phase equilibria and fluid properties in the chemical industry estimation and correlation

Phase Equilibria and Fluid
Properties in the Chemical Industry
Estimation and Correlation
Truman S. Storvick,

EDITOR

University of Missouri, Columbia

Stanley I. Sandler,

EDITOR

University of Delaware
A symposium co-sponsored
by the Engineering Foundation,
the American Institute
of Chemical Engineers,
and the National
Science Foundation at the
Asilomar Conference Grounds
Pacific Grove, CA,
January 16-21,

ACS

1977

SYMPOSIUM

SERIES

AMERICAN CHEMICAL SOCIETY
WASHINGTON, D. C.

1977

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

6
0


Library of Congress

CIP Data

Phase equilibria and fluid properties in the chemical
industry.
(ACS symposium series; 60 ISS
Includes bibliographical references and index.
1. Phase rule and equilibrium—Congresses. 2. Thermodynamics—Congresses. 3. Liquids—Congresses.
I. Storvick, Truman S., 1928. II. Sandler, Stanley
I., 1940. III. Engineering Foundation, New York.
IV. Series: American Chemical Society. ACS symposium
series; 60.
QD501.P384
ISBN 0-8412-0393-8

Copyright ©

660.2'9'63
ACSMC8

60 1-537

77-13804
(1977)

1977

American Chemical Society
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PRINTED IN T H E UNITED STATES OF AMERICA

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.;

ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


ACS Symposium Series
Robert F. Gould,

Editor

Advisory Board
D o n a l d G. Crosby
Jeremiah P. Freeman
E . Desmond G o d d a r d
Robert A . Hofstader
J o h n L . Margrave
N i n a I. M c C l e l l a n d
J o h n B. Pfeiffer
Joseph V . Rodricks
A l a n C . Sartorelli
Raymond B. Seymour
R o y L . Whistler
Aaron W o l d

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


FOREWORD
The A C S SYMPOSIUM SERIES was founded in 1974

to provide

a medium for publishing symposia quickly in book form. The
format of the SERIES parallels that of the continuing ADVANCES
IN CHEMISTRY SERIES except that in order to save time the
papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form.

As a further

means of saving time, the papers are not edited or reviewed
except by the symposium chairman, who becomes editor of
the book.

Papers published in the A C S SYMPOSIUM SERIES

are original contributions not published elsewhere in whole or
major part and include reports of research as well as reviews
since symposia may embrace both types of presentation.

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


DEDICATION
This

work is dedicated to the memory of three men who contributed

to our understanding of fluid properties.
Ping L . Chueh
Shell Development C o .

Houston, T X
Geral
M c G i l l University
Montreal, Quebec, Canada
Thomas M . Reed
University of Florida
Gainesville, FL
Illness and accident cut short their careers in 1976 and have left us
with their last contribution.

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


PREFACE
We

had two goals in organizing this conference.

T h e first was to

* * provide a forum for state-of-the-art reviews of an area of chemical
engineering often referred to as "thermodynamics and physical properties." T h e reviews should represent the work of both the academic researcher and the industrial practitioner. This we thought was both necessary and timely because there were obvious dislocations between the
current needs of the industrial chemical engineer and the research being
done at universities, on the one hand, and the slow acceptance of new
theoretical tools by the industria
Our second objective was, through these reviews and the ensuing
discussion, to develop a collection of research objectives for the next
decade. W e asked the session reporters to try to identify the important
research problems that were suggested in the presentations and discussions of the sessions, as well as to set down their thoughts in this regard.

In this way, the major papers in this volume summarize the current state
of research and industrial practice, while the reporter's summaries provide a listing of important questions and research areas that need attention now.
T h e conference was attended by 135 engineers and scientists from
North America, Europe, Asia and Africa.

They represented, in almost

equal numbers, the industrial and academic sectors. Recognized authorities, presently active in physical properties work, were chosen to be
speakers, panel members, session reporters and session chairmen. T h e
conference was held at the Asilomar Conference Grounds on the Monterey Penninsula of California, the beautiful setting matched by idyllic
weather.

W e have tried to give an accurate account of the material

presented at the conference sessions, but the printed word cannot reflect
the friendships that were established nor the extent of the academioindustrial dialogue which was initiated. Similarly, the unusual enthusiasm of the conference is not reflected here. Indeed, this enthusiasm was
so great that there were six ad-hoc sessions, continuations of scheduled
sessions and meetings packed into the four sunny afternoons of the
meeting.
Many important areas of work were identified as needing further
attention during the next decade.

Several obvious to us (in no special

order) are listed below:
• It was generally agreed that nine out of 10 requests for data by

xi

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.;

ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


design engineers were for vapor-liquid equilbrium or mixture enthalpy
data. Reduction to field-level practice of either data banks or estimating
procedures to supply this information would be very useful.
• Significant progress has been made on the group contribution
methods for estimating phase equilibrium data. Further development of
these procedures is clearly justified.
• Perturbation methods based on theory from physics and chemistry,
electronic computer simulation studies, and careful comparisons with
real fluid behavior are moving quickly toward producing an effective
equation of state for liquids. These efforts are in the hands of the theoretician today, but further development and reduction to practice should
be explored.
• F l u i d transport properties were not the primary concern at this
conference, but progress
between prediction and experiment for viscosities and thermal conductivities of gaseous mixtures was reported.

