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Equilibrium behaviour of pure substances from the convex molecule perturbation theory
Fluid Phase Equilibria, 54 (1990) 221-235 Elsevier Science Publishers B.V., Amsterdam
221 -
Printed
in The Netherlands
EQUILIBRIUM BEI-IAVIOUR OF PURE SUBSTANCES FROM THE CONVEX MOLECULE PERTURBATION THEORY TOMAS
BOUBLiK
Institute of Chemical Process Fundamentals, Czechoslooak Academy of Sciences, Prague 6 (Czechoslovakia) (Received
February
1, 1989; accepted
in final form March 1, 1989)
ABSTRACT Boublik, T., 1990. Equilibrium behaviour of pure substances perturbation theory. Fluid Phase Equilibria, 54: 221-235.
from
the convex
molecule
The recently developed perturbation theory of convex molecule systems is applied to determine the equilibrium behaviour of pure non-polar compounds. Systems with molecules of several types are considered (roughly spherical, prolate and oblate) and it is shown that their equilibrium behaviour along the coexistence curves can be described up to the reduced temperature (T, < 0.9) with fair accuracy not depending on the shape of molecules. Parameters of the Kihara pair potential of 21 compounds are given.
INTRODUCTION
Equilibrium data on the behaviour of real gases, liquids and their mixtures form the necessary basis for the rational design and process optimization of basic units of chemical technology. An accurate equation of state together with perfect gas data can provide these values in a unique way. At present, semiempirical equations of state based on the ideas of the well-known van der Waals’ equation are in predominant use. However, with the realization of limits of these equations and with the recent development of the theory of fluids the statistical-thermodynamic perturbation (and variational) methods to characterize the equilibrium behaviour of fluids became popular. These methods start from the proper characterization of intermolecular forces among molecules and yield all the residual thermodynamic functions of liquids and real gases in terms of several quite general parameters characterizing these forces. While the methods to determine the equilibrium behaviour of systems composed of spherical molecules (with central type of interaction forces) have been known for a long time, the description of non-spherical molecule
222
fluids is more recent. For the interaction site model (ISM) Kohler et al. (1979) and Fischer (1980) formulated a variant of the perturbation theory of the Weeks-Chandler-Andersen (WCA) type (Weeks et al., 1971) and applied it to two-centre (and higher-centre) Lennard-Jones models of many molecules. The main disadvantage of the ISM model consists of the increasing complexity of the description (and with it a connected increase in computer time) with the number of interaction sites involved. The situation is different in the case of the Kihara generalized pair potential of non-polar molecules for which the perturbation theory has been proposed by the present author (Boublik, 1976). The perturbation term contains an integral in one variable only and its evaluation is relatively easy for any convex molecule. Recently we proposed a variation of this method (of the Barker-Henderson (1976) type) in which the substitution of the Yukawa pair potential into a part of the first-order perturbation term makes it possible to write the perturbation expansion for the Helmholtz energy and the compressibility factor in an analytic way (Boublik, 1987). We applied the method to several mono- and diatomic fluids taking the bond length as the length of rods of convex models and adjusting two L-J parameters to properties of the saturated pure liquids. In this paper we apply the theory to describe the two-phase behaviour of 21 pure compounds. We adjust three parameters to properties of liquids and show that the vapour-liquid equilibrium (VLE) can be described with fair accuracy for all the investigated compounds, the molecules of which can be characterized by different convex models, e.g. rods, plates, spheres, tetrahedrons or octahedrons.
