Equations of State - American Chemical Society

14 Thermodynamics of Multipolar Molecules The Perturbed-Anisotropic-Chain Theory P. Vimalchand, Marc D. Donohue, and Ilga Celmins Department of Chemical Engineering, The Johns Hopkins University, Baltimore, MD 21218

The Perturbed-Hard-Chain theory (PHCT) is modified to treat rigorously dipolar and quadrupolarfluidsand their mixtures. The multi-polar interactions, which are treated explicitly, are calculated by extending the perturbation expansion of Gubbins and Twu to chain-like molecules. Moreover, the square-well potential used to characterize the spherically symmetric interactions in the PHCT has been replaced by a soft-core (Lennard-Jones) potential. Theoretical calculations and data reduction on a number of purefluidsand mixtures indicate that the above changes result in physically more meaningful pure component parameters, and the properties of even highly non-ideal mixtures can be predicted accurately without the use of a binary interaction parameter. Although considerable phase equilibrium data are available for hydrocarbon systems of low molecular weight, data for high molecular weight hydrocarbons, especially aromatic hydrocarbons, are scarce. Unfortunately, none of the theories that have been developed for the properties of lower molecular weight aliphatic compounds encountered in natural gas and oil are accurate for heavy hydrocarbons such as coal derivatives. They require large values of an adjustable binary interaction parameter, and therefore, the predictive ability of these theories becomes poor for multicomponent mixtures and for systems for which there are no experimental data. There are two main reasons why these theories fail for high molecular weight multipolar compounds. First, these theories are valid only for compounds with molecular weight below 150 because they ignore the effect of rotational and vibrational motions on thermodynamic properties. While this deficiency was overcome, in part, by the polymer theories of Flory (1) and Prigogine (2L), their theories are valid only at high density. The second deficiency in all the theories that were developed for oil and gas-refining operations concerns the nature of intermolecular forces. All these theories tacitly assume that molecules interact with London dispersion forces (also referred to as van der Waals forces). Since coal and its derivatives are generally high molecular 0097-6156/ 86/ 0300-0297S06.00/ 0 © 1986 American Chemical Society

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

298

weight (150 to 1000) compounds that have numerous benzene rings and functional groups which have strong anisotropic (dipolar and quadrupolar) potential functions, any theory for coal compounds must reflect this difference. These two deficiencies are taken into account in developing the Perturbed-Anisotropic-Chain theory (PACT) which enables calcula tion of thermodynamic properties for a wide variety of pure fluids and mixtures. The P A C T equation is based on the Perturbed-So ft-Chain theory (PSCT) and on improvements to Pople's perturbation expansion (2.) by Gubbins and Twu (!). The PSCT equation is essentially the PerturbedHard-Chain theory (PHCT) of Donohue and Prausnitz (5_), but the potential energy function used to characterize the interaction between the molecules is different. The original PHCT equation uses a squarewell intermolecular potential, while in the PSCT equation, interactions are calculated with the Lennard-Jones potential. Multipolar interactions in the P A C T are treated by combining the perturbation expansion of Gubbins and Twu for anisotropic molecules with lattice theory for chain-like molecules. The PHCT and PSCT equations have been shown to predict accu rately the properties of pure fluids and mixtures of non-polar molecules of varying size and shape jBeret and Prausnitz (£); Donohue and Prausnitz (u); Kaul et al. (Ζ) and Morris (S.)], including polymeric sys tems [Beret and Prausnitz (&) and Liu and Prausnitz (£.)]. For mixtures containing quadrupolar molecules (such as carbon dioxide - ethylene sys tem; carbon dioxide - ethane system; benzene - 1-methyl naphthalene system), the P A C T equation predicts properties better than either the PHCT or PSCT equations [Vimalchand and Donohue (ID)]. In this paper, we extend the P A C T to include dipolar interactions. The theorem ical results show that the P A C T equation also can be used for predicting accurately the properties of highly non-ideal mixtures (without the use of a binary interaction parameter) containing strongly dipolar molecules such as acetone. The Perturbed-Anisotropic- Chain Theory The canonical ensemble partition function used in the PerturbedAnisotropic-Chain theory is of form:

