Fluid Phase Equilibria 171 Ž2000. 1–10
www.elsevier.nlrlocaterfluid
Surface tension of isomers of pure hydrocarbons: a method for
estimation and prediction
Ascencion
Arturo Trejo ) , Florentino Murrieta-Guevara
´ Romero-Martınez,
´
Instituto Mexicano del Petroleo,
Eje Lazaro
Cardenas
152,
´ Area de InÕestigacion
´ en Termofısica,
´
´
´
07730 Mexico City D.F., Mexico
Received 10 June 1999; accepted 20 November 1999
Abstract
A new method to estimate and predict surface tension in the full liquid-state temperature range for isomers of
pure hydrocarbons has been developed. The method requires as input parameters the surface tension value
corresponding to the linear or normal member of a given hydrocarbon homologous series, modified by an
empirical parameter here proposed for each isomer under study. This in turn is calculated using molar volume
and solubility parameter values for both the normal and the isomer hydrocarbon. This new method was used to
calculate surface tension values for 56 isomers of n-alkane and four isomers of the 1-alkene homologous series
in the range 253–373 K. The average error obtained from a comparison between experimental and calculated
surface tension values for 497 points of the 60 isomers considered was only 1.5%. The developed method may
also be used to obtain surface tension data with high reliability above and below the temperature range for
which experimental data are available for both normal and isomer hydrocarbons. q 2000 Elsevier Science B.V.
All rights reserved.
Keywords: Estimation and prediction method; Hydrocarbon isomers; Normal hydrocarbons; Surface tension
1. Introduction
Thermophysical properties and phase equilibria behaviour of pure components and mixtures have
the utmost importance for engineers for the design, operation, and optimisation of different industrial
equipment and processes, as well as for scientists for the development and testing of theories and
models. For both engineering and scientific purposes, it is highly desirable to have the needed
information over wide ranges of temperature and pressure, and preferably obtained from experiment.
)
Corresponding author. Tel.: q52-368-8373; fax: q52-368-4203.
E-mail address: [email protected] ŽA. Trejo..
0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 8 1 2 Ž 0 0 . 0 0 3 2 1 - 6
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
2
However, experimentation for measuring any thermophysical property or phase equilibria for compounds of interest is, in most cases, time-consuming and expensive. This fact has attracted the
attention of many people to develop empirical, semi-empirical, and theoretical models, to reproduce
the phenomenological behaviour or to predict it. Nonetheless, for these endeavours a great amount of
experimental information is also necessary. Unfortunately, the experimental information for a large
number of thermophysical properties is not always available at the conditions of a specific need;
surface tension is one of such properties. Surface tension of pure and mixed compounds is an
important thermophysical property for almost every separation process involving mass transfer
phenomena between vapour and liquid phases as well as for the coating and adhesives industries.
There have been many attempts to generalize the experimental surface tension behaviour for pure
substances since the pioneering works with the temperature-independent parachor and the liquid and
vapour densities by McLeod w1x and Sugden w2x, who established the possibility of estimating the
parachor from the structure of the molecule. Le Grand and Gaines w3x proposed an equation relating
surface tension to the relative molar mass for a homologous series of compounds at a given
temperature. Grigoryev et al. w4x proposed another equation relating surface tension with temperature
in the full liquid-state region for four n-alkanes. More recently, Romero-Martınez
and Trejo w5x
´
developed two equations to correlate and predict surface tension data over a large temperature range
for n-alkanes, 1-alkenes, cycloalkanes, and aromatic hydrocarbons.
As part of our interest on surface tension, this work is a continuation of our search to generalize the
experimental behaviour for pure hydrocarbons. Thus, the results included here correspond to the
estimation and prediction of surface tension values for isomers of two different homologous series of
hydrocarbons, n-alkanes and 1-alkenes, using a new method that makes use of the corresponding
surface tension values for the normal or linear hydrocarbon, which in turn is modified by an empirical
isomer parameter. The latter is derived from solubility parameter and molar volume values for the
normal hydrocarbon and the corresponding isomer under study. The method developed in this work
can be applied in the full liquid-state temperature range; that is, reliable extrapolation of data is
possible. The method has been tested using almost 500 experimental points, in the range 253–373 K,
corresponding to 60 different isomers of the n-alkanes and 1-alkenes homologous series with an
average relative error of 1.5%.
