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A111D3 D771S3

NATL INST OF STANDARDS & TECH R.I.C.

A11 1030771 53

Rainwater, James C/Vapor-llquld equlllbr QC100 .U5753 N0.1328 1989 V198 C.1 NIST- NIST TECHNICAL NOTE 1328

NIST

PUBLICATIONS

U.S. DEPARTMENT OF COMMERCE / National Institute of Standards and Technology
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CXCl DO

Vapor-Liquid Equilibrium of ^

i^^^

Binary IVIixtures in the /^b?

Extended Critical Region t.^

I. Thermodynamic Model

James C. Rainwater

Thermophysics Division

Center for Chemical Engineering

National Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

/ V \

<?-

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kammer, Acting Director

Issued April 1989
National Institute of Standards and Technology Technical Note 1328

Natl. Inst. Stand. Technol., Tech. Note 1328, 72 pages (Apr. 1989)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1989

For sale by the Superintendent of Documents, U.S. Governnnent Printing Office, Washington, DC 20402-9325
CONTENTS

Page

Abstract 1

1. Introduction 2

2. Transformation of Thermodynamic Variables 4

3. Specification of the Thermodynamic Potential 18

4. Specification of the Functions Ci (C) 25

5. Specification of the Function H ((,t) 36

6. Representation of the Critical Locus 42

7. Summary 45

8. Acknowledgments 47

9. References 48

Appendix A

Comparison with Conventional Equations of State 52

Appendix B

Critical Divergence of the Specific Heat at Constant Volume 57

Appendix C

Comparison of the Original and "Modified" Leung—Griffiths Models 61

111

Vapor—Liquid Equilibrium of Binary Mixtures in the

Extended Critical Region

I. Thermodynamic Model

by

James C. Rainwater

The thermodynamic model of Leung and Griffiths for binary mixture

vapor—liquid equilibrium near the critical locus, as modified by Moldover,

Rainwater and co-workers, is extended to accommodate fluid pairs of greater

dissimilarity. The maximum absolute value of a dimensionless ratio of

composition difference to density difference (a2m) is shown to be a very useful measure of dissimilarity of components. New parameters are introduced into

the model with guidelines that the number of permissible parameters depends

on, and increases with, Q:2in • The quantity K in the Leung—Griffiths

fugacity ratio ( is generalized from a constant to a temperature—dependent

function. Among other consequences, the previously approximate relation

that the mole fraction x = 1—^ on the critical locus can for most mixtures be

made exact. For a2m < 0.25, an empirical principle is asserted which predicts a relationship between the critical slopes of constant—composition curves on

temperature—density plots or isotherms on density—composition plots and

derivatives of critical locus parameters. The coexistence surface is shown to

be insensitive to dx/d^ on the critical locus at the dilute limits. The range of

the model is extended to Q2m « 0.3, e.g. propane + n—octane or carbon

dioxide + n—butane. The modified model is contrasted with conventional

equation—of—state methods and with the original Leung—Griffiths formalism.

Key words: binary mixtures; coexisting densities; corresponding states; critical

region; dew—bubble curves; field variables; Leung—Griffiths model; scaling—law

exponents; vapor—liquid equilibrium
1. Introduction

The accurate correlation of vapor—liquid equilibrium (VLE) in the critical region

has long been recognized as an important and inadequately resolved problem. Rowlinson

and Swinton note at several places throughout their book Liquids and Liquid Mixtures [1] that conventional thermodynamic approaches are unsuccessful near a critical locus, and

that special mathematical forms for the thermodynamic variables must be utilized in that

region. Their final observation, just prior to their "Conclusions" section on the state of

liquid theory, is that the model of Leung and Griffiths [2] (as modified by Moldover and

Gallagher [3,4]) overcomes in principle the problems of conventional methods and, in

particular, yields an accurate fit to the sulfur hexafluoride + propane VLE data of Clegg

and Rowlinson [5]. Their last statement is that "this technique could, with advantage, be

applied to other systems."

Our objective in this report and Part II of this series is to present and to summarize

our progress to date in the fulfillment of the above suggestion by Rowlinson and Swinton.

During the decade since the publication of the articles by Moldover and Gallagher [3,4],

progress began slowly. In retrospect, this was because their model, while based on an

ingenious hypothesis that serves as an excellent starting point for a more generally

applicable technique, was in its original form too simple to account for all but a small

fraction of binary mixtures.

Within this project, we have endeavored to collect critical region VLE data from all

available sources, and have ordered the mixtures according to ease or difficulty of fitting, a concept which is made quantitative later in this report. As we have progressed from the

easier to the more difficult mixtures, parameters have been added, but only as necessary.

Our system includes a set of guidelines for how many parameters may be used on a given

mixture. Accordingly, the model fits VLE data adequately without overfitting, and in

many cases can distinguish bad from good data.

