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Influence of molecular shape on vapour-liquid equilibria
Fluid Phase Equilibria, Elsevier
95 (1994) 371-381
371
Science B.V.
Short Communication
Influence of molecular Application
of a modified
shape on vapour-liquid Christoforakos-Franck
equilibria
equation
of state
Dorde PlaEkov a,b, David E. Mainwaring a** and Richard J. Sadus b a Department of Applied Chemistry, Swinburne University of Technology, PO Box 218, Hawthorn, 3122 Vie. (Australia) b Department of Computer Science, Swinburne of Technology, PO Box 218, Hawthorn, 3122 Vie. (Australia) (Received
January
26, 1993; accepted
in final form November
University
22, 1993)
ABSTRACT PlaEkov, D., Mainwaring, D.E. and Sadus, R.J., 1994. Influence of molecular shape vapour-liquid equilibria. Application of a modified Christoforakos-Franck equation state. Fluid Phase Equilibria, 95: 371-381.
on of
The Christoforakos-Franck equation of state, originally derived for spherical molecules, is modified to account for deviations from spherical geometry. The contribution from repulsive interactions is obtained from the Boublik-Nezbeda model for the compressibility of hard bodies of arbitrary shape. The deviation from sphericity (c() is deduced from the hard convex body model of molecular geometry. Attractive intermolecular interactions are characterized by the depth ( - E) and relative width (n) of a square-well potential. The influence of molecular shape on the vapour-liquid equilibrium properties of pure fluids is investigated by using the modified equation to predict the vapour pressure of non-spherical molecules. It is concluded that incorporating molecular shape in the conventional equation of state methodology does not significantly influence the quality of the predicted vapour-liquid equilibria. Keywords:
theory,
equation
of state, vapour-liquid
equilibria,
non-spherical
molecules.
INTRODUCTION
Equations of state (EOS) have assumed an expanding role in the study of the phase equilibria of fluids and fluid mixtures. They can be applied to the gas, liquid and supercritical phases without encountering any conceptual difficulties, and all the physical fluid properties (i.e. composition, temperature, pressure and volume) can be obtained simultaneously. However, there is no universally applicable EOS. Empirical equations, which have been *Corresponding
author.
037%3812/94/$07.00 0 1994 - Elsevier SSDZ 0378-3812(93)02453-T
Science B.V. All rights reserved
372
B. PlaEkov et al. / Fluid Phase Equilibria 95 (1994) 371-381
modified to yield optimal agreement at low pressures, often fail to predict accurately the properties of fluids at high pressures, and vice versa. The development of EOS models usually conforms to the van der Waals formalism, i.e. the contribution of repulsive and attractive fluid-fluid interactions are treated separately. A dichotomy has emerged between EOS which retain the inadequate van der Waals hard-sphere repulsive term plus an empirically improved attractive term, and EOS which incorporate a theoretically better description of repulsive fluid interactions. The RedlichKwong (Redlich and Kwong, 1949), Peng-Robinson (Peng and Robinson, 1984) equations are examples of empirically 1976) and Soave (Soave, improved equations which have been used widely in chemical engineering applications, whereas the Carnahan-Starling (Carnahan and Starling, 1969, 1965) and hard convex body (HCB) 1972) Guggenheim (Guggenheim, (Svejda and Kohler, 1983) equations exemplify the latter approach. An accurate description of repulsive interaction is particularly important in order to predict high-pressure equilibria, whereas attractive interactions are a more important consideration at lower pressures. EOS at or near the critical state suffer from the limitation that the critical state is not an analytical function of energy. Nevertheless, it is important to use an accurate representation of repulsive interactions when predicting critical phenomena. Christoforakos and Franck (CF) (Christoforakos and Franck, 1986) have proposed an EOS with some promising features. The CF EOS has been applied to the critical properties of a series of binary mixtures containing water plus a small, non-polar component. Using two adjustable parameters, the predicted critical locus of the water + carbon dioxide mixture compares very well with experimental data. The results are better than can be obtained from other accurate hard-sphere equations such as the Guggenheim model (Guggenheim, 1965). It is of theoretical and engineering interest to examine whether the EOS can improve the agreement of theory with experiment for other aspects of phase equilibria. It must be emphasized that the aim of this work is not the accurate prediction of vapour pressure data. There are many empirical approaches which can be used to correlate vapour-liquid equilibrium data with a high degree of accuracy. These good results are achieved by “empirical” improvements which are devoid of physical meaning or rationale. Often they provide no insight into the fundamental weakness of the model. They “correlate” rather than “predict” vapour pressure data. Because this approach is so prevalent in the chemical literature, relatively little quantitative data is available about the failure of theoretical models. Data of this type are valuable in assessing the limitations of theory and providing insight into improved EOS. Examining the properties of pure substances is an important starting point for the phase behaviour of fluid mixtures in general. Applying
B. PlaCkov et aLlFluid Phase Equilibria 95 (1994) 371-381
313
the EOS to the vapour-liquid equilibria of a pure substance is a direct measure of the assumptions made concerning the nature of intermolecular interactions in the fluids. In contrast, the analysis of multicomponent equilibria is complicated by uncertainties introduced by the nature of mixing rules and combining rules for unlike intermolecular interactions. The CF EOS was specifically derived for spherical molecules. It has been demonstrated that molecular shape has a profound effect on such properties as virial coefficients (Kihara, 1963) and excess functions (Siddiqi et al., 1983). The vapour-liquid critical point is largely uninfluenced by molecular shape (Sadus et al., 1988), but its influence at lower pressures and temperatures remains untested. An advantage of the CF EOS is that it can be readily modified to incorporate molecular shape in a theoretically justifiable manner. In this work, the CF EOS is modified to account explicitly for molecules of non-spherical geometry, and applied to the prediction of pure component vapour-liquid equilibria of non-spherical molecules. THEORY
The Christoforakos - Franck equation of state The CF EOS may be written as p_RTV:+V;~+V,j32-f13 - T(iz - l)[exp(s/RT) - l] (1) VIII (Vnl-P)’ m where /I = b/4, and R, p, T, V, and b denote the universal gas constant, pressure, temperature, molar volume and co-volume, respectively. The first term of eqn. (1) is the Carnahan-Starling term for the interaction of hard spheres, which is an accurate description of hard-sphere repulsive interactions. Christoforakos and Franck (1986) modified the term in order to allow the parameter p to be deduced from a temperature-dependent expression, i.e. p = j3TC(TC/T)3’m
(2) where m is an arbitrary constant, typically assigned the value of 10. The second term is the contribution of attractive fluid-fluid interactions determined from the second virial coefficient of a hard-sphere fluid interacting via a square-well potential. The form of the square-well potential depends on both the depth ( -E) and the relative width (3L)of the well. Dipolar fluids exhibit attractive interactions over a narrow range of intermolecular distances; consequently, 1 is close to unity for these molecules and 1~ 1 for non-polar molecules which interact over a range of intermolecular distances. Subsequent development of the CF EOS has included a Pad6 approximation of the attractive interactions of a square-well potential (Heilig and Franck, 1989).