2. Estimation method
The method developed in this work to estimate and predict surface tension values, as a function of
temperature, for pure hydrocarbon isomers is based on the observation that the value of this property,
for a given isomer, is very close to that of the corresponding normal or linear hydrocarbon at the same
temperature. Hence, the surface tension value of the normal hydrocarbon is modified by an empirical
isomer surface tension ŽIST. parameter, according to the following relation:
g iso s gnor = IST
Ž1.
for which gnor is calculated using Eq. Ž3. , and the dimensionless IST parameter is calculated for each
isomer with:
d isoVnor
IST s
Ž2.
dnorViso
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
3
where d is the total solubility parameter and V is the molar volume, both at 298.15 K; the subscripts
iso and nor correspond to the isomer and the normal hydrocarbon, respectively.
In order to apply Eq. Ž1. to estimate and predict surface tension values, as a function of
temperature, for a large number of isomers of hydrocarbons in the full liquid-state temperature range,
we have derived values of the surface tension for the normal members, gnor , of the homologous series
of hydrocarbons studied in this work through the use of the two-parameter generalized surface tension
equation ŽGSTE2. developed in previous work Ž Romero-Martınez
´ and Trejo w5x.:
gnor s at
1.26
bt 1.76
Ž3.
M 2r3
In this equation, gnor is the surface tension for the normal alkane, a and b are the adjustable
parameters, t is a reverse reduced temperature scale Žt s 1 y TrTc ., with T the temperature in kelvin
and Tc the vapour–liquid critical temperature in kelvin, and M is the relative molar mass.
Eq. Ž3. was used in the correlation of 260 experimental points for C 5 to C 20 n-alkanes, 159 points
for C 5 to C 20 1-alkenes, 23 points for cycloalkanes, and 48 points for aromatic hydrocarbons
ŽRomero-Martınez
and Trejo w5x.. It was shown that the GSTE2, Eq. Ž 3. , with only two adjustable
´
parameters, is capable of reproducing within experimental error the experimental surface tension data
as a function of temperature, from the triple point up to the gas–liquid critical point for all the normal
members of the above mentioned homologous series of pure hydrocarbons Ž Romero-Martınez
and
´
Trejo w5x.. Hence, this equation is highly useful within the method developed here, both to estimate
and extrapolate surface tension data for the isomers included in the present study through the use of
Eq. Ž1..
The method developed in this work requires as input parameters reliable values for the gas–liquid
critical temperature of the normal member of a given homologous series, molar volume and the total
solubility parameter for both the normal and the isomer hydrocarbons at 298.15 K. The corresponding
values were obtained from different sources Ž Ambrose and Walton w6x; Dreisbach w7x; Hoy w8x; Reid et
al. w9x; Simmrock et al. w10x; Walas w11x..
q
3. Results and discussion
We have tested the method here proposed estimating surface tension values, as a function of
temperature, for 60 different isomers of the n-alkane and 1-alkene homologous series. Values are
given in Table 1 of the molar volume, total solubility parameter and the calculated dimensionless IST
parameter from Eq. Ž 2. for the 60 isomers considered as well as for the corresponding normal
homologues. It may be observed that the values of the IST parameter are all very close to one.
Substituting Eq. Ž3. for normal alkanes into Eq. Ž1., we obtain the estimated surface tension values
for the different isomer considered. The values of the parameters a and b in Eq. Ž 3. used for
calculating the surface tension as a function of temperature for the normal homologues hexane,
heptane, octane, and nonane are a s 51.01 mN P my1 and b s 81.3 mN P my1 Žg P moly1 . 2r3, whereas
for 1-pentene the values are a s 50.53 mN P my1 and b s 146.5 mN P my1 Ž g P moly1 . 2r3 w5x.