PERTURBATION
THEORY
OF CONVEX MOLECULES
The employed perturbation theory (Boublik, 1987) assumes the intermolecular forces in fluids to be characterized by the Kihara generalized pair potential in which hard convex cores are ascribed to individual molecules and the interaction potential depends only on the shortest surface-to-surface distance, s. The 12-6 function is considered, i.e. U(T, w,, w2) = U(S) = 4E (where u and Barker-Henderson
[(i,”
- (fi”]
E are the characteristic parameters). The extended (1976) perturbation expansion of the second order is
223 TABLE
1
Geometric
quantities
R, S and Y of hard convex cores of different
shapes
Core
Parameter
R
s
V
Sphere Rod Rectangular
a 1 1
a 0.25 1 0.42678 1
4ma2 _ I2
R R R R
0.64952 R 0.75 R 0.74486 R 0.83119 R
2.59807 5.19615 4.61880 6.92820
+a3 _ _
triangle
Regular triangle Regular hexagon Tetrahedron Octahedron
R2 R2 R2 R2
0.51320 R3 1.33333 R3
used, which for the Helmholtz energy, F, possesses the form F-F* NkT
=
F”-F* NkT
p + -jwu(s)so(s)S,+,+s 2kT (I
ds (2)
Here p is the number density, T is temperature, k is the Boltzmann constant, go is the average correlation function, D = kT( ap/aP)’ and S l+Z+s is the mean surface area of two convex cores with the surfaceto-surface distance, s. For pure fluids we can express S, +2+s in terms of surface areas, S, and the mean curvature integrals divided by 4?r, R, of individual cores: s 1+2+s
=
4ns* + 16nRs + 2( S + 47rR2)
(3)
Geometric quantities, R and S, together with a volume, V, for the models used are given in Table 1. From the known values of R, S, and V it is possible to evaluate the non-sphericity parameter, CY = RS/3V, which together with the packing fraction, y = pV) enables one to determine the reference term, F” - F * from the hard body equation of state (Boublik, 1981):
(4 In the perturbation term we first substitute for the distribution function, g’(s), go(s) = 1 + h’(s), where ho is the average total correlation function of the representative hard convex bodies. After substitution into eqn. (2) one obtains immediately the larger part of the perturbation integral in an analytical form as a function of (I, R and S. In the remaining part of the
224
first-order term we approximate the function ho(s) on the basis of the total correlation function of the equivalent hard spheres. After substituting the Yukawa pair potential for the L-J 12-6 function the integral can be expressed in terms of (generally) three couples of the Laplace transforms of xg’(x) in the Percus-Yevick approximation (see for example Boublik et al., 198O).Thus, the residual Helmholtz function and similarly the compressibility factor can be expressed analytically, avoiding completely the numerical integration. The details are given elsewhere (Boublik, 1987). APPLICATION
Within the framework of the convex molecule perturbation theory all the thermodynamic functions of fluids depend on the values of parameters E and u and on the quantities characterizing the size and shape of individual cores. In this treatment we take the basic information on the shape of cores from the known structural data (bond lengths and bond angles) and adjust only one parameter: radius of a sphere, u, in the case of spherical molecules, rod length, I, in the case of linear molecules and radius, R, of the circumscribed circle or sphere to the given planar figure (triangle, hexagon) or body (tetrahedron, octahedron) in the remaining cases. One of these three parameters, together with e and u, were determined from the saturated liquid densities and vapour pressures of the individual compounds in the range of reduced temperatures T,< 0.9. The equilibrium conditions P' = P " and
ApLI= Ap” + ln( p'/p"
)
0)
for the liquid (‘) and vapour ( “) phases were considered, with Ap denoting the residual chemical potential. Properties of the liquid phase were determined from the second order perturbation expansion whereas those of the gaseous phase from the virial expansion
in which the exact second virial coefficient, B, and the reference part of the third virial coefficient, Ch, were employed. The virial coefficients are given by eqns. (7) and (8) B=+J~F,(T*)+~~R~~~F,(T*)+(S+~~R~)~F~(T*)+(V+RS)
(7)
and Ch = (1 + 6i~ + 3a*)V In eqn. (7) Fi( T *)- F'JT *) are the known functions
(8) of the reduced
225
temperature; the Pade approximants for them were taken from the work of Boublik and Vosmansky (1981). Firstly, monoatomic fluids argon, krypton and xenon, together with methane (with roughly spherical molecule), were studied. It is well known that pair interactions of these systems can be characterized by the relatively complicated realistic pair potentials of the central type. The simple L-J 12-6 potential is not fully sufficient even as the effective potential; however, the introduction of a relatively small core compensates to a great extent for the shortcomings of the L-J function. We thus considered spherical cores for the above systems and adjusted the core radius, a, together with parameters 6 and CJto experimental data. The next group of fluids is formed by linear molecule systems-nitrogen, oxygen, fluorine, chlorine, carbon monoxide, carbon dioxide, acetylene, ethylene and ethane. Hard rods were assumed as cores of these molecules, with length, I, adjusted to the liquid properties. (In principle, prolate spherocylinders instead of rods could be considered; however, the relatively narrow ranges of temperatures for which orthobaric data are available do not provide evidence in favour of adjusting two parameters of the core, length and breadth). In the majority of cases the values of length, I, found are smaller than the corresponding bond length (or sum of bond lengths). In some cases this fact can be explained from the structure of the given molecules. Systems with planar molecules are represented in our study by propane, cyclopropane and benzene. Propane is modelled by a triangle with the C-C-C angle $I = 90 o and with the adjusted side lengths, 1. The cores of cyclopropane and benzene are the regular triangle and hexagon, respectively, with the adjusted radius of the circumscribed circle, R. The last group includes four systems with high symmetrical molecules of carbon tetrafluoride, carbon tetrachloride, sulphur hexafluoride and neopentane. Two kinds of cores were considered: (i) in the former case
Fig. 1. The cores of the studied compounds.