314
B. Plafkov
Modified
Christoforakos
et al. /Fluid
Phase Equilibria 95 (1994) 371-381
- Franck equation of state
For the work presented here, eqn. (1) was modified to account for the non-spherical molecular geometry by replacing the Carnahan-Starling (1969) hard-sphere term with an HCB representation (Sadus et al., 1988). The result is P=
RT[(V/U*)3 + (3a - 2)(V/v*)2 + (3c(‘- 3a + l)( V/v*) - ~(‘1 a -v’ V(V/v” - 1)’
(3)
where a = RTu*(13 - l)[exp(c/RT)
- l]
(4)
v* is the volume occupied by one mole of HCBs and M (defined below) represents the deviation from spherical geometry. The hard convex volume (v*) replaces the co-volume parameter (b) in the original CF EOS. The CF attractive term remains unaltered except for this substitution. The IJ* term can be made temperature dependent in a similar fashion to the co-volume parameter: v* = v&(T~/T)~‘~
(5)
where m = 10 as proposed by Christoforakos and Franck (1986). The first term in eqn. (3) is effectively the Boublik-Nezbeda relationship (Boublik, 1981) for the repulsive interaction of molecules of arbitrary shape. Hard convex body properties
A procedure described by Sadus et al. (1988) has been previously used to obtain the characteristic radius (R”), surface area (S*) and volume (V*) of an HCB. The physical dimensions of the molecule were examined to deduce the geometry of a non-variable hard core. This inner core was covered with an outer layer of uniform thickness (d) which determined the final shape of the HCB. The characteristic HCB dimensions (R*, V*, S*) were deduced from the corresponding core parameters, and the thickness of the hard convex layer from the formulae proposed by Kihara (1953). These data were used to determine the extent to which the molecule deviates from a sphere, i.e. c( = R=‘=S*/3V*
(6)
A value of CI= 1 indicates spherical symmetry, whereas values of c( in excess of unity indicate deviations from sphericity. When CI= 1, v* = b/4 and the repulsive terms of eqns. (1) and (3) are equivalent.
B. PlaEkov et aLlFluid Phase Equilibria 95 (1994) 371-381
315
Calculation of v* and E
Values of u* and E were evaluated conditions of a one-component fluid:
using eqn. (3) to solve the critical
(7) and p( T”, Vc) = p’. This calculation was performed and a.
at specified values of II
Vapour - liquid equilibrium calculation
The liquid phase of a pure substance will be in equilibrium with its vapour if the phases are of (i) equal temperature and pressure (mechanical equilibrium), and (ii) equal chemical potential: PI= p”
(8)
where the primes denote the vapour and liquid phases. For a pure substance, it is more convenient to replace eqn. (8) with the equivalent condition that the Gibbs function of molecules of either phase is equal: G’ = G”
(9)
The Gibbs function can be obtained from the Helmholtz function (A) via the standard thermodynamic relationship G = A +p V, where A can be directly evaluated by integrating the EOS: A=
-
s
pdV
The result of this integration A=
(10) is
-RT[&nV+(l-a2)ln(V/u*-l)-(a2+3a)(V/u*-1))’ - a”( V/u* - 1) -‘I - aV_’
(11)
The procedure employed for determining the vapour pressure of the pure substances involved using eqn. (3) to determine the volumes of the liquid and vapour phases at specified temperature and pressure, and establishing whether or not criterion (9) was satisfied. The pressure was modified until the calculated volumes of the vapour and liquid phases gave equal Gibbs functions. Because of their complexity, the integration of eqn. (lo), and the first and second derivatives of pressure with respect to volume, were verified by numerical integration and differentiation, respectively.
B. PlaCkov et al. 1 Fluid Phase Equilibria 9.5 (1994) 371-381
376 RESULTS
AND
DISCUSSION
The vapour-liquid equilibrium properties of 17 one-component fluids were calculated and compared with experimental data. The systems chosen represent a diverse range of molecular size and shape. There exist ample vapour pressure data for a variety of fluids, although the measurements are often restricted to a relatively narrow temperature range. The sources of the pure component vapour pressure data employed are given in Table 1. The critical property data were those of Ambrose and Townsend (1977), with the exception of the critical molar volumes of 1,2-dichloroethane (225 cm3), n-nonane (552 cm3) and n-decane (612 cm3) which were estimated. The values of a were those of Sadus (1987). TABLE
1
Predicted vapour pressures using the Christoforakos-Franck forakos-Franck (modified CF) equations of state Substance
T, range
N
CF 1
(CF)
Modified AAD (%)
c(
and modified
Ref.
