.64 27.19 9.77 23.27 23.77 4.78 17.61 33.06

16.00 7.78 18.22 12.95 12.00 18.50 5.95 11.21 8.48 9.91 3.35 12.43 41.38 5.03 20.63 31.19 9.06 27.91 28.90 23.98 21.65 4.19 7.48 29.79 22.00 50.84 37.74 45.21

24.31 8.06 40.23 16.60 21.03 28.26 2.73 3.78 2.16 17.70 3.51 34.15 45.60 11.33 8.39 14.68 15.88 12.12 12.39 8.60 26.02 11.84 7.14 23.24 21.70 55.22 26.61 41.60

71.04 51.15 66.70 69.79 64.46 73.87 65.38 85.24 81.42 62.10 56.49 58.17 112.83 48.08 116.86 140.57 61.49 136.63 131.65 120.85 82.53 72.64 61.05 25.58 39.07 41.98 37.54 30.23

9 6 6 5 52 11

−0.05035 −0.12521 −0.09832 −0.00670 −0.03190 −0.04438

6.31 1.88 4.48 7.21 5.71 6.19

0.6679 0.6061 0.6877 1.1687 0.5781 0.5288

1.44 40.87 35.40 67.42 19.73 −18.56

10.58 2.61 5.18 5.52 12.41 9.73

7.33 17.30 13.73 11.07 7.75 7.21

16.16 13.12 21.28 9.65 11.03 12.93

39.39 149.89 37.99 77.19 23.28 37.76

51.09 76.57 33.97 29.47 41.80 51.56

42.39 103.02 31.82 40.69 35.17 44.06

39.89 194.96 42.27 120.66 21.67 35.72

Liquid systems 2,2,4-Trimethylpentane

1,3,5Trimethylbenzene Benzene Ethylbenzene o-Xylene p-Xylene Toluene

4

−0.18230

3.22

1.0142

150.50

0.84

36.05

-

58.70

68.00

105.27

48.48

4 4 4 4 4

−0.10507 −0.14632 −0.13140 −0.07683 −0.09552

1.26 1.27 2.33 2.81 1.55

1.1336 0.8035 0.9929 0.7954 1.0099

145.53 144.09 147.66 140.69 144.99

1.61 1.33 2.24 1.94 1.58

26.78 35.01 29.54 17.09 23.72

-

48.72 55.63 51.64 35.18 43.53

43.89 60.58 56.09 39.81 43.79

116.08 107.53 102.80 79.76 98.76

3.12 39.77 35.68 21.86 4.53

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Chlorotrifluoromethane

Solute (2)

LJ-1 (this work: Eqs. (16)–(25))

Table 4 (Continued) System

LJ-1 (this work: Eqs. (16)–(25))

DHB (Eq. (B.15))

B × 107 (mol/cm s K0.5 )

VD (cm3 /mol)

Zhu (Eqs. (B.8)–(B.14))

HYS (Eqs. (B.6) and (B.7))

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

Solute (2)

NDP

k12

AARD

Cyclohexane

Argon Carbon tetrachloride Ethane Ethylene Krypton Methane Phenanthrene p-Xylene Tetrabutyltin Tetraethyltin Tetramethyltin Tetrapropyltin Toluene Xenon 12-Crown-4 15-Crown-5 18-Crown-6 Argon Carbon tetrachloride Dicyclohexano-18crown-6 Dicyclohexano-24crown-8 Krypton Methane s-Trioxane Tetrabutyltin Tetraethyltin Tetramethyltin Tetrapropyltin Xenon 1,3,5Trimethylbenzene Acetone Benzene Carbon dioxide Carbon monoxide Hydrogen Linoleic acid methyl ester m-Xylene Naphthalene n-Decane n-Hexadecane n-Octane n-Tetradecane Toluene

6 6 5 5 6 6 8 8 7 7 7 6 12 6 4 4 4 3 3 4

−0.19274 −0.30516 −0.29956 −0.32459 −0.21552 −0.25714 −0.13688 −0.14215 −0.52376 −0.46562 −0.36392 −0.48181 −0.35350 −0.26748 −0.12568 −0.13989 −0.17021 −0.13151 −0.09022 −0.20064

4.61 9.86 9.04 7.98 4.33 6.40 18.07 14.32 17.12 14.93 12.27 14.12 18.29 5.63 3.17 5.15 3.10 2.33 2.95 1.25