226
TABLE 2 Characteristic
parameters of the Kihara generalized potential
Compound
+
Argon Krypton Xenon Methane Nitrogen Oxygen Fluorine Chlorine Carbon monoxide Carbon dioxide Acetylene Ethylene Ethane Carbon disulphide Propane Cyclopropane Benzene Carbon tetrafluoride Carbon tetrafluoride Carbon tetrachloride Carbon tetrachloride Sulfur hexafluoride Sulfur hexafluoride Neopentane Neopentane
s t s t s o S
t
123.583 171.337 231.310 156.060 113.917 133.652 134.120 417.606 124.791 351.919 349.764 283.486 308.271 541.111 411.700 428.634 704.594 269.328 263.430 692.430 682.130 389.450 392.543 528.559 525.429
0 (A> 3.249 3.481 3.874 3.587 3.216 3.116 2.852 3.296 3.140 2.688 2.904 3.362 3.487 3.767 3.600 3.459 3.501 3.048 3.104 3.635 3.671 3.332 3.292 3.878 3.877
o (A) 0.081 0.079 0.042 0.077
-
_ R = 1.292 0.642 R = 0.889 0.905 R = 1.302 0.779 R = 1.018 0.894 R = 1.316
1 (A)
_ 0.930 0.605 1.014 1.821 1.218 2.862 2.821 1.885 1.967 1.665 1.658 0.886 -
spherical cores were assumed with the adjusted radius, a, similar to monoatomic fluids, (ii) in the latter case tetrahedron or octahedron, characterized by the radius, R, of the circumscribed sphere were considered. From the comparison of deviations in the liquid density and in pressure it follows that there is no distinct difference between the former and latter models. All the models used are depicted in Fig. 1. In Table 2 the calculated values of e, (T and the parameter characterizing the given core (a, I, or R) are listed. In Table 3 we compare the calculated and experimental densities of the liquid phase, p’, (mol I-‘) and calculated and experimental pressures, p, (bar) for ethane, cyclopropane and carbon tetrafluoride as representatives of the linear, planar and globular types of molecules. One can see that differences usually amount to less than one per cent in both densities and
221 TABLE 3 Orthobaric data and deviations in liquid densities and pressures p” (mol 1-l)
T (K)
p’ (mol
Ethane 100.00 120.00 150.00 180.00 210.00 240.00
21.34 20.60 19.47 18.28 16.97 15.46
0.00001 0.00035 0.00780 0.05400 0.20800 0.58000
Cyclopropane 243.15 16.09 293.15 14.87 303.15 14.52 313.15 14.15 323.15 13.75 333.15 13.31 343.15 12.83
0.31400 0.38500 0.48400 0.61200 0.77500 0.98000
1.14000 6.31000 8.27000 10.65000 13.48000 16.82000 20.73000
Carbon tetrafluoride 110.00 20.24 120.00 19.72 130.00 19.17 140.00 18.59 150.00 17.99 160.00 17.35
0.00374 0.00374 0.01140 0.06130 0.11800 0.20900
0.03720 0.11800 0.31000 0.70100 1.41000 2.60000
1-l)
o.ooooo
P (bar) 0.00011 0.00350 0.09700 0.79000 3.34000 9.67000
dev. p’ (mol 1-l)
dev. p (bar)
-
0.02207 0.03359 0.05186 0.05591 0.01733 0.09883
0.00001 0.00006 - 0.00080 - 0.00836 - 0.01335 0.01087
0.05252 0.17538 0.13601 0.08842 0.02144 0.07281 0.19453
0.00008 - 0.01607 0.00047 - 0.01521 - 0.00187 0.02650 0.11601
- 0.00022 - 0.06905 - 0.03826 0.00893 0.05963 0.12745
- 0.00022 0.00102 - 0.00047 0.00006 0.00632 - 0.01024
-
pressures. Larger differences in pressure at very low temperatures are partly due to uncertainties in the experimental data (these are taken from the work of Bohn et al. (1986), Vargaftik (1975), Boublik et al. (1984) and Dreisbach (1955)). Differences in the calculated and experimental densities and in the calculated and experimental pressures (both in per cent) as functions of the reduced temperature, T * = T/T, (where T, is the experimental critical temperature) for the studied systems are summarized in Figs. 2-4 and 5-7. From the graphs, where the interval of + 1% is marked, one can see that for systems of the first two groups deviations in the liquid density do not exceed this interval (with exception of Ar at the highest reduced temperature). The same is true also for deviations in pressure with exception of values at the lowest temperatures where variations in the experimental pressures are comparable with the given deviations. In the group of planar molecules good results are obtained for propane and cyclopropane. However, differences in both density and pressure of
228
-1
I 03
I
OL
05
06
0.7
0.8
I
I
09
1
1.
