J. of Supercritical Fluids 82 (2013) 151–157
Contents lists available at ScienceDirect
The Journal of Supercritical Fluids
journal homepage: www.elsevier.com/locate/supflu
Melting temperature depression caused by high pressure gases. Effect
of the gas on organic substances and on ionic liquids
José O. Valderrama a,b,∗ , Pedro F. Arce c
a
b
c
University of La Serena, Faculty of Engineering, Department of Mechanical Engineering, P.O. Box 554, La Serena, Chile
Center for Technological Information (CIT), c/Monseñor Subercaseaux 667, La Serena, Chile
University of São Paulo, Engineering School of Lorena, Campus I, P.O. Box 116, Lorena, SP, Brazil
a r t i c l e
i n f o
Article history:
Received 15 April 2013
Received in revised form 13 July 2013
Accepted 16 July 2013
Keywords:
Ionic liquids
Melting temperature depression
Equations of state
Supercritical gases
a b s t r a c t
The effect on the melting temperature depression (MTD) of organic substances and ionic liquids caused by
different types of pressurizing gases is analyzed. A high pressure gas produces a combined effect between
solubility and pressure that causes the melting temperature to decrease. The authors have previously
used phase equilibrium relations to correlate MTD of organic substances and ionic liquids under high
pressure carbon dioxide, but other gases were not considered. The Peng–Robinson equation of state with
the Wong–Sandler mixing rules showed to be appropriate for correlating the phase equilibrium in these
high pressurized systems and is the model used is in this work. Three organic substances (naphthalene,
biphenyl and octacosane) under high pressure produced by three gases (ethane, ethylene and carbon
dioxide) for which experimental data on MTD are available were considered in this study. Then extension
to an ionic liquid under high pressure carbon dioxide and high pressure ethylene was done. The proposed
thermodynamic method and the model used show to have the necessary flexibility to acceptably correlate
the MTD produced in these systems.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
Pure crystalline solids have a characteristic melting temperature, which is expressed as the temperature at which the solid
melts to become a liquid. Pure crystalline substances have a clear
and sharply defined melting temperature. During the phase change
process, all energy added to the substance is consumed as heat of
fusion, and the temperature remains constant, at a given pressure.
The experimental determination of the melting temperature is relatively simple and is used in many areas of chemistry to obtain a
first impression of the purity of a substance. This is because even
small quantities of impurities change the melting temperature, or
at least clearly enlarge its melting range [1]. Extensive collections of
tables and handbooks give the exact values of melting temperature
of many pure inorganic and organic substances [1,2].
The melting temperature Tm is a fundamental physicochemical
property of a molecule that is controlled by both single-molecule
properties and intermolecular interactions due to packing in the
solid state [3]. It finds applications in chemical identification,
∗ Corresponding author at: University of La Serena, Faculty of Engineering, Department of Mechanical Engineering, P.O. Box 554, La Serena, Chile. Tel.: +56 51 551158;
fax: +56 51 551158.
E-mail addresses: [email protected], [email protected] (J.O. Valderrama).
0896-8446/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.supflu.2013.07.007
purification, and calculations of a number of other physicochemical properties such as vapor pressure and aqueous solubility [4].
However because of the complex influence of energy and entropy
factors, melting temperatures are difficult to predict.
For a pure substance, if the pressure increases the melting temperature usually increases (except for water, for instance, which has
a different behavior at certain ranges of temperature). However, if
a liquid substance is in contact with a high pressure soluble gas, the
solid–liquid–gas equilibrium is established at a temperature below
the melting temperature of the pure liquid. This phenomenon is not
new and in fact may be produced by several factors such as mixture
with another component, by size reduction or by exerting pressure
with a soluble gas [5]. In the latter case, when the pressure of the
gas increases, it could also happen that the melting temperature
decreases, goes through a minimum and then increases again.
The phenomenon of MTD has been widely studied for organic
compounds such as aromatics, polymers and lipids. Hammam and
Sivik [6] tested a series of glycerides that showed a melting point
depression of 15–25 ◦ C in the presence of high pressure carbon
dioxide. Similar behavior of solid lipids was noticed by Sampaio
de Sousa and co-workers [7] who found Tm decrease up to 13 ◦ C.