Conventional methods are based on an analytic equation of state, i.e. pressure (P)

as an analytic function of temperature (T), molar density (p), and composition (x). As

described in Appendix A, such methods can at best mimic, rather than correctly represent,

thermodynamic behavior in the critical region. The Leung—Griffiths approach employs

different variables and thus is "unorthodox" in the context of traditional phase equilibrium

methodology, and to workers in the field may initially appear confusing or of questionable

legitimacy.
Details of the equations were presented in the original Leung—Griffiths paper [2],

and completely but briefly for its modified form in Appendix A of Moldover and

Gallagher [4]. Subsequent articles [6—8] have presented the model only as recipe, with

enough equations so that calculations may be reproduced, but largely without derivation or thermodynamic justification. Meanwhile, the model has undergone significant additional

development since the work of Moldover and Gallagher.

There is, therefore, a need for a current and detailed explanation of the theory and

techniques, which this report is intended to satisfy. The mostly self—contained exposition

should be useful for a novice as well as for an individual, of either scientific or engineering

background, experienced in phase equilibrium calculations. Although this first report of a

series of two contains many new results and insights not published previously, the primary

justification for the work is its wide—ranging success in correlating VLE data as described

in Part II of this series.

Furthermore, the Leung—Griffiths formalism is in part rigorous thermodynamics

and in part approximation and assumption. Some of our newly introduced parameters can

be given heuristic justification, whereas others are purely empirical. The exposition is organized to make clear the logical structure of the method, to identify what is exact and

what is approximate, and to present arguments justifying the approximations. In Sec. 2 we

define the transformed variables of the Leung—Griffiths model and translate the

fundamental laws of thermodynamics into relationships among these variables. The

description to this point is general and devoid of assumptions and approximations. With

the proper motivation, in Sec. 3 the equation specifying the independent variable

(thermodynamic potential), as a function of the dependent variables, is constructed. This

step corresponds in the conventional approach to the specification of an equation of state.

Specific emphasis is placed on the conformity with simple scaling—law behavior and the

thermodynamic consistency of the model. In Sees. 4 and 5, certain heretofore unspecified

functions of the independent variables are defined explicitly. The representation of the

critical locus in Sec. 6 completes the description of the model. New results and features are summarized in Sec. 7. In Part II of this series we will present in detail our correlations to four binary

alkane mixtures: propane + n—butane, propane + n—pentane, methane + ethane, and

(with more limited success) propane + n—octane. We will also summarize and present in

tabular form the parameters of successful fits to more than thirty additional binary

mixtures.
2. Transformation of Thermodynamic Variables

The system under consideration is a normal, nonazeotropic binary fluid mixture

with a continuous VLE critical line joining the two pure fluid critical points. Figures 1 to

4 show schematically the typical phase behavior of such a mixture as projected onto

various planes in the F—T—p—x thermodynamic volume. Isopleths, or loci of constant

composition, are shown in Figs. 1 and 2, and isotherms are shown in Figs. 3 and 4. These

figures qualitatively represent most of the VLE data currently available, as isobaric data

near critical are very scarce. Note that in P—T space (Fig. 1) the critical locus is the

envelope of the constant—composition dew—bubble curves, whereas in P—x space (Fig. 3)

the critical locus is the locus of maximum pressure points on the isothermal dew—bubble

curves. In the latter case, but not the former, the dew—bubble curves extend slightly

outside the critical locus.

Such a mixture is "Class 1" and, in the absence of liquid—liquid equilibrium,

"Type 1" according to the classification scheme of van Konynenberg and Scott [9]. The

model is also applicable, with some caveats, to azeotropic mixtures and, in some cases, to

Class 1 mixtures with liquid—liquid equilibrium (LLE), i.e. Types 2 and 6 mixtures [10], as described later.

Central to the Leung—Griffiths model is the distinction, first made by Griffiths and

Wheeler [12], between "field" and "density" variables, and the expectation that the

thermodynamics of mixtures is most efficiently represented entirely in terms of field

variables. By definition, field variables such as P, T, and the chemical potentials Hi

and /i2, have the same value for coexisting vapor and liquid, whereas density variables,

e.g. p, have different values. A crucial observation is that the composition x is a density

variable, since the vapor is rich and the liquid poor in the more volatile component. Thus

equations of state, P (T, p, x), are a mixed representation of a field variable as a function

of one field and two density variables.

In actual VLE experiments, P, T, x, and in many but not all cases p are directly

measured; the molar enthalpy, another density variable, is less frequently determined.

