State-of-the-Art Review of Phase Equilibria - ACS Symposium Series

2 State-of-the-Art Review of Phase Equilibria J. M. PRAUSNITZ

Downloaded by UNIV OF MISSOURI COLUMBIA on July 8, 2013 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch002

University of California, Berkeley, Calif. 94720

I welcome the opportunity to discuss the state of the art for calculating phase equilibria in chemical engineering first, because I consider it a high honor to have been chosen for this important assignment and second, because it may give me a chance to influence the direction of future research in this field. When I mentioned these two reasons to one of my more candid coworkers, he said "What you really mean is, that you enjoy the opportunity to go on an ego trip and that you are glad to have an audience which you can subject to your prejudices." While this restatement of my feelings is needlessly unkind, I must confess that it bears an element of truth. The assignment that Professor Sandler has given me--to review applied phase equilibrium in an hour or two--is totally impossible and it follows that in choosing material for this presentation, I must be highly selective. Since time is limited, I must omit many items which others, in exercising their judgment, might have included. At the outset, therefore, I want to apologize to all in the audience who may feel that some publications, notably their own, have received inadequate attention. While I have tried to be objective and critical in my selection, it is human nature to give preference to that work with which one is most familiar and that, all too often, tends to be one's own. Nevertheless, I shall try to present as balanced a picture as I can. After more than 20 years, I have developed a certain point of view conditioned by my particular experience and I expect that it is pervasive in what I have to say. However, I want very much to assure this audience that I present my point of view without dogmatic intent; it is only a personal statement, a point of departure for what I hope will be vigorous discussion during the days ahead. My aim in attending this conference is the same as yours: at the end of the week I want to be a little wiser than I am now, at the beginning.

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

12

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY


Thermodynamics:

Not Magic but a Tool

A l l too o f t e n , when I t a l k w i t h chemical engineers from i n d u s t r y who have l i t t l e experience i n thermodynamics, I o b t a i n the impression that they look upon me as a medicine man, a magician who i s supposed to i n c a n t obscure formulas and, i n e f f e c t , produce something out of nothing. This audience knows b e t t e r but n e v e r t h e l e s s , we must remind o u r s e l v e s that thermodynamics i s not magic, that i t i s only a u s e f u l t o o l f o r e f f i c i e n t o r g a n i z a t i o n of knowledge. Thermodynamics alone never t e l l s us the value of a d e s i r e d e q u i l i b r i u m property; i n s t e a d , i t t e l l s us how the d e s i r e d e q u i l i b r i u m property i s r e l a t e d to some other e q u i l i b r i u m property. Thus thermodynamics provides us w i t h a time-saving bookkeeping system: we do not have to measure a l l the e q u i l i b r i u m p r o p e r t i e s ; we measure only some and then we can c a l c u l a t e o t h e r s . Thus, from an engineer's p o i n t of view, the main advantage of thermodynamics i s that i t reduces experimental e f f o r t : e.g., i f we know how the Gibbs energy of mixing v a r i e s w i t h temperature, we need not measure the enthalpy of mixing s i n c e we can c a l c u l a t e i t u s i n g the Gibbs-Helmholtz equation, o r , i n a b i n a r y system, i f we know how the a c t i v i t y c o e f f i c i e n t of one component v a r i e s w i t h compos i t i o n , we can use the Gibbs-Duhem equation to c a l c u l a t e the other. We must keep reminding ourselves and others as to j u s t what thermodynamics can and cannot do. F a l s e expectations o f t e n l e a d t o c o s t l y disappointments. While the l i m i t a t i o n s of c l a s s i c a l thermodynamics a r e c l e a r enough, the p o t e n t i a l l y v a s t p o s s i b i l i t i e s opened by s t a t i s t i c a l thermodynamics a r e s t i l l f a r from r e a l i z e d . J u s t what modern p h y s i c s can do f o r us w i l l be discussed l a t e r i n the week; f o r now, I j u s t want to say that even a t t h i s e a r l y stage, simple molecular ideas can do much to s t r e t c h the range of a p p l i c a t i o n of thermodynamics. When thermodynamics i s coupled w i t h the molecular theory of matter, we can c o n s t r u c t u s e f u l models; w h i l e these only roughly approximate t r u e molecular behavior, they n e v e r t h e l e s s enable us t o i n t e r p o l a t e and e x t r a p o l a t e w i t h some confidence, thereby reducing f u r t h e r the experimental e f f o r t r e q u i r e d f o r r e l i a b l e r e s u l t s . When my n o n t e c h n i c a l f r i e n d s ask me what I , a molecular thermodynamicist do, I answer w i t h a naive but e s s e n t i a l l y accurate analogy: I am a greedy tax c o l l e c t o r . From the s m a l l e s t p o s s i b l e c a p i t a l , I t r y t o e x t r a c t the l a r g e s t p o s s i b l e revenue. Keeping i n mind that thermodynamics i s no more than an e f f i c i ent t o o l f o r o r g a n i z i n g knowledge toward u s e f u l ends, I f i n d t h a t , f o r phase-equalibrium work, thermodynamics provides us w i t h two procedures, as shown i n F i g u r e 1. Our aim i s to c a l c u l a t e f u g a c i t i e s and we can do so e i t h e r using method ( a ) , based e n t i r e l y on an equat i o n of s t a t e a p p l i c a b l e to both phases a and $, or u s i n g method (b), which uses an equation of s t a t e only f o r c a l c u l a t i n g the vaporphase f u g a c i t y and a completely d i f f e r e n t method, expressed by the a c t i v i t y c o e f f i c i e n t , f o r c a l c u l a t i n g condensed-phase f u g a c i t i e s . I now want to examine these two methods because they a r e the ones which have been used i n e s s e n t i a l l y a l l a p p l i e d p h a s e - e q u i l i b r i u m work.

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

2.

FOR EVERY COMPONENT i , Downloaded by UNIV OF MISSOURI COLUMBIA on July 8, 2013 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch002

13

Review of Phase Equilibria

PRAUSNITZ

£ l

= £ l

IN PHASES a AND 0

f = FUGACITY

EITHER

- J n

i

d

V

"

l

n

^1

= MOLES OF i ;

V = TOTAL VOLUME

fl.y.P

f^-Tx.f?

OR

(b)

fY= I

AND

I

y,x = COMPOSITION;

I

'

1

1

1

° = STANDARD STATE

0 = FUGACITY COEFFICIENT (FROM EQUATION OF STATE) T = ACTIVITY COEFFICIENT Figure 1.

Two thermodynamic methods for calculation of fluid-phase equilibria



(b)

(a)

METHOD

P - V - T - X DATA ARE SUFFICIENT; IN PRINCIPLE, NO PHASE EQUILIBRIUM DATA NEEDED.

EASILY UTILIZES THEOREM OF CORRESPONDING STATES.

CAN BE APPLIED TO CRITICAL REGION.

SIMPLE LIQUID-MIXTURE MODELS ARE OFTEN SATISFACTORY.

EFFECT OF TEMPERATURE IS IN f , NOT r .

APPLICABLE TO WIDE VARIETY OF MIXTURES, INCLUDING POLYMERS AND ELECTROLYTES.

2.

3.

4.

1.

2.

3.

PRIMARILY

NO STANDARD STATES.

1.

ADVANTAGES

CUMBERSOME FOR COMPONENTS.

2. 3.

NEED SEPARATE METHOD TO FIND v

1.

DIFFICULT TO APPLY IN CRITICAL REGION.

Q

DIFFICULT TO APPLY TO POLAR COMPOUNDS, LARGE MOLECULES, OR ELECTROLYTES.

3.

SUPER-CRITICAL

OFTEN VERY SENSITIVE TO MIXING RULES.

2.

0

NO REALLY GOOD EQUATION OF STATE AVAILABLE FOR ALL DENSITIES

1.

DISADVANTAGES



2.

PRAUSNITZ


15

When encountering a p a r t i c u l a r p h a s e - e q u i l i b r i u m problem, the very f i r s t d e c i s i o n i s to decide which of these methods i s most s u i t a b l e for the p a r t i c u l a r problem. I t i s t h e r e f o r e important t o review the r e l a t i v e advantages and disadvantages of both methods; these are summarized i n F i g u r e 2. The s t a t e of the a r t today i s such that f o r mixtures of simple, or what P i t z e r has c a l l e d "normal" f l u i d s , we can o f t e n c a l c u l a t e v a p o r - l i q u i d e q u i l i b r i a , even at high p r e s s u r e s , w i t h good success u s i n g some e m p i r i c a l equation of s t a t e . However, f o r mixtures i n c l u d i n g one or more s t r o n g l y p o l a r or hydrogen-bonding component, we must r e s o r t t o the use of a c t i v i t y c o e f f i c i e n t s and standardstate fugacities. As i n d i c a t e d i n F i g u r e 2, an equation of s t a t e f o r a l l f l u i d phases has many advantages because one very troublesome f e a t u r e , v i z . s p e c i f y i n g a standard s t a t e , i s avoided. This f e a t u r e i s troublesome because we f r e q u e n t l y are concerned w i t h multicomponent mixtures where a t l e a s t one component i s s u p e r c r i t i c a l . I n that event, the choice of a p r o p e r l y defined a c t i v i t y c o e f f i c i e n t and standard s t a t e i n t r o d u c e s formal d i f f i c u l t i e s which are o f t e n mathematically inconvenient and, f o r p r a c t i c a l implementation, r e q u i r e parameters from experimental data that are only r a r e l y available. For l i q u i d - p h a s e m i x t u r e s , p o l a r or nonpolar, i n c l u d i n g polymers and e l e c t r o l y t e s , a t low o r moderate p r e s s u r e s , the a c t i v i t y c o e f f i c i e n t provides the most convenient t o o l we have but our fundamental knowledge about i t i s sparse. Thermodynamics gives us l i t t l e h e l p ; we have three well-known r e l a t i o n s : f i r s t , the Gibbs-Duhem equation which r e l a t e s the a c t i v i t y c o e f f i c i e n t of one component i n a s o l u t i o n t o those of the o t h e r s , second, the Gibbs-Helmholtz equation which r e l a t e s the e f f e c t of temperature on the a c t i v i t y c o e f f i c i e n t to the enthalpy of mixing and f i n a l l y , an equation which r e l a t e s the p a r t i a l molar volume t o the e f f e c t of pressure on the a c t i v i t y c o e f f i c i e n t . These i l l u s t r a t e what I s a i d e a r l i e r , v i z . that c l a s s i c a l thermodynamics i s l i t t l e more than an e f f i c i e n t o r g a n i z a t i o n of knowledge, r e l a t i n g some e q u i l i b r i u m p r o p e r t i e s to o t h e r s , thereby reducing experimental work. But the p r a c t i c a l a p p l i c a t i o n s of these c l a s s i c a l thermodynamic r e l a t i o n s f o r a c t i v i t y c o e f f i c i e n t s are l i m i t e d , i n c o n t r a s t to the more powerful thermodynamic r e l a t i o n s which enable us to c a l c u l a t e f u g a c i t i e s u s i n g only v o l u m e t r i c p r o p e r t i e s . From a s t r i c t l y thermodynamic p o i n t of view, u s i n g an equation of s t a t e i s more e f f i c i e n t than using a c t i v i t y c o e f f i c i e n t s . I f we have an equation of s t a t e a p p l i c a b l e to a l l phases of i n t e r e s t , we can c a l c u l a t e not only the f u g a c i t i e s from v o l u m e t r i c data but a l s o a l l the other c o n f i g u r a t i o n a l p r o p e r t i e s such as the enthalpy, entropy and volume change on mixing. Our i n a b i l i t y to use equations of s t a t e f o r many p r a c t i c a l s i t u a t i o n s f o l l o w s from our inadequate understanding of f l u i d s t r u c t u r e and i n t e r m o l e c u l a r f o r c e s . Only f o r simple s i t u a t i o n s do we have t h e o r e t i c a l i n f o r m a t i o n on s t r u c t u r e and f o r c e s f o r e s t a b l i s h i n g an equation of s t a t e w i t h a t h e o r e t i c a l b a s i s and only f o r the more



