2298
J. Phys. Chem. 1992, 96, 2298-2301
Equilibrium Behavior of n-Alkanes from the Convex Molecule Perturbation Theory Jan Pavlkek and Tomig Boubdk* Institute of Chemical Process Fundamentals, Czechoslovak Academy of Sciences, 165 02 Prague 6, Czechoslovakia (Received: August 13, 1991)
Perturbation theory of convex molecule fluids is applied to determine the orthobaric data along the coexistence curve of n-alkanes from ethane to hexadecane. Equilibrium pressures and densities in the liquid and vapor phases are evaluated at the reduced temperatures P < 0.9 from the perturbation scheme for the Kihara molecules of the rodlike shape. Fair agreement of the calculated and experimental data is found for all the studied systems. Parameters of the Kihara pair potential obtained from the optimization procedure depend in a simple way on the number of C atoms in the respective molecules; the rod length increases linearly whereas dependencies of clk and u are nonlinear. Regularities in the dependence of the Kihara parameters on the number of C atoms in molecules allow a fair prediction of the behavior of higher n-alkanes.
Introduction Knowledge of methods enabling in a unique way the determination of thermodynamic functions of fluids in the broad range of independent variables is of great importance both in the theory of equilibrium processes and in chemical engineering applications. From methods of statistical thermodynamics the perturbation theory of fluids is the most general and widely applicable. For fluids of molecules of nonspherical shape two variants of the perturbation theory have been proposed which differ in the assumed type of pair potential describing intermolecular interactions. For systems of molecules interacting via the multicenter pair potential (e.g., the two-center Lennard-Jones), the first-order perturbation expansion has been extensively studied by Kohler and Fi~cher.'-~ We have proposed a simple perturbation methodM for systems of convex molecules interacting via the Kihara pair potential. Because the Kihara potential7 depends even for nonspherical molecules only on a single variable, the theory leads to simple integrals which, under simplifying approximations, can be expressed This simplicity enables broad applications in describing the equilibrium behavior of fluids. Besides the behavior of nonpolar fluids (see ref 9) we have recently considered also the problem of characterizing systems composed of nonspherical polar molecules.I0-l2 With the aim of studying a certain series of compounds and with the recently increasing interest in the behavior of n-alkanesI3 we have applied the variant of the perturbation theory of convex molecules to a series of n-alkanes, from ethane to n-hexadecane; for these compounds the necessary equilibrium data are available in the literature; data for higher n-alkanes (than Ck6H3.,) are scarce. We employed an optimization procedure to find the parameters of the Kihara potential from the orthobaric data. Three types of cores were considered in the preliminary studies: (i) planar combinations of (2x4triangles, (ii) hard ellipsoids of revolution, and (iii) hard rods. The rodlike cores were taken into account in the final optimization procedure and results for this choice are given. The values of the parameters for this model disclose considerable regularity in their dependence on the number of C (1) Fischer, J. J . Chem. Phys. 1980, 72, 5371. (2) Bohn, M.; Lustig, R.; Fischer, J. Fluid Phase Equilib. 1986, 25, 25 1 . ( 3 ) Bohn,M.; Fischer, J.; Kohler, F. Fluid Phase Equilib. 1986, 31, 233. (4) Boubljk, T. Mol. Phys. 1976, 32, 1737. (5) Boubljk, T. J . Chem. Phys. 1987, 87, 1751. (6) Boublik, T. J . Phys. Chem. 1988, 92, 2629. (7) Kiharg, T. Ado. Chem. Phys. 1963, 5,47. ( 8 ) Boublik, T. Fluid Phase Equilib. 1990, 54, 221. (9) Boublik, T. Pure Appl. Chem. 1989, 61, 993. (10) Boublik, T. Mol. Phys. 1990, 69, 497. (11) Boublik, T. Mol. Phys. 1990, 71, 1193. (1 2) Boublik, T. Mol. Phys., in press. (1 3) 6th International IUPAC Workshop in V-L-E in 1-alkamol + n-alkane mixtures, Liblice, 1991.
atoms which makes the extrapolation to higher n-alkanes possible.