Fig. 2. The per cent deviations in density, p’, against the reduced temperature,
T, = T/Tqexp.
benzene are relatively great. The possible explanation could be that at low temperatures benzene does not form the isotropic fluid which is the basic precondition of the theory. In the last group the theory describes well the behaviour of methane and neopentane whereas SF, and Ccl, show greater discrepancies in per cent deviations in densities while the absolute values of deviations remain close to those of the other systems. Another weighting factor in the objective function could probably improve these results. From the description of the optimization procedure it is evident that the parameters of the Kihara potential should yield better prediction of the
229
Fig. 3. See Fig. 2.
properties of dense systems (i.e. the liquid phase) than those of rarefied gas. It is thus instructive to compare values of the second virial coefficient calculated from the parameters of Table 2 with the corresponding experimental data. Such comparison is given in Fig. 8 where values of B, (cm3 mol-‘) are plotted against the reduced temperature, T, = kT/c, for nitrogen and ethane for which B, is known over a broad range of temperatures. Good agreement (deviations are approximately 2-3 times the estimated errors of the experimental data) indicates good performance of the theory even for the gaseous phase.
DISCUSSION AND CONCLUSION
In the preceding part of the paper it was shown how the convex molecule perturbation theory can be used for the description of the equilibrium
230
AQ -1 + ’ - c-C3He 0
\
-1 II 2-
-2 I a3
I
I 0.4
as
Oh
03
0.1) 09 1.
1
Fig. 4. See Fig. 2.
behaviour of pure fluids along the coexistence curve. It appears that the description is fair up to the maximum reduced temperature T/T,= 0.9. The inclusion of higher temperatures brings in a decrease in accuracy of the
231 1‘
SP
Ar
0
-1 I’
‘-
Kr
1I
Xe
l-
00 1 ; l'-
N2
0, l2.1 'cl3 0
03
a4
0.5
0.6
Q7
0.6
09
1
1'
Fig. 5. The
per cent deviationsin pressure,p, againstthe reducedtemperature,T, = T/Tc,exp.
determination of density and pressure. This is a general trait of all the different perturbation theories of molecular fluids including the theory of Kohler and Fischer and the variant of the convex molecule perturbation theory proposed recently (Kantor and Boublik, 1988). Both the last theories, which are variants of the WCA type, yield predictions of the orthobaric data with usually larger density deviations than the present theory; the maximum deviations in density or volume are approximately 3%. In the case of the former theory, this is probably a consequence of the absence of the secondorder term, in the case of the latter method, of the fact that the rod lengths
232
\_
0 AP
-1 ;
l'-
C'2
O-1 1: l-
C2H2
l=2% 0
-1t
-2 t
I __
_.
0.3
0.4
a5
06
0.7
ae
09
1
1'
Fig. 6. See Fig. 5.
(or other characteristic parameters of cores) were not adjusted but taken from the structural data of molecules. The most important trait of the theory employed here in comparison with all other perturbation methods for molecular fluids is its simplicity which is due to the analytic expressions for both the Hehnholtz function and pressure. This makes the approach especially suitable for the purposes of chemical engineers.
233 1r
0
AD
C3%
-1 ,) ’ - c-C3l$ cl -1
1 0
-1
-3
1
0 -1 2 1
0
-1 -2 2 1
a
Fig. 7. See Fig. 5.
234 0
3
82 cm3/md -Km -
-ax, -
-300 / -a
-
1:
;
a_
0.5
Fig. 8. Second virial coefficient temperature T, = kT/c.
1 10
1.5
1. of nitrogen
(-
) and ethane
(- - - - - -) vs. the reduced
REFERENCES
Barker, J.A. and Henderson, D., 1976. What is “liquid”? Understanding the states of matter. Rev. Mod. Phys., 48: 587-671. Bohn, M., Lustig, R. and Fischer, J., 1986. Description of polyatomic real substances by two-center Lennard-Jones model fluids. Fluid Phase Equilibria, 25: 251-262. Boubllk, T., 1976. Perturbation theory for fluids of rod-like molecules interacting via the Kihara potential. Mol. Phys., 32: 1737-1749. Boubllk, T., 1981. Equation of state of hard convex body fluids. Mol. Phys., 42: 209-216. Boublik, T., 1987. Simple perturbation method for convex-molecule fluids. J. Chem. Phys., 87: 1751-1756. Boublik, T., Fried, V. and Hala, E., 1984. The vapour pressures of pure substances. Elsevier, Amsterdam. Boublik, T., Nezbeda, I. and Hlavatjr, K., 1980. Statistical thermodynamics of simple liquids and their mixtures. Elsevier, Amsterdam. Boublik, T. and Vosmanskjr, J., 1981. Use of the Pade approximants for calculating the second virial coefficient. Colln. Czech. Chem. Commun., 46: 542-551. Dreisbach, R.F.G., 1955, 1959 and 1961. Physical properties of chemical compounds. Adv. Chem. Ser. 15, 22 and 29. Am. Chem. Sot., Washington, DC. Fischer, J., 1980. Perturbation theory for the free energy of two-center Lennard-Jones liquids. J. Chem. Phys., 72: 5371-5377. Fischer, J., Lustig, R., Breitenfelder-Manske, H. and Lemming, W., 1984. Influence of intermolecular potential parameters on orthobaric properties of fluids consisting of spherical and linear molecules. Mol. Phys., 52: 485-497. Kantor, R. and Boublik, T., 1988. Vapour liquid equilibria for pure fluids from the perturbation theory of the Kihara molecules. Ber. Bunsenges. Phys. Chem., 92: 1123-1129. Kohler, F., Quirke, N. and Perram, J.W., 1979. Perturbation theory with a hard dumbbell reference system. J. Chem. Phys., 71: 4128-4131.