16


common f l u i d s do we have s u f f i c i e n t experimental i n f o r m a t i o n to e s t a b l i s h r e l i a b l e e m p i r i c a l equations of s t a t e . Thanks to c o r r e s ponding s t a t e s , we can extend the a v a i l a b l e e m p i r i c a l b a s i s to a much wider c l a s s of f l u i d s but again, we are l i m i t e d here because corresponding s t a t e s cannot e a s i l y be extended to p o l a r or hydrogenbonding m a t e r i a l s . Our b i g g e s t b o t t l e n e c k i s that we have not been able to e s t a b l i s h a u s e f u l s t a t i s t i c a l mechanical treatment f o r such f l u i d s nor even to c h a r a c t e r i z e the i n t e r m o l e c u l a r f o r c e s between t h e i r molecules. At l i q u i d - l i k e d e n s i t i e s , the d i p o l e moment i s not good enough and the s t r e n g t h of a hydrogen bond depends not only on p a r t i c u l a r c o n d i t i o n s l i k e d e n s i t y and temperature but, what i s worse, a l s o on the method used to measure i t . L a t e r i n the week, when we d i s c u s s the c o n t r i b u t i o n of theory, we s h a l l h o p e f u l l y r e t u r n to some of these problems. Equations of S t a t e f o r Both F l u i d Phases Let us now see what k i n d of p r a c t i c a l p h a s e - e q u i l i b r i u m problems we can handle using nothing beyond one of the many c u r r e n t l y a v a i l a b l e equations of s t a t e . For r e l a t i v e l y simple mixtures, e.g., those found i n processing of n a t u r a l gas and l i g h t petroleum f r a c t i o n s , we do w e l l w i t h one of the many m o d i f i c a t i o n s of the BenedictWebb-Rubin equation; i n i t s o r i g i n a l v e r s i o n , t h i s equation had e i g h t constants f o r each f l u i d but i n l a t e r v e r s i o n s t h i s number had i n c r e a s e d , sometimes c o n s i d e r a b l y so. To i l l u s t r a t e , Figure 3 shows c a l c u l a t e d and observed K f a c t o r s f o r methane i n heptane at two temperatures. In these c a l c u l a t i o n s , Orye (1) f o l l o w e d the u s u a l procedure; he assumed a o n e - f l u i d theory, i . e . , he assumed t h a t the equation of s t a t e of the mixture i s the same as that of a pure f l u i d except that the c h a r a c t e r i s t i c constants depend on composition according to some more or l e s s a r b i t r a r y r e l a t i o n s known as mixing r u l e s . Experience has repeatedly shown that at l e a s t one of these mixing r u l e s must c o n t a i n an a d j u s t a b l e b i n a r y constant; i n t h i s case, that constant i s M-^j which was found by f i t t i n g to the b i n a r y data. U n f o r t u n a t e l y , the c a l c u l a t e d r e s u l t s are o f t e n h i g h l y s e n s i t i v e to the mixing r u l e s and to the value of the a d j u s t a b l e parameter. In t h i s case Orye found what many others have a l s o found, v i z . , that the a d j u s t a b l e b i n a r y parameter i s more-or-less i n v a r i a n t w i t h d e n s i t y and composition but o f t e n depends on temperature. Another example, a l s o from Orye, i s given i n F i g u r e 4 f o r the system methane-carbon d i o x i d e at -65°F. The continuous l i n e through the diamonds i s not c a l c u l a t e d but connects the experimental p o i n t s of Donnelly and Katz; the c a l c u l a t e d l i n e s are dashed and the c i r c l e s and t r i a n g l e s i n d i c a t e p a r t i c u l a r c a l c u l a t i o n s , not data. F i r s t we note that the value of M^j has a strong e f f e c t , e s p e c i a l l y on the l i q u i d u s curve; a ten percent change i n M-^j produces a l a r g e e r r o r i n the bubble pressure. When M^j i s adjusted e m p i r i c a l l y to 1.8, much b e t t e r r e s u l t s are achieved but note that Orye r e p o r t s no c a l c u l a t i o n s i n the c r i t i c a l r e g i o n . There are two good reasons f o r t h i s : f i r s t , a l l c l a s s i c a l a n a l y t i c a l equations tend to be poor i n the


2.

PKAUSNITZ

17



\- o

1200

Figure 3.

Methane-n-heptane (Orye, 1969) • O Kohn (1961); equation

Modified BWR




1000

1.0 Mole Fraction Methane

Figure 4. Methane-carbon dioxide (Orye, 1969). Temp., —65°F; 0 Donnelly and Katz (1954); O modified BWR equation, Mn = 1.8; A modified BWR equation, original mixing rule, Mn = 2.0.



2.

PRAUSNITZ


19

c r i t i c a l r e g i o n and second, computational problems are o f t e n severe because convergence i s hard to achieve. A s i m i l a r s i t u a t i o n i s shown i n F i g u r e 5, taken from S t a r l i n g and Han (2^), who used on 11-constant v e r s i o n of the BWR equation. Again, note t h a t an a d j u s t a b l e b i n a r y constant k^i i s r e q u i r e d . A l s o , note t h a t , c o n t r a r y to u s u a l p r a c t i c e , the l i n e s represent experiment and the p o i n t s represent c a l c u l a t i o n s , suggesting problems i n the c r i t i c a l r e g i o n . I t i s evident t h a t the more constants i n an equation of s t a t e , the more f l e x i b i l i t y i n f i t t i n g experimental data but i t i s a l s o c l e a r t h a t to o b t a i n more c o n s t a n t s , one r e q u i r e s more experimental i n f o r m a t i o n . For example, a twenty-constant equation of s t a t e , e s s e n t i a l l y an e x t e n s i o n of the BWR equation, was proposed by Bender (_3) who a p p l i e d i t to oxygen, argon, n i t r o g e n , and a few l i g h t hydrocarbons. For these f l u i d s , Bender i s able to o b t a i n a h i g h l y accurate r e p r e s e n t a t i o n of experimental data over a wide d e n s i t y range. To i l l u s t r a t e one u n u s u a l l y f i n e f e a t u r e of Bender's equat i o n , F i g u r e 6 shows the r e s i d u a l heat c a p a c i t y of propylene f o r s e v e r a l temperatures near the c r i t i c a l temperature, 365 K. This i s a very s e n s i t i v e t e s t and Bender's equation does a remarkable j o b . Bender has a l s o a p p l i e d h i s equation to mixtures of argon, n i t r o g e n , and oxygen, u s e f u l f o r design of a i r - s e p a r a t i o n p l a n t s . For each b i n a r y mixture, Bender r e q u i r e s 3 b i n a r y parameters. With a l l these constants and a l a r g e computer program, Bender can c a l c u l a t e not only accurate v a p o r - l i q u i d e q u i l i b r i a but a l s o heats of mixing as shown i n F i g u r e 7. The heats of mixing here are very s m a l l and agreement between c a l c u l a t i o n and experiment i s e x t r a o r d i n a r y . However, i t i s c l e a r that c a l c u l a t i o n s of t h i s s o r t are r e s t r i c t e d t o those few systems where the molecules are simple and s m a l l , where we have no s i g n i f i c a n t p o l a r i t y , hydrogen bonding or other s p e c i f i c "chemical f o r c e s " and, u n f o r t u n a t e l y , t o those cases where we have l a r g e q u a n t i t i e s of experimental data f o r both pure f l u i d s and f o r b i n a r y m i x t u r e s . I n the process i n d u s t r i e s we r a r e l y meet a l l these necessary c o n d i t i o n s . I f our accuracy requirements are not extremely l a r g e , we can o f t e n o b t a i n good approximations u s i n g c a l c u l a t i o n s based on a simple equation of s t a t e , s i m i l a r i n p r i n c i p l e t o the Van der Waals equation. The s i m p l e s t s u c c e s s f u l v a r i a t i o n of Van der Waals' equation i s t h a t by R e d l i c h and Kwong, proposed i n 1949. That equation, i n t u r n , i s to a p p l i e d thermodynamics what Helen of Troy has been to l i t e r a t u r e ; you r e c a l l t h a t i t was the b e a u t i f u l Helen who i n s p i r e d the l i n e "...the face t h a t launched a thousand s h i p s . " Ten years ago the B e a t l e s turned on an e n t i r e g e n e r a t i o n of teenagers and i n s p i r e d c o u n t l e s s v a r i a t i o n s and e x t e n s i o n s ; s i m i l a r l y , s t a r t i n g about ten years ago, the Redlich-Kwong equation i n i t i a t e d an epoch of i m i t a t i o n unequalled i n the h i s t o r y of a p p l i e d thermodynamics. The number of m o d i f i e d RK equations i s probably c l o s e to a hundred by now and, s i n c e I am amongst f r i e n d s , I must confess to having c o n s t r u c t e d a few myself. A few years ago, there was an a r t i c l e i n Chemical Engineering Science devoted e x c l u s i v e l y to v a r i a t i o n s on




Figure 5.

Predicted and experimental K-values for the methane-hydrogen sulfide system at 40°F


PRAUSNITZ


10

373 15 K

365 15 K

Measurements of Bier et al o


•

348 15 K 365 15 K 373 15 K

X

36815 K 396 15 K

•

42315 K

•

473 15 K Equation of state this work

e a. -388 15 K

398 15 K

348

Figure 6. Comparison of the residual isobaric heat capacities of propylene of Bier et al. with those predicted by the equation of state of Bender


22


60

c

t7 * / f /


i 40

20

Figure 7. Molar excess enthalpies of the binary system Ar-0 (Bender). Temp. = 84°K: O exptl, equation of state of Bender; Temp. = 86° K: + exptl, equation of state of Bender. 2

EXPERIMENTAL (BESSERER A N D ROBINSON TEMP F LIQUID VAPOR 100 • o 220 • •

/* // // // //

\ +

f

02

0.4

06

08

1973)

#

1200

PREDICTED

8, =o S,: =0130

-±-

0 2

MOLE

0 4

FRACTION

06

CARBON

-L.

08

1

>

o

DIOXIDE

Figure 8. Pressure-equilibrium phase composition diagram for isobutane-carbon dioxide system, calculations using Peng-Robinson equation of state 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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2.