««™ -MM

m hw)

hw) «

where Ν is the number of molecules at temperature Τ and volume V; A is the thermal deBroglie wavelength, and k is Boltzmann's constant. The partition function is based on the generalized van der Waals theory where molecular translational motions are governed by intermolecular attractions and repulsions. The molecular repulsions are written in terms of free volume, V}, which is the volume available to the center of mass of a single molecule as it moves about the system holding the positions of all other molecules fixed. The molecular attractions are generalized in terms of a potential field, φ/2, which is the intermolecular potential energy of one molecule due to the presence of all other molecules. For non-central force molecules (with dipolar or quadrupolar forces), the intermolecular potential energy function can be written as a sum of iso tropic and anisotropic interactions. The isotropic pair potential, which depends only on the distance between the molecules, is taken as an

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unweighted average over all orientations. Summing the intermolecular interactions, φ/2 is given by the sum of isotropic and anisotropic interac tions of one molecule with all other molecules in the system, φ /2 and φ /2, respectively. The rotational and vibrational motions, which are strongly densitydependent, are affected by intermolecular interactions. To account for deviations in equation of state due to rotational and vibrational motions, a parameter c is defined such that 3c is the total number of densitydependent degrees of freedom. This includes the three densitydependent translational degrees of freedom. Further, the non-idealities in the equation of state (i.e., due to attractive and repulsive forces) caused by each of the 3c- 3 density-de pendent rotational and vibrational degrees is approximated as equivalent to the non-idealities caused by each of translational degree of freedom. The inclusion of the rotational and vibrational motions in an approximate way through the parameter c improves the simultaneous prediction of various thermodynamic proper ties. The partition function in equation 1 approaches the correct ideal-gas limit as density approaches zero and a Prigogine- and Flory-type partition function at liquid densities. Moreover, this partition function is valid for both non-polar and anisotropic molecules. For non-polar molecules, with φ = 0, the partition function in equation 1 reduces to a form analogous to that used in the PHCT. The isotropic interactions, φ , are calculated using a Lennard-Jones potential energy function, and following Gubbins and Twu (4), the iso tropic dipole-induced dipole interactions are calculated assuming an aver age polari ζ ability for the molecule. In this work, for anisotropic interac tions, φ , we are considering only systems in which the dipolar forces are predominant, and all other anisotropic interactions are assumed to be negligible. The P A C T equation of state, obtained by differentiating the parti tion function in equation 1, is given by 180

αηι

αη%

100

αηι

Ρ

( 1 + c z** + c

=

ani

+ c z }

(2)

In the P A C T repulsions due to hard-chains are calculated using the parameter c and the equation of Carn ah an-Starling (11) for hard-sphere molecules. The attractive Lennard-Jones isotropic interactions are calcu lated using the perturbation expansion of Barker and Henderson (12). Higher-order terms in the perturbation expansion are accounted for by using a Pade' approximation for Helmholtz free energy. The perturba tion expansion results for spherical molecules are extended to chain-like molecules with the following reduced quantities: f

=

^ T*

=

— eq

(3)

and ~ Vi

_

~

_ J i _ _

vl ~

(A)

V

N rd*/V2

1

A

where N is Avagadro's number. The equation of state for pure nonpolar molecules contains three characteristic parameters. Besides c, the parameters are molecular soft-core size, ν (or the related hard-core size, v£), and characteristic temperature, T*. Values for these three A

'

300

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

parameters are obtained from data reduction, using experimental vapor pressure and liquid-density data. The other parameters e/k, q, r, and d are only necessary for mixture calculations. The Parameter q is propor tional to the molecular surface area, e is the energy per unit external sur face area of a molecule and r is the number of segments of diameter d (or σ) in a chain-like molecule. For pure fluids, the parameters e and q and parameters r and σ always appear as a product. For mixtures, σ and e are determined by correlating v* and cT for a large number of similar fluids. In PACT, both the anisotropic multipolar interactions and isotropic dipole-induced dipole interactions are calculated using the perturbation expansion of Gubbins and Twu (4.) assuming the molecules to be effectively linear. The anisotropic interactions are calculated by treating these forces as a perturbation over isotropic molecules. Higher-order terms in the perturbation expansion are accounted for by using a Pade' approximation for Helmholtz free energy. Gubbins and Twu obtained the isotropic dipole-induced dipole interactions by treating these forces as a perturbation over Lennard-Jones molecules and truncating the series after the first perturbation term. Their results for small molecules are extended to chain-like molecules by use of reduced quantities in equa tions 3 and 4 and by defining the following characteristic reduced tem peratures. For dipolar interactions, 3

and for isotropic dipole-induced dipole interactions

™

Τ x

ν* ckT

,

αλ

αμ

where ε characterizes the segment-segment dipolar interactions and a is the average polari ζ ability. For pure fluids, parameters ε and q always appear as a product which can be evaluated from known values of dipole moment, μ, and using the relation μ