Fukné-Kokot et al. [8] determined MTD of several organic solids
under high pressure CO2 while Fujiwara et al. [9] observed a
remarkable lowering of Tm for semicrystalline of poly(lactide) in
supercritical CO2 . The gas-induced MTD has been also observed
152
J.O. Valderrama, P.F. Arce / J. of Supercritical Fluids 82 (2013) 151–157
Notation
Symbols
ai , bi
am , bm
A12 , A21
f
F
GE
kij
N
ni
P
Pc
R
T
Tm
Tc
Tr
x
y
Z
constants for component “i” in the PR EoS
mixture constants in PR EoS
van Laar parameters
fugacity
objective function Eq. (7)
excess Gibbs free energy
binary interaction parameter
number of experimental data points
number of moles of component i
melting pressure
critical pressure
gas universal constant
melting temperature
normal melting temperature
critical temperature
reduced temperature
mole fraction in liquid phase
mole fraction in gas phase
compressibility factor
Abbreviations
EoS
equation of state
ionic liquid, ionic liquids
IL, ILs
MTD
melting temperature depression
PR/WS/VL Peng–Robinson/Wong–Sandler/Van Laar
WS
Wong–Sandler
Sm
entropy of fusion at Tm
%Tm
% deviation between correlated and experimental
melting temperature
|%Tm | absolute % deviation between correlated and experimental melting temperature
Greek letters
fugacity coefficient
ω
acentric factor
mole fraction (any phase)
Superindex
gas, liquid, solid
G,L,S
SCL
sub-cooled liquid
in ionic liquids (ILs) but has received special attention in the last
few years only, especially after discovering MTD higher than 100 ◦ C
for some ILs. The MTD of ILs would allow the use of ILs that has
relatively high melting temperatures (say around 70–130 ◦ C) as solvents at room temperatures. A little before the 1990s two reports
about MTD in the presence of carbon dioxide in ionic salts could
be found [10,11]. Years later, Kazarian et al. [12] observed liquidcrystal transition for [C16 mim][PF6 ] with carbon dioxide These
authors found that high pressure carbon dioxide induced melting
point depression in the range of what it was found for other organic
solids, that is not higher than 25 ◦ C. More recently, Scurto and Leitner [13] reported that high pressure carbon dioxide can induce
surprisingly high melting temperature depression, up to 120 ◦ C.
Scurto et al. [14] described the phenomenon of MTD and
presented a general pressure-temperature diagram of a highly
asymmetric system for an organic compound and a compressed
gas. The phenomenon is similar to that of organic compounds under
high pressure gases but that, according to Scurto and Leitner [13]
did not received much attention until ILs were discovered to have
exceptional characteristics as reaction media for many reactions.
The melting temperature depression caused by a high pressure gas
that dissolves in the liquid is a complex physicochemical process
in which van der Waals forces and electrostatic interaction forces
compete in some way. The impact of the two forces plays different
roles for different kinds of substances and especially for ILs. From a
thermodynamic point of view, which is of interest in this paper,
the phenomenon is a phase equilibrium situation in which the
three phases are present with two substances involved [14]. Therefore equilibrium equations can be formulated at the new melting
temperature in which the pure solid will form.
In this paper, the effects of three pressurizing gases (ethane,
ethylene and carbon dioxide) on the melting temperature depression (MTD) of three substances of organic type (naphthalene,
biphenyl and octacosane) and one ionic liquid ([TBAm][BF4 ]), hereafter named as “substances”, are analyzed. The thermodynamic
model used is described in what follows.
2. The thermodynamic model
For calculating the MTD, experimental data (melting temperature vs. pressure) were modeled using the fundamental equation
of phase equilibrium for the different phases present. That is the
equality of fugacities of each component in the different phases.