CF
1
Christo-
AAD (%)
Neon Argon Krypton Xenon
0.56-0.99 0.56-1.00 0.55-1.00 0.56-0.99
20 69 48 65
4.6 3.5 3.5 3.4
5.3 2.9 3.2 3.3
n-Pentane n -Hexane n-Heptane n-Octane n -Nonane n-Decane
0.40-1.00 0.42-0.99 0.40- 1.00 0.41-0.99 0.41-0.85 0.4330.85
47 31 45 36 34 41
1.9 1.8 1.7 1.6 1.6 1.6
16.3 19.0 22.5 26.7 26.9 33.8
1.374 1.434 1.456 1.489 1.497 1.515
3.5 3.3 3.2 2.9 2.9 2.8
14.6 17.4 19.7 23.8 25.0 31.3
3,4, 5 1,3,6 l-3, 7 1,398 1,3,7 l-9
Cyclopentane Cyclohexane Cycloheptane Benzene
0.44-0.63 0.50-1.00 0.56-0.72 0.53-0.98
22 2.1 33 2.0 15 1.9 38 2.0
14.6 9.4 7.8 9.3
1.085 1.100 1.118 1.196
3.4 3.3 3.0 3.3
14.2 8.4 8.0 8.5
7, 10 7, 11, 12 13 14, 15
Dichloromethane 1,2-Dichloroethane Hexamethyldisiloxane
0.52-1.00 0.43-1.00 0.60-0.80
4.5 21.7 13.3
1.155 1.200 1.224
3.4 3.0 2.6
3.8 21.0 12.8
16,17 18,19 20
113 83 21
2.0 1.8 1.6
1 2 1 1
‘Vargaftik (1975); 2Stewart and Jacobsen (1989); 3Carruth and Kobayashi (1973); 4Messerly and Kennedy (1940); 5Das et al. (1977); 6Letcher and Marsicano (1974); 7Boublik et al. (1984); *Wu et al. (1991); ‘Chirico et al. (1989); “Aston et al. (1943a); “Aston et al. (1943b); 12Hugill and McGlashan (1978); 13Finke et al. (1956); 14Ambrose (1981); “Ambrose (1987); 16Boublik and Aim (1972); ‘7Garcia-Sanchez et al. ( 1989); 18Pearce and Peters (1929); “Garcia-Sanchez and Trejo (1985); *OScott et al. (1961).
0.
PlaZkov et a/./Fluid Phase Equilibria 95 (1994
371-381
377
The CF EOS explicitly assumes that intermolecular interaction can be described by the interaction of hard-sphere entities. In order to test the accuracy of the EOS for spherical molecules, the vapour-liquid equilibria of the noble gases were calculated and compared with experimental data. Table 1 summarizes the values of the average absolute deviation (AAD) defined by AAD
= ( 1/N)
“t---f-‘c
g
i=
(12)
1
The CF EOS generally predicts the vapour pressure of the noble gases at temperatures ranging between the triple point and the critical temperature with an accuracy of approximately 3%. The larger deviation for neon (5.3%) can possibly be attributed to quantum effects at low temperatures. The agreement between theory and experiment deteriorates substantially when the EOS is applied to demonstrably non-spherical fluids (Table 1). An error of < 10% is typically obtained for small non-spherical or symmetric molecules, such as the cycloalkanes or dichloromethane. However, the discrepancy for the linear n-alkanes is unacceptably large ( > 20%) and increases progressively with chain length. It can be reasonably expected that the agreement between theory and experiment could be improved by accounting for the molecular geometry of the fluid. Consequently, the analysis was repeated using eqn. (3) which attempts to take molecular geometry into account. The value of the L parameter is increased because the volume of the HCB has replaced the co-volume term in the attractive part of the EOS. The error in the observed vapour pressure is reduced. However, the calculations remain quantitatively inaccurate and it is evident that molecular shape, as modelled by the HCB approximation, has only a relatively small effect on vapour-liquid equilibria. This is illustrated in Fig. 1 for the vapour pressure of n-pentane. A similar conclusion was reached by Sadus et al. (1988) who used the HCB equation to calculate the gas-liquid critical properties of binary mixtures. This insensitivity to molecular shape arises despite the demonstrably uon-spherical nature of the molecules investigated, and the important role of molecular shape in phenomena such as the excess functions of mixing. There is scope for improving the prediction of pure component vapourliquid equilibria via theoretically justified improvements to EOS. For example, van Pelt et al. (van Pelt, 1992; van Pelt et al., 1992) have recently used a simplified perturbed hard chain theory (SPHCT) EOS to correlate accurately the vapour pressure- temperature behaviour of several substances. In common with the CF EOS, the SPHCT EOS includes a contribution from the Carnahan-Starling hard-sphere term. The accuracy of their
378
a.