Except for isolated instances [13], fii and p,2 are not directly measured and, in the absence

of a reliable equation of state in the critical region, must be considered unknown. This

creates a dilemma in that a field variable description of thermodynamics necessarily must

be expressed in terms of unknown chemical potentials. The dilemma is ultimately resolved
Figure 1. Phase diagram in the P—T plane (schematic). The broken curves are the

pure—fluid vapor pressure curves, the dashed curve is the critical locus, the

solid curves are dew—bubble isopleths at, for example, x = 0.2, 0.4, 0.6 and

0.8 from right to left. In this projection the critical locus is the envelope of

the dew—bubble curves.
Figure 2. Phase diagram in the T—p plane (schematic). The broken curves are the

pure—fluid coexisting density curves, the dots are the pure-fluid critical

points, the dashed curve is the critical locus, and the solid curves are dew—bubble isopleths at, for example, x = 0.2, 0.4, 0.6 and 0.8 from top to

bottom.
Figure 3. Phase diagram in the P—x plane (schematic). The dashed curve is the

critical locus and the solid curves are dew—bubble isotherms at, for example,

four evenly spaced temperatures between Td and Tc2 with temperature

increasing from right to left. In this projection, the critical locus is the locus

of points of maximum pressure on the dew—bubble curves. In contrast to

Fig. 1, the dew—bubble curves extend slightly outside the critical locus.
X

Figure 4. Phase diagram in the /?—x plane (schematic). The dashed curve is the

critical locus and the solid curves are dew—bubble isotherms at, for example,

four evenly spaced temperatures between Td and Tc2 with temperature

increasing from right to left.
by assuming a specific parametric dependence of a particular function of chemical

potentials on P and T . The statement of this assumption, and its (essentially empirical)

justification will be made at the appropriate point in the development.

We acknowledge at the outset that there are many successive transformations of

variables which may seem confusing and tedious upon first reading. To the extent possible,

we will provide motivation for the introduction of new variables, although their significance can only be fully appreciated a posteriori. The objective is to work toward a set of

thermodynamic variables in terms of which a mixture most closely resembles a pure fluid,

and is thus most efficiently described and correlated.

Leung and Griffiths [2] first choose the following transformation of field variables:

Lj= P/RT, B = 1/RT and u^ = /^i/RT, i = 1, 2, where R is the gas constant. The

fundamental differential relationship is

da; = pi di^i + p2du — udB (1)

where p\ is the molar density of fluid i and u is the internal energy per unit volume.

We note that Eq. (1) is easily derived from the expression for the total Gibbs free

energy G,

G =Ni//i + N2/i2 = U + PV-TS (2)

where Ni is the number of moles of fluid i , U is the total (extensive) internal energy, V

is the volume and S is the entropy, together with the differential relationship for a closed

system that

dG = Ni d/zi + N2 d//2 = -S dT + V dP (3)

by eliminating the entropy from Eqs. (2) and (3).

Equation (1) is a useful representation since the differentials are of field variables,

whereas the conjugate functions are related to measurable density variables by

P = P\ + P2 (4)

X = p2/{pi + P2) (5)
and the relation for the molar enthalpy h„ ,

h„ = p-i(P + u) (6)

We can thus anticipate that the formalism will yield differential, rather than algebraic

equations for the measurable thermodynamic properties.

In the Griffiths and Wheeler formalism [12] one of the field variables, a choice which

in principle is arbitrary, is singled out as the "potential" or dependent variable; here the

choice for the potential is uj . Leung and Griffiths then introduce a second transformation

of independent field variables, C, , r and h . These are best understood as distance

variables and are, respectively, the distance from pure fluid 2, the distance from the critical

locus, and the distance from the coexistence surface.

The first new field variable is

Ki (B) e^i . C = = (7)

K2(B)e^2 + Ki(B)e^i K[B)e''^~^^ + 1

where K(B) = K2(B)/Ki(B); it can be interpreted as an activity or fugacity fraction.

This variable is absolutely central to the Leung—Griffiths representation of

thermodynamics and in many ways is a field variable analog of the density variable x . In

previous publications [2-4,6—8], K was assumed to be a constant with arbitrary value.

Here we generalize the model by assuming K can be a temperature dependent function,

K(T) or K(B), but must be independent of the chemical potentials, as was first mentioned

in Ref. 15. We first note some important properties of (,. Our (arbitrary) convention is that

X = for pure fluid 1 and x = 1 for pure fluid 2, the more volatile fluid. In the limit of

pure fluid 1, U2 has the leading—order divergence

i/2-*inx (8)

and in the limit of pure fluid 2, u^ diverges as

1^1 ^ (.n(l-x) = £nxi (9)

10
Therefore, ( = 1 for x = (pure fluid 1) and C = ^ for x = 1 (pure fluid 2), and

at any point within the mixture ( lies between and 1. Furthermore, for small x , 1 — ^

is proportional to x to leading order and is an analytic function of x , whereas for small

xi = 1 — x, ( is analytic in, and to leading order proportional to, xj

Had we chosen a different multiplicative factor in the exponent which defines ^,

i.e. [16]

RT RT

(10)

where Z is not an integer (e.g. the inverse critical compressibility factor [16]) then at the

z z respective limits 1 — C would be proportional to x and ( to x i

. While such a

redefinition would not by itself be inconsistent, a thermodynamic inconsistency would later

arise when uj is expanded in an analytic power series in the redefined (, since at the

mixture boundaries a; is analytic in x or xj. While the Leung—Griffiths formalism can

be generalized in other ways, it is important for the above reason not to alter the original

definition of (. For well—behaved mixtures and a reasonable choice for K(B), we expect that, as x

goes smoothly from to 1, C goes smoothly from 1 to 0. The critical locus is defined by

the function Pc(x), Tc(x), and Pc(x), but these can equally well be viewed as functions

of ( according to the correspondence on the critical locus. The second new field variable

introduced by Leung and Griffiths is

1 RTe(C)

-B = Be(0-B (11)

Note that each successive new field variable is defined in terms of both the old and

previously defined new field variables. Here, r is a measure of distance from the critical

line along a locus of constant (, and is negative in the two—phase region.

