Origin of the Acentric Factor

1 Origin of the Acentric Factor K E N N E T H S. P I T Z E R

Downloaded by 82.6.218.247 on April 10, 2014 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch001

University of California, Berkeley, Calif. 94720

It was a pleasure to accept Dr. Sandler's invitation to open this conference by reviewing the ideas and general point of view which led me to propose the acentric factor in 1955. Although I had followed some of the work in which others have used the acentric factor, the preparation of this paper provided the incentive to review these applications more extensively, and I was most pleased to find that so much has been done. I want to acknowledge at once my debt to John Prausnitz for suggestions in this review of recent work as well as in many discussions through the years. Beginning in 1937, I had been very much interested in the thermodynamic properties of various hydrocarbon molecules and hence of those substances in the ideal gas state. This arose out of work with Kemp in 1936 on the entropy of ethane (1) which led to the determination of the potential barrier restricting internal rotation. With the concept of restricted internal rotation and some advances in the pertinent statistical mechanicsitbecame possible to calculate rather accurately the entropies of various light hydrocarbons (2). Fred Rossini and I collaborated in bringing together his heat of formation data and my entropy and enthalpy values to provide a complete coverage of the thermodynamics of these hydrocarbons in the ideal gas state (3). As an aside I cite the recent paper of Scott (4) who presents the best current results on this topic. But real industrial processes often involve liquids or gases at high pressures rather than ideal gases. Hence it was a logical extension of this work on the ideal gases to seek methods of obtaining the differences in properties of real fluids from the respective ideal gases without extensive experimental studies of each substance. My first step in this direction came in 1939 when I was able to provide a rigorous theory of corresponding states (5) on the basis of intermolecular forces for the restricted group of substances, argon, kryptron, xenon, and in good approximation also methane. This pattern of behavior came to be called that of a simple fluid. It is the reference pattern from which the acentric factor measures the departure. Possibly we should recall the key ideas. The 1 In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


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PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