μ

For mixtures, the segmental anisotropic dipolar interaction energy can be obtained from q [ = cT /(e/&)] and equation 7. Complete equations for the configurational Helmholtz energy and the equation of state involving dipolar and isotropic dipole-induced dipole interactions are given by Vimalchand (12.). Mixtures. The pure-component partition function is extended to mix tures using a one-fluid approximation, however, without the usual random-mixing assumption. Following Donohue and Prausnitz, the mixing rules for both the isotropic and anisotropic terms are derived using a Lattice theory model. The pure component partition function given by equation 1 is extended to mixtures of chain molecules satisfying the following conditions: (i) For mixtures of both small and large (polymeric) molecules, mixture properties should be based on surface and volume fractions rather than on mole fractions, (ii) Both the isotro pic and anisotropic part of the second virial coefficient should have a quadratic dependence on mole fraction, (iii) The pressure and other

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excess thermodynamic properties must remain finite when the chain length becomes infinitely large, (iv) The athermal entropy of mixing must reduce to the well-known Flory-Huggins entropy of mixing when the reduced volume of the pure components and solution are identical, (v) For correct prediction of Henry's constant, the residual chemical potential should be proportional to the product q^j rather than g-r-. (vi) For molecules with significantly different intermolecular potential ener gies, nonrandom-mixing due to molecular clustering becomes important. For mixtures of spherical molecules, the mixing rules should agree with the nonrandom-mixing theory of Henderson (14.). Details of the derivation of mixing rules satisfying the above condi tions are given by Vimalchand (12.). Complete expression for the configurational Helmholtz free energy is given in the Appendix. t

t

Results and Discussion Pure Fluids. The Perturbed-Anisotropic-Chain theory has been tested with a wide variety of fluids in which molecules interact with substantial quadrupolar [Vimalchand and Donohue (lu)] and dipolar forces. The properties of pure fluids were predicted with parameters which are independent of temperature and pressure. The pure-component partition function was fitted to available experimental data and then purecomponent parameters were correlated to ensure reliability of data and data reduction. Pure-component parameters have been obtained for eleven fluids with appreciable dipofe moments and these are tabulated in Table I. Table I., Pure-Component Parameters

T*K

lOOv mol

c

6

Τ* Κ

Τ

D ipolar Fluids Carbon monoxide Diethyl ether Chloroform Methyl acetate Dimethyl ether Hydrogen sulfide Sulfur dioxide Methyl iodide Methyl chloride Acetone Acetonitrile

101.9 297.2 351.9 324.0 278.6 271.9 279.1 417.3 280.5 325.3 272.1

2.0274 5.9382 4.6651 4.5508 3.6650 2.1695 2.4765 3.8332 3.2858 4.4500 3.4482

1.0275 1.8054 1.6158 1.8314 1.3214 1.1181 1.5037 1.0002 1.1194 1.3768 1.0013

105 115 140 145 115 150 105 160 178 145 180

225.1 311.4 305.8 326.4 371.3 439.9 377.8

3.1947 6.2948 6.4009 7.3984 6.3798 3.7143 5.7982

1.1636 1.6501 1.6078 1.8224 1.5507 1.0045 1.4883

105 105 105 105 118 142 118

Nonpolar fluids Ethane Pen tan e Isopentane Hexane Cyclohexane Carbon disulfide Carbon tetrachloride

2.5 42.8 49.5 105.7 106.0 127.2 214.7 216.4 315.7 411.9 1094.3


302

Parameters for eight non-polar fluids are also given in Table I. For dipo lar fluids, the product ε ς was determined using the dipole moments reported by McCellan (UI). The remaining three pure-component parameters ( T*, v*, and c) for each compound were found by fitting the partition function to experimental liquid-density and vapor pressure data. For mixture calculations, either parameter e or q need to be determined independently. The segmental energy parameter, e, was determined by examining the predicted binary mixture data where one of the com ponents is a C or higher alkane. Following Kaul et al. (Ζ.), e for normal and branched alkanes was assigned a value of 105 Κ. With the parameters given in Table I, errors in calculated vapor pressure and liquid-density are typically within 2% over a wide range of temperature and pressure. Figure 1 shows the experimental and calcu lated vapor pressures for hydrogen sulfide, sulfur dioxide and acetone from triple point to critical point. The average error is less than 2%. Liquid densities were also calculated over a wide range of temperature and présure with similar accuracy. Also, the saturated liquid and vapor volumes usually were predicted within 2% error (up to 0.95 of the critical temperature) as shown in Figure 2 for hexane and for the dipolar fluids, sulfur dioxide, acetone and methyl acetate. Since both acetone and acetonitrile have large dipole moments, we included the isotropic dipole-induced dipole interactions assuming an average polari ζ ability [LandoltrBornstein (lu.)] for these molecules. The parameters obtained by fitting the experimental vapor pressure and liquid-density data are given in Table II. The average errors in vapor pressure and liquid-density are similar to those reported above. The inclusion of isotropic induction forces yields a characteristic energy parameter T* of 303.4 Κ compared to T* = 325.3 Κ obtained without induction forces while the other parameters (v and c) are nearly identi cal in both cases. Similarly for acetonitrile Τ decreases to 226.4 Κ from a high value of 272.1 obtained without induction forces. The values of characteristic dispersive energy ( = cT*) of acetone and acetonitrile, when induction forces are included may be the more realistic and as a result parameters for these fluids can be expected to correlate well with other fluids. μ