2.6698 1.3613 2.0214 2.1435 2.1405 2.8585 0.9775 1.2210 0.7438 1.0441 1.2553 0.8573 1.3305 1.8186 0.5689 0.4514 0.4654 1.8070 0.7923 0.3810

96.95 102.47 100.00 99.75 98.04 99.81 101.00 100.22 103.12 102.93 102.25 102.94 100.99 99.39 183.45 182.06 184.73 179.89 183.18 185.79

2.44 1.11 2.40 2.30 1.26 0.68 4.31 3.88 2.35 2.16 1.25 2.10 2.85 1.15 2.87 5.22 3.14 0.32 0.14 1.16

43.30 76.72 106.17 102.01 66.64 40.39 6.34 28.73 13.61 57.13 94.10 21.82 55.83 85.28 40.09 22.23 4.80 47.60 71.32 84.24

-

21.29 6.41 25.57 25.19 8.49 28.46 39.49 40.98 6.48 5.80 4.80 4.89 26.14 5.62 24.21 28.12 28.17 3.65 17.93 27.03

36.48 5.52 32.30 33.48 23.56 39.35 30.69 36.75 16.43 7.45 5.16 12.46 24.81 15.35 27.07 35.97 40.23 26.93 10.97 49.82

49.69 26.29 17.02 21.04 63.83 24.92 27.25 24.95 12.44 20.03 40.12 14.80 24.36 60.65 62.14 59.91 54.71 115.17 73.12 43.17

55.63 10.65 49.22 50.76 45.49 56.38 29.80 38.43 21.90 6.75 3.53 14.08 26.52 37.97 7.17 17.23 23.10 56.51 24.32 37.56

4

−0.22161

1.71

0.3344

185.84

1.71

137.91

-

29.14

57.42

42.19

47.41

3 3 4 4 3 4 4 4 4

−0.13060 −0.21158 −0.13736 −0.22885 −0.14314 −0.18046 −0.19791 −0.12199 −0.13037

1.74 1.26 2.00 3.24 4.82 3.51 3.92 2.66 3.03

1.3923 1.8223 0.8812 0.4353 0.5949 0.7287 0.4903 1.1823 0.4609

180.80 181.67 182.90 185.39 183.97 183.47 184.67 181.74 213.97

0.78 0.03 0.42 2.16 1.42 1.43 1.55 0.62 0.79

69.75 46.00 50.22 20.57 47.46 71.35 24.65 68.96 119.21

-

16.32 12.44 25.93 23.40 19.59 30.51 23.25 23.29 16.71

9.82 31.11 12.59 38.68 21.59 21.83 32.67 4.84 13.82

139.73 75.42 100.54 45.00 57.36 93.97 51.39 132.95 52.67

45.21 57.78 25.52 23.71 6.30 17.31 15.35 37.54 19.88

5 4 9 9 9 4

−0.16929 −0.14203 −0.15144 −0.21802 −0.39095 −0.23215

1.39 1.42 2.65 7.33 7.86 2.08

0.8244 0.6766 1.2167 1.4392 3.9660 0.2853

215.52 214.82 210.03 212.70 215.14 217.39

1.68 1.30 2.20 5.90 5.29 0.92

102.53 122.54 23.83 27.26 47.80 41.86

-

16.62 16.31 10.06 17.45 59.75 13.66

3.20 5.03 30.15 38.00 71.88 29.88

86.12 74.30 93.72 72.63 13.38 24.46

34.68 30.83 58.09 62.82 83.78 18.23

4 5 5 5 9 5 4

−0.09820 −0.06931 −0.08227 −0.14672 −0.08135 −0.15173 −0.09938

2.36 2.91 4.61 4.44 2.27 6.03 1.68

0.5477 0.5064 0.5822 0.4466 0.6640 0.4729 0.6374

214.25 214.04 219.24 221.99 218.97 221.14 214.93

1.54 1.41 2.98 6.69 1.48 7.84 2.01

108.51 81.24 29.97 19.30 25.45 17.19 110.79

-

9.33 12.27 21.54 14.84 21.35 18.00 9.90

5.86 8.59 18.92 12.56 21.56 15.58 4.09

48.13 48.57 12.00 12.69 12.11 15.38 55.84

27.84 23.94 30.49 14.84 34.17 19.31 30.91

n-Decane

n-Dodecane

AARD

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Solvent (1)

915

916

Table 4 (Continued) System

LJ-1 (this work: Eqs. (16)–(25))

DHB (Eq. (B.15))

VD (cm3 /mol) B × 107 (mol/cm s K0.5 )