Theory In the present application of the perturbation theory of convex molecule fluidsS it is assumed that molecules interact via the Kihara pair potential7 u(s) = 4t[(u/s)'Z
- (u/s)6]
(1)
where s is the surface-to-surface distance ( s s ) between cores ascribed to the pair of interacting molecules. Rods of variable length, I, are taken as the cores of the molecules of n-alkanes; t and u are two characterivtic parameters. In the present version of the perturbation theory we adopt the BarkaHenderson-like of the pair potential into repulsive (s < a) and attractive (s > u ) branches. With this division the definition of the reference ( 0 ) and perturbation (p) potentials is connected. We consider the second-order perturbation expansion for the residual Helmholtz function, (A - A*)/NkT = (Ao - A*)/NkT
+ A f / N k T + Ag/NkT (2)
The reference term (Ao - A*)/NkT of the system of soft convex bodies follows from the equation of state of the representative hard convex bodies, in our case that of hard prolate spherocylinders of the same core (rod) as that ascribed to the studied molecule and with thickness of the spherocylinder, S; obtained from the condition (for details see refs 5 and 6) J(exp[-uhcb/kr] - exp[uO/kr]l Si+s+ids = 0
(3)
In (3) Si+s+i stands for the mean surface area of a pair of a r e s (of the type i) with the s-s distance s; in our case
Si+s+i= 87rR:
+ 167rR,s + 4us2
(4)
(where Ri is the 1 / 4 r multiple of the mean curvature integral; Rj = 1/4). To evaluate the reference term we employ the hard convex body equation of state15 pV/NkT = 1/(1 - q ) + 3aq/(l - 7)' [ 3 a -~ 27)
+ + 5 ~ 1 + / ( 1 - 7)3
(5)
which yields (Ao - A*)/NkT in the form (Ao - A*)/NkT = (6a2 - 5a - 1) In (1 - q ) + (15a2 - 9a)q/2(1 - q ) + (5a - 3a3)q/2(1 - q)* (6) (14) Barker, J.; Henderson, D. J . Chem. Phys. 1967, 47, 4714. (15) Boublik, T. Mol. Phys. 1981, 42, 209.
0022-365419212096-2298%03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2299
Equilibrium Behavior of +Alkanes In ( 5 ) and (6) 7 stands for the packing fraction, 9 = NV,/V, and a for the parameter of nonsphericity, a = R&/3VC, with R,, S,, and V, related to Ri and { through
+{ S, = 8rRj{ + 4u5? V, = 4rRic2 + 477t3/3
TABLE I: Saturated Liquid Densities and Pressures vs Temperature for Propane and n-Pentadecane PI, mol/L
T,K
Apl, mol/L
R, = Ri
(7)
In the first-order perturbation term (8)
the average distribution function of hard convex bodies (hcb) is expressed in terms of the average total correlation function, hhcb, [pb(s)= 1 hhcb(s)]so that the perturbation integral, Q, splits into two parts the greater of which possesses the form
+
+
Q = -l6atu3(l2R,2/55uZ 3Ri/5u
+ 2/91
(9)
To evaluate the smaller part of the integral the knowledge of the course of the average total correlation function hkb is needed. We approximated it in terms of the total correlation function of the equivalent hard spheresI6 (ehs) hhcb(s/u) = (Shcb/Schs)hchs(x)
(10)
where
+ C U S / U ( ~+ 2Rj)
(11)
and Shcb and sChsare surface areas of the convex body and of its equivalent hard sphere (of the same volume). For the total Correlation function of ehs we employ the expression resulting from the corrected Percus-Yevick relationship for the radial distribution function, &y. In spite of the use of hehs in (10) the average correlation function hhcbis a noncentral function. The approximation has proved to describe well the structure of hard prolate spherocylinders with the length-to-breadth ratio 1.6,2, and 3 (ref 16) and oblate spherocylinders of 4 = 2 (ref 5); it can be conjectured that (10) and ( 1 1 ) will yield a fair description of the structure of all the models considered here. Substituting the corresponding double Yukawa potential for the original Kihara function and employing the Laplace transform of &y, it is possible to express this part of the perturbation integral in an analytic form (for details see ref 5 ) . To determine the second-order perturbation term we employ the macroscopic compressibility approximation of Barker and Henderson.I4 Thus
Aq/NkT = ( j 1 / 4 k ’ P ) D ’ ~ ~ ~ ~ ghcb(s)Sj+s+i [ u ( s ) ] ~ ds
(12)
where
Dhcb= k T(8p / 8 p )hcb
(13)
For higher densities the whole second-order term is small due to the factor Pcb; it is thus legitimate to use an approximation Pb = 1 for the entire range of densities and distances. Then,
AqlNkT =
+
( 1 6 s p * / ~ 2 ) D b C b ~ m ( r--z1”2) ( 2 R ~ / u Z 4 R j z / u
+ z2)d r (14)
(where p* = pa3 and T+ = k T / t ) . It is evident that AP/NkT can be written in a simple analytic form similar to the first-order term. Differentiation of the relationship for the residual Helmholtz function with respect to density leads to the compressibility factor which, due to the explicit dependence of the Laplace transform on density, can be written in the analytic form, too. Thus, both the chemical potential and pressure can be easily determined, this (16) Boublik, T. Mol. Phys. 1984, 51, 1429.