The third new field variable introduced by Leung and Griffiths is

h = In Ki(B)e^i + K2(B)e^2 -in Ki(B)e''i + K2(B)e^2

= in Ki(B)e^i.+ K2(B)e^2 -H(C,r)

(12)

11
where the superscript (t refers to the value of v\ on the vapor—liquid coexistence surface

for the same (C, t) as the thermodynamic point for which h is being evaluated. The

second line of Eq. (12) defines the function H (^, r). While C and h are first defined in

terms of two functions Ki(B) and K2(B), they only depend on K = K2/K1, but H as defined above can depend on Kj and K2 separately.

The critical line and coexistence surface have a very simple mathematical form in

the variable space ((, r, h); the critical line is simply the straight line segment h = 0, r = 0, < C

< 1, and the coexistence surface is the strip h = 0, r < 0, < C < 1- Note

that r < on the coexistence surface so that further specification of h is required for r > 0. Leung and Griffiths state that for r > the superscript a refers to the value of 1^1 on the smooth extension of the coexistence surface beyond the critical line, although in

practice this extension is a function of the model to be constructed. However, since the

primary concern of this work is the VLE surface for which r < 0, the problem of defining h

for r > is of no immediate consequence.

At this point the partial differential relationships specified by Eq. (1) are related to

the partial differential relationships of a; with respect to the new field variables, i.e.

du)

dv I l^2,B = Pi =

dijj

r,h

+

dix;

C.h

dr

i^2,B

+

A. (,r 1^2,^

(13)

and similarly for

du)

du^. = P2 and

l/l,B

duj

dB = —u. We eschew the shorthand notation

i^l,^2

of Leung and Griffiths, e.g. u^ = [du/dQ^^Y^ , because we shall subsequently eliminate r

in favor of yet another variable t and such shorthand notation could lead to confusion

since, for example, {du/dQ^^-^ ^ (^cj/^^t.h • While the resulting equations in principle can be quite complicated, a remarkable

simplification takes place for the total molar density.

P = Pi-^P2 =

dh

(14)

(J

12
which is a consequence of the following equations,

K

dvi

+

1^2,

B

K

du' = (15)

l/l,B

dr

+

J^2,B

dr = (16)

i/l,B

5h

^^1,

+

i^2,B

dti dV; = 1 (17)

Z/l,B

Equation (15) is a direct consequence of the fact that C

is a function only of (1/2 — ^i)-

Equation (16) follows from Eq. (11), where B is held constant.

dr

dui

U;,B dC

(18)

z/i,B

Finally, because of Eqs. (15) — (16)

dR

du I +

Z^2,B

5H

5^^ = (19)

^l,B

and Eq. (17) follows directly from Eqs. (12) and (19).

Expressions for the other measurable density variables x and u are somewhat

more complicated. For x , we require p2 which is given by Eq. (13) with subscripts 1 and

2 reversed. Since

A.

duo = -C(i-C) (20)

Z/l,B

13
and, from Eqs. (12), (18) and (20)

dE

dv = i-C + C(i-C)

2Jz/i,B

dE

dC

I dB,(0

dC

dE

dr

d

(21)

it can be shown that

X = P2/P = l-C-ai-QQ/p (22)

where the function Q is defined by

Q =

du

Hi T,h

I dB,(0

dC

dijj

dr.

-P

dE , dB,(0

dC

BE

dr

C

(23)

Two important observations are emphasized at this point. First, since px and P2» or p and x , involve only partial derivatives with B (or T) held constant, their formal

expressions are unchanged when K , which appears in the definition of C » is generalized

from a constant to a temperature—dependent function. However, the formal expression for

the energy density will be changed, as discussed below.

Second, from Eq. (20) the partial derivatives of C, required to calculate x are algebraic functions of ( itself. This would not be true in general, since transformation to an arbitrary function of /ii, ^2) T and parameters (e.g. K) would lead, through the analogs

of Eq. (13), to partial derivatives that could depend differently on p,^, p2, T and K . This

property of ( is reminiscent of similar properties of the Fermi—Dirac distribution function

[17], which ( closely resembles.