0 0.4
0.8
0.6
PIaEkov et al. / Fluid Phase Equilibria 95 (1994) 371-381
1.0
T/TC
Fig. 1. The temperature dependence of the relative deviation of pressure for n-pentane the modified Christoforakos-Franck equation of state.
with
analysis partly relied on using the critical volume as an arbitrary adjustable parameter. CONCLUSION
Despite improvements made to both the repulsive and attractive contributions of intermolecular interactions, the CF model and the modified CF EOS cannot be used to predict accurately the vapour pressures of medium or large non-spherical molecules. It is well known that the Carnahan-StarIing and Boublik-Nezbeda potentials are accurate models of the interaction of hard bodies, even at the densities encountered for the vapour pressure curve. It appears likely that the incorporation of molecular shape in the conventional equation of state methodology does not make an important contribution to the quality of the prediction of vapour-liquid equilibria. This conclusion is consistent with earlier work on the gas-liquid critical properties of binary mixtures. LIST OF SYMBOLS
a A b m
parameter incorporating the attractive term (proportional der Waals EOS) molar Helmholtz energy co-volume hard-sphere parameter arbitrary constant
to the van
a. Plaekov et aLlFluid Phase Equilibria 95 (1994) 371-381
N P
R R* S*
T V V’” V*
379
number of data pressure universal gas constant mean radius of a hard convex body surface area of a hard convex body temperature volume volume of a hard convex body volume of one mole of hard convex bodies
Greek letters
& rz P
molecule sphericity parameter in the CF model depth of square-well potential relative width of square-well potential chemical potential
Subscripts
talc exp m
calculated experimental molar
Superscript C
critical
REFERENCES Ambrose, D., 1981. Reference values of vapour pressure. The vapour pressures of benzene and hexafluorobenzene. J. Chem. Thermodyn., 13: 1161- 1167. Ambrose, D., 1987. Vapour pressures of some aromatic hydrocarbons. J. Chem. Thermodyn., 19: 1007-1008. Ambrose, D. and Townsend, R., 1977. Vapour-liquid critical properties. National Physical Laboratory, Teddington. Aston, J.G., Fink, H.L. and Schumann, SC., 1943a. The heat capacity and entropy, heats of transition, fusion and vaporization and the vapour pressures of cyclopentane. Evidence for a non-planar structure. J. Am. Chem. Sot., 65: 341-346. Aston, J.G., Szasz, G.J. and Fink, H.L., 1943b. The heat capacity and entropy, heats of transition, fusion and vaporization and the vapour pressures of cyclohexane. The vibrational frequencies of alicychc ring systems. J. Am. Chem. Sot., 65: 1135- 1139. Boublik, T., 1981. Statistical thermodynamics of nonspherical molecule fluids. Ber. Bunsenges. Phys. Chem., 85: 1038-1041. Boublik, T. and Aim, K., 1972. Heat of vaporization of simple non-spherical molecule compounds. Collect. Czech. Chem. Commun., 37: 3513-3521.
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B. PlaCkov et al. / Fluid Phase Equilibria 95 (1994) 371-381
Boublik, T., Fried, V. and Hala, E.. 1984. The vapour pressures of pure substances. Elsevier, Amsterdam. Carnahan. N.F. and Starling, K.E., 1969. Equation of state for nonattracting rigid spheres. J. Chem. Phys., 51: 6355636. Carnahan, N.F. and Starling, K.E., 1972. Intermolecular repulsions and the equation of state for fluids. AIChE J., 18: 1184-l 189. Carruth, G.F. and Kobayashi. R., 1973. Vapor pressure of normal paraffins ethane through n-decane from their triple points to about 10 mm Hg. J. Chem. Eng. Data, 18: 115126. Chirico, R.D., Nguyen, A., Steele, W.V., Strube, M.M. and Tsonopoulos, C., 1989. Vapor pressure of n-alkanes revisited. New high-precision vapor pressure data on n-decane, n-eicosane, and n-octacosane. J. Chem. Eng. Data, 34: 149-156. Christoforakos, M. and Franck, E.U., 1986. An equation of state for binary fluid mixtures to high temperatures and high pressures. Ber. Bunsenges. Phys. Chem., 90: 780-789. Das, T.R., Reed, Jr., C.O. and Eubank, P.T., 1977. PVT surface and thermodynamic properties of n-pentane. J. Chem. Eng. Data, 22: 3-9. Finke, H.L., Scott, D.W., Gross, M.E., Messerly, J.F. and Waddington, G., 1956. Cycloheptane, cyclooctane and 1,3,5cycloheptatriene. Low temperature thermal properties, vapor pressure and derived chemical thermodynamic properties. J. Am. Chem. Sot., 5: 54695476. Garcia-Sanchez, F. and Trejo, A., 1985. Vapour pressure and critical constants of 1,2dichloroethane. J. Chem. Thermodyn., 17: 981-983. Garcia-Sanchez, F., Romero-Martinez, A. and Trejo, A., 1989. Vapour pressure, critical temperature, and critical pressure of dichloromethane. J. Chem. Thermodyn., 21: 823-826. Guggenheim, E.A., 1965. Variations on van der Waals’ equation of state for high densities. Mol. Phys., 9: 199-200. Heilig, M. and Franck, E.U., 1989. Calculation of thermodynamic properties of binary fluid mixtures to high temperatures and high pressures. Ber. Bunsenges. Phys. Chem., 93: 898-905. Hugill, J.A. and McGlashan, M.L., 1978. The vapour pressure from 451 K to the critical temperature, and the critical temperature and critical pressure, of cyclohexane. J. Chem. Thermodyn., 10: 95-100. Kihara, T.. 1953. Virial coefficients and models of molecules in gases. Rev. Mod. Phys., 25: 831-843. Kihara, T., 1963. Convex molecules in gaseous and crystalline states. Adv. Chem. Phys. 5: 147-188. Letcher, T.M. and Marsicano, F., 1974. Vapour pressure and densities of some unsaturated C, acyclic and cyclic hydrocarbons between 300 and 320 K. J. Chem. Thermodyn., 6: 509-514. Messerly, G.H. and Kennedy, R.M., 1940. The heat capacity and entropy, heats of fusion and vaporization and the vapor pressure of n-pentane. J. Am. Chem. Sot., 62: 2988-299 I. Pearce, J.N. and Peters, P.E., 1929. The vapor pressure of ethylene chloride between - 30” and 100”. J. Phys. Chem., 33: 873-878. Peng, D.-Y. and Robinson, D.B., 1976. A new two-constant equation of state. Ind. Eng. Chem., Fundam., 15: 59-64. Redlich, 0. and Kwong, J.N.S., 1949. On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem. Rev., 44: 233-244. Sadus, R.J., 1987. Phase behaviour of binary and ternary mixtures. Doctoral Dissertation, University of Melbourne, Australia.