235 Vargaftik, N.B., 1975. Tables on the thermodynamical properties of liquids and gases. Wiley, New York. Weeks, J.D., Chandler, D. and Andersen, H.C., 1971. Role of repulsive forces in determining the equilibrium structure of simple fluids. J. Chem. Phys., 54: 5237-5247.
221 -
Printed
in The Netherlands
EQUILIBRIUM BEI-IAVIOUR OF PURE SUBSTANCES FROM THE CONVEX MOLECULE PERTURBATION THEORY TOMAS
BOUBLiK
Institute of Chemical Process Fundamentals, Czechoslooak Academy of Sciences, Prague 6 (Czechoslovakia) (Received
February
1, 1989; accepted
in final form March 1, 1989)
ABSTRACT Boublik, T., 1990. Equilibrium behaviour of pure substances perturbation theory. Fluid Phase Equilibria, 54: 221-235.
from
the convex
molecule
The recently developed perturbation theory of convex molecule systems is applied to determine the equilibrium behaviour of pure non-polar compounds. Systems with molecules of several types are considered (roughly spherical, prolate and oblate) and it is shown that their equilibrium behaviour along the coexistence curves can be described up to the reduced temperature (T, < 0.9) with fair accuracy not depending on the shape of molecules. Parameters of the Kihara pair potential of 21 compounds are given.
INTRODUCTION
Equilibrium data on the behaviour of real gases, liquids and their mixtures form the necessary basis for the rational design and process optimization of basic units of chemical technology. An accurate equation of state together with perfect gas data can provide these values in a unique way. At present, semiempirical equations of state based on the ideas of the well-known van der Waals’ equation are in predominant use. However, with the realization of limits of these equations and with the recent development of the theory of fluids the statistical-thermodynamic perturbation (and variational) methods to characterize the equilibrium behaviour of fluids became popular. These methods start from the proper characterization of intermolecular forces among molecules and yield all the residual thermodynamic functions of liquids and real gases in terms of several quite general parameters characterizing these forces. While the methods to determine the equilibrium behaviour of systems composed of spherical molecules (with central type of interaction forces) have been known for a long time, the description of non-spherical molecule
222
fluids is more recent. For the interaction site model (ISM) Kohler et al. (1979) and Fischer (1980) formulated a variant of the perturbation theory of the Weeks-Chandler-Andersen (WCA) type (Weeks et al., 1971) and applied it to two-centre (and higher-centre) Lennard-Jones models of many molecules. The main disadvantage of the ISM model consists of the increasing complexity of the description (and with it a connected increase in computer time) with the number of interaction sites involved. The situation is different in the case of the Kihara generalized pair potential of non-polar molecules for which the perturbation theory has been proposed by the present author (Boublik, 1976). The perturbation term contains an integral in one variable only and its evaluation is relatively easy for any convex molecule. Recently we proposed a variation of this method (of the Barker-Henderson (1976) type) in which the substitution of the Yukawa pair potential into a part of the first-order perturbation term makes it possible to write the perturbation expansion for the Helmholtz energy and the compressibility factor in an analytic way (Boublik, 1987). We applied the method to several mono- and diatomic fluids taking the bond length as the length of rods of convex models and adjusting two L-J parameters to properties of the saturated pure liquids. In this paper we apply the theory to describe the two-phase behaviour of 21 pure compounds. We adjust three parameters to properties of liquids and show that the vapour-liquid equilibrium (VLE) can be described with fair accuracy for all the investigated compounds, the molecules of which can be characterized by different convex models, e.g. rods, plates, spheres, tetrahedrons or octahedrons.