Table 2 shows a comparison between experimental and calculated surface tension values for four
isomers of n-hexane at several temperatures in the range 283–333 K. It can be observed that the
agreement is quite good since the absolute mean difference for the 25 points is of the same order of
4
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
Table 1
Values of the molar volume, total solubility parameter and the isomer surface tension parameter ŽEq. Ž2.. for all the different
isomers considered in this work
Isomer
V Žcm3 moly1 .
d Žcal1r2 cmy3r2 .
IST parameter
n-hexane
2-methylpentane
3-methylpentane
2,2-dimethylbutane
2,3-dimethylbutane
n-heptane
3-ethylpentane
2-methylhexane
3-methylhexane
2,2-dimethylpentane
2,3-dimethylpentane
2,4-dimethylpentane
3,3-dimethylpentane
2,2,3-trimethylbutane
n-octane
3-ethylhexane
2-methylheptane
3-methylheptane
4-methylheptane
2,3-dimethylhexane
2,4-dimethylhexane
2,5-dimethylhexane
3,3-dimethylhexane
3,4-dimethylhexane
3-ethyl-2-methylpentane
3-ethyl-3-methylpentane
2,2,4-trimethylpentane
2,3,3-trimethylpentane
2,3,4-trimethylpentane
n-nonane
3-ethylheptane
4-ethylheptane
2-methyloctane
3-methyloctane
4-methyloctane
2,2-dimethylheptane
2,3-dimethylheptane
2,4-dimethylheptane
2,5-dimethylheptane
2,6-dimethylheptane
3,3-dimethylheptane
3,4-dimethylheptane
3,5-dimethylheptane
4,4-dimethylheptane
3-ethyl-2-methylhexane
3-ethyl-3-methylhexane
131.61
132.88
130.62
133.17
131.16
147.47
144.39
148.58
146.71
149.65
145.02
149.94
145.41
146.09
163.54
161.00
164.61
162.77
163.05
161.31
164.07
165.70
161.80
159.72
159.70
157.87
166.08
158.13
159.75
179.67
177.39
176.65
180.76
178.92
179.12
181.50
177.61
180.13
180.38
181.94
177.88
176.29
178.37
177.88
176.41
174.02
7.27
7.02
7.12
6.69
6.95
7.50
7.35
7.21
7.28
6.91
7.23
6.96
7.08
6.94
7.54
7.44
7.32
7.38
7.36
7.31
7.18
7.15
7.20
7.28
7.32
7.30
6.86
7.25
7.27
7.64
7.53
7.57
7.51
7.55
7.54
7.28
7.49
7.33
7.38
7.33
7.38
7.51
7.40
7.34
7.46
7.45
1.0000
0.9564
0.9868
0.9094
0.9593
1.0000
1.0009
0.9542
0.9757
0.9079
0.9803
0.9127
0.9574
0.9341
1.0000
1.0023
0.9645
0.9834
0.9791
0.9829
0.9492
0.9359
0.9652
0.9886
0.9942
1.0029
0.8959
0.9944
0.9871
1.0000
0.9983
1.0078
0.9771
0.9924
0.9899
0.9433
0.9918
0.9570
0.9622
0.9474
0.9757
1.0018
0.9756
0.9704
0.9945
1.0068
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
5
Table 1 Ž continued .
Isomer
V Žcm3 moly1 .
d Žcal1r2 cmy3r2 .