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et aLlFluid Phase Equilibria 95 (1994
371-381
381
Sadus, R.J., Young, C.L. and Svejda, P., 1988. Application of hard convex body and hard sphere equations of state to the critical properties of binary mixtures. Fluid Phase Equilibria, 39: 89-99. Siddiqi, M.A., Svejda, P. and Kohler, F., 1983. A generalized van der Waals equation of state II. Excess heat capacities of mixtures containing cycloalkanes (C,, C,), methylcycloalkanes (C,, C,) and n-decane. Ber. Bunsenges. Phys. Chem., 87: 1176-l 181. Scott, D.W., Messerly, J.F., Todd, S.S., Guthrie, G.B., Hossenlopp, LA., Moore, R.T., Osborn, A., Berg, W.T. and McCullough, J.P., 1961. Hexamethyldisiloxane: Chemical thermodynamic properties and internal rotation about the siloxane linkage. J. Phys. Chem., 65: 1320- 1326. Soave, G., 1984. Improvement of the van der Waals equation of state. Chem. Eng. Sci., 39: 357-369. Stewart, R.B. and Jacobsen, R.T., 1989. Thermodynamic properties of argon from the triple point to 1200 K with pressure to 1000 MPa. J. Phys. Chem. Ref. Data, 18: 639-798. Svejda, P. and Kohler. F., 1983. A generalised van der Waals equation of state I. Treatment of molecular shape in terms of the Boublik-Nezbeda equation. Ber. Bunsenges. Phys. Chem., 87: 672-680. Van Pelt, A., 1992. Critical phenomena in binary fluid mixtures. Classification of phase equilibria with the simplified-Perturbed-Hard-Chain-theory. Doctoral Dissentation, Delft University of Technology, The Netherlands. Van Pelt, A., Peters, C.J. and de Swaan Arons, J., 1992. Application of the simplified-Perturbed-Hard-Chain Theory for pure components near the critical point. Fluid Phase Equilibria, 74: 67 - 83. Vargaftik, N.B., 1975. Handbook of physical properties of liquids and gases. Hemisphere, Washington, DC. Wu, H.S., Locke, III, W.E. and Sandler, S.I., 1991. Isothermal vapor-liquid equilibrium of binary mixtures containing morpholine. J. Chem. Eng. Data, 36: 127-130.
95 (1994) 371-381
371
Science B.V.
Short Communication
Influence of molecular Application
of a modified
shape on vapour-liquid Christoforakos-Franck
equilibria
equation
of state
Dorde PlaEkov a,b, David E. Mainwaring a** and Richard J. Sadus b a Department of Applied Chemistry, Swinburne University of Technology, PO Box 218, Hawthorn, 3122 Vie. (Australia) b Department of Computer Science, Swinburne of Technology, PO Box 218, Hawthorn, 3122 Vie. (Australia) (Received
January
26, 1993; accepted
in final form November
University
22, 1993)
ABSTRACT PlaEkov, D., Mainwaring, D.E. and Sadus, R.J., 1994. Influence of molecular shape vapour-liquid equilibria. Application of a modified Christoforakos-Franck equation state. Fluid Phase Equilibria, 95: 371-381.
on of
The Christoforakos-Franck equation of state, originally derived for spherical molecules, is modified to account for deviations from spherical geometry. The contribution from repulsive interactions is obtained from the Boublik-Nezbeda model for the compressibility of hard bodies of arbitrary shape. The deviation from sphericity (c() is deduced from the hard convex body model of molecular geometry. Attractive intermolecular interactions are characterized by the depth ( - E) and relative width (n) of a square-well potential. The influence of molecular shape on the vapour-liquid equilibrium properties of pure fluids is investigated by using the modified equation to predict the vapour pressure of non-spherical molecules. It is concluded that incorporating molecular shape in the conventional equation of state methodology does not significantly influence the quality of the predicted vapour-liquid equilibria. Keywords:
theory,
equation
of state, vapour-liquid
equilibria,
non-spherical
molecules.
INTRODUCTION
Equations of state (EOS) have assumed an expanding role in the study of the phase equilibria of fluids and fluid mixtures. They can be applied to the gas, liquid and supercritical phases without encountering any conceptual difficulties, and all the physical fluid properties (i.e. composition, temperature, pressure and volume) can be obtained simultaneously. However, there is no universally applicable EOS. Empirical equations, which have been *Corresponding
author.