PRAUSNiTz


23

the RK equation but i t i s now h o p e l e s s l y out of date and even then, between the time the paper was w r i t t e n and the time i t was p u b l i s h e d , seven new v a r i a t i o n s had appeared. (I get much of t h i s i n f o r m a t i o n d i r e c t l y from Otto R e d l i c h , who keeps a c l o s e eye on " c h i l d r e n " of h i s 1949 a r t i c l e . I n c i d e n t a l l y , I am happy to r e p o r t that Otto, aged 80, i s w e l l , a c t i v e and very pleased about the recent p u b l i c a t i o n of h i s thermodynamics book by E l s e v i e r . Whenever Otto has a need to f e e l young a g a i n , he t a l k s w i t h the i n c r e d i b l e J o e l Hildebrand who, at 95, i s h a l e , h e a r t y , i n good humor and busy w r i t i n g a monograph on transport properties i n l i q u i d s . ) Perhaps the most s u c c e s s f u l v a r i a t i o n on the RK equation i s that proposed by Soave (4) who expresses the RK constant a by an e m p i r i c a l f u n c t i o n of reduced temperature and a c e n t r i c f a c t o r . This e m p i r i c a l f u n c t i o n was determined from vapor-pressure data f o r p a r a f f i n s and t h e r e f o r e , when Soave*s equation i s used w i t h reasonable mixing r u l e s and one a d j u s t a b l e b i n a r y parameter, i t gives good Κ f a c t o r s f o r t y p i c a l l i g h t - h y d r o g e n m i x t u r e s ; however, i t p r e d i c t s poor l i q u i d d e n s i t i e s . This i l l u s t r a t e s a p o i n t known to a l l workers i n the e q u a t i o n - o f - s t a t e f i e l d ; i t i s not d i f f i c u l t to represent any one thermodynamic property but i t i s d i f f i c u l t , w i t h one equation of s t a t e , to represent them a l l . A comparatively recent v a r i a t i o n on the RK equation was proposed by Peng and Robinson ( 5 ) ; i t i s s i m i l a r to Soave s equation but appears to have b e t t e r behavior i n the c r i t i c a l r e g i o n ; an example i s given i n F i g u r e 8 f o r the isobutane-carbon d i o x i d e system. I n t h i s case the c r i t i c a l r e g i o n i s p r e d i c t e d w e l l and the a d j u s t a b l e b i n a r y parameter i s independent of temperature i n the r e g i o n 100 to 220°F. C a l c u l a t i n g phase e q u i l i b r i a from v o l u m e t r i c data does not n e c e s s a r i l y r e q u i r e an a n a l y t i c a l equation of s t a t e . The v o l u m e t r i c data can be s t o r e d i n t a b u l a r or a n a l y t i c a l form f o r an a r b i t r a r i l y chosen r e f e r e n c e substance and then, u s i n g corresponding s t a t e s , these data can be used to p r e d i c t p r o p e r t i e s of other f l u i d s , i n c l u d i n g m i x t u r e s . This procedure, o f t e n c a l l e d the p s e u d o - c r i t i c a l method o r , i n a more elegant form, the theory of conformai s o l u t i o n s , has been a p p l i e d by numerous authors. Here time permits me t o c a l l a t t e n t i o n to only one example, a p a r t i c u l a r l y u s e f u l one, i n i t i a t e d by Rowlinson and Mollerup and e x t e n s i v e l y developed by Mollerup i n recent years ( 6 ) . Using Goodwin's e x c e l l e n t experimental data f o r methane as a r e f e r e n c e , Mollerup c a l c u l a t e s w i t h h i g h accuracy thermodynamic p r o p e r t i e s of mixtures encountered i n the n a t u r a l - g a s i n d u s t r y . To do so, he uses the o l d Van der Waals mixing r u l e s but he pays very c l o s e a t t e n t i o n to the a l l - i m p o r t a n t b i n a r y constants. F i g u r e 9 shows e x c e l l e n t agreement between c a l c u l a t e d and e x g e r i mental r e s u l t s f o r the system methane-ethane from 130 to 200 K, u s i n g only one temperature-independent b i n a r y constant. Even more impressive i s the e x c e l l e n t r e p r e s e n t a t i o n f o r carbon monoxidemethane shown i n F i g u r e 10 where the c r i t i c a l r e g i o n i s reproduced almost w i t h i n experimental e r r o r . F i n a l l y , F i g u r e 11 shows that the corresponding-states method a l s o g i v e s e x c e l l e n t e n t h a l p i e s of 1


24


METHANE


199.9? K

J

»

J

i_

1

5

10

50

PRE S S U R E . A T M

Figure 9. K-values vs. pressure for methane-ethane mixtures; corresponding-states method of Mollerup and Rowlinson


2.

25


PRAUSNITZ

50


CARBON MONOXIDE

10

•J \9067k

1 2 3

178K

^k

05

01

005

METHANE —

Predicted

O Experimental 05

1

5

10

50

PRESSURE, ATM

Figure 10. K-values vs. pressure for methane-carbon monoxide mixtures (Mollerup, 1975)


26



O

X

N

20

Figure 11.

=

0.44

40 60 P R E S S U R E , ATM

80

100

Excess enthalpy of methane-nitrogen mixtures at 201.2°K (Mollerup, 1975)



2.

PRAUSNITZ


27

mixing f o r gaseous mixtures at h i g h p r e s s u r e s , c o r r e c t l y r e p r o ducing the observed maxima when the excess enthalpy i s p l o t t e d a g a i n s t pressure. These few i l l u s t r a t i o n s should be s u f f i c i e n t to o u t l i n e our present p o s i t i o n w i t h respect to phase e q u i l i b r i u m c a l c u l a t i o n s using an equation of s t a t e f o r both phases. I have e a r l i e r pointed out some of the advantages of t h i s type of c a l c u l a t i o n but I want now to add one more: i f we can c o n s t r u c t an equation of s t a t e a p p l i c a b l e to normal f l u i d s and t h e i r b i n a r y m i x t u r e s , then we need not worry about how to c a l c u l a t e e q u i l i b r i a i n t e r n a r y (or h i g h e r ) m i x t u r e s . For mixtures of normal f l u i d s , pure-component parameters and b i n a r y parameters are almost always s u f f i c i e n t f o r c a l c u l a t i n g e q u i l i b r i a i n multicomponent m i x t u r e s . For multicomponent mixtures of normal f l u i d s , the o n e - f l u i d theory i s u s u a l l y s a t i s f a c t o r y u s i n g only pure-component and b i n a r y c o n s t a n t s . This happy f a c t i s of tremendous importance i n chemical technology where multicomponent mixtures are much more common than b i n a r i e s . P r e d i c t i n g m u l t i component e q u i l i b r i a u s i n g only pure-component and b i n a r y data i s perhaps one of the g r e a t e s t triumphs of a p p l i e d thermodynamics. Having p r a i s e d the uses o f equations of s t a t e , I must a l s o p o i n t out t h e i r contemporary l i m i t a t i o n s which f o l l o w from our i n a b i l i t y to w r i t e s e n s i b l e equations of s t a t e f o r molecules t h a t are very l a r g e o r very p o l a r , or both. That i s where the f r o n t i e r l i e s . I see l i t t l e p o i n t i n pursuing f u r t h e r the obsession of modif y i n g the Redlich-Kwong equation. We must i n t r o d u c e some new p h y s i c s i n t o our b a s i c n o t i o n s of how to c o n s t r u c t an equation of s t a t e and there we must r e l y on suggestions s u p p l i e d by t h e o r e t i c a l p h y s i c i s t s and chemists. U n f o r t u n a t e l y most of these are "argon people" although, I am happy t o say, i n the l a s t few years a few brave t h e o r i s t s have s t a r t e d to t a c k l e n i t r o g e n . Some computer-type t h e o r i s t s have spent a l o t of time on water and on p r o t e i n s but these h i g h l y complicated s t u d i e s are s t i l l f a r removed from e n g i neering a p p l i c a t i o n s . N e v e r t h e l e s s , there are some new t h e o r e t i c a l ideas which could be used i n f o r m u l a t i n g new equations of s t a t e s u i t a b l e f o r those f l u i d s that cannot now be d e s c r i b e d by the u s u a l equations of s t a t e . Not t h i s morning, but perhaps l a t e r i n t h i s conference, I hope to have an o p p o r t u n i t y to say a few words about that. Vapor-Phase F u g a c i t y C o e f f i c i e n t s I now t u r n to what I have e a r l i e r c a l l e d Method ( b ) , that i s , f u g a c i t y c o e f f i c i e n t s f o r the vapor phase only and a c t i v i t y c o e f f i c i e n t s f o r a l l condensed phases. Method (b) i s used whenever we d e a l w i t h mixtures c o n t a i n i n g molecules t h a t are l a r g e or p o l a r or hydrogen-bonded or e l s e when a l l components are s u b c r i t i c a l and the pressure i s low. At modest vapor d e n s i t i e s , our most u s e f u l t o o l f o r vapor-phase f u g a c i t y c o e f f i c i e n t s i s the v i r i a l equation of s t a t e truncated a f t e r the second term. For r e a l f l u i d s , much i s known about second v i r i a l


PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY BP

c

_

j(o)

-

J (T )

j ( l ) +

R

J (T ) R

B = SECOND VIRIAL COEFFICIENT; P

- CRITICAL PRESSURE;

c (j) « ACENTRIC FACTOR


j

AND f

T

R

j(2)

J (T )

+

R

=* T/T

c

T = CRITICAL TEMPERATURE c

ARE KNOWN FUNCTIONS SIMILAR TO THOSE

FIRST PROPOSED BY PITZER AND CURL.

FOR NONPOLAR FLUIDS

TSONOPOULOS PROPOSES

a - b - 0.

FOR POLAR (NONHYDROGEN-BONDED) FLUIDS

a * 0

BUT

b * 0.

Figure 12. Correlation of second virial coefficients (Tsonopoulos)

0

100

200

REDUCED DIPOLE MOMENT, / /

300 R

Figure 13. Dependence of a on reduced dipole moment for nonhydrogen bonding compounds (Tsonopoulos, 1974)



2.

PRAUSNITZ


29

c o e f f i c i e n t s ; l i t t l e i s known about t h i r d v i r i a l c o e f f i c i e n t s and n e a r l y nothing i s known about higher v i r i a l c o e f f i c i e n t s . Therefore, a p p l i c a t i o n i s l i m i t e d to moderate d e n s i t i e s , t y p i c a l l y d e n s i t i e s up to about 1/2 the c r i t i c a l . There are two major advantages of the t h e o r e t i c a l l y - d e r i v e d v i r i a l equation: f i r s t , the v i r i a l c o e f f i c i e n t s can be q u a n t i t a t i v e l y r e l a t e d to the i n t e r m o l e c u l a r f o r c e s and second, extension to mixtures r e q u i r e s no a d d i t i o n a l assumptions. For e n g i n e e r i n g , the f i r s t advantage i s important because i t enables us to i n t e r p r e t , c o r r e l a t e and m e a n i n g f u l l y e x t r a p o l a t e l i m i t e d v i r i a l c o e f f i c i e n t data, and the second i s important because we do not have to guess a t a r b i t r a r y mixing r u l e s f o r expressing the composition dependence of the v i r i a l c o e f f i c i e n t s . Any standard thermodynamics t e x t t e l l s us how t o c a l c u l a t e f u g a c i t y c o e f f i c i e n t s from the v i r i a l equation of s t a t e . The most important problem i s to estimate the v i r i a l c o e f f i c i e n t s and here we can u t i l i z e an extended form of corresponding s t a t e s , i l l u s t r a t e d by the c o r r e l a t i o n of Tsonopoulos (7) shown i n F i g u r e 12. The f i r s t term on the r i g h t holds f o r simple f l u i d s ; the second term c o r r e c t s for a c e n t r i c i t y and the t h i r d term c o r r e c t s f o r p o l a r i t y and hydrogen bonding. The constants a and b cannot be completely g e n e r a l i z e d but good estimates are o f t e n p o s s i b l e by observing trends w i t h i n chemical f a m i l i e s . F i g u r e 13 shows r e s u l t s f o r constant a p l o t t e d a g a i n s t a dimensionless d i p o l e moment; s i n c e p o l a r i t y i n c r e a s e s a t t r a c t i v e f o r c e s , we f i n d , as shown, that constant a becomes more negative as the reduced d i p o l e moment r i s e s , g i v i n g a more negative second v i r i a l c o e f f i c i e n t . F i g u r e 14 gives some r e s u l t s f o r constant b f o r a l c o h o l s , again p l o t t e d a g a i n s t reduced d i p o l e moment. Note t h a t the p o s i t i o n of the OH r a d i c a l has a n o t i c e a b l e e f f e c t . For these f l u i d s cons t a n t a i s s l i g h t l y p o s i t i v e because the hydrogen-bonding nature expressed by constant b dominates, e s p e c i a l l y at lower temperatures. To estimate c r o s s v i r i a l c o e f f i c i e n t s B ] ^ ' one must make some assumptions about the i n t e r m o l e c u l a r f o r c e s between molecules 1 and 2 and then s u i t a b l y average the molecular parameters appearing i n the c o r r e l a t i o n . Only f o r simple cases can any general r u l e s be used; whenever we have p o l a r components, we must l o o k c a r e f u l l y a t the molecular s t r u c t u r e and use judgment which, u l t i m a t e l y , i s based on experience. The v i r i a l equation i s u s e f u l f o r many cases but, when there i s strong a s s o c i a t i o n i n the vapor phase, the t h e o r e t i c a l b a s i s of the v i r i a l equation i s not v a l i d and we must r e s o r t to what i s commonly c a l l e d a "chemical treatment", u t i l i z i n g a chemical e q u i l i b r i u m constant f o r d i m e r i z a t i o n . D i m e r i z a t i o n i n the vapor phase i s e s p e c i a l l y important f o r organic a c i d s and even a t low p r e s s u r e s , the vapor-phase f u g a c i t y c o e f f i c i e n t s of mixtures c o n t a i n i n g one (or more) o r g a n i c a c i d are s i g n i f i c a n t l y removed from u n i t y . F i g u r e 15 shows some r e s u l t s based on the c o r r e l a t i o n of Hayden and O C o n n e l l (8) c a l c u l a t e d by Tom Anderson f o r the system p r o p i o n i c a c i d methyl i s o b u t y l ketone a t 1 atm along the v a p o r - l i q u i d s a t u r a t i o n curve. When the mole f r a c t i o n of a c i d i s very low, the f u g a c i t y c o e f f i c i e n t s a r f