For the variables given, three equilibrium equations are needed:
gas–liquid equilibrium for both components (the gas and the substance) and solid–gas equilibrium for the substance (hydrocarbons
or ionic liquid). It is assumed that no gas is dissolved in the solid
phase of the substance. The concentration of the substance in the
gas phase (y2 ) was maintained as a variable to be calculated, so
the solid–gas equilibrium which includes the melting temperature
could be applied and the system of equations solved together with
the other equilibrium relations.
In the equilibrium relations, expressed through the equality of
fugacities, an equation of state was used to calculate the required
fugacities. The Peng–Robinson (PR) equation of state (EoS) was used
[15] with the Wong–Sandler (WS) mixing rules [16] The van Laar
model (VL) [17] was incorporated into the mixing rules of the WS
model. The model is designated hereafter as PR/WS/VL and is summarized in Table 1. In the first column the PR equation and the WS
mixing rules are shown while in the second column the equations
that describe the fugacity coefficient are presented.
Therefore, the following three phase equilibrium equations are
written:
f2G = f2S
(1)
f1G = f1L
(2)
f2G = f2L
(3)
In these equations f represents the fugacity, subscript 1 represents the pressurizing gas and subscript 2 represents the substance.
The superscripts S, L and G represent the solid, liquid and gas,
respectively. The fugacities are expressed in terms of the fugacity
coefficients as follows:
a) for the solid–gas equilibrium of component 2 (the substance to
de solidified) the fugacity of the solid can be written to a good
approximation as follows [17]:
y2 2G P = f2SCL exp
S m
R
1−
Tm
T
(4)
b) for the gas–liquid equilibrium the following equation can be
written for the high pressure gas (designated as component 1),
Eq. (2) is:
y1 1G = x1 1L
(5)
J.O. Valderrama, P.F. Arce / J. of Supercritical Fluids 82 (2013) 151–157
153
Table 1
The PR/WS/VL model and equations for the fugacity coefficient.
Peng–Robinson EoS and Wong–Sandler MR
Fugacity coefficient for the model PR/WS
RT
a
P=
−
V −b
V (V + b) + b(V − b)
ln 2 =
a = 0.4572 (R2Tc2/Pc) ˛(Tr )
∂nb
1
=
1−D
∂n2
b = 0.0778 (RTc/Pc)
1
n
a(Tr )
0.5
= [1 + F(1 − Tr0.5 )]
Q =
∂nb/∂n2
a
(Z − 1) − ln (Z − B) + √
b
2 2bRT
∂n2 a
∂n2
bm =
˙˙i j (b − a/RT )ij
1 − ˙i ai /bi RT −
am = bm
i
ai
bi
(GE /RT )/˝
GE /RT
+
˝
(A12 )1 2
GE
=
RT
1 (A12 /A21 ) + 2
1 ∂n2 a
n ∂n2
= RTD
2
F = 0.3746 + 1.5423ω − 0.2699ω2
−
∂nb
∂n2
x2 xj b −
∂n2 Q
∂n2
a RT
j
(1 − D)
2
+ RTb
1−
∂nD
∂n2
∂nD
∂n2
∂nb/∂n2
1/n(∂n2 a/∂n2 )
−
a
b
ln
√
2)B
√
Z + (1 + 2)B
Z + (1 −
2j
GE
a2
+ x2
D=
b2 RT
˝RT
1
n
Q
=2
xj b −
a RT 2j
j
ln (2 )
a2
∂nD
=
+
b2 RT
∂n2
˝
ln 2 = A21
A12 x2
A12 x1 + A21 x2
2
˝ = −0.62323 and the van Laar model is used for GE
and for component 2 (the substance to be solidified), Eq. (3) is:
y2 2G
=
x2 2L
(6)
In these equations P is the system pressure, iG and iL are
the fugacity coefficients of component “i” in the gas (G) and liquid (L) phases. f2SCL is the fugacity of the subcooled liquid at the
melting temperature, Sm is the entropy of fusion of the substance, Tm is the normal melting temperature of the pure substance
(in absence of the high pressure gas). The fugacity coefficients,
required for the phase equilibrium calculations (Eq. (4) to Eq. (6)),
are determined from exact thermodynamic relationships using the
PR/WS/VL model [18]. The model equations are shown in Table 1.