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

NDP

k12

n-Eicosane

Carbon dioxide Carbon monoxide Hydrogen n-Dodecane n-Hexadecane n-Octane n-Decane n-Dodecane n-Hexadecane n-Octane n-Tetradecane Carbon dioxide Carbon monoxide Hydrogen n-Decane n-Dodecane n-Octane n-Tetradecane 1,3,5Trimethylbenzene Acetone Acetonitrile Benzene Carbon disulphide Indole m-Xylene Naphthalene Phenanthrene p-Xylene Toluene Carbon dioxide Carbon monoxide Hydrogen n-Dodecane n-Hexadecane n-Octane 1,3,5Trimethylbenzene Argon Benzene Carbon tetrachloride Ethyl benzene Krypton Methane o-Xylene p-Xylene Tetrabutyltin Tetraethyltin Tetramethyltin Tetrapropyltin Toluene Xenon

5 5 5 5 5 5 5 5 8 4 5 10 10 10 5 5 10 5 20

−0.28359 −0.36788 −0.63053 −0.16067 −0.18720 −0.14837 −0.01037 −0.00291 −0.04914 0.00047 −0.01066 −0.27704 −0.37970 −0.67791 −0.18681 −0.16438 −0.16926 −0.16297 −0.03642

4.12 4.82 6.38 4.74 5.13 4.92 3.09 4.65 3.77 3.45 4.50 3.08 5.40 3.00 3.82 1.38 3.16 2.41 3.51

0.8724 1.0324 3.0212 0.3222 0.2678 0.4003 0.9631 0.8855 0.7499 1.1427 0.8028 0.9379 1.0499 2.1314 0.4751 0.3976 0.5159 0.3774 0.8743

352.03 354.30 357.65 361.84 362.04 359.67 132.45 132.67 133.82 133.32 133.05 278.97 280.25 268.40 291.18 288.61 289.46 290.23 101.17

2.07 2.43 4.65 4.28 3.44 4.78 1.65 2.18 3.09 1.92 2.96 1.91 3.08 7.32 6.01 2.43 3.65 5.42 14.54

21.16 31.74 36.80 60.70 55.72 56.15 6.81 15.55 24.56 5.00 29.65 34.82 51.49 37.24 62.45 55.76 64.04 42.01 5.88

-

16.80 19.36 62.00 14.60 13.47 16.80 34.06 30.61 33.68 35.23 29.40 12.21 13.21 55.28 13.07 14.96 13.88 13.59 48.09

37.40 44.76 76.22 18.07 13.51 25.59 22.77 19.05 22.78 26.79 18.67 33.40 38.83 70.82 14.69 15.22 18.00 12.80 46.23

108.39 84.64 13.99 8.99 9.14 12.53 18.52 18.51 24.44 20.30 18.71 102.80 87.01 23.96 13.91 16.29 16.54 14.22 46.11

64.95 69.09 87.20 43.31 26.38 50.47 30.40 23.25 24.34 35.41 19.23 61.01 64.22 83.58 39.13 24.68 42.79 21.72 47.59

5 7 36 10 4 5 20 15 17 28 5 5 5 5 5 5 4

−0.02365 0.04660 −0.02810 0.08914 −0.01577 −0.02762 −0.00039 0.00207 −0.00956 −0.00821 −0.52119 −0.63474 −0.91913 −0.43090 −0.44961 −0.40864 −0.03237

2.62 15.08 11.86 25.91 4.32 2.43 4.62 4.58 5.21 12.01 4.73 6.87 6.72 2.39 1.66 2.73 2.16

1.4801 1.1551 1.1371 1.1409 0.4384 1.3601 0.8828 0.6767 0.9088 1.0354 0.5620 0.7097 1.8034 0.1909 0.1543 0.2457 0.8117

111.01 102.14 103.51 102.23 84.37 115.15 98.35 81.65 93.68 102.80 485.36 489.28 483.00 498.98 499.57 496.79 148.56

3.03 5.93 14.54 4.66 2.83 2.00 15.15 11.81 16.35 16.17 2.08 3.79 2.26 3.77 4.40 2.65 0.28

7.37 22.16 26.17 64.04 10.84 5.49 10.14 12.83 8.34 18.14 36.43 50.03 32.37 142.06 149.34 123.16 22.30

-

5.42 115.32 80.10 178.18 10.40 6.24 49.57 63.76 56.45 78.86 173.15 131.14 12.28 183.91 200.39 166.97 8.13

