The Generalized van der Waals Partition Function as a Basis for

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Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 19, 2016 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

The Generalized van der Waals Partition Function as a Basis for Equations of State Mixing Rules and Activity Coefficient Models S. I. Sandler, K.-H. Lee, and H. Kim Department of Chemical Engineering, University of Delaware, Newark, DE 19716

The generalized van der Waals (GVDW) partition function provides a basis for understanding the molecular level assumptions underlying presently used equations of state, their mixing rules including the new local composition (density dependent) models, and activity coefficient models. Further, our Monte Carlo computer simulation results for mixtures of squarewell fluids provide a method of testing these assumptions, many of which are found to be incorrect. Using the combination of the GVDW partition function and computer simulation results, we have formulated new equations of state and local composition models which are simple and accurate. Thermodynamic modelling of r e a l f l u i d s and mixtures, using both equations of state and a c t i v i t y c o e f f i c i e n t s , i s an area i n which there i s a strong temptation to introduce empiricism. The various modifications of the van der Waals equation, which now number i n the hundreds, i s one example of t h i s . S t a t i s t i c a l mechanics, on the other hand, leads to the detailed calculation of the properties of model f l u i d s with very simple intermolecular potential functions, but few generalizations to r e a l f l u i d s . For the l a s t several years we have been pursuing a d i f f e r e n t approach wherein we start from a firm s t a t i s t i c a l mechanical basis, but seek results of general v a l i d i t y , rather than to calculate numbers for an idealized i n t e r action potential model. In p a r t i c u l a r , we are seeking answers to such questions as: (1) Why are present equations of state and their mixing rules not applicable to mixtures of molecules of greatly d i f f e r i n g f u n c t i o n a l i t y or size?, and (2) What i s the molecular basis for l o c a l composition and density dependent mixing rules, and how can such rules be improved? This communication i s a progress report on our e f f o r t s . The basic idea i n our work i s the use of the rigorous generalized van der Waals p a r t i t i o n function, which we consider i n the following sections, to understand the molecular l e v e l assumptions imbedded i n presently used equations of state, their 0097-6156/ 86/ 0300-0180$06.50/ 0 © 1986 American Chemical Society

Chao and Robinson; Equations of State ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

SANDLER ET AL.

181

mixing rules, and a c t i v i t y c o e f f i c i e n t models, and then to test these assumptions using computer simulation. In addition, we have been using the results of our simulations to formulate new and better molecular thermodynamic models. Our progress to date w i l l be reviewed here. Pure Fluids


The s t a r t i n g point for our study of pure f l u i d s i s a new form of the generalized van der Waals p a r t i t i o n function we have derived [1]: 3 N

Q(N,V,T)

A- (q q q ) r

v

N

V

e

e x

f P

etc. [The extension to nonspherical p a r t i c l e s i s considered b r i e f l y i n reference 1.] F i n a l l y , EÔNF^ the energy of interaction, for a pair-wise additive system i s given by CONF

M

E

J (r)g(r;N,V,T)clr u

(4)

and the mean potential

ψ

T

J ^

dT

(5)

i s the free energy of bringing the system from T=« (where only hard core repulsive forces are important) to the temperature of i n t e r e s t . In these l a s t equations, u(r) i s the two-body intermolecular poten t i a l and g(r;N,V,T) i s the two-body r a d i a l d i s t r i b u t i o n function, which i s a function of density, temperature and intermolecular separation distance. Once the p a r t i t i o n function i s known, a l l thermodynamic properties can be found. For example,


182

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

kT(4^) 3V

P(N,V,T) =

(6) N,T

and (7)


A(N,V,T) = -kTlnQ

The f i r s t of these equations provides a method of obtaining the volumetric equation of state from the p a r t i t i o n function, while from the second the fundamental equation of state i n the sense of Gibbs i s obtained. The importance of equations (1-5) i s that they focus attention on the two quantities which are needed f o r thermodynamic modelbuilding, the temperature and density dependence of the free volume Vf and the configurâtional energy, EÔNF Indeed, these are central to obtaining good equations of state, as w i l l be seen shortly. As an aside, we note that f o r the square-well potential e