30

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY i

1

1

1

r Ethanol

0.06-

2-Propanol \ ^

tert-Butanol

Methanol

?

2-Butanol s ^ V / ^ Isobutanol 1-Propanol

b = 0.00908 +0.0006957

f

J 0

10

( 2 )

= 0.0878/T

Figure 14.

R

6

- b/T

^ ) R

R

8

L

20

30

40

50

60

REDUCED DIPOLE MOMENT, / /

70

80

90

100

R

Dependence of b on reduced dipole moment for alcohols (Tsonopoulos, 1974)

Dew-Point Temperature, °C 1 —

ient at Satui

o o

M4.9 2.0

124.2 i

129.3 1

!33.'2 1

136.9 i

140.0

-

1.5

1.0

'

-

*2

o

Coeff


1-Butanol

-

0.7

-

o 0.5 o o» if

i i

0.3 0

0.2

1 0.4

1 0.6

i 0.8

1.0

Vapor-Phase Mole Fraction Propionic Acid Figure 15.

Fugacity coefficients for saturated propionic acid (1)—methyl isobutyl ketone (2) at 1 atm



2.

PRAUSNITZ


31

near u n i t y because d i m e r i z a t i o n between a c i d molecules i s n e g l i g i b l e . As the mole f r a c t i o n of a c i d r i s e s , d i m e r i z a t i o n becomes i n c r e a s i n g l y l i k e l y and t h e r e f o r e , on the r i g h t s i d e of the diagram, the f u g a c i t y c o e f f i c i e n t s of both components are w e l l removed from u n i t y even though the temperature i s reasonably high (140C) and the pressure i s only 1 atm. At high p r e s s u r e s , where the v i r i a l equation i s no longer u s e f u l , e m p i r i c a l equations must be used t o c a l c u l a t e f u g a c i t y c o e f f i c i e n t s . However, contrary to Method ( a ) , the equation of s t a t e now need not hold f o r both the vapor phase and the l i q u i d phase; v a l i d i t y i n the vapor phase i s s u f f i c i e n t . To i l l u s t r a t e , I now show some r e s u l t s using an equation d e v e l oped by de S a n t i s and Breedveld (9) f o r gases a t h i g h pressures c o n t a i n i n g water as one of the components. As i n d i c a t e d i n Figure 16, the equation i s l i k e that of Redlich-Kwong except that the a t t r a c t i v e f o r c e constant a i s d i v i d e d i n t o a p o l a r and a nonpolar component. For mixtures of water w i t h nonpolar components, the c r o s s - c o e f f i c i e n t a-]^ i s found from the geometric-mean assumption, but i n t h i s assumpt i o n only the nonpolar p a r t of constant a f o r water i s used because the nonaqueous component i s nonpolar. Figure 17 shows that the modified RK equation gives good f u g a c i t y c o e f f i c i e n t s f o r aqueous water but t h i s i s hardly s u r p r i s i n g s i n c e the constants a and b were determined from steam-table data. More g r a t i f y i n g are the r e s u l t s shown i n Figure 18 which show that c a l c u l a t e d v o l u m e t r i c p r o p e r t i e s a t high pressures are i n e x c e l l e n t agreement w i t h e x p e r i ment f o r gaseous mixtures of water and argon. The equation of de S a n t i s and Breedveld has r e c e n t l y been a p p l i e d by Heidemann to the problem of w e t - a i r o x i d a t i o n . When vapor-phase f u g a c i t y c o e f f i c i e n t s are c a l c u l a t e d from t h i s equation of s t a t e , and l i q u i d - p h a s e f u g a c i t i e s are c a l c u l a t e d from the p r o p e r t i e s of pure water c o r r e c t e d f o r s o l u b i l i t y of gases i n the water, i t i s p o s s i b l e to c a l c u l a t e the saturated water content and other e q u i l i b r i u m prope r t i e s of combustion gases. Figure 19 shows the saturated water content i n n i t r o g e n and F i g u r e 20 shows how that water content i s enhanced when CO2 i s present i n the gas phase; r e s u l t s are shown f o r two molar compositions (dry b a s i s ) : 20% CO2, 80% N2 and 13% CO2, 87% N2. E s p e c i a l l y at moderate temperatures, the pressure of CO2 s u b s t a n t i a l l y r a i s e s the saturated water content. Figure 21 shows enthalpy c a l c u l a t i o n s , again based on the equation of s t a t e of de S a n t i s and B r e e d v e l t , u s e f u l f o r designing a w e t - a i r o x i d a t i o n process. In v a p o r - l i q u i d e q u i l i b r i a according to Method (b), f u g a c i t y c o e f f i c i e n t s c o n s t i t u t e only h a l f the s t o r y , u s u a l l y (but not always. ) the l e s s important h a l f , w h i l e i n l i q u i d - l i q u i d e q u i l i b r i a f u g a c i t y c o e f f i c i e n t s p l a y no r o l e at a l l . We now must t u r n our a t t e n t i o n to the l a s t and i n some respects the most d i f f i c u l t t o p i c , v i z . , the activity coefficient. 1


PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY P

a(T)

RT

=

" ^

B

' T

(WATER) "

'

u

1 / 2

cm3 6

(o)

a(T)

v(v+b)

a

ra

/ °i

(1) a(T)

+

(NONPOLAR)


e

/TABULATED VALUES OBTAINED\ I FROM STEAM TABLES. J

(POLAR)

FOR BINARY MIXTURES CONTAINING WATER(1) AND NONPOLAR GAS(2)

b

=

y

l

b

l

+

Y

a = Y& 1

a

B

2 2

+ y

1

(o) = (a a )

1 2

x

A

2

+

1

/

2

2

2 y

Y

A

l 2 12

2

(o) TO FIND a

x

, USE B

1 2

(SECOND VIRIAL COEFFICIENT) DATA FOR

MIXTURES OF WATER WITH N , Ar, CH , ETC. 2

4

Figure 16. Vapor-phase equation of state for mixtures containing water (de Santis and Breedveld)

Figure 17. Fugacity coefficients for gaseous water. Calculations using equation of state proposed by de Santis et al.




PRAUSNITZ

Figure 19. Water content of nitrogen; comparison with experiment (Heidemann)




0

100

Figure 20.

Effect of C0 /N

200 TEMPERATURE ° C

Figure 21.

2

2

300

ratio on saturated water content (Heidemann)

Enthalpy of saturated combustion gases (Heidemann)


2.

PRAUSNITZ


35


Liquid-Phase A c t i v i t y C o e f f i c i e n t s In p r i n c i p l e , everybody knows that an a c t i v i t y c o e f f i c i e n t has no s i g n i f i c a n c e unless there i s a c l e a r d e f i n i t i o n of the standard s t a t e to which i t r e f e r s . In p r a c t i c e , however, there i s a l l too o f t e n a tendency to n e g l e c t p r e c i s e s p e c i f i c a t i o n of the standard s t a t e and i n some cases f a i l u r e t o g i v e t h i s exact s p e c i f i c a t i o n can lead t o s e r i o u s d i f f i c u l t i e s . This problem i s e s p e c i a l l y important when we consider s u p e r c r i t i c a l components or e l e c t r o l y t e s i n l i q u i d mixtures and, a l i t t l e l a t e r , I s h a l l have a few comments on t h a t s i t u a t i o n . But f o r now, l e t us consider mixtures of t y p i c a l n o n e l e c t r o l y t e l i q u i d s a t a temperature where every component can e x i s t as a pure l i q u i d . I n that event, the s t a n d a r d - s t a t e f u g a c i t y i s the f u g a c i t y of the pure l i q u i d a t system temperature and pressure and that f u g a c i t y i s determined p r i m a r i l y by the pure l i q u i d ' s vapor pressure. F i g u r e 22 reviews some well-known r e l a t i o n s between a c t i v i t y c o e f f i c i e n t s and excess f u n c t i o n s . A l l t h i s i s s t r i c t l y c l a s s i c a l thermodynamics and the e n t i r e aim here i s that of c l a s s i c a l thermodynamics; v i z . , to organize our knowledge of e q u i l i b r i u m p r o p e r t i e s i n an e f f i c i e n t way so t h a t , by r e l a t i n g v a r i o u s q u a n t i t i e s to one another, we can minimize the amount of experimental e f f o r t r e q u i r e d f o r engineering design. There are three noteworthy f e a t u r e s i n F i g u r e 22: 1. The excess f u n c t i o n s used here are i n excess of those which apply to a p a r t i c u l a r k i n d of i d e a l s o l u t i o n v i z . that ( e s s e n t i a l l y ) given by Raoult's law. This choice of i d e a l i t y i s a r b i t r a r y and f o r some s i t u a t i o n s a d i f f e r e n t d e f i n i t i o n of i d e a l s o l u t i o n may be more s u i t a b l e . F u r t h e r , choosing ( e s s e n t i a l l y ) R a o u l t s law as our d e f i n i t i o n of an i d e a l s o l u t i o n , we are n a t u r a l l y l e d to the use of mole f r a c t i o n x as our c h o i c e of composition v a r i a b l e . That i s not n e c e s s a r i l y the best c h o i c e and there are s e v e r a l cases ( n o t a b l y , polymer s o l u t i o n s and s o l u t i o n s of e l e c t r o l y t e s ) where other measures of composition are much more convenient. 2. Equation (2) can be d e r i v e d from Equation (1) only i f we use the Gibbs-Duhem equation. Therefore, i f we organize our e x p e r i mental i n f o r m a t i o n a c c o r d i n g t o the scheme suggested by F i g u r e 22, we assure that the f i n a l r e s u l t s obey at l e a s t a c e r t a i n degree of thermodynamic c o n s i s t e n c y . 3. The excess Gibbs energy g^ i s a combination of two terms as shown i n Equation ( 3 ) . When we t r y t o c o n s t r u c t models f o r g^, we o f t e n do so d i r e c t l y but, i f we want our model to have p h y s i c a l s i g n i f i c a n c e we should i n s t e a d make models f o r h^ and s^ because these are the p h y s i c a l l y s i g n i f i c a n t q u a n t i t i e s t h a t can be r e l a t e d to molecular behavior; g^ i s only an o p e r a t i o n a l combination of them. The excess enthalpy i s concerned p r i m a r i l y w i t h e n e r g e t i c i n t e r a c t i o n s between molecules w h i l e excess entropy i s concerned p r i m a r i l y w i t h the s t r u c t u r e of the s o l u t i o n , i . e . , the s p a t i a l arrangements of the molecules which leads us to concepts l i k e randomness and segreg a t i o n . U n f o r t u n a t e l y , excess enthalpy and excess entropy are not f


36


(1)

x

E

T

±


(2)

(USES RAOULT S LAW FOR IDEALITY)

g = RT Y, ± l i

1

lnT

= ACTIVITY COEFFICIENT;

RT I n * = f T 6 n

E\ g

)

(BASED ON GIBBS-DUHEM EQUATION)

= TOTAL NO MOLES;

(3)

g

x = MOLE FRACTION

n

±

= NO MOLES OF COMPONENT i

Ts

E

DETERMINED PRIMARILY BY INTERMOLECULAR FORCES

DETERMINED PRIMARILY BY MOLECULAR STRUCTURE (SIZE, SHAPE, POSITION, DEGREES OF FREEDOM)

Figure 22.