5

Mixtures. Fluid-mixture properties are calculated with pure-component parameters and a single binary interaction parameter defined by

tu = v ^ 7 ^ ( i - % )

(8)

Experimental K-factor (ratio of vapor phase mole fraction to liquid phase mole fraction) data are used to determine the binary interaction parame ter, kjj, which is independent of temperature, density, and composition. Table II. Parameters including average polarizability for compounds with large dipole moments Dipolar fluids Acetone Acetonitrile

Τ* Κ 303.4 226.4

lOOv mo I 4.5515 3.7124

c 1.3628 1.0000

e

Τ 145 180

τ* κ μ

à

406.8 1017.9

0.0592 0.0514

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Figure 1. Comparison of experimental and predicted vapor pressures of dipolar fluids from their triple point to critical point.


304

points are expt. PACT

0.021

I 200

.

I

300

.

1

400

1

1

500

TEMPERATURE , 'K

Figure 2. Comparison of calculated and experimental saturated liquid molar volumes (up to 0.95 of the critical temperature).

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The binary interaction parameter corrects for inadequacies in theory and in mixing rules. To predict properties of multicomponent mixtures with ease and reasonable accuracy, it is desirable to develop an equation to predict mixture properties without the use of k - or at least, with small values of & ·. The PHCT, compared to many other equations of state, predicts accurately the properties of non-polar mixtures with small values of kjj (less than 0.05). The PHCT, however, still requires large values of for fluid mixtures containing a dipolar and a quadrupolar sub stance. Consideration of dipolar and quadrupolar forces in the P A C T allows fairly accurate prediction of mixture properties from purecomponent parameters alone. Usually, the P A C T requires small values of to predict mixture properties with accuracy. Binary interaction parameters for 9 dipolar fluid mixtures and 13 nonpolar - dipolar fluid mixtures are given in Table III. The inclusion of isotropic dipole-induced dipole interaction does not change k^ significantly and therefore it is neglected except for the system acetone pentane and acetone - cyclohexane. For these two systems, the acetone parameters given in Table II were used. Like many widely used equations of state [such as the Peng - Robin son equation (11.)], both the PHCT and P A C T have (three) adjustable parameters for pure-components which are fitted to experimental data. As a result, each fits the pure-component properties rather well. How ever, a much more stringent test of the theory is the prediction of mix ture properties from pure-component parameters alone. Such a predic tion is made in Figures 3 to 6. The K-factors calculated from the P A C T with hij = 0 are compared with experimental values for mixtures con taining dipolar fluids. Also for comparison, K-factors calculated with k^ = 0 using the PHCT and the Peng - Robinson equation of state (PR) are shown. The inclusion of dipolar interactions improves the prediction of Kfactors significantly for the systems sulfur dioxide - acetone (Figure 3), hydrogen sulfide - pentane (Figure 4), and methyl iodide - acetone (Fig ure 5). Sulfur dioxide, acetone, hydrogen sulfide, and methyl iodide are all dipolar fluids. Acetone forms a weak complex with sulfur dioxide and yet the P A C T gives a good prediction of K-factors. In the hydrogen sulfide - pentane system, where molecular sizes differ considerably, both the P A C T and the PHCT predict better K-factors than the Peng - Robin son equation. In all these figures, it can be seen that the P A C T predicts the experimental data much more closely than either the PHCT or the Peng - Robinson equation. The y-x phase diagram for the systems and acetone - cyclohexane (Figure 6) show the effect of including, explicitly, the dipole interactions of acetone, rather than the use of the equivalent attractive dispersion interaction as was done in the PHCT. The P A C T predicts the azeotrope and closely follows the experimental data, however, the PHCT fails to predict the azeotrope and in addition, poorly follows the experimental values. This may be explained as follows: In the alkane-rich liquid phase each acetone molecule is surrounded by alkane molecules and therefore there are no ace to ne-ace to ne (dipole-dipole) interactions. However, when the dipole interaction between pure acetone molecules is empirically replaced by an equivalent dispersion interaction (as was done in the PHCT), the equation predicts an erroneously large attractive energy between acetone and the surrounding alkane molecules. This additional attractive energy incorrectly lowers the predicted mole fraction {j