AI’, 0.1 MPa
250.00 265.00 280.00 295.00 305.37
12.660 12.230 11.770 11.270 10.920
2.180 3.670 5.820 8.780 11.440
0.0041 -0.0089 -0,0177 0.0121 0.0344
493.16 513.16 533.16 553.16 573.16 593.16
2.928 2.848 2.764 2.675 2.588 2.501
-0.172 -0.106 -0.021 0.090 0.161
n-Pentadecane
A P / N ~ =T ( p / 2 k n S m ~ ( Pb(s)si+s+i s) ds
x=1
P, 0.1 MPa
Propane
-0.053 -0.034 -0.014 0.010 0.029 0.046
0.2918 0.4999 0.8139 1.268 1.900 2.754
0.0023
-0.00 13 -0.0064 -0.0100 -0,0028 0.0340
TABLE II: Comparison of Experimental and Calculated Second Vinal Coefficients for Propane BCXP,
BCA.
T, K
mL/mol
mL/mol
240.0 250.0 260.0 270.0 285.0 300.0 315.0 330.0 350.0 375.0 400.0 430.0 470.0 500.0 550.0
-640.0 -584.0 -526.0 -478.0 -424.0 -382.0 -344.0 -313.0 -276.0 -238.0 -208.0 -177.0 -143.0 -124.0 -97.0
-597.1 -550.8 -510.0 -473.7 -426.2 -385.5 -350.2 -3 19.3 -283.7 -246.3 -215.1 -183.7 -149.0 -128.6 -99.7
Bcal
- Bcrpr
mL/mol 42.9 33.2 16.0 4.3 -2.2 -3.5 -6.2 -6.3 -7.7 -8.3 -7.1 -6.7 -6.7 -4.6 -2.7
fact makes it possible to describe in a simple way the phase equilibrium of the one-component system.
Application The described method has been used to correlate thermodynamic data of pure n-alkanes along their coexistence curve and to evaluate the Kihara potential parameters elk, u as well as geometric characteristics of hard cores ascribed to the molecules under consideration. Alkanes from ethane to hexadecane have been considered; for methane with the pointwise core the results of the data fitting and values of the Kihara parameters are given elsewhere.* For individual n-alkanes the set of vapor pressures, liquid densities, and, when available, vapor densities at given temperatures (for the reduced temperatures < 0.9) were taken of phase equilibria were considered into a c c o ~ n t . ~ ~Conditions -’~ and the optimized values of three parameters were determined from an objective function defined as a sum of squares of deviations in the relative pressures and relative liquid densities. As mentioned in the introductory part, three types of cores were considered in the beginning, namely, the combination of triangles, hard ellipsoids of revolution, and hard rods. There was no essential difference in the magnitude of deviations in pressure and liquid density between models of the second and third type whereas models of the first type yielded substantially worse results. For the sake of simplicity we have taken rodlike cores and optimized the length, I, of the respective rods in addition to elk and u. In the optimization procedure the behavior of the liquid phase is described (17) Vargaftik, N . B. Handbook on Thermophysical Properties of Gases and Liquids; Nauka: Moscow, 1972. (18) ’Selected Values of Properties of Hydrocarbons and Related
Compounds”; API Res. Project 44, Thermodynamics Research Center, Texas A&M University, College Station, TX (loose-leaf data sheets, extant, 1972). (19) Cibulka, I., private communication.