Excess functions and activity coefficients

0.02 9 - (*i V

x v)

E

2

2

agreement between c a l c u l a t e d and experimental r e s u l t s i s much improved and, as shown here f o r a l i m i t e d c l a s s of mixtures, i t i s possible to correlate molecular s t r u c t u r e . The parameter Z^_2 ^ H i a b s o l u t e v a l u e but i t has a pronounced e f f e c t , as i n d i c a t e d i n F i g u r e 24. The b a s i c i d e a of s o l u b i l i t y parameter was f i r s t d e s c r i b e d by Hildebrand about 50 years ago. I t was b a r e l y known to chemical e n g i neers u n t i l about 20 years ago but s i n c e then i t has been both used and abused e x t e n s i v e l y f o r a v a r i e t y of purposes, both l e g i t i m a t e and otherwise, f a r beyond J o e l H i l d e b r a n d s w i l d e s t dreams. I t shows up i n the p a i n t and v a r n i s h i n d u s t r y , i n m e t a l l u r g y , p h y s i o l o g y , c o l l o i d chemistry and pharmacology and r e c e n t l y I have seen i t m u t i l a t e d i n a magazine a r t i c l e on " s c i e n t i f i c " a s t r o l o g y . Before l e a v i n g the s o l u b i l i t y parameter, I want to p o i n t out one use which, w h i l e not new, has perhaps not r e c e i v e d as much a t t e n t i o n as i t deserves. I r e f e r t o the use of the s o l u b i l i t y parameter f o r d e s c r i b i n g the s o l v e n t power of a dense gas, w i t h p a r t i c u l a r r e f e r e n c e to high-pressure gas e x t r a c t i o n , c e r t a i n l y not a n o v e l process but one which i s r e c e i v i n g renewed a t t e n t i o n i n c o a l l i q u e f a c t i o n and i n food processing. E

E

E


E

E

E

E

E

E

E

w

s s

m

a

i

t

n

n

!


PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY 1

1

r

0.6

0.8

EXPERIMENTAL


o

1

0

0.2 MOLE

0.4 FRACTION

1.0

BENZENE

Figure 24. Experimental and calculated excess Gibbs energies for two binaries containing benzene at 50°C (Funk)



2.

PRAUSNITZ

39


Using the well-known t a b l e s by P i t z e r , i t i s e a s i l y p o s s i b l e to c o n s t r u c t a g e n e r a l i z e d s o l u b i l i t y parameter diagram as shown i n F i g u r e 25. I n t h i s p a r t i c u l a r case the a c e n t r i c f a c t o r 0.075 i s c l o s e to that of ethylene because when t h i s chart was prepared over ten years ago, a p p l i c a t i o n was d i r e c t e d a t i n t e r p r e t i n g s o l u b i l i t y data f o r naphthalene i n ethylene at high pressure. J u s t to o r i e n t o u r s e l v e s , when the temperature i s around 20 C and the pressure i s about 400 atm, the s o l u b i l i t y parameter i s i n the r e g i o n 5 o r 6 (cal/cm^)!/^, only about one o r two u n i t s lower than that of a l i q u i d p a r a f f i n l i k e hexane. When s o l u b i l i t y data i n compressed ethylene are used w i t h the Scatchard-Hildebrand equation to back-out a s o l u b i l i t y parameter f o r l i q u i d naphthalene, we f i n d r e s u l t s shown i n Figure 26. The remarkable f e a t u r e s of t h i s f i g u r e are f i r s t , that the s o l u b i l i t y parameter obtained i s i n good agreement w i t h what one would o b t a i n by e x t r a p o l a t i n g the known s o l u b i l i t y parameter of l i q u i d naphthalene to temperatures below the m e l t i n g p o i n t and second, that the backed-out s o l u b i l i t y parameter i s n e a r l y constant w i t h pressure. This i n d i c a t e s that the Hildebrand r e g u l a r s o l u t i o n equat i o n i s u s e f u l f o r mixtures of nonpolar f l u i d s r e g a r d l e s s of whether these are l i q u i d s or gases, provided only that the d e n s i t y i s s u f f i ciently large. F i n a l l y , a u s e f u l f e a t u r e of the s o l u b i l i t y parameter i s shown i n F i g u r e 27 f o r n i t r o g e n . S i m i l a r diagrams can be constructed f o r any f l u i d ; n i t r o g e n i s here shown only as an example. Note that the s o l u b i l i t y parameter i s s t r o n g l y s e n s i t i v e to both pressure and temperature i n the c r i t i c a l r e g i o n . H i g h l y s e l e c t i v e e x t r a c t i o n can t h e r e f o r e be c a r r i e d out by s m a l l changes i n temperature and pressure. F u r t h e r , such s m a l l changes can be e x p l o i t e d f o r e f f i c i e n t s o l v e n t r e g e n e r a t i o n i n continuous s e p a r a t i o n processes. L o c a l Composition to Describe Nonrandomness For mixtures c o n t a i n i n g p o l a r and hydrogen-bonded l i q u i d s , equations f o r g which emphasize h ( r a t h e r than s ) tend to be u n s a t i s f a c t o r y because i n t h e i r b a s i c f o r m u l a t i o n such equations give l i t t l e a t t e n t i o n to the d i f f i c u l t problem of nonrandomness. In a r e g u l a r s o l u t i o n , the molecules are " c o l o r - b l i n d " which means that they arrange themselves i n a manner d i c t a t e d only by the r e l a t i v e amounts of the d i f f e r e n t molecules that are present. It i s easily p o s s i b l e to add a c o r r e c t i o n which takes i n t o account the e f f e c t of molecular s i z e , as given by the Flory-Huggins expression. (Size c o r r e c t i o n s are e s s e n t i a l f o r polymer s o l u t i o n s . ) V a r i a t i o n s on that expression [e.g., Staverman or Tompa (12)] can a l s o account, i n p a r t , f o r d i f f e r e n c e s i n molecular shape. But, f o r s t r o n g l y i n t e r a c t i n g molecules, r e g a r d l e s s of s i z e and shape, there are l a r g e d e v i a t i o n s from random mixing; such molecules are f a r from " c o l o r - b l i n d " because t h e i r choice of neighbors i s h e a v i l y i n f l u e n c e d by d i f f e r e n c e s i n i n t e r m o l e c u l a r f o r c e s . An i n t u i t i v e idea toward d e s c r i b i n g t h i s i n f l u e n c e was introduced by Wilson w i t h h i s n o t i o n of l o c a l compos i t i o n , shown s c h e m a t i c a l l y i n F i g u r e 28 (13). Viewed m i c r o s c o p i c a l l y , E

E

E


40



I

Figure 25.

2

3

4

5

6

7

8

9

10

Solubility parameters for dense gases with an acentric factor of 0.075



2.

PRAUSNITZ

41


200

220

240

280

PRESSURE, atm. Figure 26. Solubility parameter of naphthalene calculated from solubility data in gaseous ethylene

TEMPERATURE, *K

Figure 27. Solubility parameter for nitrogen



42


15 of type 1 15 of type 2

Overall mole fractions: Xi = x — \ Local mole fractions: 2

*

2 1

_ Molecules of 2 about a central molecule 1 ~~ Total molecules about a central molecule 1 1, as shown

*2i+*n

=

*12+*22

= 1

X21

~ f

Figure 28. Local compositions and the concept of local mole fractions (Cukor)


2.


PRAUSNITZ

a s o l u t i o n i s not homogeneous because molecules have d e f i n i t e p r e f e r ences i n choosing t h e i r immediate environment, l e a d i n g to very s m a l l r e g i o n s , sometimes c a l l e d domains, which d i f f e r i n composition. There i s no obvious way to r e l a t e l o c a l composition to o v e r a l l ( s t o i c h i o m e t r i c ) composition but the use of Boltzmann f a c t o r s provides us w i t h one reasonable method f o r doing so. I n the l a s t ten y e a r s , Wilson's equation f o r g has enjoyed much p o p u l a r i t y and, f o r c e r t a i n m i x t u r e s , notably alcohol-hydrocarbon s o l u t i o n s , i t i s remarkably good. U n f o r t u n a t e l y , i t has one major f l a w : i t i s not a p p l i c a b l e to part i a l l y m i s c i b l e mixtures and, as a r e s u l t , there has been a f l u r r y of a c t i v i t y to extend and modify Wilson's equation; new m o d i f i c a t i o n s appear almost monthly. Time does not permit me to d i s c u s s any of these m o d i f i c a t i o n s but I want to p o i n t out an important development which only r e c e n t l y has become i n c r e a s i n g l y evident. In many cases the c h o i c e of model f o r g i s not as important as the method chosen f o r data r e d u c t i o n ; that i s , the procedure used to o b t a i n model parameters from l i m i t e d experimental data i s o f t e n more important than d e t a i l s w i t h i n the model. When reducing experimental data t o o b t a i n model parameters, a t t e n t i o n must be given to the e f f e c t of experimental e r r o r s . Not a l l experimental measurements are e q u a l l y v a l u a b l e and t h e r e f o r e , a proper s t r a t e g y f o r weighting i n d i v i d u a l experimental p o i n t s i s needed to o b t a i n "best" parameters (14, 15). Any r e a l i s t i c s t r a t e g y shows a t once that f o r any given set of b i n a r y data, there i s no unique s e t of model parameters. I n a t y p i c a l case, there are many s e t s of parameters which are e q u a l l y good, as i l l u s t r a t e d i n Figure 29 prepared by Tom Anderson f o r the system ethanol-cyclohexane. I n t h i s p a r t i c u l a r case, the model used was Wilson's but that i s not important here. The important message i s t h a t any p o i n t i n the e l l i p s e s shown can represent the experimental data e q u a l l y w e l l ; there i s no s t a t i s t i c a l s i g n i f i c a n c e i n p r e f e r r i n g one p o i n t over another. The e l l i p t i c a l r e s u l t s shown i n F i g u r e 29 are t y p i c a l ; the parameters have a tendency to be c o r r e l a t e d but, without a d d i t i o n a l i n f o r m a t i o n , i t i s not p o s s i b l e to say which p o i n t w i t h i n the e l l i p s e i s the b e s t . Since the two e l l i p s e s do not o v e r l a p , we are j u s t i f i e d i n a s s i g n i n g a temperature dependence to the parameters. When we do so, we o b t a i n the pressure-composition diagrams shown i n F i g u r e 30. I n t h i s case we assumed that the temperature dependence of one parameter i s p a r a l l e l to that of the other ( i . e . , we used t h r e e , not f o u r , a d j u s t a b l e parameters s i n c e b has no s u b s c r i p t s ) but that procedure i s not always s u c c e s s f u l ; f r e q u e n t l y f o u r parameters are r e q u i r e d . On the other hand, i f the two e l l i p s e s i n F i g u r e 29 had a r e g i o n of o v e r l a p , there would be no good reason to use temperature-dependent parameters; two temperature-independent parameters would be sufficient. Another example prepared by Tom Anderson i s shown i n F i g u r e 31 f o r the system butanol-water; i n t h i s case the UNIQUAC model was used r a t h e r than Wilson's because we are concerned w i t h v a p o r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a . The l e f t s i d e shows t h a t when vaporl i q u i d and l i q u i d - l i q u i d e q u i l i b r i a are reduced s e p a r a t e l y , we o b t a i n E


43

E


In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977. — X22 i

* A

2

- in

* Xi2

2

i

= a « + b/T = a + b/T

• Data of Scatchard and Satkiewicz, 1964

Figure 30. Vapor-liquid equilibrium using Wilsons equation for ethanol(l)-cyclohexane(2)


| *

tg


300

~

1

\

Figure 31.