Only because of this property, knowledge of K will turn out to be unnecessary for

calculation of the P—T—p—x coexistence surface. It is evident already that if we can

"model", i.e. hypothesize, forms for uj at h = and H as functions of ( and r, as well

as the critical properties as functions of ( and the leading—order dependence of a; on h , then p and x can be calculated explicitly without knowledge of K . Since field variables

are continuous functions of each other but have discontinuous derivatives at a phase

boundary, Eq. (14) will yield a liquid or vapor density according as h approaches zero from above or below. Substitution of these densities into Eqs. (22) and (23) will yield

respectively the liquid and vapor compositions.

14
We next examine the internal energy density u. The analog of Eq. (13) is

—u =

duj

VUV2

du)

T,h

+

du

Ch

dT

+

du}

P\ (J

dh

VU^2

VU^2

(24)

Furthermore, from Eq. (7),

dB

^b^2

= -C(l-C)— ^nK

dB

(25)

dB

^1,^2

dC dB

(26)

These terms were simply zero and minus one, respectively, in previous derivations where K

was held constant. In addition.

dh

VuV2

dB

pT.

+ -^ tn K j dB

i-C + C(i-C) dB

H\

, dB,(C)

dC

dB

dT

+ — tnKi

dB

Combining these results, we find that

(27)

u =

duj

PT.

-p

dB

PT.

-p^— inKi-p2— ^nK2

dB dB

(28)

which reproduces Eq. (2.20) of Leung and Griffiths except, of course, for the last two

additional terms. Equation (28) can be differentiated to obtain the specific heats, as

described for the special case of a pure fluid in App. B.

15
An interesting connection can now be made with the reference states of the chemical

potentials. It is well known that the zeroes of chemical potential are arbitrary, and the

transformations /^i - //i + /^lo and /i2 - /*2 + M20) a,re constant over all thermodynamic

space, leave the equation of state P (p, T, x) unchanged. However, the internal energy

density is shifted by {pi //lo + p2 ^20)- From Eqs. (7) and (12), such a shift can be effected

without altering ( or h, if Kj is simultaneously multiplied by e

^^° and K2 by

e ^^°. Then under such a shift, Eq. (28) yields precisely the required shift in u , whereas

the expressions for p and x are appropriately left unchanged.

Therefore, the temperature dependence of K absorbs the degrees of freedom

present in the reference states for p,i and fi2 , but also introduces additional degrees of

freedom to the model. As an aside, in an earlier paper [6] it was incorrectly implied that

such a shift in zeroes is equivalent to a change in K from one constant to another.

Rather, such a shift would make a constant K become temperature—dependent.

At this point, for the first time in our discussions we come to a distinction between

the "original" and "modified" Leung—Griffiths formalisms, as contrasted in App. C. In the

latter version, first introduced by Moldover and Gallagher [3,4], the variable r is eliminated in favor of t , where

t = T -Te(C) ^ r Tc(C) Be(C) - r

(29)

A disadvantage of the variable r is that is not dimensionless. While it could be

made dimensionless [16] by multiplication by RTd or RTc2, where T^ is the critical

temperature of pure fluid i , such a redefinition would destroy the symmetry of the model

between the labels of two fluids. The variable t is conveniently both dimensionless and

symmetric in the fluid labels. Partial derivatives then transform as follows:

(,h

t + t' (30)

C.t

d(jj K T,h

du

'^k r dC

du}

di

(31)

C,h

and partial derivatives of cj with respect to h are unchanged.

16
The modified version also introduces a function Q, the first two terms of Eq. (23).

Q

duj

[Hi r,h

I dB.(C) du)

dr

H,^

du

H.

dto

+

t,h

t dB.(0

r dC

du

dt

C.h

l^C t,h Bc(C) dC

du

dt

C,h

(32)

and Eq. (22) is rewritten as

X = i-C-C(i-0 Q(C,t) Q(C,t=0) fl^^^^^

. P PciC)

(33)

where the function H is defined by

a (c,t) =

dE

[dC

+

dB,(C) dE

dr - Q(C >t-o)

Pc(C)

C

(34)

The addition and subtraction performed in the construction of Eqs. (33) — (34)

isolates those terms which cause x to deviate from (1—C) along the critical line. In previous

work [3,4,6—8], the hypothesis that x= 1 — ( along the critical locus was employed, but it was assumed that this was an approximate relation that could never be made exact.

However, with our present generalization of the model to admit a temperature—dependent

K , we can make a stronger statement. A rearrangement of Eq. (7) gives

tnK = (/ii-/i2)/RT + 2tanh-i(l-20 (35)

Let us consider the consequences of assuming x = 1 ~ C on the critical locus, in

which case Eq. (35) becomes

UK = (/xi-/i2)/RT + 2 tanh-i(2x-l) (36)

17
On the interval [0,1] in x , each term on the right—hand side is smooth and finite except

for singularities at the boundaries. However, from Eqs. (8) and (9) these singularities

cancel, and the full expression is always finite.