PERTURBATION
THEORY
OF CONVEX MOLECULES
The employed perturbation theory (Boublik, 1987) assumes the intermolecular forces in fluids to be characterized by the Kihara generalized pair potential in which hard convex cores are ascribed to individual molecules and the interaction potential depends only on the shortest surface-to-surface distance, s. The 12-6 function is considered, i.e. U(T, w,, w2) = U(S) = 4E (where u and Barker-Henderson
[(i,”
- (fi”]
E are the characteristic parameters). The extended (1976) perturbation expansion of the second order is
223 TABLE
1
Geometric
quantities
R, S and Y of hard convex cores of different
shapes
Core
Parameter
R
s
V
Sphere Rod Rectangular
a 1 1
a 0.25 1 0.42678 1
4ma2 _ I2
R R R R
0.64952 R 0.75 R 0.74486 R 0.83119 R
2.59807 5.19615 4.61880 6.92820
+a3 _ _
triangle
Regular triangle Regular hexagon Tetrahedron Octahedron
R2 R2 R2 R2
0.51320 R3 1.33333 R3
used, which for the Helmholtz energy, F, possesses the form F-F* NkT
=
F”-F* NkT
p + -jwu(s)so(s)S,+,+s 2kT (I
ds (2)
Here p is the number density, T is temperature, k is the Boltzmann constant, go is the average correlation function, D = kT( ap/aP)’ and S l+Z+s is the mean surface area of two convex cores with the surfaceto-surface distance, s. For pure fluids we can express S, +2+s in terms of surface areas, S, and the mean curvature integrals divided by 4?r, R, of individual cores: s 1+2+s
=
4ns* + 16nRs + 2( S + 47rR2)
(3)
Geometric quantities, R and S, together with a volume, V, for the models used are given in Table 1. From the known values of R, S, and V it is possible to evaluate the non-sphericity parameter, CY = RS/3V, which together with the packing fraction, y = pV) enables one to determine the reference term, F” - F * from the hard body equation of state (Boublik, 1981):
(4 In the perturbation term we first substitute for the distribution function, g’(s), go(s) = 1 + h’(s), where ho is the average total correlation function of the representative hard convex bodies. After substitution into eqn. (2) one obtains immediately the larger part of the perturbation integral in an analytical form as a function of (I, R and S. In the remaining part of the
224
first-order term we approximate the function ho(s) on the basis of the total correlation function of the equivalent hard spheres. After substituting the Yukawa pair potential for the L-J 12-6 function the integral can be expressed in terms of (generally) three couples of the Laplace transforms of xg’(x) in the Percus-Yevick approximation (see for example Boublik et al., 198O).Thus, the residual Helmholtz function and similarly the compressibility factor can be expressed analytically, avoiding completely the numerical integration. The details are given elsewhere (Boublik, 1987). APPLICATION
Within the framework of the convex molecule perturbation theory all the thermodynamic functions of fluids depend on the values of parameters E and u and on the quantities characterizing the size and shape of individual cores. In this treatment we take the basic information on the shape of cores from the known structural data (bond lengths and bond angles) and adjust only one parameter: radius of a sphere, u, in the case of spherical molecules, rod length, I, in the case of linear molecules and radius, R, of the circumscribed circle or sphere to the given planar figure (triangle, hexagon) or body (tetrahedron, octahedron) in the remaining cases. One of these three parameters, together with e and u, were determined from the saturated liquid densities and vapour pressures of the individual compounds in the range of reduced temperatures T,< 0.9. The equilibrium conditions P' = P " and
ApLI= Ap” + ln( p'/p"
)
0)
for the liquid (‘) and vapour ( “) phases were considered, with Ap denoting the residual chemical potential. Properties of the liquid phase were determined from the second order perturbation expansion whereas those of the gaseous phase from the virial expansion
in which the exact second virial coefficient, B, and the reference part of the third virial coefficient, Ch, were employed. The virial coefficients are given by eqns. (7) and (8) B=+J~F,(T*)+~~R~~~F,(T*)+(S+~~R~)~F~(T*)+(V+RS)
(7)
and Ch = (1 + 6i~ + 3a*)V In eqn. (7) Fi( T *)- F'JT *) are the known functions
(8) of the reduced
225
temperature; the Pade approximants for them were taken from the work of Boublik and Vosmansky (1981). Firstly, monoatomic fluids argon, krypton and xenon, together with methane (with roughly spherical molecule), were studied. It is well known that pair interactions of these systems can be characterized by the relatively complicated realistic pair potentials of the central type. The simple L-J 12-6 potential is not fully sufficient even as the effective potential; however, the introduction of a relatively small core compensates to a great extent for the shortcomings of the L-J function. We thus considered spherical cores for the above systems and adjusted the core radius, a, together with parameters 6 and CJto experimental data. The next group of fluids is formed by linear molecule systems-nitrogen, oxygen, fluorine, chlorine, carbon monoxide, carbon dioxide, acetylene, ethylene and ethane. Hard rods were assumed as cores of these molecules, with length, I, adjusted to the liquid properties. (In principle, prolate spherocylinders instead of rods could be considered; however, the relatively narrow ranges of temperatures for which orthobaric data are available do not provide evidence in favour of adjusting two parameters of the core, length and breadth). In the majority of cases the values of length, I, found are smaller than the corresponding bond length (or sum of bond lengths). In some cases this fact can be explained from the structure of the given molecules. Systems with planar molecules are represented in our study by propane, cyclopropane and benzene. Propane is modelled by a triangle with the C-C-C angle $I = 90 o and with the adjusted side lengths, 1. The cores of cyclopropane and benzene are the regular triangle and hexagon, respectively, with the adjusted radius of the circumscribed circle, R. The last group includes four systems with high symmetrical molecules of carbon tetrafluoride, carbon tetrachloride, sulphur hexafluoride and neopentane. Two kinds of cores were considered: (i) in the former case
Fig. 1. The cores of the studied compounds.