037%3812/94/$07.00 0 1994 - Elsevier SSDZ 0378-3812(93)02453-T
Science B.V. All rights reserved
372
B. PlaEkov et al. / Fluid Phase Equilibria 95 (1994) 371-381
modified to yield optimal agreement at low pressures, often fail to predict accurately the properties of fluids at high pressures, and vice versa. The development of EOS models usually conforms to the van der Waals formalism, i.e. the contribution of repulsive and attractive fluid-fluid interactions are treated separately. A dichotomy has emerged between EOS which retain the inadequate van der Waals hard-sphere repulsive term plus an empirically improved attractive term, and EOS which incorporate a theoretically better description of repulsive fluid interactions. The RedlichKwong (Redlich and Kwong, 1949), Peng-Robinson (Peng and Robinson, 1984) equations are examples of empirically 1976) and Soave (Soave, improved equations which have been used widely in chemical engineering applications, whereas the Carnahan-Starling (Carnahan and Starling, 1969, 1965) and hard convex body (HCB) 1972) Guggenheim (Guggenheim, (Svejda and Kohler, 1983) equations exemplify the latter approach. An accurate description of repulsive interaction is particularly important in order to predict high-pressure equilibria, whereas attractive interactions are a more important consideration at lower pressures. EOS at or near the critical state suffer from the limitation that the critical state is not an analytical function of energy. Nevertheless, it is important to use an accurate representation of repulsive interactions when predicting critical phenomena. Christoforakos and Franck (CF) (Christoforakos and Franck, 1986) have proposed an EOS with some promising features. The CF EOS has been applied to the critical properties of a series of binary mixtures containing water plus a small, non-polar component. Using two adjustable parameters, the predicted critical locus of the water + carbon dioxide mixture compares very well with experimental data. The results are better than can be obtained from other accurate hard-sphere equations such as the Guggenheim model (Guggenheim, 1965). It is of theoretical and engineering interest to examine whether the EOS can improve the agreement of theory with experiment for other aspects of phase equilibria. It must be emphasized that the aim of this work is not the accurate prediction of vapour pressure data. There are many empirical approaches which can be used to correlate vapour-liquid equilibrium data with a high degree of accuracy. These good results are achieved by “empirical” improvements which are devoid of physical meaning or rationale. Often they provide no insight into the fundamental weakness of the model. They “correlate” rather than “predict” vapour pressure data. Because this approach is so prevalent in the chemical literature, relatively little quantitative data is available about the failure of theoretical models. Data of this type are valuable in assessing the limitations of theory and providing insight into improved EOS. Examining the properties of pure substances is an important starting point for the phase behaviour of fluid mixtures in general. Applying
B. PlaCkov et aLlFluid Phase Equilibria 95 (1994) 371-381
313
the EOS to the vapour-liquid equilibria of a pure substance is a direct measure of the assumptions made concerning the nature of intermolecular interactions in the fluids. In contrast, the analysis of multicomponent equilibria is complicated by uncertainties introduced by the nature of mixing rules and combining rules for unlike intermolecular interactions. The CF EOS was specifically derived for spherical molecules. It has been demonstrated that molecular shape has a profound effect on such properties as virial coefficients (Kihara, 1963) and excess functions (Siddiqi et al., 1983). The vapour-liquid critical point is largely uninfluenced by molecular shape (Sadus et al., 1988), but its influence at lower pressures and temperatures remains untested. An advantage of the CF EOS is that it can be readily modified to incorporate molecular shape in a theoretically justifiable manner. In this work, the CF EOS is modified to account explicitly for molecules of non-spherical geometry, and applied to the prediction of pure component vapour-liquid equilibria of non-spherical molecules. THEORY
The Christoforakos - Franck equation of state The CF EOS may be written as p_RTV:+V;~+V,j32-f13 - T(iz - l)[exp(s/RT) - l] (1) VIII (Vnl-P)’ m where /I = b/4, and R, p, T, V, and b denote the universal gas constant, pressure, temperature, molar volume and co-volume, respectively. The first term of eqn. (1) is the Carnahan-Starling term for the interaction of hard spheres, which is an accurate description of hard-sphere repulsive interactions. Christoforakos and Franck (1986) modified the term in order to allow the parameter p to be deduced from a temperature-dependent expression, i.e. p = j3TC(TC/T)3’m
(2) where m is an arbitrary constant, typically assigned the value of 10. The second term is the contribution of attractive fluid-fluid interactions determined from the second virial coefficient of a hard-sphere fluid interacting via a square-well potential. The form of the square-well potential depends on both the depth ( -E) and the relative width (3L)of the well. Dipolar fluids exhibit attractive interactions over a narrow range of intermolecular distances; consequently, 1 is close to unity for these molecules and 1~ 1 for non-polar molecules which interact over a range of intermolecular distances. Subsequent development of the CF EOS has included a Pad6 approximation of the attractive interactions of a square-well potential (Heilig and Franck, 1989).
314
B. Plafkov
Modified
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et al. /Fluid
Phase Equilibria 95 (1994) 371-381
- Franck equation of state
For the work presented here, eqn. (1) was modified to account for the non-spherical molecular geometry by replacing the Carnahan-Starling (1969) hard-sphere term with an HCB representation (Sadus et al., 1988). The result is P=
RT[(V/U*)3 + (3a - 2)(V/v*)2 + (3c(‘- 3a + l)( V/v*) - ~(‘1 a -v’ V(V/v” - 1)’
(3)
where a = RTu*(13 - l)[exp(c/RT)
- l]
(4)
v* is the volume occupied by one mole of HCBs and M (defined below) represents the deviation from spherical geometry. The hard convex volume (v*) replaces the co-volume parameter (b) in the original CF EOS. The CF attractive term remains unaltered except for this substitution. The IJ* term can be made temperature dependent in a similar fashion to the co-volume parameter: v* = v&(T~/T)~‘~
(5)
where m = 10 as proposed by Christoforakos and Franck (1986). The first term in eqn. (3) is effectively the Boublik-Nezbeda relationship (Boublik, 1981) for the repulsive interaction of molecules of arbitrary shape. Hard convex body properties
A procedure described by Sadus et al. (1988) has been previously used to obtain the characteristic radius (R”), surface area (S*) and volume (V*) of an HCB. The physical dimensions of the molecule were examined to deduce the geometry of a non-variable hard core. This inner core was covered with an outer layer of uniform thickness (d) which determined the final shape of the HCB. The characteristic HCB dimensions (R*, V*, S*) were deduced from the corresponding core parameters, and the thickness of the hard convex layer from the formulae proposed by Kihara (1953). These data were used to determine the extent to which the molecule deviates from a sphere, i.e. c( = R=‘=S*/3V*
(6)
A value of CI= 1 indicates spherical symmetry, whereas values of c( in excess of unity indicate deviations from sphericity. When CI= 1, v* = b/4 and the repulsive terms of eqns. (1) and (3) are equivalent.