IST parameter
4-ethyl-2-methylhexane
4-ethyl-3-methylhexane
2,2,3-trimethylhexane
2,2,4-trimethylhexane
2,2,5-trimethylhexane
2,3,3-trimethylhexane
2,3,4-trimethylhexane
2,3,5-trimethylhexane
2,4,4-trimethylhexane
3,3,4-trimethylhexane
3,3-diethylpentane
3-ethyl-2,2-dimethylpentane
3-ethyl-2,3-dimethylpentane
2,2,3,3-tetramethylpentane
1-pentene
cis-2-pentene
trans-2-pentene
2-methyl-1-butene
3-methyl-1-butene
178.37
173.78
176.80
180.18
182.38
174.73
174.40
178.65
178.11
172.98
171.00
175.44
171.00
170.32
110.39
107.83
109.06
107.84
111.86
7.33
7.54
7.31
7.11
7.09
7.37
7.44
7.25
7.19
7.46
7.54
7.30
7.48
7.44
7.07
7.31
7.28
7.17
6.70
0.9664
1.0204
0.9723
0.9280
0.9142
0.9919
1.0033
0.9544
0.9494
1.0142
1.0369
0.9785
1.0287
1.0273
1.0000
1.0585
1.0423
1.0381
0.9352
magnitude as the experimental uncertainty, which is reported to be "0.1 mN P my1 Ž Jasper w12x.. This
clearly means that the method is accurate since it is capable of reproducing the experimental data with
a relative error slightly less than 1% for the isomers considered.
The comparison of calculated and experimental values using the Macleod and Sugden and the
Brock and Bird methods w9x for 2,3-dimethylbutane gives relative percent errors of y0.9 and y0.2 at
293 K, and 0.6 and y0.5 at 313 K, which are higher than the errors obtained with the method of this
work. No other results from reported procedures to estimate surface tension values for the isomers
here studied were found to compare with.
Fig. 1 shows experimental and estimated values of surface tension for four isomers of n-hexane,
including previously calculated values for the latter in the whole liquid region w5x. In order to illustrate
the extrapolating capabilities of the reported method Fig. 1 also includes calculated surface tension
values for the isomers above and below the experimental temperature range. It is clear that the
procedure is capable of yielding reliable values independently of the position and number of
substituting methyl groups in the isomer molecule in the whole liquid-state-temperature range.
The results for the hexane isomers indicate that the dimensionless IST parameter used in Eq. Ž 1. is
capable of discerning the effect on the surface tension of the differences in structure amongst the
isomers themselves and also between that for the isomers and the structure of the normal hydrocarbon. That is, the parameter is acting as a scaling or correction factor in order to adequately modify the
surface tension value of the normal hydrocarbon to generate accurate values, both estimated and
extrapolated, for the different isomers considered. Thus, it may be observed from Table 1 that the
value of the IST parameter for each one of the four isomers is lower than one, since the surface
tension for the isomers is always lower than that for the n-hexane ŽFig. 1. in the temperature range
considered.
6
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
Table 2
Comparison between experimental and calculated surface tension values, as function of temperature, for isomers of n-hexane
Relative error Ž%. s Žcalc.yexp..rexp.4=100.
Isomer
T ŽK.
gexp mN my1
gcal mN my1
Relative error Ž%.
2-methylpentane
283.15
293.15
298.15
303.15
313.15
323.15
333.15
283.15
293.15
298.15
303.15
313.15
323.15
333.15
283.15
293.15
298.15
303.15
313.15
283.15
293.15
298.15
303.15
313.15
323.15
18.37
17.38
16.87
16.37
15.36
14.39
13.39
19.20
18.14
17.61
17.08
16.02
14.96
13.90
17.39
16.31
15.81
15.32
14.33
18.38
17.38
16.88
16.38
15.38
14.38
18.41
17.37
16.85
16.33
15.32
14.31
13.33
19.00
17.92
17.38
16.85
15.80
14.77
13.75
17.51
16.51
16.02
15.53
14.57
18.47
17.42
16.90
16.38
15.36
14.36
0.2
y0.1
y0.1
y0.2
y0.3
y0.6
y0.4
y1.0
y1.2
y1.3
y1.3
y1.4
y1.3
y1.1
0.7
1.2
1.3
1.4
1.7
0.5
0.2
0.1
0
y0.1
y0.1
3-methylpentane
2,2-dimethylbutane
2,3-dimethylbutane
Further test of the method was carried out by estimating surface tension values for eight isomers of
n-heptane, 14 isomers of n-octane, 30 isomers of n-nonane, and four isomers of 1-pentene in the
range 253–373 K. A total of 472 experimental points for these hydrocarbons were used in this stage
for the evaluation of the technique. Table 1 contains the input data of the method for the isomers
studied.