In the equations shown in Table 1 a common simplification is
introduced into the original WS mixing rules. In the original model
the excess Helmholtz free energy at infinite pressure AE∞ is included
in the mixing rules. However the model can be simplified by the
following approximation: AE∞ ≈ AEo ≈ GE . For the excess Gibbs free
energy GE any of the liquid phase models available in the literature and formulated for low pressure mixtures could be used, for
instance the van Laar model. Also in the table, the symbol represents the mole fraction ( = x for the liquid phase and = y for the
gas phase).
As seen in Table 1, the model includes three adjustable parameters: an interaction binary parameter k12 and the van Laar
parameters A12 and A21 . If the entropy of fusion Sm that appears
in Eq. (4) is unknown, this property is also considered as an additional adjustable parameter. To estimate the parameters k12 , A12 ,
A21 and Sm of the model, the Eq. (4) to Eq. (6) were simultaneously solved using experimental data of melting temperature
at different pressures for each of the solid–liquid–gas systems.
The optimization routine to find the model parameters uses the
Levenberg–Marquardt algorithm as described by Reilly [19] using
the temperature deviation as the objective function:
T calc − T exp F=
T exp (7)
i
The Levenberg–Marquardt algorithm is used to adjust the
parameters of the model to fit the experimental data by minimizing the sum of squares of residuals. Of the several local minima
that the method finds, the optimum solutions are those that make
minimum the sum of the absolute deviation given by Eq. (7)
A previous study by the authors [20] demonstrated the
capability of the PR/WS/VL model to correlate with acceptable
accuracy the qualitative and quantitative behavior of the melting
temperature curve as the gas-pressure increases. That study however did not consider the effect of different pressurizing gases as
done in this work.
3. Data used
Three organic substances (naphthalene, biphenyl and octacosane) under high pressure carbon dioxide, ethylene and ethane,
for which experimental data on MTD are available in the literature were considered for the study. Then extension to an ionic
liquid system was done by studying the systems [TBAm][BF4 ] under
high pressure carbon dioxide and ethylene. To the best of the
author’s knowledge this is the only ionic liquid for which experimental MTD data are available at several pressures and for cases
with at least two pressurizing gases. The Table 2 describes the
ranges of pressure and temperature of the data used in this work,
including the references from where the data were obtained. The
necessary properties of the organic substances and of the gases
were obtained from the DIPPR database [24] and from Valderrama
and Robles [25] for the ionic liquid and the values are shown in
Table 3.
The information required for correlating the MTD was organized
in a data file named mtd.dot that is provided as supplementary
material (see Table 4). Eleven sets of data are included in the
file (three organic substances with the three gases and the ionic
liquid with two gases). For the systems naphthalene + CO2 and
biphenyl + CO2 , data on the solubility of the gas in the liquid phase
are provided and listed as input data. For all the other systems such
data are not available in the literature where the MTD is given. However, this information is needed in the iteration procedure as the
starting point for the calculations. When the solubility is not given,
this value must be estimated in some way and included in the input
data as seen in the mtd.dot file. In this paper the ideal solubility
“x” is determined and used as the starting point for the correlation procedure. This ideal solubility can be calculated following the
expression reported by McHugh et al. [26]:
ln (x) = −
Hm
R
1
T
−
1
Tm
+
Pv
RT
(8)
154
J.O. Valderrama, P.F. Arce / J. of Supercritical Fluids 82 (2013) 151–157
Table 2
Details on the experimental data for the systems considered in this study. In the table the temperature values have been rounded to the closest integer.
Systems
N
T (K)
CO2 (1) + naphthalene
CO2 (1) + biphenyl
CO2 (1) + octacosane
C2 H4 + naphthalene
C2 H4 + biphenyl
C2 H4 + octacosane
C2 H6 + naphthalene
C2 H6 + biphenyl
C2 H6 + octacosane
CO2 + [TBAm][BF4 ]
C2 H6 + [TBAm][BF4 ]
13
18
12
15
16
8
16
13
6
4
11
333–346
326–336
325–334
334–350
314–339
316–327
329–346
321–337
312–337
346–382
300–373
P (MPa)
x (1)
3.1–24.3
2.9–34.0
1.4–33.6
1.1–7.6
1.1–20.5
3.3–20.5
2.4–12.4
1.2–19.3
0.1–8.4
10.0–25.0
7.8–33.5
Ref.