Clearly, much work needs to

be done, especially for liquids.
• Real difficulties remain when attempts are made to predict, to
extrapolate, or even to interpolate data for multicomponent mixtures containing hydrocarbons, alcohols, acids, etc.

Such systems were affection-

ately identified as a "Krolikowski mess" at the conference.

Multicom-

ponent mixtures of this kind may include more than one liquid and/or
solid phase and with components that "commit chemistry" as well as
physically distribute between the phases are commonly encountered in
industrial practice.
from

nightmare

to

T h e goal for the future is to reduce these problems
headache

proportions

in industrial applications,

though they may continue to remain an enigma for the theoretician.
• Cries for more experimental data were often heard. Special needs
include high pressure vapor-liquid equilibrium data; data on several
properties for mixtures with very light, volatile components in heavy
hydrocarbon mixtures; ionic solutions; acid gases in hydrocarbons; and
certainly more emphasis on mixtures containing aromatic hydrocarbons.
Data with intrinsic value for design work and accurate enough for discriminating theoretical comparisons should have high priority.

Signifi-

cantly, several conferees stated that their primary sources of new experimental data are rapidly shifting to laboratories outside the United States.
A n important measure of the success of a conference is its long-term
impact.

It remains to be seen whether this conference results in any

permanent interchange of ideas between academic and industrial engineers and whether the ideas expressed influence research in the coming
years.

xii

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


A cknou/ledgments
This volume is based on the Engineering Foundation Conference,
"The Estimation and Correlation of Phase Equilibria and F l u i d Properties in the Chemical Industry," convened at the Asilomar Conference
Grounds, Pacific Grove, C A , on Jan. 16-21, 1977. T h e views presented
here are not necessarily those of the Engineering Foundation, 345 East
47th St., New York, N.Y. 10017. T h e advice, financial and moral support,
and the concern for local arrangements, publicity, registration by Sandford Cole, Harold Commerer, Dean Benson and their staff permitted us
to concentrate on the technical aspects of the meeting. Manuscript typing
was done by the University of Missouri, Stenographic Services Department.
Major funding for the conference by the National Science Foundation was a key ingredien
for many American and
have been otherwise unable to participate.

T h e interest and support of

Marshall L i h and William Weigand of the National Science Foundation
were especially appreciated.
The American Institute of Chemical Engineers made important contributions by co-sponsoring and publicizing the conference.

W e also thank the members of the Organizing Committee: Stanley
Adler, Pullman-Kellogg Co.; Howard Hanley of the National Bureau of
Standards; Robert Reid of the Massachusetts Institute of Technology;
and L y m a n Yarborough of the Amoco Production C o . They brought
focus and structure to the general concept of the conference we brought
to them.
Finally, and most important we thank the speakers, session reporters,
and chairman who d i d their work diligently and in the best scientific
tradition; and the conferees for their enthusiastic participation and important discussion contributions that made this conference special.
T. S. STORVICK

STANLEY I. SANDLER

University of Missouri—Columbia

University of Delaware

June 1977

xiii

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


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

University of California, Berkeley, Calif. 94720

It was a pleasure t
Sandler' invitatio
this conference by reviewin
which led m to propose the acentric factor in 1955. Although I had
e
followed s m of the work in which others have used the acentric
oe
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
m debt to John Prausnitz for suggestions in this review of recent
y
work as well as in m n discussions through the years.
ay
Beginning in 1937, I had been very m c interested in the
uh
thermodynamic properties of various hydrocarbon molecules and hence
of those substances in the ideal gas state. This arose out of work
with K m in 1936 on the entropy of ethane (1) which led to the
ep
determination of the potential barrier restricting internal rotation.
With the concept of restricted internal rotation and s m advances
oe
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 m entropy and enthalpy values to provide a
y
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) w o presents the best current results on this topic.
h
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.
M first step in this direction came in 1939 when I was able to
y
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 c m to be called that of a simple fluid. It is
ae
the reference pattern from which the acentric factor measures the
departure. Possibly w should recall the key ideas. T e
e
h
1
In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.;
ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


2

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 correspondin
s t a t e i n terms of the reduce
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.


1.

PITZER

3

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 th K i h a r model w i t h s p h e r i c a l
the
the r e l a t i v e core s i z e ca
t i o n t o the energy and d i s t a n c
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

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


4

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

1

11

1

1

1
-

r

!

_^*- *^-

if
1

0

\J

0.
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

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

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

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

6

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

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
mentary. He chose f o r h i
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
=
—
RT = z

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

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


1.

Origin of the Acentric Factor

PITZER

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

3 8
H

CH groups in C H —hence the name
acentric factor for the forces arising from
points other than molecular centers.
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

01
.

02
.
CJ.

03
.

04
.

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

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

16


8

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

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 suggeste
for the normal p a r a f f i n s bot
y
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

9

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


1.