Table 3 gives the temperature interval and the number of experimental points for each one of the
different isomers considered together with the average relative error obtained from comparing the
experimental and calculated surface tension values. These results show that the method reproduces
with high accuracy the surface tension experimental data w12x for all the isomers included in this
comparison, regardless of the position and number of asymmetric carbon atoms as in the case of the
hexane isomers. It may be observed that there are only four isomers with relative error larger than
3%. We believe this may be due to either low precision of the available input data used in our method
or to experimental problems arisen from contaminants present in those samples. The average relative
error for all the isomers of heptane is 1.5%, for the isomers of octane is also 1.5%, for the isomers of
nonane is 1.4%, whereas for the four isomers of pentene is 2.4%.
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
7
Fig. 1. Surface tension as a function of temperature for hexane isomers. Points represent experimental data and lines are both
estimated and extrapolated values.
From Table 1 it may be observed that the IST parameter for each of the 56 isomers considered in
the results of Table 3 is very close to one, which once again indicates that such parameter may be
interpreted as a correcting factor of the surface tension values of the corresponding normal homologue
hydrocarbon.
Table 4 gives a summary of the global results about the reliability of the technique for the almost
500 experimental points, for 60 isomers of hydrocarbons for which the corresponding surface tension
value was estimated using the method developed in this work. This table includes the number of
isomers for each normal homologue studied, the total number of experimental points, the temperature
range for which experimental surface tension values were found, the maximum difference between
experimental and calculated surface tension value for a given isomer, the average standard deviation,
8
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
Table 3
Results of the comparison between experimental and calculated surface tension values for isomers of n-heptane, n-octane,
n-nonane, and 1-pentene
Relative error Ž%. s Žcalc.yexp..rexp.4=100.
Isomer
DT ŽK.
Number of points
Relative error Ž%.
3-ethylpentane
2-methylhexane
3-methylhexane
2,2-dimethylpentane
2,3-dimethylpentane
2,4-dimethylpentane
3,3-dimethylpentane
2,2,3-trimethylbutane
3-ethylhexane
2-methylheptane
3-methylheptane
4-methylheptane
2,3-dimethylhexane
2,4-dimethylhexane
2,5-dimethylhexane
3,3-dimethylhexane
3,4-dimethylhexane
3-ethyl-2-methylpentane
3-ethyl-3-methylpentane
2,2,4-trimethylpentane
2,3,3-trimethylpentane
2,3,4-trimethylpentane
3-ethylheptane
4-ethylheptane
2-methyloctane
3-methyloctane
4-methyloctane
2,2-dimethylheptane
2,3-dimethylheptane
2,4-dimethylheptane
2,5-dimethylheptane
2,6-dimethylheptane
3,3-dimethylheptane
3,4-dimethylheptane
3,5-dimethylheptane
4,4-dimethylheptane
3-ethyl-2-methylhexane
3-ethyl-3-methylhexane
4-ethyl-2-methylhexane
4-ethyl-3-methylhexane
2,2,3-trimethylhexane
2,2,4-trimethylhexane
2,2,5-trimethylhexane
2,3,3-trimethylhexane
253–353
253–363
253–363
253–353
253–353
253–353
253–353
283–363
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–373
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–373
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
11
12
12
11
11
11
11
10
11
11
11
11
11
11
11
11
11
11
11
11
11
11
7
7
7
7
7
7
7
7
7
11
7
7
7
7
7
7
7
7
7
7
7
7
2.0
1.0
1.5
1.3
1.9
1.2
2.2
0.6
0.5
0.6
0.9
0.7
0.5
1.7
1.5
0.4
2.5
1.6
3.6
2.9
2.3
0.7
1.0
0.1
1.3
0.7
0.4
2.7
0.4
1.8
2.4
2.7
0.2
0.8
1.4
0.3
1.3
2.0
0.7
5.6
0.6
2.4
3.7
0.2
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
9
Table 3 Ž continued .