0

1

1 9

, cal/mole

-300

1

1

0

1

FROM VAPOR-LIQUID/ AND LIQUID-LIQUID DATA 20-118°C

1

UNIQ UAC parameters for butanol(l)-water(2) (—99% Confidence elipses)

Au

300

1

\ \ \ ^

FROM LIQUID-LIQUID

FROM VAPOR-LIQUID/ DATA 93-118 C

r\

NT"

1

>


300


46

PHASE

EQUILIBRIA

AND

FLUID

PROPERTIES

IN

CHEMICAL

INDUSTRY

two e l l i p s e s w i t h no o v e r l a p . When both sets of data are reduced s i m u l t a n e o u s l y , we o b t a i n the e l l i p s e shown on the r i g h t . F i g u r e 32 compares c a l c u l a t e d w i t h experimental r e s u l t s using temperatureindependent parameters. Agreement i s f a i r but not as good as we would l i k e i t to be. Further a n a l y s i s shows that s i g n i f i c a n t l y b e t t e r agreement cannot be obtained by a l l o w i n g the parameters to vary l i n e a r l y w i t h 1/T. For t r u l y s a t i s f a c t o r y agreement i t i s necessary f o r t h i s system to a s s i g n a quadratic dependence on 1/T. Once we have obtained good d e s c r i p t i o n s of b i n a r y l i q u i d mixt u r e s , we can o f t e n p r e d i c t the p r o p e r t i e s of multicomponent l i q u i d mixtures using only b i n a r y data. This procedure saves much e x p e r i mental e f f o r t and i t i s usually- s u c c e s s f u l f o r multicomponent vaporl i q u i d e q u i l i b r i a but o f t e n i t i s not f o r multicomponent l i q u i d liquid equilibria. Group-Contributions

for Activity Coefficients

The v a r i e t y of equations based on the l o c a l composition concept has given us an improved t o o l f o r handling s t r o n g l y n o n i d e a l s o l u t i o n s but, perhaps more important, these equations have s t i m u l a t e d another development which, i n my view, i s p a r t i c u l a r l y promising f o r chemical engineering a p p l i c a t i o n . I r e f e r to the g r o u p - c o n t r i b u t i o n method f o r e s t i m a t i n g a c t i v i t y c o e f f i c i e n t s , a technique where a c t i v i t y c o e f f i c i e n t s can be c a l c u l a t e d from a t a b l e of groupi n t e r a c t i o n parameters. The fundamental i d e a , d a t i n g back over 50 years to Langmuir, i s that i n a l i q u i d s o l u t i o n of polyatomic molec u l e s , i t i s not the i n t e r a c t i o n s of molecules, but the i n t e r a c t i o n s of f u n c t i o n a l groups comprising the molecules (e.g., CH^, 2 > > e t c . ) which are important; F i g u r e 33 i l l u s t r a t e s the general i d e a . About 15 years ago, Deal, Derr and Wilson developed the ASOG g r o u p - c o n t r i b u t i o n method based on W i l s o n s equation where the important composition v a r i a b l e s are not the mole f r a c t i o n s of the components but the mole f r a c t i o n s of the f u n c t i o n a l groups (16, 17). In chemical technology the number of d i f f e r e n t f u n c t i o n a l groups i s much s m a l l e r than the number of molecular s p e c i e s ; t h e r e f o r e , the g r o u p - c o n t r i b u t i o n method provides a very powerful scale-up t o o l . With a r e l a t i v e l y s m a l l data base to c h a r a c t e r i z e group i n t e r a c t i o n s , i t i s p o s s i b l e to p r e d i c t a c t i v i t y c o e f f i c i e n t s f o r a very l a r g e number of systems, i n c l u d i n g those f o r which no experimental data are a v a i l a b l e . There i s no time now to d i s c u s s g r o u p - c o n t r i b u t i o n methods; we s h a l l have an o p p o r t u n i t y l a t e r t h i s week to go i n t o some d e t a i l s . Here I j u s t want to mention that some of the d i f f i c u l t i e s and l i m i t a t i o n s of the ASOG method have been overcome by a s i m i l a r method, c a l l e d UNIFAC (18), based on the UNIQUAC equation. Very r e c e n t l y , Fredenslund and Rasmussen i n Denmark and Gmehling and Onken i n Germany have s i g n i f i c a n t l y extended the e a r l i e r UNIFAC work; i n a p u b l i c a t i o n now i n p r e s s , the UNIFAC data base has been much enlarged and, t h e r e f o r e , the range of a p p l i c a t i o n i s now much i n c r e a s e d . The l a t e P r o f e s s o r R a t c l i f f at M c G i l l has a l s o developed a g r o u p - c o n t r i b u t i o n method and, d u r i n g the l a s t few years, P r o f e s s o r s Chao and N0

1


C 0 0 H


2.

PRAUSNiTz

Review

of Phase

Equilibria

47

Figure 32. Temperature-equilibrium phase composition diagram for butanol(l)— water(2) system. Calculations are based on UNIQUAC equation with temperature-independent parameters.

American Chemical Society Library 16thinSt., N.W. Industry; Storvick, T., et al.; In Phase Equilibria and Fluid1155 Properties the Chemical ACS Symposium Series; American Chemical Washington, D . C . Society: 20036Washington, DC, 1977.

48

P H A S E EQUILIBRIA

E.G.

A N D FLUID PROPERTIES

(1)

(2)

ACETONE

TOLUENE

IN CHEMICAL

(CH)


in

Y

i

=

In

Y

i

(COMBINATORIAL)

C

F (x,

F (X , K

MOLE

a,

Q/

R, a ,

FRACTION

GROUP

Figure 33.

• *n y .

Q, R)

C

R

(

MN

C-Qh

(RESIDUAL)

R »

GROUP VOLUME

0 *

6R0UP AREA

X »

MOLE FRACTION

T)

OF GROUP

INTERACTION

R

INDUSTRY

K;

PARAMETER

Group contributions to activity coefficients y and y t

2

Q2 ( - )

301 OA

r = 0.502 b = 2.62

-0.6

0.2

0.4

( - ) (cal/cc) 0.6

0.8

1.0

Figure 34. Activity coefficients for ethanol (A)-triethy Limine (B) system at 34.85°C (Nitta and Katayama, 1973)


2.

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49

Greenkorn at Purdue have s t a r t e d to work i n t h i s area. The groupc o n t r i b u t i o n method n e c e s s a r i l y provides only an approximation but f o r many a p p l i c a t i o n s that i s s u f f i c i e n t . For p r a c t i c a l - m i n d e d chemical engineers t h i s new r e s e a r c h i n a p p l i e d thermodynamics r e p r e sents perhaps the most e x c i t i n g development s i n c e P i t z e r ' s a c e n t r i c factor.


Chemical Theory f o r A c t i v i t y C o e f f i c i e n t s While the l o c a l composition concept has been h i g h l y u s e f u l f o r s t r o n g l y n o n i d e a l m i x t u r e s , i t i s a l s o p o s s i b l e to represent data f o r such s o l u t i o n s by assuming t h a t molecules a s s o c i a t e o r s o l v a t e t o form new molecules. I t f o l l o w s from t h i s viewpoint that a b i n a r y mixture of A and B i s r e a l l y not a b i n a r y m i x t u r e , but i n s t e a d , a multicomponent mixture c o n t a i n i n g , i n a d d i t i o n to A and B monomers, a l s o polymers A2, A3 ... and B2, B3 ... as w e l l as copolymers c o n t a i n i n g both A and B i n v a r i o u s p o s s i b l e s t o i c h i o m e t r i c p r o p o r t i o n s . D e v i a t i o n s from i d e a l behavior are then explained quant i t a t i v e l y by a s s i g n i n g e q u i l i b r i u m constants to each of the postul a t e d chemical e q u i l i b r i a . This i s a Pandora's box because, i f we assume a s u f f i c i e n t number of e q u i l i b r i a , a d j u s t the s t o i c h i o m e t r y of the polymers and copolymers and a l s o a d j u s t the e q u i l i b r i u m c o n s t a n t s , we can o b v i o u s l y f i t anything. N e v e r t h e l e s s , the chemical method makes sense provided we have independent chemical i n f o r m a t i o n (e.g., s p e c t r o s c o p i c data) which a l l o w s us to make s e n s i b l e a p r i o r i statements concerning what chemical species are present. For example, we know that a c e t i c a c i d forms dimers, t h a t a l c o h o l s polymerize to dimers, t r i m e r s , e t c . and that chloroform and acetone are l i n k e d through a hydrogen bond. Thus an " e n l i g h t e n e d " chemical theory can o f t e n be used to represent experimental data w i t h only a few parameters where a s t r i c t l y e m p i r i c a l equation r e q u i r e s many more parameters t o g i v e the same f i t . The l i t e r a t u r e i s r i c h i n examples of t h i s s o r t ; a recent one by N i t t a and Katayama (19) i s given i n F i g u r e 34 which shows a c t i v i t y c o e f f i c i e n t s f o r the system e t h a n o l t r i e t h y l a m i n e . Here A stands f o r a l c o h o l and B f o r amine. S u b s c r i p t C denotes chemical c o n t r i b u t i o n ; i n a d d i t i o n to the chemical e f f e c t s , there are p h y s i c a l f o r c e s between the " t r u e " molecules and these are taken i n t o account through the parameter b which has u n i t s of energy d e n s i t y . E q u i l i b r i u m constant K^, f o r continuous p o l y m e r i z a t i o n of e t h a n o l , i s obtained independently from alcohol-hydrocarbon mixture data. Parameter r i s the r a t i o of the e q u i l i b r i u m constant f o r A + B"Z5£AB t o K^. The e x c e l l e n t f i t i s , t h e r e f o r e , obtained w i t h two a d j u s t a b l e parameters, b and r . While chemical t h e o r i e s are o f t e n u s e f u l f o r d e s c r i b i n g s t r o n g l y n o n i d e a l l i q u i d m i x t u r e s , they are n e c e s s a r i l y s p e c i f i c , l i m i t e d to a p a r t i c u l a r type of s o l u t i o n . I t i s d i f f i c u l t to c o n s t r u c t a general theory, a p p l i c a b l e to a wide v a r i e t y of components, without i n t r o d u c i n g complicated a l g e b r a and, what i s worse, a p r o h i b i t i v e l y l a r g e number of parameters. This d i f f i c u l t y a l s o makes chemical theory i m p r a c t i c a l f o r multicomponent mixtures and indeed, w h i l e the


50

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l i t e r a t u r e i s r i c h w i t h a p p l i c a t i o n of chemical theory to b i n a r i e s , there are few a r t i c l e s which apply chemical theory to t e r n a r y (or higher) m i x t u r e s . For p r a c t i c a l chemical e n g i n e e r i n g , t h e r e f o r e , the chemical theory of l i q u i d s o l u t i o n s has l i m i t e d u t i l i t y .