i n t e r m o l e c u l a r p o t e n t i a l must be given by a u n i v e r s a l f u n c t i o n w i t h s c a l e f a c t o r s of energy and d i s t a n c e f o r each substance. By then i t was well-known that the dominant a t t r a c t i v e f o r c e f o l l o w e d an i n v e r s e sixth-power p o t e n t i a l f o r a l l of these substances. A l s o the r e p u l s i v e f o r c e s were known to be very sudden. Thus the i n v e r s e s i x t h , power term w i l l dominate the shape of the p o t e n t i a l curve at longer d i s t a n c e s . Even without d e t a i l e d t h e o r e t i c a l reasons f o r exact s i m i l a r i t y of shorter-range terms, one could expect that a u n i v e r s a l f u n c t i o n might be a good approximation. I n a d d i t i o n one assumed s p h e r i c a l symmetry (approximate f o r methane), the v a l i d i t y of c l a s s i c a l s t a t i s t i c a l mechanics, and that the t o t a l energy was determined e n t i r e l y by the v a r i o u s i n t e r m o l e c u l a r d i s t a n c e s . I should r e c a l l that i t was not f e a s i b l e i n 1939 to c a l c u l a t e the a c t u a l equation of s t a t e from t h i s model. One could o n l y show that i t y i e l d e d corresponding s t a t e s , i . e . , a u n i v e r s a l equation of s t a t e i n terms of the reduced v a r i a b l e s of temperature, volume, and pressure. One could p o s t u l a t e other models which would y i e l d a c o r r e s ponding-states behavior but d i f f e r e n t from that of the simple f l u i d . However, most such molecular models were s p e c i a l and d i d not y i e l d a s i n g l e f a m i l y of equations. Rowlinson (6) found a somewhat more general case; he showed that f o r c e r t a i n types of a n g u l a r l y dependent a t t r a c t i v e f o r c e s the net e f f e c t was a temperature dependent change i n the r e p u l s i v e term. From t h i s a s i n g l e f a m i l y of funct i o n s arose. I had observed e m p i r i c a l l y , however, that the f a m i l y r e l a t i o n ship of equations of s t a t e was much broader even than would f o l l o w from R o w l i n s o n s model. I t included g l o b u l a r and e f f e c t i v e l y s p h e r i c a l molecules such as tetramethylmethane (neopentane), where no a p p r e c i a b l e angular dependence was expected f o r the i n t e r m o l e c u l a r p o t e n t i a l , and f o r elongated molecules such as carbon d i o x i d e the angular dependence of the r e p u l s i v e f o r c e s seemed l i k e l y to be at l e a s t as important as that of the a t t r a c t i v e f o r c e s . Thus the core model of K i h a r a (7) appealed to me; he assumed that the LennardJones 6-12 p o t e n t i a l a p p l i e d to the s h o r t e s t d i s t a n c e between cores i n s t e a d of the d i s t a n c e between molecular centers. He was a b l e to c a l c u l a t e the second v i r i a l c o e f f i c i e n t f o r v a r i o u s shapes of core. And I was a b l e to show that one obtained i n good approximation a s i n g l e f a m i l y of reduced second v i r i a l c o e f f i c i e n t f u n c t i o n s f o r cores of a l l reasonable shapes. By a s i n g l e f a m i l y I mean that one a d d i t i o n a l parameter s u f f i c e d to d e f i n e the equation f o r any p a r t i c u l a r case. While t h i s d i d not prove that a l l of the complete equations of s t a t e would f a l l i n t o a s i n g l e f a m i l y , i t gave me enough encouragement to go ahead w i t h the numerical w o r k — o r more a c c u r a t e l y to persuade s e v e r a l students to undertake the n u m e r i c a l work. Let me emphasize the importance of f i t t i n g g l o b u l a r molecules i n t o the system. I f these molecules are assumed to be s p h e r i c a l i n good approximation, they are easy to t r e a t t h e o r e t i c a l l y . Why aren't they simple f l u i d s ? Many t h e o r e t i c a l papers ignore t h i s q u e s t i o n . In f l u i d p r o p e r t i e s neopentane departs from the simple f l u i d p a t t e r n much more than propane and almost as much as n-butane. But propane f

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


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Origin of the Acentric Factor

i s much l e s s s p h e r i c a l than neopentane. The e x p l a n a t i o n l i e s i n the narrower a t t r a c t i v e p o t e n t i a l w e l l . The i n v e r s e - s i x t h - p o w e r a t t r a c t i v e p o t e n t i a l now operates between each p a r t of the molecule r a t h e r than between molecular c e n t e r s . Thus the a t t r a c t i v e term i s steeper than i n v e r s e s i x t h power i n terms of the d i s t a n c e between molecular c e n t e r s . This i s shown i n F i g u r e 1, taken from my paper (8) i n 1955. We need not bother w i t h the d i f f e r e n c e s between the models y i e l d i n g the dotted and dashed curves f o r the g l o b u l a r molecule. The important f e a t u r e i s the narrowness of the p o t e n t i a l w e l l f o r e i t h e r of these curves as compared t o the s o l i d curve f o r the molecules of a simple f l u i d . I t was easy t o show that the i n t e r m o l e c u l a r p o t e n t i a l curves f o r s p h e r i c a l molecules would y i e l d a s i n g l e f a m i l y of reduced equations of s t a t e . I f one takes the K i h a r a model w i t h s p h e r i c a l c o r e s , then the r e l a t i v e core s i z e can be taken as the t h i r d parameter i n a d d i t i o n t o the energy and d i s t a n c e s c a l e f a c t o r s i n the t h e o r e t i c a l equation of s t a t e . With an adequate understanding of g l o b u l a r molecule b e h a v i o r , I then showed as f a r as was f e a s i b l e that the p r o p e r t i e s of other nonp o l a r or weakly p o l a r molecules would f a l l i n t o the same f a m i l y . I t was p r a c t i c a l a t that time only to c o n s i d e r the second v i r i a l c o e f f i cent. The K i h a r a model was used f o r nonpolar molecules of a l l shapes w h i l e R o w l i n s o n s work provided the b a s i s f o r d i s c u s s i o n of p o l a r molecules. F i g u r e 2 shows the reduced second v i r i a l c o e f f i c i e n t f o r s e v e r a l cases. Curves l a b e l e d a / p r e f e r t o s p h e r i c a l - c o r e molecules w i t h a i n d i c a t i n g the core s i z e , c o r r e s p o n d i n g l y £/p i n d i c a t e s a l i n e a r molecule of core l e n g t h £, w h i l e y r e f e r s to a d i p o l a r molecule w i t h y = u /e ^r where u i s the d i p o l e moment. The non-polar p o t e n t i a l i s 1