»

r v > N l

^

N 2 # e e )

_£

Τ kT / [

(26)

E i

CONF (

T > V i

,Ni)

i kT

dT

2

T=«

EX

EX ^T,V

ECONF( ^ N ,N ...)-jE T

V f

X

2

CONF iT,V ,N ) ±

±

±

dT T

T=oo

/ t-

(27)

T=oo

The f i r s t term on the righthand side of this equation i s the excess free energy of mixing when only repulsive forces are important. This term accounts f o r size and shape e f f e c t s , and may be modelled by the Flory-Huggins 11 l j , Guggenheim-Stave man [12] or other suitable expressions. The second term i s the excess free energy of mixing for bringing the system from T=eij

,Oij)

(37)

where f i s any function of Ν, V, Τ and the potential parameters of a l l species, but independent of composition, lead to different forms of the a t t r a c t i v e term i n the equation of state, i n each case the van der Waals one-fluid mixing rule of Equation (34) applies at a l l densities. Defining N ^ to be the t o t a l coordination number for a species i molecule i n the mixture, from Equation (33a) we have (for similar s i z e molecules) that c

Nci

« »il

+

Nji -

f

C = N

cl

(38)

so that ο x

«il - i N

c i

ο

= xjN

c i


(39)

8.


S A N D L E R ET A L .

193

where N £ i s the coordination number for pure species i molecules at the same density. Using these results i n Equation (2 7) y i e l d s C

A

EX T-oo +

T,V


where 0 - j

1

N

° cl^ ll~ 12^ e

e

+

J

(40)

ΝΧΙΧ Θ 2

1 N

° c2^ 22" 12^ · ε

ε

F

o

r

R e c u l e s of s i m i l a r

size and shape, the f i r s t term on the right-hand side of Equation (40) vanishes, and we have the one constant Margules expansion. I f , on the other hand, Equation (33b) i s assumed, we then find that each species coordination number i s a l i n e a r function of mole fraction, N,

N

Nij

x

+

cj

(41)

Xiôj

that N

i cj

X j

N

(42)

c j

and that EX A

_ ^

EX

A

T,V

with Θ j

- Θ - j

1

(43)

!

+ Νχ^Χ2Θ +ΝχΐΧ2(χι~Χ2)Φ

T,V

1 where v± i s some measure of the molecular volume of species i molecules X

v

x

v


194



If we assume that Equation (35) i s applicable to each species i n the mixture, then Equation (41) i s again obtained, and the two-constant Margules expansion results, though with s l i g h t l y d i f f e r e n t expressions for the Margules coeffcients. Indeed, whenever we s t a r t from l o c a l coordination number models of the type we have been considering, the excess Helmholtz (or Gibbs) free energy i s always of the Margules form, and the equation of state mixing rules are of the van der Waals one-fluid form at a l l densities. We now consider another class of l o c a l composition assumptions. Suppose, instead of Equations (33) we assume N

N

lj

C

- i

ij

x

, i

C

ij

(45)

where C-y i s some constant. [A common choice f o r the constants C i s some measure of the molecular volume v, so that = C - M v-^, and C j j = C j i = V j . In this case the r a t i o of the l o c a l coordination numbers, that i s the l o c a l composition r a t i o , i s equal to the ratio of volume fractions.] By i t s e l f , Equation (45) i s i n s u f f i c i e n t , since each species coordination number ( i . e . , the separate Njj and Njj) i s needed i n the generalized van der Waals p a r t i t i o n function analysis. Implicit i n many a c t i v i t y c o e f f i c i e n t models i s the a d d i t i o n a l and quite separate assumption that the t o t a l coordination number for each species i n the mixture, N j = N - J J ^jj> * independent of mole f r a c t i o n , and therefore equal to tne pure component coordination number. That i s , the assumption i s β