S u p e r c r i t i c a l Components i n the L i q u i d Phase I have p r e v i o u s l y s t r e s s e d the d i f f i c u l t y of standard s t a t e s when we d e a l w i t h s u p e r c r i t i c a l components. For these components, e.g., methane or n i t r o g e n , a t o r d i n a r y temperatures, i t has been common p r a c t i c e to i g n o r e the problem simply by e x t r a p o l a t i n g purel i q u i d f u g a c i t i e s to temperatures above the c r i t i c a l . This i s convenient but u l t i m a t e l y u n s a t i s f a c t o r y because there i s no unambiguous way to perform an e x t r a p o l a t i o n f o r a h y p o t h e t i c a l q u a n t i t y . The most common method i s to assume that a s e m i - l o g a r i t h m i c p l o t of the f u g a c i t y versus r e c i p r o c a l temperature i s a s t r a i g h t l i n e . Experience has shown t h a t at temperatures f a r above the c r i t i c a l , t h i s i s a bad assumption but r e g a r d l e s s of what shape the p l o t i s assumed to be, on semilog paper, the t h i c k n e s s of the p e n c i l can a l r e a d y make a s i g n i f i c a n t d i f f e r e n c e . The only p o s s i b l e s a t i s f a c t o r y procedure f o r proper use of a c t i v i t y c o e f f i c i e n t s of s u p e r c r i t i c a l components i s to use Henry's constants as the s t a n d a r d - s t a t e f u g a c i t y . Henry's constants are not h y p o t h e t i c a l but are e x p e r i m e n t a l l y a c c e s s i b l e ; a l s o , at l e a s t i n p r i n c i p l e , they can be c a l c u l a t e d from an equation of s t a t e . Remarkably l i t t l e a t t e n t i o n has been given to the formal thermodynamics of l i q u i d mixtures c o n t a i n i n g s u p e r c r i t i c a l components. Using Henry's constants i n t r o d u c e s a v a r i e t y of problems but they are by no means insurmountable. Y e t , chemical engineers have stubbornly r e s i s t e d u s i n g Henry's constants f o r s t a n d a r d - s t a t e f u g a c i t i e s ; whenever I have t r i e d to i n t e r e s t my i n d u s t r i a l colleagues i n t h i s p o s s i b i l i t y I f e l t l i k e a gun-control e n t h u s i a s t t a l k i n g to the N a t i o n a l R i f l e A s s o c i a t i o n . As long as the s o l u t i o n i s d i l u t e , Henry's constant i s s u f f i c i e n t but as the c o n c e n t r a t i o n of s o l u t e r i s e s , unsymmetrically normalized a c t i v i t y c o e f f i c i e n t s must be introduced and at present we have l i t t l e experience w i t h these. While b i n a r y mixtures can be handled w i t h r e l a t i v e ease, major formal d i f f i c u l t i e s a r i s e when we go to multicomponent mixtures because, u n f o r t u n a t e l y , Henry's constant depends on both s o l u t e and s o l v e n t and, t h e r e f o r e , when we have s e v e r a l s o l v e n t s present, we must be very c a r e f u l to d e f i n e our s t a n dard s t a t e s and corresponding a c t i v i t y c o e f f i c i e n t s i n a thermodynamically c o n s i s t e n t way. About ten years ago the l a t e Ping Chueh and I wrote a monograph on the use of unsymmetric a c t i v i t y c o e f f i c i e n t s f o r c a l c u l a t i n g K f a c t o r s i n hydrocarbon and n a t u r a l - g a s m i x t u r e s , but i t never caught on. About f i v e years ago I was window shopping i n the Time Square s e c t i o n of New York and to my amazement I saw a copy of our monograph on a t a b l e i n a used book s t o r e , completely surrounded by books on pornography. I t appeared that my colleagues were t r y i n g to t e l l me something.



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However, times change and now that pornography i s accepted w i t h l i t t l e o p p o s i t i o n , maybe Henry's constants f o r standard-state fugac i t i e s can be accepted too. John O'Connell a t F l o r i d a has been working on t h i s and i n Figure 35 we see some r e s u l t s f o r excess Henry's constants f o r ethylene, carbon d i o x i d e and carbon monoxide i n b i n a r y s o l v e n t mixtures (20). To a f i r s t approximation, the l o g a r i t h m of Henry's constant f o r a gas i n a mixed s o l v e n t i s given by a simple m o l e - f r a c t i o n average; F i g u r e 35 shows d e v i a t i o n s from that f i r s t approximation. F i g u r e 36 presents another example, given by N i t t a and Katayama (21); i t shows Henry's constant f o r n i t r o g e n i n mixtures of n-propanol and iso-octane. Two p l o t s a r e shown, one against mole f r a c t i o n and the other a g a i n s t volume f r a c t i o n of the s o l v e n t mixture. Since i s o octane i s a much l a r g e r molecule than propanol, i t i s not s u r p r i s i n g that the v o l u m e - f r a c t i o n p l o t i s more n e a r l y l i n e a r than the molef r a c t i o n p l o t , but, n e v e r t h e l e s s , there i s n o t i c e a b l e departure from s t r a i g h t - l i n e behavior. Katayama a p p l i e s h i s chemical model f o r e x p l a i n i n g the d e v i a t i o n w i t h r e s u l t s shown i n Figure 37. One c o n t r i b u t i o n of the F l o r y Huggins type c o r r e c t s f o r s i z e d i f f e r e n c e s , another (chemical) c o n t r i b u t i o n c o r r e c t s f o r a s s o c i a t i o n of a l c o h o l molecules and f i n a l l y , a physical contribution corrects for differences i n intermolecular f o r c e s . The sum of the c o r r e c t i o n s gives good agreement w i t h e x p e r i ment. Since the c o r r e c t i o n s f o r s i z e and a s s o c i a t i o n were c a l c u l a t e d from other data, only one a d j u s t a b l e parameter was used i n preparing the f i n a l p l o t . Aqueous S o l u t i o n s of Weak V o l a t i l e E l e c t r o l y t e s I have i n d i c a t e d e a r l i e r that the chemical theory of l i q u i d mixtures presents some d i f f i c u l t i e s and that the use of Henry's constants a l s o gives us headaches. However, when we come to s o l u t i o n s of v o l a t i l e e l e c t r o l y t e s we are r e a l l y i n a bad way because now we must use not only the awkward chemical theory but i n a d d i t i o n , those unpleasant Henry's constants. We have no r e a l choice here because i n d i l u t e aqueous s o l u t i o n , weak v o l a t i l e e l e c t r o l y t e s (e.g., ammonia, hydrogen s u l f i d e , s u l f u r d i o x i d e ) d i s s o c i a t e i n t o ions and thus there i s r e a l chemistry going on which we cannot ignore. Further, s i n c e ions a r e n o n v o l a t i l e , we must use unsymmetrically normalized a c t i v i t y c o e f f i c i e n t s ; the f u g a c i t y of a pure v o l a t i l e e l e c t r o l y t e l i q u i d which i s not i o n i z e d doesn't t e l l us anything that would be u s e f u l f o r a d i l u t e aqueous s o l u t i o n where the s o l u t e i s , a t l e a s t i n p a r t , i n i o n i c form. The s i t u a t i o n we must d e s c r i b e i s shown s c h e m a t i c a l l y i n Figure 38. The h o r i z o n t a l e q u i l i b r i u m i s chemical, c h a r a c t e r i z e d essent i a l l y by a d i s s o c i a t i o n constant, and the v e r t i c a l e q u i l i b r i u m i s p h y s i c a l , c h a r a c t e r i z e d e s s e n t i a l l y by Henry's constant. Detailed development toward q u a n t i t a t i v e r e s u l t s a l s o r e q u i r e s unsymmetrically normalized a c t i v i t y c o e f f i c i e n t s , i . e . , those a c t i v i t y c o e f f i c i e n t s which go t o u n i t y not as the composition approaches the pure s o l v e n t ,


EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L

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PHASE

ure 35.

Deviation of Henrys constant from that in an ideal solution (O'Connell)



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Figure 36. Henry's constants for nitrogen in n-propanol (A)-isooctane (B) mixture vs. mole fraction and volume fraction (Katayama et al., 1973)

Figure 37. Experimental and calculated In K-values vs. volume fraction for n-propanol (A)-isooctane (B) mixtures (Katayama et al, 1973) lnic = ln H ,

N2 m

— %®j In H ^ j

j


54

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P

T

Vapor Phase


Mol ecular Elec trolyte

Mol ocular Elec rolyte ~

Figure 38. Vapor-liquid equilibrium in a single-solute system

(1)

^ Ions

Liquid Phase

MASS BALANCE m

A "

m

a

\

+

(m

+

+

m

-

)

m f MOLALITY SUBSCRIPT A = STOICHIOMETRIC SUBSCRIPT a - MOLECULAR

(2)

DISSOCIATION

K -

EQUILIBRIUM

V -

T ^ l

(3)

AS m ^ O

ELECTRONEUTRALITY m

(4)

ACTIVITY

+

*m

VAPOR-LIQUID EQUILIBRIA

^ a

P

~ a a m

T

H ( P C

>

H = HENRY'S CONSTANT PC - POYNTING CORRECTION (P = VAPOR-PHASE FUGACITY COEFFICIENT

Figure 39.