ι;

306


of acetone in the vapor phase as shown in the Figure 6. Similar arguments indicate that the PHCT when compared to the P A C T would predict an increase in mole fraction of acetone in the vapor phase in equilibrium with an acetone-rich liquid phase where alkane molecules are surrounded by acetone molecules. Mixture predictions by the PACT with pure-component parameters alone are quantitatively correct but usually not within experimental error. Hov/ever, mixture properties usually can be fit with small errors using small values of a binary interaction parameter given in Table III. Phase equilibria of even complex hydrogen-bonding systems like chloroform acetone and chloroform - diethyl ether are predicted well with small values of interaction parameters. Conclusion. A new theoretical equation of state, the PerturbedAnisotropic-Chain theory has been developed. This equation takes into account the effects of differences in molecular size, shape, and intermolecular forces including anisotropic dipolar and quadrupolar forces. While prediction of properties of pure fluids is no better for the P A C T than many other equations of state, the prediction of mixture properties, especially highly non-ideal mixtures is improved significantly. The P A C T accurately predicts mixture properties from pure-component parameters alone or with small values of a binary interaction parameter. Table III. Binary Interaction Parameters Binary mixtures

100%

Dipolar fluid mixtures Sulfur dioxide - Acetone Sulfur dioxide - Methyl acetate Sulfur dioxide - Chloroform Acetone - Methyl acetate Acetone - Diethyl ether Acetone - Chloroform Acetone - Methyl iodide Methyl acetate - Chloroform Diethyl ether - Chloroform

-0.75 -1.50 1.74 -1.24 -1.00 -5.00 0.96 -2.55 -4.18

Non- polar and Dipolar fluid mixtures Carbon monoxide - Ethane Ethane - Hydrogen sulfide Ethane - Methyl acetate Ethane - Diethyl ether Hydrogen sulfide - Pentane Acetone - Pentane Acetone - Carbon disulfide Acetone - Cyclohexane Acetone - Carbon tetrachloride Isopentane - Diethyl ether Hexane - Chloroform Chloroform - Carbon tetrachloride Methyl iodide - Carbon tetrachloride

0.20 3.40 3.78 0.82 2.90 0.62 1.60 0.11 0.55 0.60 0.50 0.21 0.96

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308


Figure 4. Comparison of experimental K-factors for hydrogen sulfide pentane system and calculated values (with k^ = 0).

VIMALCHANDETAL.


310

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

X acetone

Figure 6. Comparison of experimental y - χ phase equilibrium data of acetone in acetone - cyclohexane system and calculated values (with

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311

Because of this success, the P A C T may prove very valuable in predict ing, to a fair degree of accuracy, mixture properties of systems contain ing intermediate molecular weight compounds for which no experimental data exist. We are currently extending the P A C T for systems involving a dipolar and a quadrupolar molecule as well as for complex systems involving polymeric and hydrogen-bonding molecules. Acknowledgments Support of this research by the Chemical Sciences division of the Office of Basic Energy Sciences, U . S. Department of Energy under contract number DE-AC02-81ER10982-A004 is gratefully acknowledged. Appendix A summary of various terms in the Helmholtz free energy expression used for calculating the mixture properties with the PerturbedAnisotropic-Chain theory is given below. IG

A - A (T, V)

Γ

= Α

Ϊ

Η ^

Λ

Α

+

μ

+ Α

μ

Repulsions: 2

{r

r

d

d

re

A *> = NkT 1

r d

where < · · · > represents a mixture property, r =0.7405 and c

< >

=

Σ

-^J-

χ

;

= Σί

The ratio of segmental diameters, d (hard-core) and σ (soft-core), is evaluated as a function of temperature and fitted to a polynomial in reduced temperature [Vimalchand and Donohue (lu)]. Lennard-Jones Attractions: -1

u

A

J

A\ jNkT

NkT

J

NkT

A[ /NkT

wher ,4

l m

m _ 1

NkT

!

C

l m

Equations of State - American Chemical Society

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