Let us further consider a mixture for which Tg is a monotonic function of x . This

restriction includes most normal mixtures but excludes most azeotropic ones. Again, it is emphasized that the chemical potentials along the critical line, except for their dilute—limit

singularities, must be regarded as unknown. However, at each temperature between the

critical temperatures of the two pure fluids, the right—hand side of Eq. (36) has a

particular, finite value. If we set in K equal to this value, we have constructed a temperature—dependent K such that x = 1 — ( on the critical locus exactly.

While we cannot determine this particular K(T) quantitatively, we can exploit the

knowledge that it exists and, once the zeroes of chemical potential have been specified, is unique. On primarily empirical grounds, we have found that this particular choice of K(T)

appears to be optimal for correlation of VLB data for the majority of mixtures. This choice

imposes the constraint in Eq. (34) that H(C,0) = 0, as seen from Eq. (33). The same

general argument can apply to any smooth function Xc(C) that satisfies the boundary

conditions Xc(0) = 1 and Xc(l) = 0, although forms that deviate too much from linearity

will probably violate stability criteria [12].

For most azeotropic mixtures, Tc has a minimum in x, and thus over a certain

interval two distinct compositions possess the same critical temperature. Consequently, in

general a K(T) does not exist such that x = l—( for the entire critical locus. However, in

view of the success in applying earlier versions of this model to azeotropic mixtures, it can be presumed that ( can still be made approximately linear in x along the critical locus

for such mixtures.

3. Specification of the Thermodynamic Potential

The preceding section has described the definition of a new set of thermodynamic

variables and the conversion of the basic laws of thermodynamics into relationships among

these new variables. As a consequence, the equations derived there are perfectly general

and are independent of any particular assumptions or approximations.

18
To proceed, we now construct an explicit mathematical form for a; as a function of

(, t and h . This step is analogous to the imposition of an equation of state P (T, p, x) in

the conventional thermodynamic formalism. Here u [(,t,h) is a. "generalized equation of

state" in terms of field variables, in contrast to the conventional mixed set of field and

density variables.

The development in this section follows that of Moldover and Gallagher [4], which is a slight modification of that of Leung and Griffiths [2]. For the steps described in this

section, we have essentially left the Moldover—Gallagher formalism intact in the current

version of the model. However, changes and additional features should prove to be useful

in future work, and such possible changes are discussed where appropriate.

We first take note of two important boundary conditions that the generalized

equation must satisfy. For h = and C = or ^ = 1, the equation must reduce to that of

the vapor pressure equation for pure fluid 1 or 2 respectively, which may be conveniently

represented as

_ P _ Pc

''^pure — — RT RTp

l + C3(-t)2-Ot+C4t + C5t2+C6t3 (37)

where the Ci are dimensionless constants that are characteristic of the particular pure

fluid.

Equation (37) introduces the critical exponent a , which is defined as the exponent

characterizing the divergence of the specific heat at constant volume Cy , i.e. in the

asymptotic limit near the critical point,

Cy (X t^ (38)

The connection between Eqs. (37) and (38) is derived in App. B. The exponent a has a

theoretical value of 0.110 largely confirmed by experiment [18], but for an "effective"

exponent providing a best fit over an extended critical region we shall use a = 0.1

.

Classical equations of state (see App. A) yield the result a = 0, i.e. no divergence in Cy • In addition, for (" = or 1 the coexisting densities should reduce to an appropriate

form for a pure fluid; a convenient such form is

Ppure — Pc l±Ci(-t)^ + C2t (39)

19
where Ppure = (^'-i^/^)(;,t evaluated at h=0 and (=0 or 1, and the plus sign refers to the

saturated liquid and the minus to the saturated vapor.

Equation (39) introduces a second critical exponent P, which has a theoretical

value of 0.325 also confirmed by experiment [19], but as an "effective" exponent over an

extended critical region is best represented by 0.355. Classical equations of state yield the

result P = 0.5, a significant difference which in our view is largely responsible for the

deficiencies of such classical equations.

In the study of critical phenomena, a large number of critical exponents were

originally introduced. However, according to Widom's scaling hypothesis [20], only two of

these exponents are independent, and fixing the values of two exponents determines all other exponents. Here we take the independent exponents to be a and P

.

Both Eqs. (37) and (39) are approximations which, for pure fluids, leave something

to be desired, although they should be adequate for our generalization to mixtures over an

extended critical region where the experimental data are typically less precise. In practice,

the vapor pressure curve is more efficiently fitted by an Antoine equation [21], or a

representation that combines an Antoine equation with the necessary asymptotic

expansion, e.g.

tn

P

lPc. = Ai

T^

+ A2 (-t)2^ + A3 t + A4 t2 + ... (40)

where the A\ are relatively small for i > 2. The (—t)2-ot term is not typically required

from a statistical analysis of vapor pressure data; in fact some correlators [22], on empirical

grounds, prefer in Eq. (40) an exponent near 1.3 instead of 1.9 = 2—a. While Eq. (40) is a more efficient representation of the vapor pressure curve than Eq. (37), we do not know of a mechanism to incorporate Eq. (40) into a scaling—law formalism, and here Eq. (37) has

been used instead.