0.13–0.71
0.12–0.68
–
–
–
–
–
–
–
–
–
[10]
[10]
[5,21]
[22]
[5,22]
[5,22]
[23]
[5]
[5]
[14]
[14]
Table 3
Properties for all substances involved in this study.
Components
MW
Tc (K)
Pc (MPa)
ω
Tm (K)
Psat (MPa)
Vsat (L/mole)
Carbon dioxide
Ethane
Ethylene
Naphthalene
Biphenyl
Octacosane
[TBAm][BF4]
44.01
30.07
28.05
128.17
124.21
394.77
329.27
304.2
305.32
282.34
748.0
773.0
832.0
676.0
7.38
4.87
5.04
4.05
3.38
0.85
1.17
0.2236
0.0995
0.0863
0.3020
0.4029
1.2380
1.1030
–
–
–
353.43
342.20
334.35
429.15
–
–
–
9.90 × 10−4
1.90 × 10−4
1.15 × 10−10
1.65 × 10−3
–
–
–
0.1310
0.1556
0.5076
0.3294
In this equation, T is the melting temperature under the pressurizing gas, Tm is the melting temperature of the pure substance
without any pressurizing gas, Hm is the normal heat of melting,
P is the total pressure, R is the ideal gas constant and v is the difference between the volume of the pure liquid and the pure solid
of the organic substances or the ionic liquid.
user chooses the most appropriate one, usually based on physical concepts. For instance, a solution with a value of Sm far away
from experimental values or a solution with a very high value (negative or positive) of k12 , although can provide low average absolute
deviation of Tm cannot be considered optimum physical solutions
for the problem.
4. Data input for the correlating program
5. Results and discussion
As said, the data needed for correlating the MTD were organized
in an mtd.dot data file. The mtd.dot file also includes the critical
properties of the substance and of the gas. Additionally, the model
constants for the saturation pressure and for the liquid density must
be given. The Table 4 clarifies the form in which these constants
must be included. The equations for the saturation pressure and
for the liquid density of the substance are those recommended in
the DIPPR database [24]:
The Table 5 presents the optimum parameters and the deviations for the temperature and for x1 , for those cases in which x1
is available (naphthalene and biphenyl). The Table 5 also shows
three statistical parameters that according to the authors are the
most representative of the accuracy of the method, as previously
discussed in the literature [27].
The differences in phase behavior of the systems that are related
to the physical–chemical properties of the solvents and to the
e
D
(mol/L) = A/B(1+(1−T/C)
)
(9)
30
(10)
If the constants in the above equations are not available, an
approximate constant value for these properties must be provided,
in the following form: (i) for the saturation pressure: a = Ln [Psat ]
(at the normal melting temperature) and b = c = d = 0; and (ii) for the
density: A = liquid density of the substance at a temperature close
to Tm and B = C = D = 1.
After the experimental data are listed, the searching range for
all unknown parameters (k12 , A12 , A21 and Sm ) must be given.
These parameters must be carefully chosen to avoid divergence.
Additionally, some dumping coefficients for the variables x, y and
T involved in the calculations must be provided to guarantee fast
convergence. It is recommended not to modify these parameters
unless it is necessary, when other more complex systems are analyzed and convergence is not achieved. For the cases treated in this
work, the values included are adequate enough.
The program also includes two reporting files: (i) the mtd.tot file,
that includes all results that succeeded the convergence tests (local
optima); and (ii) the mtd.opt file, that includes the best 20 solutions,
meaning those solutions that give the lowest average absolute deviations in the correlated Tm . From all optimum local solutions the
25
Naphthalene - Ethane (exp)
Naphthalene - Ethane (this work)
20
Naphthalene - Ethylene (exp)
Pressure (MPa)
P sat (Pa) = ea+b/T +c∗log T +dT
Naphthalene - Ethylene (this work)
Naphthalene - CO2 (exp)
15
Naphthalene - CO2 (this work)
10
5
0
325
330
335
340
345
350
355
Melting temperature (K)
Fig. 1. Melting temperature of naphthalene as a function of pressure exerted by the
three gases considered in the study. In the figure, the symbols are: (, ) ethylene,
(, 䊉) carbon dioxide; (, ) ethane.