5.90 110.08 82.83 168.35 2.18 11.30 45.58 56.82 54.15 82.91 68.01 41.98 40.01 149.00 175.99 119.42 16.08

64.29 246.91 114.18 362.18 19.65 36.87 44.12 53.53 49.43 106.45 524.73 430.14 175.33 231.40 227.40 246.24 43.28

25.75 60.48 74.29 98.69 9.11 6.26 49.47 60.56 56.94 76.21 10.33 22.17 68.36 68.82 92.39 43.11 1.08

4 4 4 4 4 4 4 4 4 5 4 4 4 4

−0.07673 0.02089 −0.02667 −0.02758 −0.08222 −0.11889 0.01904 0.05529 −0.14964 −0.07103 −0.11523 −0.12798 0.00987 −0.07319

3.16 3.60 3.22 4.07 3.96 1.62 1.42 1.60 3.08 10.03 5.72 4.45 3.78 3.79

1.9181 1.0124 1.0157 0.7797 1.5181 2.0640 1.0323 1.0964 0.5544 0.7116 0.8621 0.6115 0.9051 1.3238

142.44 144.89 148.85 145.55 143.52 145.64 149.57 149.51 150.12 148.73 147.09 149.11 145.10 145.23

2.94 0.22 0.59 1.55 2.86 2.95 0.87 0.86 1.13 6.31 1.35 1.10 1.51 1.84

19.90 16.11 34.74 22.88 37.40 18.87 13.46 6.18 13.75 17.69 36.01 8.07 15.73 42.84

-

2.39 0.98 12.51 6.06 16.13 12.50 1.66 8.17 16.40 20.42 19.45 16.00 1.81 20.04

20.76 2.07 11.62 10.97 5.08 27.42 3.84 3.69 28.68 26.20 18.35 24.91 2.10 5.65

117.57 48.93 65.23 44.99 135.75 72.82 36.40 25.17 29.94 53.34 76.82 36.66 41.37 124.07

51.97 32.43 22.48 6.40 41.27 54.70 12.53 18.66 21.47 13.76 19.03 16.32 28.80 34.70

n-Hexane

n-Octacosane

n-Octane

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Solute (2)

n-Hexadecane

AARD

HYS (Eqs. (B.6) and (B.7))

Solvent (1)

n-Heptane

AARD

Zhu (Eqs. (B.8)–(B.14))

Table 4 (Continued) System

Solvent (1)

Solute (2)

Propane

1-Octene 1-Tetradecene

LJ-1 (this work: Eqs. (16)–(25))

DHB (Eq. (B.15))

NDP

VD (cm3 /mol) B × 107 (mol/cm s K0.5 )

8 8

k12 0.00487 0.05334

AARD 1.75 1.78

1.5602 1.3094

57.26 58.74

AARD

Zhu (Eqs. (B.8)–(B.14))

HYS (Eqs. (B.6) and (B.7))

WC (Eq. (B.1))

TC (Eqs. (B.2) and (B.3))

Sch (Eq. (B.4))

RD (Eq. (B.5))

AARD

AARD

AARD

AARD

AARD

AARD

1.91 1.92

9.91 40.78

-

1.72 2.29 1.33 1.49 1.24 5.46 0.78 4.79 0.28 0.16 0.16 1.14 0.95 3.53 0.38 5.16 2.91 2.38 2.92 2.29 1.72 0.11 1.12 1.25 0.63 0.78 2.11 1.25 1.81 0.96 0.74 1.36 0.5 0.12 0.32 0.86 0.92

12.61 72.01 71.9 22.49 67.17 35.22 38.79 9.12 73.95 69.03 70.85 35.8 11.79 9.95 8.64 51.24 32.57 81.76 53.32 9.25 19.26 110.69 46.96 57.07 61.1 61.51 66.53 72.28 18.82 75.74 52.52 66.7 72.82 2.5 11.66 37.93 51.75

-

7.32 16.66

27.75 25.61

34.70 15.79

19.51 24.66

Gas systems Argon

Carbon monoxide Deuterium Ethane Ethylene Helium Krypton

Methane Neon

Nitrogen

Oxygen Sulfur hexafluoride Tetrafluoromethane

9 5 8 9 8 25 9 48 7 7 7 5 14 49 7 17 6 6 17 8 10 5 5 24 5 6 8 29 7 5 6 8 13 5 5 5 5