B. PlaEkov et aLlFluid Phase Equilibria 95 (1994) 371-381
315
Calculation of v* and E
Values of u* and E were evaluated conditions of a one-component fluid:
using eqn. (3) to solve the critical
(7) and p( T”, Vc) = p’. This calculation was performed and a.
at specified values of II
Vapour - liquid equilibrium calculation
The liquid phase of a pure substance will be in equilibrium with its vapour if the phases are of (i) equal temperature and pressure (mechanical equilibrium), and (ii) equal chemical potential: PI= p”
(8)
where the primes denote the vapour and liquid phases. For a pure substance, it is more convenient to replace eqn. (8) with the equivalent condition that the Gibbs function of molecules of either phase is equal: G’ = G”
(9)
The Gibbs function can be obtained from the Helmholtz function (A) via the standard thermodynamic relationship G = A +p V, where A can be directly evaluated by integrating the EOS: A=
-
s
pdV
The result of this integration A=
(10) is
-RT[&nV+(l-a2)ln(V/u*-l)-(a2+3a)(V/u*-1))’ - a”( V/u* - 1) -‘I - aV_’
(11)
The procedure employed for determining the vapour pressure of the pure substances involved using eqn. (3) to determine the volumes of the liquid and vapour phases at specified temperature and pressure, and establishing whether or not criterion (9) was satisfied. The pressure was modified until the calculated volumes of the vapour and liquid phases gave equal Gibbs functions. Because of their complexity, the integration of eqn. (lo), and the first and second derivatives of pressure with respect to volume, were verified by numerical integration and differentiation, respectively.
B. PlaCkov et al. 1 Fluid Phase Equilibria 9.5 (1994) 371-381
376 RESULTS
AND
DISCUSSION
The vapour-liquid equilibrium properties of 17 one-component fluids were calculated and compared with experimental data. The systems chosen represent a diverse range of molecular size and shape. There exist ample vapour pressure data for a variety of fluids, although the measurements are often restricted to a relatively narrow temperature range. The sources of the pure component vapour pressure data employed are given in Table 1. The critical property data were those of Ambrose and Townsend (1977), with the exception of the critical molar volumes of 1,2-dichloroethane (225 cm3), n-nonane (552 cm3) and n-decane (612 cm3) which were estimated. The values of a were those of Sadus (1987). TABLE
1
Predicted vapour pressures using the Christoforakos-Franck forakos-Franck (modified CF) equations of state Substance
T, range
N
CF 1
(CF)
Modified AAD (%)
c(
and modified
Ref.
CF
1
Christo-
AAD (%)
Neon Argon Krypton Xenon
0.56-0.99 0.56-1.00 0.55-1.00 0.56-0.99
20 69 48 65
4.6 3.5 3.5 3.4
5.3 2.9 3.2 3.3
n-Pentane n -Hexane n-Heptane n-Octane n -Nonane n-Decane
0.40-1.00 0.42-0.99 0.40- 1.00 0.41-0.99 0.41-0.85 0.4330.85
47 31 45 36 34 41
1.9 1.8 1.7 1.6 1.6 1.6
16.3 19.0 22.5 26.7 26.9 33.8
1.374 1.434 1.456 1.489 1.497 1.515
3.5 3.3 3.2 2.9 2.9 2.8
14.6 17.4 19.7 23.8 25.0 31.3
3,4, 5 1,3,6 l-3, 7 1,398 1,3,7 l-9
Cyclopentane Cyclohexane Cycloheptane Benzene
0.44-0.63 0.50-1.00 0.56-0.72 0.53-0.98
22 2.1 33 2.0 15 1.9 38 2.0
14.6 9.4 7.8 9.3
1.085 1.100 1.118 1.196
3.4 3.3 3.0 3.3
14.2 8.4 8.0 8.5
7, 10 7, 11, 12 13 14, 15
Dichloromethane 1,2-Dichloroethane Hexamethyldisiloxane
0.52-1.00 0.43-1.00 0.60-0.80
4.5 21.7 13.3
1.155 1.200 1.224
3.4 3.0 2.6
3.8 21.0 12.8
16,17 18,19 20
113 83 21
2.0 1.8 1.6
1 2 1 1
‘Vargaftik (1975); 2Stewart and Jacobsen (1989); 3Carruth and Kobayashi (1973); 4Messerly and Kennedy (1940); 5Das et al. (1977); 6Letcher and Marsicano (1974); 7Boublik et al. (1984); *Wu et al. (1991); ‘Chirico et al. (1989); “Aston et al. (1943a); “Aston et al. (1943b); 12Hugill and McGlashan (1978); 13Finke et al. (1956); 14Ambrose (1981); “Ambrose (1987); 16Boublik and Aim (1972); ‘7Garcia-Sanchez et al. ( 1989); 18Pearce and Peters (1929); “Garcia-Sanchez and Trejo (1985); *OScott et al. (1961).
0.