PITZER

Origin of the Acentric Factor

9

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
approac
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.
f

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.

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


10

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.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

P i t z e r , K. S., J . Chem. Phys., (1937) 5, 469, 473, 752; (1940)
8, 711; Chem. Rev. (1940) 27, 39.
Rossini, F. D., P i t z e r , K. S., Arnett, R. L., Braun, R. M. and
Pimentel, G. C., "Selected Values of the Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,"
Carnegie Press, Pittsburgh (1953).
Scott, D. W , J . Chem. Phys. (1974) 60, 3144.
.
P i t z e r , K. S., J . Chem. Phys. (1939) 7, 583.
Rowlinson, J . S., Trans. Faraday Soc. (1954) 50, 647; "Liquids
and Liquid Mixtures," 2nd ed. Chapter 8, Butterworth, London
(1969).
Kihara, T., Rev. Mod. Phys. (1953) 25, 831 and papers there
cited.
P i t z e r , K. S., J . Am. Chem. Soc. (1955) 77, 3427.
Riedel, L., Chem. Ing
27, 209, 475; (1956
P i t z e r , K. S., Lippman, D. Z., Curl, J r . , R. F., Huggins, C. M.
and Petersen, D. E., J . Am. Chem. Soc. (1955) 77, 3433.
Benedict, M., Webb, G. B. and Rubin, L. C., J . Chem. Phys.
(1940) 8, 334.
Opfell, J . B., Sage, B. H. and P i t z e r , K. S., Ind. Eng. Chem.
(1956) 48, 2069.
Curl, J r . , R. F., and P i t z e r , K. S., Ind. Eng. Chem. (1958)
50, 265.
P i t z e r , K. S. and Curl, J r . , R. F., J . Am. Chem. Soc., (1957)
79, 2369.
P i t z e r , K. S. and Hultgren, G. O., J . Am. Chem. Soc. (1958)
80, 4793.
Danon, F. and P i t z e r , K. S., J . Chem. Phys. (1962) 36, 425.

Lee, B. I . and Kesler, M. G., A.I.Ch.E. Journal (1975) 21, 510.
Tsonopoulos, C., A.I.Ch.E. Journal (1974) 20, 263.
Lyckman, E. W , Eckert, C. A. and Prausnitz, J . M., Chem. Engr.
.
S c i . (1965) 20, 703.
Barner, H. E. and Quinlan, C. W , I . and E.C. Proc. Des. Dev.
.
(1969) 8, 407.
Chueh, P. L. and Prausnitz, J . M., A.I.Ch.E. Journal (1969)
15, 471.
Reid, R. C. and Sherwood, T. K., "The Properties of Gases and
Liquids," 2nd ed., McGraw-Hill Book Co., New York (1966).
Barker, J . A. and Henderson, D., Rev. Mod. Phys. (1976) 48, 587.
Rigby, M., J . Phys. Chem (1972) 76, 2014.
Beret, S. and Prausnitz, J . M., A.I.Ch.E. Journal (1975) 21,
1123.
Rogers, B. L. and Prausnitz, J . M., Trans. Faraday Soc. (1971)
67, 3474.
Tee, L. S., Gotoh, S., and Stewart, W E., Ind. Eng. Chem. Fund.
.
(1966) 5, 363.

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


2
State-of-the-Art Review of Phase Equilibria
J. M. PRAUSNITZ
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 m y give m a chance to influence
a
e
the direction of future research in this field. W e I mentioned
hn
these two reasons to one of m more candid coworkers, he said " h t
y
Wa
you really m a is, that you enjoy the opportunity to go on an ego
en
trip and that you are glad to have an audience which you can subject
to your prejudices."
While this restatement of m feelings is needlessly unkind, I
y
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 m n items which others, in exeray
cising their judgment, might have included. At the outset, therefore,
I want to apologize to all in the audience who may feel that s m
oe
publications, notably their own, have received inadequate attention.
While I have tried to be objective and critical in my selection,
it is h m n nature to give preference to that work with which one is
ua

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 m particular experience and I expect that it is
y
pervasive in what I have to say. However, I want very much to assure
this audience that I present m point of view without dogmatic intent;
y
it is only a personal statement, a point of departure for what I hope
will be vigorous discussion during the days ahead. M aim in
y
attending this conference is the s m as yours: at the end of the
ae
week I want to be a little wiser than I am now, at the beginning.

1
1
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 a
tage of thermodynamics i
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.

13

Review of Phase Equilibria

PRAUSNITZ

FOR EVERY COMPONENT i ,
£
l

= £

l

IN PHASES a AND 0

f = FUGACITY

EITHER

- J
n

d

V

"

l

n

^1

= MOLES OF i ;

V = TOTAL VOLUME

fl.y.P

i

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

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


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

(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

Review of Phase Equilibria

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 standar
troublesome because we f r e q u e n t l y
multicomponen
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

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

16

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

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 Bot
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

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


2.

PKAUSNITZ

17

Review of Phase Equilibria

\- o

1200

Figure 3.

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

Modified BWR

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