+

s

c

N

cj -

N

(46) cj

Equation (46) has i t o r i g i n i n l a t t i c e theory where a molecule i s presumed to have a fixed coordination number; i t may be v a l i d for s i m i l a r size molecules at high density, but i s incorrect for molecules of d i f f e r e n t s i z e , or at low and moderate densities. Solving Equations (45 and 46) yields X±A

y*i and

x

=

c

N

j cj Xj+xiAij

3i

H

Xj+xiAij

(47)

c

where A i j ij/ jj« [Equations f o r Nj£ and N are gotten from the expressions above by an interchange of indices.] Assuming that A y " i / j yields iA

v

v

ΔΕΧ GEX

=

Τ,Ρ

AEX

Α

= τ

^T,V

τ,ν

Ι [Τ-»

x

t

x

Ν JVi jVj 2 XiVi+XjVj

(48)

where m

(eji-ejj)

n

^

+

(ejj-sjj)

If the Τ-» term i s set equal to zero, Equation (48) i s the excess free energy expression which gives r i s e to the van Laar and Hildebrand-Scatchard Regular Solution a c t i v i t y c o e f f i c i e n t models.


8.


SANDLER ET A L .

195

A l t e r n a t i v e l y , using Equation (44) the Flory-Huggins a c t i v i t y c o e f f i c i e n t model i s obtained. One could instead choose the molecular surface areas f o r the C^j's instead of molecular volumes, to obtain a variant of these models i n which surface area fractions rather than volume fractions appear. I f , instead, we use Equation (47) and the assumption that the t o t a l coordination number i s a l i n e a r function of density i n Equation (22) we obtain the following equation of state mixing rule for the a parameter


A

=

^f-

· Σ Î M

x

3

This mixing rule does not s a t i s f y the low density van der Waals one-fluid boundary condition (unless =* V j ) , which i s not surprising since Equation (47) on which i t i s based i s incorrect at low density. Next consider yet another l o c a l composition model

£ii = £i£ii < e

e i r e j j

)/kT

( 5 0 )

which i s the ratio form of Equation (35). If we choose C j ^ = Cjj v-£, we have the l o c a l composition assumption of Wilson. If one further assumes that the t o t a l coordination number i s constant (Equation (46)) then Equations (47) are again obtained, but with e

c

(e

i r e j j

)/kT

(

5

1

β

)

Using these results i n Equation (2 7) leads to the Wilson a c t i v i t y c o e f f i c i e n t model; when used i n Equation (22) with the assumption that the t o t a l coordination number i s a l i n e a r function of density we obtain an a parameter mixing rule i n the form of Equation (49) which, as already mentioned, does not s a t i s f y the low density one-fluid mixing rule boundary condition. While the various l o c a l composition models considered so f a r lead to different equation of state mixing rules and a c t i v i t y c o e f f i c i e n t models, none result i n density dependent mixing rules which have been of much interest l a t e l y . From the analysis using the generalized van der Waals p a r t i t i o n function, i t i s evident that density dependent mixing rules can only result from a density dependent l o c a l composition model. Two such density dependent mixing rules have been suggested recently. The f i r s t , due to Whiting and Prausnitz [18J i s of the form |u N

=aî (e e

jj

N

i r e j j

)N

c j

/2kT

(

5

2

)

C

j JJ

where the t o t a l coordination number f o r eacn species N j = Njj + Njj i s assumed to be l i n e a r l y dependent on density, but independent of mole f r a c t i o n . This l o c a l composition model leads to the following mixing rule c



196

" Λ jV Xi

j

EXP

(

*^Γ$ *

E X P

( 5 3 )

where b i s a size parameter and » b^^N/4V. Also an a c t i v i t y c o e f f i c i e n t model i s obtained which i s similar to the three parameter Wilson equation. However, this model should not be expected to be very good since i t predicts that the effect of a t t r a c t i v e forces (the exponential term i n Equation (53)) vanishes at low density (where N j > 0), and i s largest at high density. This i s opposite to what should be expected since Equation (50) i s correct at low density, and as a t t r a c t i v e forces are of l i t t l e importance at high densities, the exponential term should approach unity (rather than increase i n value) i n this l i m i t . The recent " p r a c t i c a l " l o c a l composition model of Hu et a l . [19]


c

N

ij

s

3

f ^ R i j - ^ ^

3

ψ- expiaey/kT]

(54)

where 0.1865 a - 0.60-0.58 ( ρ ^ σ ϋ ) i 3

has a density dependence the observations above. density dependent mixing l o c a l composition models Table I I I .