PlaZkov et a/./Fluid Phase Equilibria 95 (1994
371-381
377
The CF EOS explicitly assumes that intermolecular interaction can be described by the interaction of hard-sphere entities. In order to test the accuracy of the EOS for spherical molecules, the vapour-liquid equilibria of the noble gases were calculated and compared with experimental data. Table 1 summarizes the values of the average absolute deviation (AAD) defined by AAD
= ( 1/N)
“t---f-‘c
g
i=
(12)
1
The CF EOS generally predicts the vapour pressure of the noble gases at temperatures ranging between the triple point and the critical temperature with an accuracy of approximately 3%. The larger deviation for neon (5.3%) can possibly be attributed to quantum effects at low temperatures. The agreement between theory and experiment deteriorates substantially when the EOS is applied to demonstrably non-spherical fluids (Table 1). An error of < 10% is typically obtained for small non-spherical or symmetric molecules, such as the cycloalkanes or dichloromethane. However, the discrepancy for the linear n-alkanes is unacceptably large ( > 20%) and increases progressively with chain length. It can be reasonably expected that the agreement between theory and experiment could be improved by accounting for the molecular geometry of the fluid. Consequently, the analysis was repeated using eqn. (3) which attempts to take molecular geometry into account. The value of the L parameter is increased because the volume of the HCB has replaced the co-volume term in the attractive part of the EOS. The error in the observed vapour pressure is reduced. However, the calculations remain quantitatively inaccurate and it is evident that molecular shape, as modelled by the HCB approximation, has only a relatively small effect on vapour-liquid equilibria. This is illustrated in Fig. 1 for the vapour pressure of n-pentane. A similar conclusion was reached by Sadus et al. (1988) who used the HCB equation to calculate the gas-liquid critical properties of binary mixtures. This insensitivity to molecular shape arises despite the demonstrably uon-spherical nature of the molecules investigated, and the important role of molecular shape in phenomena such as the excess functions of mixing. There is scope for improving the prediction of pure component vapourliquid equilibria via theoretically justified improvements to EOS. For example, van Pelt et al. (van Pelt, 1992; van Pelt et al., 1992) have recently used a simplified perturbed hard chain theory (SPHCT) EOS to correlate accurately the vapour pressure- temperature behaviour of several substances. In common with the CF EOS, the SPHCT EOS includes a contribution from the Carnahan-Starling hard-sphere term. The accuracy of their
378
a.
0 0.4
0.8
0.6
PIaEkov et al. / Fluid Phase Equilibria 95 (1994) 371-381
1.0
T/TC
Fig. 1. The temperature dependence of the relative deviation of pressure for n-pentane the modified Christoforakos-Franck equation of state.
with
analysis partly relied on using the critical volume as an arbitrary adjustable parameter. CONCLUSION
Despite improvements made to both the repulsive and attractive contributions of intermolecular interactions, the CF model and the modified CF EOS cannot be used to predict accurately the vapour pressures of medium or large non-spherical molecules. It is well known that the Carnahan-StarIing and Boublik-Nezbeda potentials are accurate models of the interaction of hard bodies, even at the densities encountered for the vapour pressure curve. It appears likely that the incorporation of molecular shape in the conventional equation of state methodology does not make an important contribution to the quality of the prediction of vapour-liquid equilibria. This conclusion is consistent with earlier work on the gas-liquid critical properties of binary mixtures. LIST OF SYMBOLS
a A b m
parameter incorporating the attractive term (proportional der Waals EOS) molar Helmholtz energy co-volume hard-sphere parameter arbitrary constant
to the van
a. Plaekov et aLlFluid Phase Equilibria 95 (1994) 371-381
N P
R R* S*
T V V’” V*
379
number of data pressure universal gas constant mean radius of a hard convex body surface area of a hard convex body temperature volume volume of a hard convex body volume of one mole of hard convex bodies
Greek letters
& rz P
molecule sphericity parameter in the CF model depth of square-well potential relative width of square-well potential chemical potential
Subscripts
talc exp m
calculated experimental molar
Superscript C
critical
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Sadus, R.J., Young, C.L. and Svejda, P., 1988. Application of hard convex body and hard sphere equations of state to the critical properties of binary mixtures. Fluid Phase Equilibria, 39: 89-99. Siddiqi, M.A., Svejda, P. and Kohler, F., 1983. A generalized van der Waals equation of state II. Excess heat capacities of mixtures containing cycloalkanes (C,, C,), methylcycloalkanes (C,, C,) and n-decane. Ber. Bunsenges. Phys. Chem., 87: 1176-l 181. Scott, D.W., Messerly, J.F., Todd, S.S., Guthrie, G.B., Hossenlopp, LA., Moore, R.T., Osborn, A., Berg, W.T. and McCullough, J.P., 1961. Hexamethyldisiloxane: Chemical thermodynamic properties and internal rotation about the siloxane linkage. J. Phys. Chem., 65: 1320- 1326. Soave, G., 1984. Improvement of the van der Waals equation of state. Chem. Eng. Sci., 39: 357-369. Stewart, R.B. and Jacobsen, R.T., 1989. Thermodynamic properties of argon from the triple point to 1200 K with pressure to 1000 MPa. J. Phys. Chem. Ref. Data, 18: 639-798. Svejda, P. and Kohler. F., 1983. A generalised van der Waals equation of state I. Treatment of molecular shape in terms of the Boublik-Nezbeda equation. Ber. Bunsenges. Phys. Chem., 87: 672-680. Van Pelt, A., 1992. Critical phenomena in binary fluid mixtures. Classification of phase equilibria with the simplified-Perturbed-Hard-Chain-theory. Doctoral Dissentation, Delft University of Technology, The Netherlands. Van Pelt, A., Peters, C.J. and de Swaan Arons, J., 1992. Application of the simplified-Perturbed-Hard-Chain Theory for pure components near the critical point. Fluid Phase Equilibria, 74: 67 - 83. Vargaftik, N.B., 1975. Handbook of physical properties of liquids and gases. Hemisphere, Washington, DC. Wu, H.S., Locke, III, W.E. and Sandler, S.I., 1991. Isothermal vapor-liquid equilibrium of binary mixtures containing morpholine. J. Chem. Eng. Data, 36: 127-130.