Pavlkek and BoubGk
2300 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 TABLE III: Comparison of Experimental and Calculated Second Virial Coefficients for n-Octane 150
325.0 350.0 375.0 400.0 425.0 450.0 475.0 500.0 550.0 600.0 700.0
-3 139.0 -2508.0 -2050.0 -1708.0 -1445.0 -1237.0 -1071.0 -935.0 -727.0 -578.0 -380.0
-258 1.3 -2164.1 -1847.3 -1598.7 -1398.7 -1234.2 -1096.8 -980.3 -793.5 -650.6 -446.7
TABLE I V Parameters of the Kihara Potential n 1, A elk, K 2 1.578 302.6 3 2.588 396.2 4 3.479 480.9 5 4.279 554.6 6 5.278 626.0 7 6.166 688.6 8 7.022 745.4 9 7.931 800.5 852.6 10 8.937 11 9.797 898.8 12 10.691 945.0 13 11.712 988.5 14 12.379 1024.8 15 13.313 1061.2 16 14.057 1098.0
557.7 343.9 202.7 109.3 46.3 2.8 -25.8 -45.3 -66.5 -72.6 -66.7
100
L
I1 1 4 1
50
1 I
i 0,
A
I
"
"
l
"
"
5
1
'
I
~
15
10
10 of C - a t o m s Figure 1. Dependence of the rod length, L (in
A), on the number of C
atoms in n-alkanes.
1
I
-I-
,0004
by the perturbation expansion for the compressibility factor and chemical potential whereas properties of the vapor phase were determined from the vinal expansion in which the complete Kihara second virial coefficient and the repulsive part of the third virial coefficient were retained. In Table I results of the optimization procedure are presented for propane and n-pentadecane. The comparison of the calculated and experimental pressures and liquid densities reveals very good agreement for the both data sets. Standard deviations in pressure and liquid density in the case of propane amount 0.27 and 1.17% whereas for pentadecane 0.84 and 1.39%. It is apparent that there is no substantial difference in the description of both the systems. The resulting parameters of the Kihara potential can be used to evaluate the second virial coefficient, B; comparison of the calculated and smoothed experimental data of B for propane and octane (we are not aware of reliable data for the higher n-alkanes) is given in the subsequent tables. It is evident that the respective effective Kihara parameters for these two compounds yield values of the virial coefficient that are in reasonable agreement with experimental dataZo(within 3 times the experimental error estimates); see Table I1 and 111. In Table IV the parameters of the Kihara potential, t/k, u, and I are summarized for individual n-alkanes. It is worth noting that ethane and propane were studied in our previous work.* While in ref 8 equilibrium data were employed from the broad range T* > 0.3 here data at higher temperatures T* E (0.6,O.g) and pressures were considered; instead of a triangle the rodlike core was assumed in this study for propane; this caused changes in the parameter values. The obtained parameters disclose considerable regularity of their dependencies on the number of C atoms in molecules. This is especially striking in the case of the length parameter, I , where the dependence is linear. If n denotes the number of C atoms in the given molecule, then The dependencies of the other two parameters, e / k and
,
i
0
3.633 3.799 3.924 4.051 4.122 4.188 4.265 4.309 4.346 4.403 4.422 4.447 4.516 4.540 4.580
1 = 0.8976n - 0.1323
I
00
(15) u,
are
(20) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures; Clarendon: Oxford, U.K., 1980.
./ ,
2301
c
,
, 5
15
10
' 16 possible. Q
Conclusion The present study of the behavior of n-alkanes was initiated by (i) the prmnt interest inthe behavior Of these (and successOf the their mixtures with theory Of convex fluids in describing the behavior Of compounds formed by relatively small nonspherical nonpolar molecules, and (iii) finding that the hard body models of the flexible chain molecules2' can be treated as linear bodies. We have considered orthoba,-ic data in similar ranges of the r e d u d temperatures for (21) Boublik, T. J . Chem. Phys. 1990, 93, 730.