Aqueous solutions of weak volatile electrolytes



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but i n s t e a d , as the aqueous s o l u t i o n becomes i n f i n i t e l y d i l u t e . F i g u r e 39 i n d i c a t e s the four c o n d i t i o n s t h a t must be s a t i s f i e d . For most s o l u t e s of i n t e r e s t , chemical e q u i l i b r i u m constant K i s known as a f u n c t i o n of temperature; the major d i f f i c u l t y l i e s ^ i n c a l c u l a t i n g H and y*. F i g u r e 40 shows two equations f o r y . Both s t a r t out w i t h the Debye-Hdckel term which depends p r i m a r i l y on i o n i c s t r e n g t h but t h a t term alone i s a p p l i c a b l e only to very d i l u t e s o l u t i o n s . Guggenheim adds an e s s e n t i a l l y e m p i r i c a l f i r s t - o r d e r c o r r e c t i o n and t h i s i s s u f f i c i e n t f o r i o n i c strengths to about 1 or 2 molar. For more concentrated s o l u t i o n s , P i t z e r has proposed a semit h e o r e t i c a l equation which, however, has many parameters and a l l of these depend on temperature (22). Time does not permit a d e t a i l e d d i s c u s s i o n but, i n view of the importance of these s o l u t i o n s i n chemical e n g i n e e r i n g , l e t me q u i c k l y show a few r e s u l t s . F i g u r e 41, from Edwards e t a l . (23),shows how experimental data i n the d i l u t e r e g i o n f o r aqueous ammonia can be reduced to y i e l d Henry's constants (the i n t e r c e p t ) and Guggenheim's c o e f f i c i e n t 3 (the s l o p e ) . Note t h a t the a b s c i s s a g i v e s the molecular m o l a l i t y of ammonia, not the t o t a l m o l a l i t y ; t h e r e f o r e , F i g u r e 41 i m p l i c i t l y i n c l u d e s the e f f e c t of i o n i z a t i o n as d e t e r mined by the independently-measured chemical d i s s o c i a t i o n constant. A s i m i l a r a n a l y s i s was made f o r s o l u t i o n s of CO2 i n water. F i g u r e 42 g i v e s p a r t i a l pressures f o r the t e r n a r y system ammoniacarbon d i o x i d e when the t o t a l ammonia m o l a l i t y i s 0.128; these r e s u l t s were p r e d i c t e d u s i n g only b i n a r y data; no t e r n a r y data were used. I n t h i s example the s o l u t i o n i s d i l u t e and Guggenheim's equat i o n i s adequate; f o r higher c o n c e n t r a t i o n s , P i t z e r ' s equation i s r e q u i r e d as shown i n F i g u r e 43 based on very recent (and as yet unpublished) work by Renon and coworkers. The l i n e on the r i g h t i s the same as the one shown i n the p r e v i o u s f i g u r e ; the m o l a l i t y i s low. The l i n e on the l e f t i s a t higher ammonia c o n c e n t r a t i o n and, as we proceed to higher r a t i o s of carbon d i o x i d e t o ammonia, the t o t a l m o l a l i t y goes w e l l above 2. We see that Edwards' l i n e , based on Guggenheim's equation, i s s a t i s f a c t o r y at f i r s t but shows i n c r e a s i n g d e v i a t i o n s as the t o t a l m o l a l i t y r i s e s . The r e s u l t s shown here are again based on b i n a r y data alone. A t 20°C, experimental data are r e l a t i v e l y p l e n t i f u l and i t was p o s s i b l e to evaluate a l l the parameters i n P i t z e r ' s equation but at higher temperatures, where good data are s c a r c e , i t i s not easy to use P i t z e r ' s equation u n t i l some r e l i a b l e method can be found to estimate how temperature a f f e c t s the parameters. F i n a l l y , I should mention t h a t the c a l c u l a t i o n s shown here are based on simultaneous s o l u t i o n of 14 equations. A good computer program i s an a b s o l u t e n e c e s s i t y . Conclusion I have t r i e d t h i s morning to present a survey of the present s t a t u s of a p p l i e d p h a s e - e q u i l i b r i u m thermodynamics. In one sense, the survey i s much too long because I am sure your p a t i e n c e has been pushed w e l l beyond i t s e l a s t i c l i m i t . In another sense, i t i s much


56

P H A S E EQUILIBRIA


IN CHEMICAL

INDUSTRY

GUGGENHEIM

In

Ve.

r.

4-

1+/T WHERE:

.m. *J J

z = CHARGE A = KNOWN CONSTANT I = IONIC STRENGTH = -

) z.m. J J

2 V


PITZER

In

Az

r.

2

l57T

+

!

l

n

(°) _,_ a

+

ZE i c

j k

m

J

m

(

1

+

b

/

r

)

i.i I

k

WHERE: a=2 AND b = l . 2 IF

i AND j ARE MOLECULAR SPECIES,

0.

Figure 40. Activity coefficients in electrolyte solutions


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2.

Figure 42. Vapor-liquid equilibria at 20°C for ammonia-carbon dioxide-water containing excess ammonia


58

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IN CHEMICAL

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PHASE

Figure 43.

Calculated and experimental partial pressures of CO at 20°C for the C0 -H 0 system: Effect of total concentration (Renon et al.) 2

2

2


NH 3


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too s h o r t because I have had t o omit many worthwhile c o n t r i b u t i o n s . I a l r e a d y f e a r the hurt and i n s u l t e d looks that I am l i k e l y to r e c e i v e from some of you f o r the r e s t of the week, i f not l o n g e r ! Let me q u i c k l y summarize what to me are the main i m p l i c a t i o n s of t h i s f r a n k l y p e r s o n a l i z e d survey. F i r s t , those of us who are i n the u n i v e r s i t i e s must get over our a r g o n - f i x a t i o n and s t a r t t h i n k i n g b o l d l y about molecules that are l a r g e , n o n - s p h e r i c a l , p o l a r and hydrogen bonded. In other words, l e t us pay more a t t e n t i o n to the r e a l w o r l d . For chemical engineers i t i s b e t t e r , I t h i n k , t o s o l v e approximately new and r e a l problems than to improve m a r g i n a l l y s o l u t i o n s to o l d problems. I am hopeful that t h i s conference w i l l c o n t r i b u t e toward that end. Second, we must stop the game of composing v a r i a t i o n s on o l d themes. The Redlich-Kwong equation, the BWR equation, the Wilson equation, a l l these represent f i n e moments i n our h i s t o r y . However, we should not honor them by minor i m i t a t i o n . Rather, we should regard them as great monuments, i n s p i r i n g us toward t a c k l i n g new frontiers. Where, then, are these new f r o n t i e r s which demand our a t t e n t i o n ? I can here mention only s i x that I f i n d p a r t i c u l a r l y c h a l l e n g i n g i n t e l l e c t u a l l y and i n d u s t r i a l l y important: 1. C o n s t r u c t i o n of approximate, but p h y s i c a l l y s e n s i b l e , equat i o n s of s t a t e a p p l i c a b l e to complex molecules i n both gaseous and l i q u i d phases. 2. Vapor-phase experimental work (PVT and g a s - s a t u r a t i o n measurements) to provide fundamental i n f o r m a t i o n on i n t e r m o l e c u l a r f o r c e s i n asymmetric b i n a r y m i x t u r e s , i . e . , those mixtures where the two components are s t r o n g l y d i f f e r e n t , e i t h e r i n molecular s i z e , o r p o l a r i t y , or both. 3. V a p o r - l i q u i d e q u i l i b r i u m experiments on mixtures of complex molecules, i n c l u d i n g p o l y n u c l e a r aromatics, polymers and h i g h l y p o l a r s o l v e n t s such as g l y c o l s , p h e n o l i c s and other "nasty" l i q u i d s . The systems water-ethanol and benzene-cyclohexane have each been s t u d i e d about 50 times. Enough of t h a t . L e t ' s measure e q u i l i b r i a i n systems where we cannot now estimate the r e s u l t s w i t h i n even an order of magnitude. 4. More a t t e n t i o n must be given to data r e d u c t i o n methods. Some data are c l e a r l y more v a l u a b l e than others and we must i n c o r porate t h a t d i s t i n c t i o n i n t o our experimental p l a n s . 5. We can u s u a l l y do a p r e t t y good job c a l c u l a t i n g vaporl i q u i d e q u i l i b r i a f o r multicomponent mixtures of t y p i c a l n o n e l e c t r o l y t e s . However, f o r multicomponent l i q u i d - l i q u i d e q u i l i b r i a the s i t u a t i o n i s much l e s s f a v o r a b l e and we should g i v e more a t t e n t i o n to those e q u i l i b r i a . 6. F i n a l l y , l e t us l e a r n to use more the powerful methods of s t a t i s t i c a l mechanics; l e t us overcome our f e a r of p a r t i t i o n funct i o n s and l e t us not h e s i t a t e to i n t r o d u c e some e n l i g h t e n e d empiricism into t h e i r construction. This assembly of over 100 s c i e n t i s t s and engineers represents a wide v a r i e t y of knowledge, i n t e r e s t s and experiences. Our meeting


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here p r o v i d e s us w i t h a unique, unprecedented o p p o r t u n i t y to exchange views, s t i m u l a t i n g us a l l to new achievements. I have presented t h i s r a p i d overview of previous accomplishments w i t h the c o n v i c t i o n that these accomplishments must serve us not as ground f o r s e l f - c o n g r a t u l a t i o n but as a firmament on which to b u i l d toward a b r i g h t e r f u t u r e . My f e e l i n g — a n d I hope i t i s yours, t o o — m u s t be that of the French philosopher who s a i d , "From the a l t a r s of the past l e t us c a r r y the f i r e , not the ashes."


Acknowledgment For f i n a n c i a l support extending over many y e a r s , the author i s g r a t e f u l to the N a t i o n a l Science Foundation, the Gas Processors A s s o c i a t i o n , the American Petroleum I n s t i t u t e , the Donors of the Petroleum Research Fund (administered by the American Chemical S o c i e t y ) , Union Carbide C o r p o r a t i o n and Gulf O i l Chemicals Company. S p e c i a l thanks a r e due to Mr. Thomas F. Anderson f o r e x t e n s i v e assistance i n preparing t h i s report.

Literature Cited 1. Orye, R. V., Ind. Eng. Chem. Process Des. Dev., (1969) 8, 579. 2. Starling, K. E., and Han, M. S. Hydrocarbon Processing, (1972), 51, 107. 3. Bender, E., Cryogenics, (1973) 13, 11; (1975) 15, 667. 4. Soave, G., Chem. Eng. Sci., (1972) 27, 1197. 5. Peng, D., and Robinson, D. B., Ind. Eng. Chem. Fundam., (1976) 15, 59. 6. Mollerup, J., and Rowlinson, J. S., Chem. Eng. Sci., (1974) 29, 1373; Mollerup, J., Advan. Cryog. Eng., (1975) 20, 172. 7. Tsonopoulos, C., A.I.Ch.E. Journal, (1974) 20, 263. 8. Hayden, J. G., and O'Connell, J. P., Ind. Eng. Chem. Process Des. Dev., (1975) 14, 209. 9. de Santis, R., Breedveld, G. J. F., and Prausnitz, J. M., Ind. Eng. Chem. Process Des. Dev., (1974) 13, 374. 10. Wohl, K., Trans. A.I.Ch.E., (1946) 42, 215. 11. Funk, E. W., and Prausnitz, J. M., Ind. Eng. Chem., (1970) 62, 8. 12. Tompa, H., Trans. Faraday Soc., (1952) 48, 363; Staverman, A. J., Rec. Trav. Chim. Pays-bas, (1950) 69, 163; Donohue, M. D., and Prausnitz, J. M., Can. J. Chemistry, (1975) 53, 1586. 13. Cukor, P. M., and Prausnitz, J. M., Intl. Chem. Eng. Symp. Ser. No. 32 (Instn. Chem. Engrs., London) 3:88 (1969). 14. Anderson, T. F., Abrams, D. S., Grens, E. A., and Prausnitz, J. M., paper presented at the 69th Annual A.I.Ch.E. Meeting, Chicago, I l l i n o i s , 1976; submitted to A.I.Ch.E. Journal. 15. Fabries, J., and Renon, H., A.I.Ch.E. Journal, (1975) 21, 735. 16. Derr, E. L., and Deal, C. H., Intl. Chem. Eng. Symp. Ser. No. 32 (Instn. Chem. Engrs., London) 3:40 (1969).



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17. Wilson, G. M . , and Deal, C. H., Ind. Eng. Chem. Fundam., (1962) 1, 20. 18. Fredenslund, Aa., Jones, R. L., and Prausnitz, J . M., A.I.Ch.E. Journal, (1975) 21, 1086; Fredenslund, Aa., Michelsen, M. L . , and Prausnitz, J . M., Chem. Eng. Progr., (1976) 72, 67; Fredenslund, Aa., Gmehling, J., Michelsen, M. L., Rasmussen, P. and Prausnitz, J . M., Ind. Eng. Chem. Process Des. Dev. (in press). 19. Nitta, T., and Katayama, T., J . Chem. Eng. Japan, (1973) 6, 1. 20. Orye, R. V . , Ind. Eng. Chem. Process Des. Dev., (1969) 8, 579. 21. Nitta, T., and Katayama, T., J . Chem. Eng. Japan (1973) 6, 1. 22. Pitzer, K. S., and Kim, J . J., J . Amer. Chem. Soc. (1974) 96, 5701. 23. Edwards, T. J., Newman, J., and Prausnitz, J . M., A.I.Ch.E. Journal (1975) 21, 248.


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