226
TABLE 2 Characteristic
parameters of the Kihara generalized potential
Compound
+
Argon Krypton Xenon Methane Nitrogen Oxygen Fluorine Chlorine Carbon monoxide Carbon dioxide Acetylene Ethylene Ethane Carbon disulphide Propane Cyclopropane Benzene Carbon tetrafluoride Carbon tetrafluoride Carbon tetrachloride Carbon tetrachloride Sulfur hexafluoride Sulfur hexafluoride Neopentane Neopentane
s t s t s o S
t
123.583 171.337 231.310 156.060 113.917 133.652 134.120 417.606 124.791 351.919 349.764 283.486 308.271 541.111 411.700 428.634 704.594 269.328 263.430 692.430 682.130 389.450 392.543 528.559 525.429
0 (A> 3.249 3.481 3.874 3.587 3.216 3.116 2.852 3.296 3.140 2.688 2.904 3.362 3.487 3.767 3.600 3.459 3.501 3.048 3.104 3.635 3.671 3.332 3.292 3.878 3.877
o (A) 0.081 0.079 0.042 0.077
-
_ R = 1.292 0.642 R = 0.889 0.905 R = 1.302 0.779 R = 1.018 0.894 R = 1.316
1 (A)
_ 0.930 0.605 1.014 1.821 1.218 2.862 2.821 1.885 1.967 1.665 1.658 0.886 -
spherical cores were assumed with the adjusted radius, a, similar to monoatomic fluids, (ii) in the latter case tetrahedron or octahedron, characterized by the radius, R, of the circumscribed sphere were considered. From the comparison of deviations in the liquid density and in pressure it follows that there is no distinct difference between the former and latter models. All the models used are depicted in Fig. 1. In Table 2 the calculated values of e, (T and the parameter characterizing the given core (a, I, or R) are listed. In Table 3 we compare the calculated and experimental densities of the liquid phase, p’, (mol I-‘) and calculated and experimental pressures, p, (bar) for ethane, cyclopropane and carbon tetrafluoride as representatives of the linear, planar and globular types of molecules. One can see that differences usually amount to less than one per cent in both densities and
221 TABLE 3 Orthobaric data and deviations in liquid densities and pressures p” (mol 1-l)
T (K)
p’ (mol
Ethane 100.00 120.00 150.00 180.00 210.00 240.00
21.34 20.60 19.47 18.28 16.97 15.46
0.00001 0.00035 0.00780 0.05400 0.20800 0.58000
Cyclopropane 243.15 16.09 293.15 14.87 303.15 14.52 313.15 14.15 323.15 13.75 333.15 13.31 343.15 12.83
0.31400 0.38500 0.48400 0.61200 0.77500 0.98000
1.14000 6.31000 8.27000 10.65000 13.48000 16.82000 20.73000
Carbon tetrafluoride 110.00 20.24 120.00 19.72 130.00 19.17 140.00 18.59 150.00 17.99 160.00 17.35
0.00374 0.00374 0.01140 0.06130 0.11800 0.20900
0.03720 0.11800 0.31000 0.70100 1.41000 2.60000
1-l)
o.ooooo
P (bar) 0.00011 0.00350 0.09700 0.79000 3.34000 9.67000
dev. p’ (mol 1-l)
dev. p (bar)
-
0.02207 0.03359 0.05186 0.05591 0.01733 0.09883
0.00001 0.00006 - 0.00080 - 0.00836 - 0.01335 0.01087
0.05252 0.17538 0.13601 0.08842 0.02144 0.07281 0.19453
0.00008 - 0.01607 0.00047 - 0.01521 - 0.00187 0.02650 0.11601
- 0.00022 - 0.06905 - 0.03826 0.00893 0.05963 0.12745
- 0.00022 0.00102 - 0.00047 0.00006 0.00632 - 0.01024
-
pressures. Larger differences in pressure at very low temperatures are partly due to uncertainties in the experimental data (these are taken from the work of Bohn et al. (1986), Vargaftik (1975), Boublik et al. (1984) and Dreisbach (1955)). Differences in the calculated and experimental densities and in the calculated and experimental pressures (both in per cent) as functions of the reduced temperature, T * = T/T, (where T, is the experimental critical temperature) for the studied systems are summarized in Figs. 2-4 and 5-7. From the graphs, where the interval of + 1% is marked, one can see that for systems of the first two groups deviations in the liquid density do not exceed this interval (with exception of Ar at the highest reduced temperature). The same is true also for deviations in pressure with exception of values at the lowest temperatures where variations in the experimental pressures are comparable with the given deviations. In the group of planar molecules good results are obtained for propane and cyclopropane. However, differences in both density and pressure of
228
-1
I 03
I
OL
05
06
0.7
0.8
I
I
09
1
1.