−0.02405 0.00898 −0.07484 −0.00946 −0.06396 −0.01116 −0.03458 −0.01413 −0.01758 0.07603 0.01337 0.07457 −0.03094 0.02316 −0.01717 0.09643 0.00316 0.05413 −0.00475 0.00835 −0.00074 −0.08171 0.04500 0.06670 0.05089 −0.00315 0.02676 0.03686 0.00762 −0.09621 −0.09232 0.03223 0.04380 −0.05083 −0.05699 −0.02521 −0.06511

1.23 2.95 2.46 2.70 1.79 1.85 0.86 6.77 0.34 0.22 0.20 1.94 0.99 4.63 0.53 2.94 3.14 1.57 2.71 2.86 1.63 0.24 2.39 1.65 1.76 0.88 2.63 1.24 2.34 3.38 2.61 1.43 1.21 0.48 0.85 0.93 2.24

4.7507 23.0413 2.6722 7.7281 2.8191 10.3605 3.6775 2.9026 19.8027 22.1377 22.1377 34.7676 4.8313 3.0975 5.0043 49.5883 5.6514 22.0575 9.0618 3.3325 5.9479 2.8076 24.0282 28.0397 31.7987 6.4366 21.4938 23.3826 7.4175 2.6491 3.2798 21.0509 26.2335 1.1504 1.1703 1.447 1.0459

7729.39 3938.78 5654.17 8889.2 6520.36 6724.28 7706.29 −3.87 5898.7 5018.43 5018.43 2840.99 6847.25 5.29 5395.35 5467.93 11563.83 9617.38 8120.62 13066.53 6501.16 7640.69 3212.22 2125.92 2881.11 4241.29 6427.19 4968.86 6561.84 3362.34 4492.5 4501.28 6132.89 8141.63 10547.92 7233.94 3426.96

-

-

-

-

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Carbon dioxide

Ethane Hydrogen i-Butane Methane n-Butane Neon Propane Ethylene Hydrogen Helium Hydrogen Hydrogen Nitrogen Carbon dioxide Nitrogen Hydrogen Argon Helium Neon Xenon Carbon dioxide Tetrachloroethene Deuterium Helium Hydrogen Xenon Helium Hydrogen Methane n-Butane Propane Helium Hydrogen Cyclohexane Methylcyclohexane 1,1,1-Trichloroethane Tetrachloroethene

Note: An hyphen means the equations are not applicable.

917

69.79 24.71 19.14 17.23 8.78 37.68 3.85 4.26 5341 Total

309

73.85 42.68 -

RD (Eq. (B.5)) Sch (Eq. (B.4))

17.37 73.67 15.88 40.86 -

TC (Eqs. (B.2) and (B.3)) WC (Eq. (B.1))

13.86 39.74 8.78 -

HYS (Eqs. (B.6) and (B.7)) Zhu (Eqs. (B.8)–(B.14))

37.26 39.03 39.95 3.78 5.30 2.33

DHB (Eq. (B.15)) LJ-1 (this work: Eqs. (16)–(25))

4.12 6.22 2.63 171 101 37

NS

(7)

4278 641 422

(6)

NDP

(5)

Supercritical Liquid Gas

(4)