More importantly, Eq. (39) is representative of so—called "simple scaling", whereas

current theoretical models employ so-called "revised [23] and extended [24] scaling".

Simple scaling is characterized by a single non—analytic term (with coefficient Ci) for the

vapor—liquid density difference, and a linear rectilinear diameter from the term with C2

.

In revised scaling [23], the rectilinear diameter possesses a "hook" near the critical point,

and in extended scaling, as first introduced by Wegner [24], there is a second term in

Eq. (39), proportional to (—t)0'^ with A « 0.5, with opposite signs for liquid and vapor.

20
Incorporation of revised and/or extended scaling would considerably increase the

complexity of the expressions for w (C, t, h). Almost certainly, as discussed in App. C,

revised scaling will be required for a proper description of the supercritical one—phase

region, h ^ 0. Furthermore, as will be discussed in the propane + n—octane correlation in

Part II of this series, extended scaling may well be the next feature introduced for VLE

within the evolution of the present model. However, for a wide range of mixtures the VLE

surface is accurately represented by simple scaling, so revised scaling and extended scaling are not at present necessary for the formalism.

Equations (37) and (39) can be rearranged as expressions written entirely in terms

of reduced units, i.e. thermodynamic variables divided by their critical values. If the

principle of corresponding states were obeyed exactly, i.e. if all fluids were "conformal",

Ci , i = 1 ... 6, would be universal constants, the same for all pure fluids. In fact, the C\

are nearly constant from fluid to fluid except for C2, the slope of the rectilinear diameter,

and Ce, the coefficient of the final term in a truncated polynomial series. (The helium

isotopes, probably because of residual quantum effects, are an exception to this rule.) As

the molecules become more acentric, the vapor pressure curve becomes steeper at the

critical point, and, as a result, fluids of highly acentric molecules [4] possess a larger value

of C4 . We require an explicit mathematical representation for the thermodynamic

potential consistent with the behavior postulated by Griffiths and Wheeler [12] near

critical. In particular, the potential must be a continuous function of the independent field

variables, but must have appropriately discontinuous derivatives, cf. Eq. (14).

Furthermore, the magnitude of the discontinuity must vary with distance from critical

(i.e., |t| ) according to the appropriate critical exponent.

To accomplish this, we follow Moldover and Gallagher [4] and utilize the Schofield

linear model [25]. For present purposes, the Schofield model may be regarded simply as a mathematical device to construct a continuous function with discontinuous derivatives.

While other such constructions exist, the Schofield model is particularly convenient in that

the critical exponents can be chosen at will. Yet another transformation (our last one)

from t and h to new variables r and 9 is required. An analytic background component

in the untransformed variables can be added to this singular function, thereby changing the

overall thermodynamics but not changing the essential structure of the phase transition

surface^

21
Given Eqs. (37) and (39) as boundary conditions, the thermodynamic potential is thus written as the sum of an analytic potential and a singular potential,

UJ (C, t, h) = CJan (C, t, h) + (x;sing (C, t, h) , (41)

where the two terms of Eq. (41) generate, respectively, the analytic and nonanalytic terms

of Eqs. (37) and (39). The functional form for the analytic part of the potential is

a;an (C, t, h) = ^^(^ [l + C4 (0 t + C5 (C) t2 + Ce (C) t^l

(42)

RTe(C)

+ Pc (C) l + C2(C)t

where Pc(C)> Tc(C) and Pc(C) are the critical pressure, temperature and density for a mixture with composition x corresponding to the given value of C, on the critical line, as discussed in the previous section. At this point in the development, Ci((') can be any

function of C, which satisfies the boundary conditions

Ci(0) = cl'' (43)

Ci(l) = c'i" (44)

(1) (2)

where C\ and Ci are the coefficients in Eqs. (37) and (39) for pure fluids 1 and 2

respectively, in which case the correct pure—fluid limits are clearly attained.

It is worth noting that we could add terms proportional to h^, h^, etc. to Eq. (42)

and the coexistence surface would not be changed. However, such terms would change the

thermodynamic behavior in the one—phase region. For the study of one—phase

thermodynamics, which is beyond the scope of the present work, this is a useful degree of

freedom within the Leung—Griffiths model; different thermodynamic potentials can be

constructed which lead to different one—phase equations of state but the same VLE surface.

For the singular potential, we transform t and h to two new variables r and 9. At the critical point r = 0, and r is (crudely speaking) a measure of distance from the

critical point. On the coexistence surface, ^ = 1 on the liquid side and ^ = —1 on the vapor

side, while ^ = is approximately the one—phase critical isochore (see App. B).