Fig. 2. The per cent deviations in density, p’, against the reduced temperature,
T, = T/Tqexp.
benzene are relatively great. The possible explanation could be that at low temperatures benzene does not form the isotropic fluid which is the basic precondition of the theory. In the last group the theory describes well the behaviour of methane and neopentane whereas SF, and Ccl, show greater discrepancies in per cent deviations in densities while the absolute values of deviations remain close to those of the other systems. Another weighting factor in the objective function could probably improve these results. From the description of the optimization procedure it is evident that the parameters of the Kihara potential should yield better prediction of the
229
Fig. 3. See Fig. 2.
properties of dense systems (i.e. the liquid phase) than those of rarefied gas. It is thus instructive to compare values of the second virial coefficient calculated from the parameters of Table 2 with the corresponding experimental data. Such comparison is given in Fig. 8 where values of B, (cm3 mol-‘) are plotted against the reduced temperature, T, = kT/c, for nitrogen and ethane for which B, is known over a broad range of temperatures. Good agreement (deviations are approximately 2-3 times the estimated errors of the experimental data) indicates good performance of the theory even for the gaseous phase.
DISCUSSION AND CONCLUSION
In the preceding part of the paper it was shown how the convex molecule perturbation theory can be used for the description of the equilibrium
230
AQ -1 + ’ - c-C3He 0
\
-1 II 2-
-2 I a3
I
I 0.4
as
Oh
03
0.1) 09 1.
1
Fig. 4. See Fig. 2.
behaviour of pure fluids along the coexistence curve. It appears that the description is fair up to the maximum reduced temperature T/T,= 0.9. The inclusion of higher temperatures brings in a decrease in accuracy of the
231 1‘
SP
Ar
0
-1 I’
‘-
Kr
1I
Xe
l-
00 1 ; l'-
N2
0, l2.1 'cl3 0
03
a4
0.5
0.6
Q7
0.6
09
1
1'
Fig. 5. The
per cent deviationsin pressure,p, againstthe reducedtemperature,T, = T/Tc,exp.
determination of density and pressure. This is a general trait of all the different perturbation theories of molecular fluids including the theory of Kohler and Fischer and the variant of the convex molecule perturbation theory proposed recently (Kantor and Boublik, 1988). Both the last theories, which are variants of the WCA type, yield predictions of the orthobaric data with usually larger density deviations than the present theory; the maximum deviations in density or volume are approximately 3%. In the case of the former theory, this is probably a consequence of the absence of the secondorder term, in the case of the latter method, of the fact that the rod lengths
232
\_
0 AP
-1 ;
l'-
C'2
O-1 1: l-
C2H2
l=2% 0
-1t
-2 t
I __
_.
0.3
0.4
a5
06
0.7
ae
09
1
1'
Fig. 6. See Fig. 5.
(or other characteristic parameters of cores) were not adjusted but taken from the structural data of molecules. The most important trait of the theory employed here in comparison with all other perturbation methods for molecular fluids is its simplicity which is due to the analytic expressions for both the Hehnholtz function and pressure. This makes the approach especially suitable for the purposes of chemical engineers.
233 1r
0
AD
C3%
-1 ,) ’ - c-C3l$ cl -1
1 0
-1
-3
1
0 -1 2 1
0
-1 -2 2 1
a
Fig. 7. See Fig. 5.
234 0
3
82 cm3/md -Km -
-ax, -
-300 / -a
-
1:
;
a_
0.5
Fig. 8. Second virial coefficient temperature T, = kT/c.
1 10
1.5
1. of nitrogen
(-
) and ethane
(- - - - - -) vs. the reduced
REFERENCES
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235 Vargaftik, N.B., 1975. Tables on the thermodynamical properties of liquids and gases. Wiley, New York. Weeks, J.D., Chandler, D. and Andersen, H.C., 1971. Role of repulsive forces in determining the equilibrium structure of simple fluids. J. Chem. Phys., 54: 5237-5247.