Q

Q

0

Q

(i) where p i s the s h o r t e s t d i s t a n c e between c o r e s . For the p o l a r molecules I omitted the core, thus p = r . While the curves i n F i g u r e 2 appear t o f a l l i n t o a s i n g l e f a m i l y , t h i s i s i n v e s t i g a t e d more r i g o r o u s l y i n F i g u r e 3 where the reduced second v i r i a l c o e f f i c i e n t a t one reduced temperature i s compared w i t h the same q u a n t i t y a t another temperature. Tg i s the Boyle temperature which i s a convenient r e f e r e n c e temperature f o r second v i r i a l c o e f f i c i e n t s . One sees that the non-polar core molecules f a l l a c c u r a t e l y on a s i n g l e curve (indeed a s t r a i g h t l i n e ) . While the p o l a r molecules d e v i a t e , the d i f f e r e n c e i s o n l y 1% a t y = 0.7 which I took as a reasonable standard of accuracy a t that time. For comparison the y values of c h l o r o f o r m , e t h y l c h l o r i d e , and ammonia are 0.04, 0.16, and 4, r e s p e c t i v e l y . Thus the f i r s t two f a l l w e l l below the 0.7 v a l u e f o r agreement of p o l a r w i t h non-polar e f f e c t s w h i l e ammonia i s beyond that v a l u e . The next q u e s t i o n was the c h o i c e of the experimental b a s i s f o r the t h i r d parameter. The vapor pressure i s the property most s e n s i t i v e t o t h i s t h i r d parameter; a l s o i t i s one of the p r o p e r t i e s most


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PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY 1

11

1

1

1 -

r

!

_^*- *^-


if 1

0

0.5

\J

i

1.0

1.5

i 2.0

2.5

r/r . 0

Figure 1. Intermolecular potential for molecules of a simple fluid, solid line; and for globular molecules such as C(CH ) dashed lines 3

T

B

If>

/ T .

Figure 2. Reduced second virial coefficients for several models: solid curve, simple fluid; curves labeled by a./p , spherical cores of radius a; curves labeled by l/p , linear cores of length 1; curves labeled by y, molecules with dipoles 0


0


Figure 3. Check on family relationship of curves of Figure 2. Comparison of deviations from simple fluid at (T /TJ = 3.5 with that at (T /T) = 2.0 B

B


6


w i d e l y measured at l e a s t near the normal b o i l i n g p o i n t . Thus both the a v a i l a b i l i t y of data and the accuracy of the data f o r the purpose s t r o n g l y i n d i c a t e d a vapor p r e s s u r e c r i t e r i o n . Since the c r i t i c a l data have to be known f o r a reduced equation of s t a t e , the reduced vapor pressure near the normal b o i l i n g p o i n t was an easy choice f o r the new parameter. The a c t u a l d e f i n i t i o n a) = -£og P

r

- 1.000

(2)