parameter seems sufficient for good representation of tracer diffusivities of all the systems studied, which makes LJ-1 a confident correlation. It is remarkable the good performance achieved for systems completely unknown, in light of the fact that their LJ parameters were calculated from critical constants also estimated: e.g., systems containing 1,1 -dimethylferrocene, cobalt(III) acetylacetonate, copper(II) trifluoroacetylacetonate, ferrocene, palladium(II) acetylacetonate, tetraethyltin, tetramethyltin, tetrapropyltin and tetrabutyltin. In these cases, the unique properties already known were the molecular weight and normal boiling point. It is worth noting that most group contribution methods available to estimate Tc , Pc and Vc do not comprehend metallic atoms like Co, Cu, Fe, Pd, Sn, etc. Hence, the critical constants have been calculated by Klincewicz method [1,117]. Even so, the average AARDs found for each systems were surprisingly good, 3.90%. Fig. 4 shows that the calculated diffusivities plotted against the experimental ones concentrate along the diagonals for the three physical states. Such systematic behaviour, along with the low AARDs found, assures the consistency and the statistically desirable behaviour of our model. Three distinct graphics were presented due to the very different orders of magnitudes of the diffusivities for gases, liquids and supercritical fluids. From Table 4 it may be observed that the absolute values of k12 are lower for gases than for liquids and supercritical fluids. Furthermore, k12 is frequently negative for liquids and generally positive for supercritical systems, which implies that such corrections tend to decrease the starting values provided by D12,LJ (Eq. (16)) for liquids, and to increase their estimates for supercritical fluids. Such results could be taken as a guideline when k12 or experimental data are not available. For these cases, one may propose the following first guesses: k12 = 0.08 for supercritical, k12 = −0.17 for liquids, and k12 = 0.00 for gaseous systems. It is important to emphasize that many experimental points of our D12 database lie outside the validity range of our F12 HS correction factor (Eq. (23)), particularly the size ratio  2 / 1 = 0.25 − 1.00. Notwithstanding the solutes diameters are frequently higher than those of the solvents (see Table 3), our model provides very good results globally. Another advantage of the model is that it is also able to represent systems containing polar compounds like alcohols, although it was derived on the basis of the LJ model. With respect to the models adopted for comparison, Table 5 points out that the hydrodynamic equations of Tyn–Calus [1,130], Scheibel [1,131] and Reddy–Doraiswy [1,132] performs deficiently with AARDs between 19.14 and 69.79%; the Wilke–Chang [1,3,129] equation offers interesting results, if one takes into account it has no parameters and furnishes AARDwc = 17.23%. The predictive He–Yu–Su model [133] has been only utilized for supercritical solvents, because it was designed for them, giving rise to AARDHYS = 8.78%. The Zhu et al. [6] expression performs more poorly, as its average absolute relative deviation demonstrates (AARDZhu = 37.68%). Despite of possessing two parameters, the Dymond equation [2,9,134] exhibits results comparable to those of our one-parameter correlation, AARDDHB = 3.85% and AARDLJ−1 = 4.26%. Besides, this reference equation has presented two physically meaningless results: quite different minimum diffusive free volumes (VD ) for the same solvent, which is not correct, and even negative values (please see Table 4) [4,9]. Moreover, the DHB equation should be used only for interpolation, which limits its application outside the fitting interval.

System

(3)

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

Table 5 Relative deviations for the supercritical, liquid and gas systems.

918

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

6. Conclusions In this work, a new correlation is proposed for the tracer diffusivity of hard spheres, more specifically the correction factor F12 given by Eqs. (23)–(25). This F12 provides a very good representation of MD data taken from literature (AARD = 4.44%), whereas the well known expressions of Sung and Stell [16], Sun and Chen [19], Easteal and Woolf [18], and Eaton and Akgerman [17] reach 63.89, 24.46, 18.20 and 67.78%. With this F12 factor it was possible to develop a new model for tracer diffusion coefficients of real fluids (LJ-1): Eq. (16) and subsidiary Eqs. (2) and (17)–(25). The proposed LJ-1 model involves only one parameter (k12 ) and requires the temperature, solvent density, and solute and solvent molecular weight and LJ force constants (these are estimated as function of Tc and Vc ). Results calculated for 309 systems and 5341 data points evidence the excellent performance of our correlation, for which AARD = 4.26%. Moreover it is important to emphasize that the new model interprets equally well the diffusive phenomena of gases, liquids and SCFs.

919

The first term in the right hand side of Eq. (A.1) is the Enskog limit, while the second is the hydrodynamic limit; the ratio D12,H /D12,E is given by (Sun and Chen [23]):



D12,H 2 = 2.881 1 + D12,E 1



1+

m1 m2

−1/2 ϕ g( ) 1 12

(A.3)

1,HS /01

where 1,HS /01 is the ratio between the viscosities of the HS solvent and ideal gas. Taking into account Enskog formula [2,9], it is possible to write: 1,HS 01



= ×

1,HS 1,E



  ×

1,E 01





=

1,HS 1,E



1 + 3.2ϕ1 + 12.761ϕ12 g(11 ) g(11 )

(A.4)

Using the MD simulations of Alder et al. [135] for the HS fluid, Liu and Ruckenstein [5] proposed the following accurate expression: 1,HS = 1 + 0.0078251∗0.1 exp(6.00371∗3 ) 1,E

(A.5)

Acknowledgments

Appendix A.