which i s q u a l i t a t i v e l y i n agreement with This l o c a l composition model leads to the rule given i n reference 19. Many of the discussed above are summarized i n

Which Local Composition Model Is Best? In the previous section we considered numerous l o c a l composition models, and i t i s reasonable to ask which i s best? To answer this question we have been calculating l o c a l compositions i n mixtures of square-well f l u i d s using Monte Carlo simulation and i n t e g r a l equation methods. We report b r i e f l y some of our simulation results here. In Figure 3 we have plotted the quantity N

»21 l ll 2 N

N

a n

d

N N

1 2

22

N Νχ

2

as a function of density and composition for a mixture of squarewell molecules of equal diameter σ, but different well depths. The points represent our simulation results, and the arrows are the exact low density l i m i t s . For completely random mixing, both these ratios would be unity, which i s clearly not the case. Since the molecules are of equal diameter, = v , and the l o c a l composition ratios of Equation (45) also reduce to a unity at a l l densities. The Wilson l o c a l composition model of Equation (50) i s independent of density, and i s seen to be correct at zero density, but to 2


8.



197

Table III Local Coordination Number Model

Equation of State Mixing Rule


Ni Nlj - V - C

N

N

ij

vdW

1-fluid

1-constant Margules

vdW

1-fluid

2-constant Margules

vdW

1-fluid

2-constant Margules

i

ν " °±3

=

Free Energy

Ν-; N

=

j

J

v

"a

i/kT

N

jj

- y -

ebrand Regular Solution and FIory=HuggIns

if N

I

+

J

Ν

NJJ

ν

N

i

j

" ^ ^

. < ' * ™

)

/

Κ

C

J

a = (Ix

i V i

)

Π

Φ ΐ Φ ^

Wilson

if N^

+ Njj -

N

cj N

cj - ν J C




198

Ζ CM

Ζ

Ζ

Z~

Ζ

ζ

Ζ

Ζ

(VI CVJ

0.7 0.0

0.1

0.2

0.3

0.4

REDUCED

Figure 3.

0.5

DENSITY,

0.6 ρσ

0.8

0.7

3

Extent of l o c a l composition i n square-well mixture f o r εχι/kT - 0.4, 622/ °· > o"ll 22> l " 2 (·, 0 and +) Monte Carlo simulation; (····) Wilson (1964); ( ) Whiting and Prausnitz ( r e f . 18); ( ) Hu et a l . ( r e f . 19); ( ) Lee, Sandler and Patel (ref. 21); ( ) random mixing. (-• shows the theoretical low density l i m i t ) kT

β

8

=

R

R

=

U

5


:


8.



199

overpredict l o c a l composition effects at a l l other densities, random mixing at low density, and strong l o c a l composition effects at high density. As can be seen, this i s completely opposite from the computer simulation results which are i n agreement with both the exact low density result and the Weeks et a l . [20J conclusion that a t t r a c t i v e forces are of l i t t l e importance i n determining l o c a l composition at high density. Also, the Whiting-Prausnitz model shows no composition dependence. Since the underlying l o c a l composition mixing rule i s incorrect, i t i s not surprising that i n application, the density dependent mixing rule of Equation (53) i s found to be of l i t t l e u t i l i t y . Recently, Hu et a l . [19J have proposed the l o c a l composition model of Equation (54); the results of this model also appear i n Figure 3. There we see that for equalsize molecules, the Hu et a l . model predicts l o c a l composition e f f e c t s which q u a l i t a t i v e l y have the correct density dependence, but are too small at a l l densities. The model of Hu et a l . , l i k e the other models considered previously, shows no composition dependence i n the ratios we have plotted. Also shown i n Figure 3 are the predictions for the l o c a l composition ratios of Equation (2 0) from a model we have proposed recently [21J· This model has the correct low density and high density l i m i t s , a s l i g h t composition dependence, and i s i n better agreement with simulation results than the other models considered here. At present we are considering the extension of this model to molecules of unequal s i z e , and this work w i l l be presented separately [22]. Also, we consider elsewhere [23J the composition v a r i a t i o n of the t o t a l coordination number i n mixtures of unequal size molecules. Conclusions F i r s t , we have shown the u t i l i t y of the generalized van der Waals p a r t i t i o n function i n that i t allows us to i n t e r r e l a t e molecular l e v e l assumptions to applied thermodynamic models. This was used, i n Section II, to ascertain the coordination number models used i n a number of equations of state. These coordination number models were tested against our Monte Carlo simulation data for a square-well f l u i d , and none were found to be s a t i s f a c t o r y . A new coordination number model was proposed, and this was found to lead to an equation of state, with no adjustable parameters, which i s more accurate for the square-well f l u i d than multiconstant equations currently i n use. In Section I I I , the generalized van der Waals p a r t i t i o n function for mixtures was used to i d e n t i f y the molecular-level l o c a l composition and other assumptions imbedded i n commonly used equation of state mixing rules and a c t i v i t y c o e f f i c i e n t models. We then compared these assumptions for l o c a l composition effects with the results of our own Monte Carlo simulation studies for mixtures of square-well molecules i n Section IV. There we found that the models currently i n use do not properly account for nonrandom mixing due to a t t r a c t i v e energy e f f e c t s . This suggests that better l o c a l composition models are needed. Once they are obtained, the generalized van der Waals p a r t i t i o n function w i l l again be useful i n developing the macroscopic thermodynamic models which results from these molecular l e v e l assumptions.