Isomer
DT ŽK.
Number of points
Relative error Ž%.
2,3,4-trimethylhexane
2,3,5-trimethylhexane
2,4,4-trimethylhexane
3,3,4-trimethylhexane
3,3-diethylpentane
3-ethyl-2,2-dimethylpentane
3-ethyl-2,3-dimethylpentane
2,2,3,3-tetramethylpentane
cis-2-pentene
trans-2-pentene
2-methyl-1-butene
3-methyl-1-butene
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–333
283–303
283–303
283–303
283–298
7
7
7
7
7
7
7
7
4
4
4
3
0.5
1.6
1.2
1.3
1.6
0.9
2.7
1.0
3.4
2.3
0.5
3.6
Table 4
Global results of the estimation of surface tension values for isomers of different homologues of the n-alkane and 1-alkene
series of hydrocarbons
Isomers of
Number of
isomers
Number of
experimental points
DT ŽK.
Maximum difference
ŽmN my1 .
s
ŽmN my1 .
Average error
error Ž%.
n-hexane
n-heptane
n-octane
n-nonane
1-pentene
4
8
14
30
4
25
89
154
214
15
283–333
253–363
283–373
283–333
283–303
0.24
0.53
0.79
1.33
0.66
0.14
0.29
0.33
0.39
0.44
0.7
1.5
1.5
1.4
2.4
and the average relative error in percent. It can be observed from the information given in Table 4 that
the estimation of surface tension values for 60 different isomers of hydrocarbons of two homologous
series included in this study, in the temperature range 253–373 K, can be performed with an average
standard deviation of 0.32 mN P my1 which yields an average relative error of only 1.5%. This value
is indeed very close to the uncertainty of most of the experimental data of surface tension for the
systems considered in this work.
The present method will continue to be tested as soon as more experimental data become available
in the open literature for the isomers here considered in a larger temperature range or as soon as new
experimental data for new hydrocarbon isomers are reported.
4. Conclusions
The method here proposed to estimate surface tension values for isomers of hydrocarbons has been
comprehensively tested with almost 500 experimental points for 60 different isomers of the n-alkane
and 1-alkene homologous series in a relatively large range of temperatures. The method is simple and
the input parameters are readily available in the literature. It was shown that the estimation errors are
10
A. Romero-Martınez
´ et al.r Fluid Phase Equilibria 171 (2000) 1–10
in the same range as the experimental uncertainty; hence, the estimation method is accurate.
Considering the good agreement observed between our estimated values and experimental data for
normal alkanes in a large temperature range, we anticipate reliable extrapolation of surface tension
data above and below the temperature range of the experimental data for branched alkanes using the
method developed in this work.
List of symbols
a, b
Adjustable parameters in Eq. Ž3.
M
Relative molar mass
T
Temperature
Tc
Vapour–liquid critical temperature
V
Molar volume
IST
Isomer surface tension parameter, Eq. Ž 2.
Subscripts
nor
Normal member of a homologous series of compounds
iso
Isomer of a normal hydrocarbon homologue
Greek letters
d
Solubility parameter
g
Surface tension
t
Reverse reduced temperature scale Žt s 1 y TrTc .
s
Standard deviation
D
Range of values of a given variable
Acknowledgements
We gratefully acknowledge the financial support for this research from the Fondo de Apoyo al
Desarrollo de Proyectos de Investigacion
y Tecnologica
con Instituciones de Educacion
´ Basica
´
´
´
Ž
.
Superior FIES del Instituto Mexicano del Petroleo,
under Research Project FIES-96-45-II.
´
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