w i t h P the reduced vapor pressure a t T = 0.700 seemed convenient, but the a c t u a l d e t e r m i n a t i o n of a) can be made from any vapor pressure v a l u e well-removed from the c r i t i c a l p o i n t . Here I should note the work of R i e d e l (9) which was substant i a l l y simultaneous w i t h mine but whose f i r s t paper preceded s l i g h t l y . His work was p u r e l y e m p i r i c a l , but was e x c e l l e n t and f u l l y complementary. He chose f o r h i s t h i r d parameter a l s o the slope of the vapor pressure c u r v e , but i n h i s case the d i f f e r e n t i a l s l o p e a t the c r i t i c a l p o i n t . That seemed to me to be l e s s r e l i a b l e and a c c u r a t e , e m p i r i c a l l y , although e q u i v a l e n t o t h e r w i s e . F o r t u n a t e l y R i e d e l and I chose to emphasize d i f f e r e n t p r o p e r t i e s as our r e s p e c t i v e programs proceeded; hence the f u l l area was covered more q u i c k l y w i t h l i t t l e d u p l i c a t i o n of e f f o r t . A l s o I needed a name f o r t h i s new parameter, and that was d i f f i c u l t . The term " a c e n t r i c f a c t o r " was suggested by some f r i e n d l y reviewer, p o s s i b l y by a r e f e r e e ; I had made a l e s s s a t i s f a c t o r y choice i n i t i a l l y . The conceptual b a s i s i s i n d i c a t e d i n F i g u r e 4. The i n t e r m o l e c u l a r f o r c e s between complex molecules f o l l o w a simple e x p r e s s i o n i n terms of the d i s t a n c e s between the v a r i o u s p o r t i o n s of the molecule. Since these f o r c e s between n o n - c e n t r a l p o r t i o n s of the molecules must be c o n s i d e r e d , the term " a c e n t r i c f a c t o r " seemed appropriate. I t i s assumed that the c o m p r e s s i b i l i t y f a c t o r and other propert i e s can be expressed i n power s e r i e s i n the a c e n t r i c f a c t o r and that a linear expression w i l l usually s u f f i c e .


r

r

pv = — = z RT

i

z

( 0 )

(0) z

= z -

z

r

r

( 1 )

The preference of P over V as the second independent v a r i a b l e i s p u r e l y e m p i r i c a l ; the c r i t i c a l pressure i s much more a c c u r a t e l y measurable than the c r i t i c a l volume. The e m p i r i c a l e f f e c t i v e n e s s of t h i s system was f i r s t t e s t e d w i t h v o l u m e t r i c data as shown on F i g u r e 5. Here pv/RT a t a p a r t i c u l a r r

r


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Ar

Ar

i Q \

CH


C

CH

4

4

Figure 4. Intermolecular forces operate between the centers of regions of substantial electron density. These centers are the molecular centers for Ar and (approximately) for CH but are best approximated by the separate CH and CH groups in C H —hence the name acentric factor for the forces arising from points other than molecular centers.

3 8 H

h

3

2

3

8

i.O

0.8-

1.30 .25 1.20

1.15

0.6 PV RT'

' 1.10

0.4

• - 1.05

1

0

#f 1.00

0.2-

A Xe

CH

4

C H HS 2

C(CH ) n-C H, 3

6

4

2

C

3

7

0

l6

2

6 6 H

C0 C H

n-C H H0

4

2

NH

8

3

I

0.1

0.2 CJ.

0.3

0.4

Figure 5. Compressibility factor as a function of acentric factor for reduced pressure 1.6 and reduced temperature shown for each line. Where several substances have approximately the same acentric factor, the individual points are indistinguishable except for n-C H C) and H O(Q). 7