A.1.2. Eaton and Akgerman [17] These authors proposed an alternative expression for the correction factor of the HS tracer diffusivity, based on the MD simulations of Easteal and Woolf [18] at different m2 /m1 , ␴2 /␴1 , and V1 /V0,1 values: m1 /m2 in the range 0.6–10,  1 / 2 in the range 1.0–2.0, and V1 /V0,1 in the range 1.5–2.0. Their model is:

A.1. Expressions for the HS correction factor, F12

F12 = a

This appendix summarizes the hard sphere correction factors adopted for comparison with the new model proposed in this work (Eq. (23)), namely: Sung and Stell [16], Eaton and Akgerman [17], Easteal and Woolf [18], and Sun and Chen [19]. The Sung and Stell [16] expression was derived from first principles at solvent–solute microscale level, and can be considered an improvement of the Enskog model at low densities and simultaneously fulfils the correct hydrodynamic limit enclosed in the Stokes–Einstein law at high density limit. It is an analytic expression requiring viscosity. The Sung and Stell equation has theoretical fundament, but does not satisfy the restriction imposed by Eq. (12). The development of the expression of Eaton and Akgerman [17] is based on the smooth–hard sphere model theory and molecular dynamics data from Easteal and Woolf [18]. These authors also published an empirical expression for F12 [18] (Eq. (A.10)), whose dependence on the size and mass ratios is logarithmic as in our case. The correlation by Sun and Chen [19] is given by Eq. (A.12), which behaves well at higher reduced number densities of solvent, but fails the low density restriction imposed by Eq. (10).

where:

Ana L. Magalhães wishes to thank PhD grant provided by Fundac¸ão para a Ciência e a Tecnologia (Portugal) (SFRH/BD/46776/2008).

A.1.1. Sung and Stell [16] Sung and Stell [16] derived analytically an expression which embodies the correct hydrodynamic limit (Stokes–Einstein behaviour) and the Enskog theory:

 F12 =

1 + B (1 /2 ) 1 + (1 /2 )

2



where and B



1 1 − 3 ϕ1

= 1 + 4ϕ1



B ≡ lim

 3

1 + (1 /2 ) D12,H g(12 ) (A.1) + 1 + 4ϕ1 g(11 ) 1 + (1 /2 ) D12,E

1 →0

1 − 3 ϕ1 +

1 − ϕ1 /2 (1 − ϕ1 )3

 1 − 3 ϕ1 +



V1 V0,1

3 ϕ1 1 + 1 /2



(A.2)





− b12 g(12 )

1 1 − 2 3

b12 =

m

1

(A.6)

(A.7)

0.03587 

0.6001 + 0.8491

m2

  1

2



−0.244

1

2 

2

(A.8)

a is a function of the molecular size ratio  2 / 1 , and g( 12 ) can be calculated √ according to Mansoori et al. [11] (see Eq. (5)), and V0,1 /V1 = 1∗ / 2. Eq. (A.7) was previously determined by Akgerman et al. [75]. Using the same MD simulations, Liu and Ruckenstein [5] obtained:

  2.0674

a = 1.689

2

(A.9)

1

A.1.3. Easteal and Woolf [18] Based on MD simulations for tracer diffusion coefficients of HS systems, Easteal and Woolf proposed an empirical model for the molecular-weight ratio (m2 /m1 ) dependence of F12 /F11 involving three coefficients (a0 , a1 and a2 ). Later, Salim and Trebble [136] published the coefficients of the model in terms of 1∗ and  2 / 1 : ln

F

12

F11



× g −1 (12 )

= a0 + a1 ln

a0 = −1.025641∗4 ln

 

m  2

m1

+ a2 ln2

m  2

m1

(A.10)

2

1 a1 = −0.24107 + 1.27589h12 − 1.35439h212 + 0.62393h312 a2 = −0.056



are given by:

2 3 ϕ1 1 + 1 /2

˛=

V0,1 V1

(A.11)



where: h12 = 1∗2 2 /1 , and g( 12 ) may be calculated according to Mansoori et al. [11] (see Eq. (5)). A.1.4. Sun and Chen [19] The F12 expression of Sun and Chen has been derived on the basis of the MD simulations of Herman and Alder [22] and Alder et al. [12], which cover the following ranges: 0.5 <  2 / 1 < 1.6,

920

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

0.5 < m2 /m1 < 4.0, and 1.5 < V1 /V0,1 < 3.0. Once more, g( 12 ) may be calculated by Eq. (5). F12 = 1.0514



2 12

1 2



2 1 + m1 /m2

× 1 − 0.97791∗

1/2 g (12 )

 m 0.167 1

m2

 m 0.0165   0.129  1

2

m2

1

(A.12)

B.1.5. He–Yu–Su correlation [133] A × 10−7 (V1k − B)T D12 (cm2 /s) = M2 k = 1, r,1 ≥ 1.2 (r,1 − 1.2) k =1+ , r,1 < 1.2 M1

(B.6)

The authors found that parameters A and B are only approximately dependent on the properties of the solvent:



Appendix B.