J.O. Valderrama, P.F. Arce / J. of Supercritical Fluids 82 (2013) 151–157
155
Table 4
Structure of the data file required for correlating the melting temperature depression. For space reasons only five experimental data for each mixture are shown.
NAME
M
Tc (K) Pc(atm)
w
Zc
SAT. PRESSURE (Pa)
SAT. DENSITY (mole/L)
a
b
c
d
e
A
B
C
D
------------------------------------------------------- ----------------------------------- ------------------------------------------- ---------------------------------------'CARBON DIOXIDE ' 44.0 304.2 72.87 0.2236 0.2746
'NAPHTHALENE ' 128.2 748.4 39.97 0.3020 0.2650 62.96 -8137.5 -5.632 2.27E-18 6.0 0.635 0.258 748.4 0.277
------------------------------------------------------- ----------------------------------- ------------------------------------------- ----------------------------------------EXPERIMENTAL SLG DATA
-------------------------------------------13
! number of experimental data points
-------------------------------------------T (K)
X1 P (atm)
---------------------------------346.44
! EXPERIMENTAL DATA
0.13 30.30
0.185 46.78
343.87
341.27
0.246 60.89
337.71
0.332 82.90
0.37 92.97
335.83
-------------------------------------------353.434
! melting temperature of the pure solid, Tm (K)
4
! number of parameter to optimize
! lower limit, upper limit, increment of parameter 1 (kij)
-0.1 0.1 0.02
! lower limit, upper limit, increment of parameter 2 (A12)
0.0 10.0 1.0
0.0 10.0 1.0
! lower limit, upper limit, increment of parameter 3 (A21)
! lower limit, upper limit, increment of parameter 4 (dSm)
0.0 20.0 1.0
! dumping coefficient for x
0.01
0.01
! dumping coefficient for y
0.0001
! dumping coefficient for T
----------------------------------------------------------- --------------------------------------------------------------------------------- ---------------------------------REFERENCE FOR EXPERIMENTAL DATA: Cheong P.L., et al. Fluid Phase Equilib. 29 (1986) 555-562
----------------------------------------------------------- --------------------------------------------------------------------------------- ---------------------------------*******************************************************************************************************************************************************
----------------------------------------------------------- ----------------------------------------------------------------------- -------- ---------------------------------'ETHYLENE ' 28.05 282.3 49.75 0.0862 0.2810
'[TBAm][BF4] ' 329.27 676.0 11.58 1.1030 0.2368
7.408 0.0
0.0
0.0
0.0
3.036
1.0 1.0
1.0
----------------------------------------------------------- -------------------- -------------------------------------------------- --------- ---------------------------------EXPERIMENTAL SLG DATA
--------------------------------------------! number of experimental data points
8
--------------------------------------------X1
P (atm)
T (K)
---------------------------------0.6000 246.73 346.15
! EXPERIMENTAL DATA
0.5375 222.06 350.15
0.4750 197.38 354.25
0.4125 172.71 358.15
0.3500 148.04 362.95
----------------------------------------------------------- --------------------------------------------------------------------------------- ---------------------------------429.15
! melting temperature of the pure solid, Tm (K)
4
! number of parameter to optimize
! lower limit, upper limit, increment of parameter 1 (kij)
-0.1 0.1 0.01
0.0 10.0 1.0
! lower limit, upper limit, increment of parameter 2 (A12)
! lower limit, upper limit, increment of parameter 3 (A21)
0.0 10.0 1.0
! lower limit, upper limit, increment of parameter 4 (dSm)
0.0 20.0 1.0
0.01
! dumping coefficient for x
! dumping coefficient for y
0.01
0.0001
! dumping coefficient for T
----------------------------------------------------------- --------------------------------------------------------------------------------- ---------------------------------REFERENCE FOR EXPERIMENTAL DATA: Scurto A.M., et al, Ind. Eng. Chem. Res. 47, 3 (2008) 493-501
----------------------------------------------------------- --------------------------------------------------------------------------------- ----------------------------------
intermolecular interactions between the gas and the substance
being pressurized are well represented by the equation of state
model employed. The different parameters involved in the mixing
rules (k12 , A12 , A21 ) and in the equilibrium equations (Sm ) seem to
compensate the complex three phase phenomenon and to represent the phase behavior and the MTD in an appropriate and accurate
way.