t


16

8


reduced temperature and pressure i s p l o t t e d a g a i n s t u). The most important r e s u l t appears only by i m p l i c a t i o n ; the r e s u l t s f o r C(CH^)^, n-CifiiQ 6 6> C02 are so n e a r l y equal t h a t they appear as s i n g l e p o i n t s on these p l o t s . Here we have f o u r w i d e l y d i f f e r e n t shapes of molecules which happen to have about the same a c e n t r i c f a c t o r , and they f o l l o w corresponding s t a t e s a c c u r a t e l y among themselves. A l s o to be noted from F i g u r e 5 i s the f a c t that the h i g h l y p o l a r molecules NH3 and H2O depart from the system. Furthermore the dependence on a) i s l i n e a r except f o r the c r i t i c a l r e g i o n . My immediate r e s e a r c h group used g r a p h i c a l methods i n d e a l i n g w i t h the experimental data and r e p o r t e d a l l of our r e s u l t s i n numeric a l t a b l e s (10). At t h a t time the best a n a l y t i c a l equation of s t a t e was t h a t of B e n e d i c t , Webb and Rubin (11) which employed e i g h t parameters and s t i l l f a i l e d to f i t v o l u m e t r i c data w i t h i n experimental accuracy. Bruce Sage suggested f i t t i n g t h i s equation to the data for the normal p a r a f f i n s both d i r e c t l y f o r each substance and w i t h i n the a c e n t r i c f a c t o r system. T h i s work (12) was done p r i m a r i l y by J . B. O p f e l l a t C a l Tech. The r e s u l t s showed that the a c e n t r i c f a c t o r system was a great advance over the simple p o s t u l a t e of corresponding s t a t e s , but the f i n a l agreement was i n f e r i o r to that obtained by g r a p h i c a l and numerical methods. Thus we continued w i t h numerical methods f o r the f u g a c i t y , entropy, and enthalpy f u n c t i o n s (13), although we d i d present an e m p i r i c a l equation f o r the second v i r i a l c o e f f i c i e n t (14). This work was done by Bob C u r l ; he d i d an e x c e l l e n t job but found the almost i n t e r m i n a b l e g r a p h i c a l work very tiresome. Thus I was pleased t h a t the B r i t i s h I n s t i t u t i o n of Mechanical Engineers i n c l u d e d C u r l i n the award of t h e i r C l a y t o n P r i z e f o r t h i s work. A f i f t h paper w i t h H u l t g r e n (15) t r e a t e d mixtures on a p s e u d o c r i t i c a l b a s i s , and a s i x t h w i t h Danon (16) r e l a t e d K i h a r a core s i z e s to the acentric factor. N a t u r a l l y , I am v e r y pleased to note t h a t o t h e r s have extended the accuracy and range of our t a b l e s and equations w i t h c o n s i d e r a t i o n of more recent experimental r e s u l t s . Of p a r t i c u l a r l y broad importance i s the 1975 paper by Lee and K e s s l e r (17) which presents both improved t a b l e s and a n a l y t i c a l equations f o r a l l of the major f u n c t i o n s i n c l u d i n g vapor p r e s s u r e s , v o l u m e t r i c p r o p e r t i e s , e n t h a l p i e s , e n t r o p i e s , f u g a c i t i e s , and heat c a p a c i t i e s . Their equation i s an e x t e n s i o n of that of Benedict, Webb, and Rubin now c o n t a i n i n g twelve parameters. They considered more recent experimental data as w e l l as a number of papers which had a l r e a d y extended my e a r l i e r work i n p a r t i c u l a r areas. I r e f e r to t h e i r b i b l i o g r a p h y (17) f o r most of t h i s more d e t a i l e d work, but I do want to note the improved equation of Tsonopoulos (18) f o r the second v i r i a l c o e f f i c i e n t . This equation deals a l s o w i t h e f f e c t s of e l e c t r i c a l p o l a r i t y . In a d d i t i o n to r e f e r e n c e s c i t e d by Lee and K e s s l e r there i s the work Lyckman, E c k e r t , and P r a u s n i t z (19) d e a l i n g w i t h l i q u i d volumes; they found i t necessary to use a q u a d r a t i c e x p r e s s i o n i n u). A l s o Barner and Quinlan (20) t r e a t e d mixtures at high temperatures and p r e s s u r e s , and Chueh and P r a u s n i t z (21) t r e a t e d the c o m p r e s s i b i l i t y C