A = 0.29263 + 1.6736 exp

B.1. Tracer diffusion models for real systems

B = 0.077Tc,1 ,



−0.75832

M1 Vc,1



Pc,1

r,1 ≥ 0.21

,

r,1 ≥ 0.21 (B.7)

This appendix contains the set of D12 equations used for comparison in this work. Here we include the classical predictive hydrodynamic models of Wilke and Chang [1,3,129], Tyn–Calus [1,130], Scheibel [1,131], and Reddy and Doraiswamy [1,132], whose parameters have been originally correlated and may be taken as universal values for all systems. The predictive He–Yu–Su correlation [133] is specific for supercritical systems and once more their parameters are universal. The 2-parameter equation of Dymond [2,9,134] is also studied since it may be taken as the benchmark model with who all molecular models should be compared. Finally, the predictive model of Zhu et al. [6] developed for gas, liquid and supercritical systems is also included, and may be taken as the most similar theoretical approach to our model.

B.1.6. Model of Zhu et al. [6]

 D12

3 = √ 8 

2 12,LJ ε12,LJ

×

∗ T12



 −

∗ 12



1−



∗ 12 ∗0.165377 1.029079T12

⎞⎤





⎢1 + ∗0.126978 ⎜ ⎣ ⎝ 12

× exp



∗ 12

m1





∗ 0.596103 12 −1









∗ 0.539292 12 − 1 + T12

0.400152−0.41054∗



⎟⎥

+ 0.68856⎠⎦

12

∗ 2T12

(B.8) B.1.1. Wilke–Chang equation [1,3,129] T

M1 D12 (cm2 /s) = 7.4 × 10−8 0.6 1 Vbp,2

(B.1)

where is a dimensionless association factor of the solvent, (for CO2 , = 1), 1 is the solvent viscosity (cP); M1 is solvent molecular weight (g/mol); Vbp,2 is solute molar volume at its normal boiling point (cm3 /mol). B.1.2. Tyn–Calus equation [1,130]



D12 (cm2 /s) = 8.93 × 10−8

∗ T12 =

T , ε12,LJ /kB

1/6

Vbp,2

P  T 1

2 Vbp,1

P2

1

0.267 Vbp,1 T V 0.433 1

∗ 3 12 = 1 12,LJ

(B.9)

The combining rules adopted to determine binary parameters are: ε12,LJ = kB 12,LJ



ε2,LJ ε1,LJ × kB kB  + 2,LJ d ) 1,LJ = (1 − k12 ; 2

where

d = 0.7926 k12

(B.2)

P identifies parachors, which are related to the liquid surface tension and may be estimated by additive group contributions. For most organic solvents, the approximation may be used: D12 (cm2 /s) = 8.93 × 10−8

∗ is calculated as before, but distinct reduced density is Here, T12 introduced, as  12,LJ is implied instead of  2,LJ :

2,LJ − 1,LJ 1,LJ + 2,LJ (B.10)

The LJ parameters for the solvent and solute are estimated by distinct expressions: ε1,LJ Tc,1 (K) = ∗ [1 + 0.47527332r,1 Tc,1 kB + (0.06300484 + 0.12374707r,1 )Tr,1 ]

(B.3)

(B.11)

bp,2

 B.1.3. Scheibel equation [1,131]  D12 (cm2 /s) =

−8

8.2 × 10

1/3

1 Vbp,2

T

1+



3Vbp,1

1,LJ (cm) =

2/3 

Vbp,2

B.1.4. Reddy–Doraiswamy equation [1,132] T M1 2 D12 (cm /s) = ˇ ×  1/3 1 Vbp,1 Vbp,2 Vbp,1 ≤ 1.5 ⇒ ˇ = 10 × 10−8 Vbp,2 Vbp,1 > 1.5 ⇒ ˇ = 8.5 × 10−8 Vbp,2

(B.4)

∗ c,1

c,1

[1 − 0.0368868r,1

+ (0.00006945 + 0.01089228r,1 )Tr,1 ] ε2,LJ Tc,2 = 1.313 kB



2,LJ = (B.5)

1/3

3

0.13ε2,LJ Pc,2

(B.12)

(B.13)

(B.14)

Such equations are based on the principle of corresponding states, and on the critical point computed by Johnson et al. [137] for the LJ fluid (Pc∗ = 0.13, Tc∗ = 1.313, c∗ = 0.31).

A.L. Magalhães et al. / J. of Supercritical Fluids 55 (2011) 898–923

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