2301
all the studied systems of n-alkanes and adjusted three parameters of the Kihara pair potential to these data. It is found that the which employs the pefiurbation approach in the variant for convex molecule fluids, it to fit the experimental data with high accuracy; this accuracy remains practically unchanged with the increasing number of carbon atoms in molecules. Deviations in the liquid density and the second virial coefficient changes their signs within the considered range of temperatures; this behavior and limits of the temperature range to P < 0.9 are connected with the fact that perturbation methods do not yield correct prediction of the critical point. We limited our interest to class of n-alkanes up to C16H34 because reliable experimental data for higher n-alkanes are less and less available. For this reason, we have been unable to find limits of applicability of the proposed method. Registry No. Ethane, 74-84-0; propane, 74-98-6; butane, 106-97-8; Pentane, 109-66-0;hexane, 110-54-3;heptane, 142-82-5; Octane, 1 1 165-9; nonane, 111-84-2;decane, 124-18-5; undecane, 1120-21-4;dodecane, 112-40-3;tridecane, 629-50-5; tetradecane, 629-59-4; pentadecane, 629-62-9; hexadecane, 544-76-3.
Electronic Structure and Electronic Excitations of Solid Neon from an ab Initio Atom-in-the-Lattice Approach A. Mar& Pendis, E. Francisco, V. Luaiia, and L.Pueyo* Departamento de Quimica Flsica y A n a l h a , Facultad de Quimica, Uniuersidad de Oviedo, E - 33006 Oviedo, Spain (Received: August 22, 1991)
The electronic structure of the ground state and the lowest electronic excited state of face-centeredcubic neon has been calculated with an ab initio atom-in-the-lattice approach consistent with the theory of electronic separability of many-electron systems. The Hartree-Fock-Roothaan equations are solved for a reference atom in the field created by the rest of the lattice. The solutions of these equations are then used to describe the quantum effects of the lattice atoms in an iterative process leading to atom-lattice consistency. The equilibrium geometry, the cohesive energy, and the lowest electronic transition energy have been computed in agreement with the experimental data. The calculation gives also atomic orbitals for the neon atom that are consistent with the crystalline environment. These crystal orbitals show a contraction that increases with applied pressure, with respect to the gas-phase orbitals. The crystal stability may be described in terms of a long-range attraction, identified with the atom-lattice exchange energy, and a short-range repulsion derived from the atom-lattice orthogonality. The partition of the binding interactions into orbital contributions shows that the stability comes essentially from the 2s density. The 2p6-lS 2 ~ ~ 3 s ' -transition ~P energy involves very large differential electron correlation and increases with external pressure up to 60 GPa.
-
I. Introduction A large variety of quantum-mechanical approaches have been developed for the calculation of the electronic structure of rare-gas crystals. Observable properties like cohesive energy, equilibrium geometry, elastic constants, equation of state, etc. can be deduced from the computed electronic energy and wave function. Most of these approaches are based on the theory of electron band structure.I4 There is also a large body of work based on the use of atom-atom pair potential^.^-^ (1) Euwema, R. N.; Wepfer, G. G.; Surratt, G. T.; Wilhite, D. L. Phys. Rev. B 1974, 9, 5249. (2) Boettger, J. C.; Trickey, S.B. Phys. Rev. B 1984, 29, 6425. (3) Boettger, J. C. Phys. Rev. B 1986, 33, 6788. (4) Bacalis, N. C. Phys. Rev. B 1988, 38, 6218. (5) Barker, J. A. J . Chem. Phys. 1987,86, 1509. (6) McLean, A. D.; Liu, B.; Barker, J. A. J . Chem. Phys. 1988,89, 6339. (7) Loubeyre, P.Phys. Rev. Lett. 1987, 58, 1857. (8) Loubeyre, P. Phys. Rev. B 1988, 37, 5432.
0022-3654/92/2096-2301$03.00/0
LeSar has recently shownI0J1that localized schemes based on the atom-in-the-lattice approach can also be useful in the description of rare-gas crystals. Using density functional theory, M a r computes the crystal potential and examines its perturbative effects on the electronic density of the crystal units. The resulting self-consistent-field solutions of these units are then used to compute the lattice potential until a consistent picture of the crystal is obtained. The success of this approach supports a localized picture of the raregas solids in which the crystal behavior is mainly determined by the properties of the constituent atoms. Clearly, this picture can only hold well below the onset of metallization where long-range phenomena are predominant. This view requires, however, further analysis on several questions that are important for a correct understanding of the atom-inthe-lattice approximation. Among them, we may quote the (9) Derbyshire, M.; Etters, R. D. J . Chem. Phys. 1983, 79, 831. (10) LeSar, R. Phys. Rev. B 1983, 28, 6812. (11) LeSar, R. Phys. Rev. Lett. 1988,61, 2121.
0 1992 American Chemical Society