H

a n d


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of l i q u i d s . Reid and Sherwood (22) g i v e an e x t e n s i v e t a b l e i n c l u d i n g a c e n t r i c f a c t o r s as w e l l as c r i t i c a l constants f o r many substances. On the t h e o r e t i c a l s i d e , one great advance has been i n the development of p e r t u r b a t i o n t h e o r i e s of a g e n e r a l i z e d van der Waals type. Here one assumes t h a t the molecular d i s t r i b u t i o n i s d e t e r mined p r i m a r i l y by r e p u l s i v e f o r c e s which can be approximated by hard cores. Then both the s o f t n e s s of the cores and the a t r a c t i v e f o r c e s a r e t r e a t e d by p e r t u r b a t i o n methods. Barker and Henderson (23) have r e c e n t l y reviewed t h e o r e t i c a l advances i n c l u d i n g t h e i r own outstanding work. Rigby (24) a p p l i e d these modern Van der Waals methods t o n o n - s p h e r i c a l molecules which represent one type of molecules w i t h non-zero a c e n t r i c f a c t o r s . I n a somewhat s i m i l a r manner Beret and P r a u s n i t z (25) developed equations a p p l i c a b l e even to h i g h polymers and r e l a t e d the i n i t i a l departures from simple f l u i d s t o the a c e n t r i c f a c t o r . But i n my view the most e f f e c t i v e approach would concentrate f i r s t on g l o b u l a r molecules. These could be modeled by K i h a r a potent i a l s w i t h s p h e r i c a l cores or by other p o t e n t i a l s a l l o w i n g the w e l l to be narrowed. The great advantage would be the r e t e n t i o n of spheri c a l symmetry and i t s t h e o r e t i c a l s i m p l i c i t y . Rogers and P r a u s n i t z (26) made an important beginning i n t h i s area w i t h c a l c u l a t i o n s based on K i h a r a models a p p r o p r i a t e f o r argon, methane, and neopentane w i t h e x c e l l e n t agreement f o r the p r o p e r t i e s s t u d i e d . While they do not d i s c u s s these r e s u l t s i n terms of the a c e n t r i c f a c t o r , the t r a n s formation of s p h e r i c a l core r a d i u s t o a c e n t r i c f a c t o r i s w e l l e s t a b l i s h e d (16, 27), Rogers and P r a u s n i t z were a l s o able to t r e a t mixtures very s u c c e s s f u l l y although those c a l c u l a t i o n s were burdensome even w i t h modern computers. I b e l i e v e f u r t h e r t h e o r e t i c a l work using s p h e r i c a l models f o r g l o b u l a r molecules would be f r u i t f u l . The move to an a n a l y t i c a l equation by Lee and K e s s l e r was undoubtedly a wise one i n view of the marvelous c a p a c i t y of modern computers to d e a l w i t h complex equations. I would expect f u t u r e work to y i e l d s t i l l b e t t e r equations. There remains the q u e s t i o n of the u l t i m a t e accuracy of the a c e n t r i c f a c t o r concept. How a c c u r a t e l y do molecules of d i f f e r e n t shapes but w i t h the same a c e n t r i c f a c t o r r e a l l y f o l l o w corresponding s t a t e s ? Apparently t h i s accuracy i s w i t h i n experimental e r r o r f o r most, i f not a l l , present data. Thus the a c e n t r i c f a c t o r system c e r t a i n l y meets engineering needs, and i t i s p r i m a r i l y a matter of s c i e n t i f i c c u r i o s i t y whether d e v i a t i o n s a r e p r e s e n t l y measurable. I t has been a pleasure to review these aspects of the " a c e n t r i c f a c t o r " w i t h you and I look forward to your d i s c u s s i o n of recent advances i n these and other areas.


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Literature Cited

1.

Kemp, J . D. and P i t z e r , K. S., J . Chem. Phys., (1936) 4, 749; J. Am. Chem. Soc. (1937) 59, 276.


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2. 3.

4. 5. 6.


7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.


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Origin of the Acentric Factor

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