The Figs. 1–3 compare the MTD of the three organic substances
caused by the three gases considered (carbon dioxide, ethylene and
ethane). According to the experimental data, higher pressures are
needed to produce similar MTD for any of the organic substances
using CO2 and lower pressures using ethane. In the case of the ionic
liquid, shown in Fig. 4, the situation is different but the model again
is able to explain the behavior as shown by the results.
Although this fact seems to be contradictory from an experimental point of view, it has been explained by Scurto et al. [14]
mainly based on concepts of solution non-ideality. According to
the authors the phenomenon is most probably due to specific
156
J.O. Valderrama, P.F. Arce / J. of Supercritical Fluids 82 (2013) 151–157
Table 5
Optimum parameters for all systems considered in this study.
Systems
Global optimum parameters
Temperature deviations
Mole fract. deviations
% x|
|% x|
|% x|max
CO2 + naphthalene
CO2 + biphenyl
CO2 + octacosane
0.0032
0.0745
0.0984
0.82
7.53
0.98
7.48
17.5
6.93
10.14
9.82
44.66
0.39
0.02
0.12
0.39
0.59
0.12
0.96
3.57
0.16
4.58
5.39
–
6.99
5.39
–
16.03
8.96
–
C2 H4 + naphthalene
C2 H4 + biphenyl
C2 H4 + octacosane
0.0580
0.0895
0.0678
9.96
10.02
5.33
1.88
14.01
7.85
12.68
13.07
44.58
0.07
-0.16
0.03
0.07
0.37
0.03
0.98
1.44
0.18
–
–
–
–
–
–
–
–
–
C2 H6 + naphthalene
C2 H6 + biphenyl
C2 H6 + octacosane
0.0465
0.0638
0.0521
5.23
2.17
4.27
3.18
7.82
5.74
12.49
12.19
42.81
0.29
0.16
0.16
0.29
0.16
0.16
0.50
0.62
0.87
–
–
–
–
–
–
–
–
–
CO2 + [TBAm][BF4 ]
C2 H6 + [TBAm][BF4 ]
0.0615
0.0413
5.13
3.03
5.94
2.12
5.62
5.43
-0.01
0.12
0.72
0.66
2.38
1.37
–
–
–
–
–
–
k12
A12
A21
Sm
% T
|% T|max
activity coefficient in different forms, far from ideal behavior. The
excess Gibbs free energy is included in the Wong–Sandler mixing
rules [16] for the parameters am and bm (as shown on Table 1):
40
35
|% T|
Byphenyl - Ethane (exp)
Byphenyl - Ethane (this work)
Pressure (MPa)
30
bm =
Byphenyl - Ethylene (exp)
Byphenyl - Ethylene (this work)
25
˙˙i j (b − a/RT )ij
Byphenyl - CO2 (exp)
Byphenyl - CO2 (this work)
am = bm
20
15
(11)
1 − ˙i ai /bi RT − (GE /RT )/˝
ai
i
bi
+
GE /RT
˝
(12)
For determining the excess Gibbs free energy GE , the van Laar
equation that contains two parameters (A12 and A21 ) is considered:
10
GE
(A12 )1 2
=
RT
1 (A12 /A21 ) + 2
5
0
310
315
320
325
330
335
340
345
Melting temperature (K)
Fig. 2. Melting temperature of biphenyl as a function of pressure exerted by the
three gases considered in the study. In the figure, the symbols are: (, ) ethylene,
(, 䊉) carbon dioxide;(, ) ethane.