200


Acknowledgments This work was supported, i n part, with funds provided by a National Science Foundation Grant (No. CPE-8316913) to the University of Delaware.


Literature Cited 1. Sandler, S. I. Fluid Phase Eq. 1985, 19, 233. 2. Alder, B. J.; Young, D. Α.; Mark, M. A. J. Chem. Phys. 1972, 56, 3013. 3. Redlich, O.; Kwong, J. N. S. Chem. Rev. 1949, 44, 233. 4. Peng, D. Y.; Robinson, D. B. I&EC Fund. 1976, 15, 59. 5. Beret, S.; Prausnitz, J. M. AIChE J. 1975, 21, 1123. 6. Donohue, M. D.; Prausnitz, J. M. AIChE J. 1978, 24, 849. 7. Lee, K.-H.; Lombardo, M.; Sandler, S. I. Fluid Phase Eq. 1985, 21, 177. 8. Carnahan, N. F.; Starling, Κ. E. J. Chem. Phys. 1969, 51, 635. 9. Aim, K.; Nezbeda, I. Fluid Phase Eq. 1983, 12, 235. 10. Ponce, L.; Renon, J. J. Chem. Phys. 1976, 64, 638. 11. Flory, P. J. "Principles of Polymer Chemistry," Ithaca, N.Y., Cornell University Press 1953. 12. Staverman, A. J. Rec. Trav. Chim. Pays-bas 1950, 69, 163. 13. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. "Regular and Related Solutions," Van Nostrand Reinhold Co. 1970. 14. Gierycz, P.; Nakanishi, K. Fluid Phase Eq. 1984, 16, 225. 15. Toukubo, K.; Nakanishi, K. J. Chem. Phys. 1976, 65, 1937. 16. Hoheisel, C.; Deiters, U. Molec. Phys. 1979, 37, 95. 17. See, for example, Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. "Molecular Theories of Gases and Liquids," J. Wiley & Sons, Inc., N.Y., 1953. 18. Whiting, W. B.; Prausnitz, J. M. Fluid Phase Equilibria 1982, 9, 119. 19. Hu, Y.; Ludecke, D.; Prausnitz, J. M. Fluid Phase Eq. 1984, 17, 217. 20. Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys. 1971, 54, 5237. 21. Lee, K.-H.; Sandler, S. I.; Patel, N. C. Fluid Phase Equilibria, accepted for publication. 22. Lee, K.-H.; Sandler, S. I. Fluid Phase Eq., to be submitted. 23. Lee, K.-H.; Sandler, S. I.; Monson, P. A. Ninth Thermophysical Properties Meeting, Boulder, CO., June 1985. RECEIVED

November 5, 1985


The Generalized van der Waals Partition Function as a Basis for

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