interactions that depend on both the characteristics of the gases
and of the substances. These different types of interactions could
give important deviations from Raoult’s law, the excess Gibbs free
energy will be different from zero and therefore will affect the
(13)
In the method used in this work the phase equilibrium relations
are expressed in terms of the fugacity coefficient for the PR/WS/VL
model. The expression for the activity coefficient derived from van
Laar model also includes the parameters A12 and A21 :
ln 2 = A21
A12 x2
A12 x1 + A21 x2
2
(14)
The complex behavior described by Scurto et al. [14] in terms
of important variations of the activity coefficient must be absorbed
by the two parameters A12 and A21 included in the above equations. That is probably the reason for the very different values that
40
CO2 + [TBAm][BF4] (exp)
CO2 + [TBAm][BF4] (this work)
C2H4 + [TBAm][BF4] (exp)
30
Pressure (MPa)
30
Pressure (MPa)
40
Octacosane - Ethane (exp)
Octacosane - Ethane (this work)
Octacosane - Ethylene (exp)
Octacosane - Ethylene (this work)
Octacosane - CO2 (exp)
Octacosane - CO2 (this work)
20
10
C2H4 + [TBAm][BF4] (this work)
20
10
0
0
305
315
325
335
Melting temperature (K)
Fig. 3. Melting temperature of octacosane as a function of pressure exerted by the
three gases considered in the study. In the figure, the symbols are: (, ) ethylene,
(, 䊉) carbon dioxide; (, ) ethane.
270
310
350
390
430
470
Melting temperature (K)
Fig. 4. Melting temperature of [TBAm][BF4] as a function of pressure exerted by the
two gases considered in the study. In the figure, the symbols are: (, ) ethylene,
(, 䊉) carbon dioxide.
J.O. Valderrama, P.F. Arce / J. of Supercritical Fluids 82 (2013) 151–157
assume these parameters for the various mixtures studied as presented in Table 5. As observed, A12 varies from around 1 to 10
(0.82 for CO2 + naphthalene and 10.02 for C2 H4 + biphenyl) and A21
varies from around 2 to 18 (1.88 for C2 H4 + naphthalene and 17.5
for CO2 + biphenyl). These parameters, together with k12 and Sm ,
give acceptable values of the MTD for all cases studied. However,
deviations between experimental and calculated melting temperatures observed in the Figs. 1–4 deserved some additional
comments.
As observed in Figs. 1–3 deviations between correlated and
experimental data present a systematic negative shift for most of
the organic systems. This means that the model usually calculates
values of the melting temperature lower than the experimental
data (Figs. 1–3). This shift, however, is positive for the system
biphenyl + ethylene, meaning that the calculated melting temperature is higher than the experimental values. However, for the ionic
liquids systems [TBAm][BF4 ] + ethylene and [TBAm][BF4 ] + carbon
dioxide (Fig. 4) deviations between calculated and experimental
melting temperatures are randomly positive and negative. Most
probably this different behavior is related to the different forms of
the curves P vs. Tm which are correlated with the same thermodynamic model (PR/WS/VL).
6. Conclusions
The proposed thermodynamic method is considered to give reasonable values of the MTD providing the correct dependency with
pressure for the three organic substances (naphthalene, biphenyl
and octacosane) and for the ionic liquid [TBAm][BF4 ] pressurized
by different type of gases (ethane, ethylene and carbon dioxide).
The relative and absolute deviations between experimental data
and correlated values of melting temperature are below 0.8% and
maximum deviations are 4% (and even lower for ionic liquids). This
is considered acceptable enough for a relative simple correlating
method such as PR/WS/VL employed in this work.
Acknowledgements
The authors thank the National Council for Scientific and
Technological Research (CONICYT), through the research grant
FONDECYT 1120162 and the Center for Technological Information
of La Serena-Chile for especial support. JOV thanks the University
of La Serena-Chile and PFA thanks the Engineering School of Lorena
of the University of Sao Paulo-Brazil.
Appendix A. Supplementary data
Supplementary material related to this article can be found,
in the online version, at http://dx.doi.org/10.1016/j.supflu.
2013.07.007.
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