22
Here r and 6 somewhat resemble a polar coordinate system with the critical point

at the origin, although loci of constant r are not circular on a P—T plot. The discontinuity

in p = (5a;/5h)A J. at the coexistence surface is introduced by the discontinuous jump in 6

for +1 to —1 as the coexistence surface is crossed, whereas a;sing, t and h are smooth

functions of r and 9 . The particular transformation and parametric representation of a;sing is as follows [4]:

^ ^ r(l-b2 e^)

b2-l

(45)

h = _MQ QA^ r2-a^^(i_^2) (46)

RTe(C)Pc(C)Ci(C)aT

UJ, (r,^,0 = Pc(()C3(C) r2-a

RTeCOa^

ao + a2 ^2 -I- a4 9^ (47)

where Ci(() and €3(0, at this point in the development, can be any functions of ( which

satisfy the boundary conditions of Eqs. (43)—(44). Note that on the coexistence surface

^ = ± 1, we have t = —r, h = and a;sing reduces to the term proportional to C3 in

Eq. (37), since, by definition,

a^ = a^ + a2 + a^ (48)

The parameters b and ai have yet to be specified. Following Moldover and

Gallagher [4], we employ the so—called "restricted Schofield model" in which the

parameters are functions of the critical exponents [26]. As stated previously, there are only

two independent critical exponents, which have been taken as a and p. Two other

important critical exponents are 7, which characterizes the critical divergence of the

specific heat at constant pressure

Cp « t"" (49)

23
and 6, which characterizes the asymptotic behavior of the density and chemical potential

of a pure fluid,

A-/^ « ip-Pc)^ (50)

According to the scaling—law hypothesis [20]

7 = 2-Q-2/5 (51)

and

6 = l-\-^/P (52)

With our choice of critical exponents (a = 0.1, P= 0.355), the consistent values for the

above critical exponents are 7 = 1.19 and 6 = 4.3521.

Within the restricted Schofield model,

b2 = —^^-^

(53)

{6-l){l-2p)

where, for our choice of critical exponents, b^ = 1.3909. The coefficients ai are expressed

below both algebraically in terms of the critical exponents and numerically for the choices

Q = 0.1 and i5= 0.355

ao = P^-^P-^'ocj ^ Q 4753 (54J

2b4(2-Q)(l-a)a

a^ = -^-3/3-bM2^-l) ^ _ 0.7561 (55)

2b2 {l-a)a

a4 = ^K^lI = 0.45 (56)

2a

a = a, + a, + a = {^'-^)Pi^-m = 0.1692 (57)

^ 0^4 b2(2-a)(l-<i)a

These terms were given only numerically in Appendix A of Moldover and Gallagher [4].

24
The singular term in density is obtained from the ratio of Jacobians

duhin SJJig

dh _ ^^sing,t)/g(r,g)

Jt,C

d{h,t)/d{T,e)

(58)

Evaluation of this expression for ^ = ± 1 yields

<9a^sin ?ing

dh = />c(C)C,(C)r ^ t,C

(2-^)b2 a^ + (l-b2)(a2 + 23^^)

b2 - 1 (59)

Finally, although the algebra is tedious, it can be shown from Eqs. (51)—(57) that

the expression in brackets is unity. Thus, from Eqs. (42) and (59), the coexisting densities on a locus of constant ( are given by

P(C,t) = PciO l±Ci(0(-t)P + C2(C)t] , (60)

i.e. the generalization of Eq. (39), where the plus sign refers to the liquid (^ = 1) and the

minus sign to the vapor (^ = —1). The full potential for h = is the generalization of

Eq. (37),

a;(C,t, h = 0)=^£^

RTe(C)

1 + C3 (C) (-t)2-> + C4 (C) t + C5 (C) t2 + Ce (C) t3 (61)

This completes the generic construction of a continuous potential with discontinuous

derivatives. The development has followed very closely that of a pure fluid, with (

playing the role of a "spectator" variable. We expect that revised scaling [23], where t is replaced by t + qh, with q a mixing parameter, or extended scaling [24] could be

included similarly with ( playing a similar spectator role. The explicit mathematics has

yet to be constructed and tested, however.

4. Specification of the Functions Cj (C)

In the preceding section, we have constructed a thermodynamic potential expanded

about t = with phase transition properties consistent with the postulates of Griffiths and

Wheeler [12]. Within the simplifications of simple scaling and the restricted Schofield

2t
linear model, as well as the truncation of the analytic component at the cubic level in t, the

potential is still perfectly general.

This point in the development reaches a limit, beyond which further specification of

the model cannot be based on well—established scientific principles. The basic laws of

thermodynamics have been incorporated in Sec. 2, and modern understanding of critical

behavior, in simplified form, has been incorporated in Sec. 3. In the absence of a microscopic theory which might predict nonuniversal amplitudes about the critical locus

from an intermolecular potential, further specification of the model must necessarily be

largely empirical.

One possible strategy at this juncture is to allow all parameters and functions as yet

undetermined to be adjustable, and to fit them to the VLE data. Thus the Ci(0 could be

written as truncated polynomials in ( (consistent with Eqs. (43) and (44) as boundary

conditions), x on the critical locus could also be written as a truncated polynomial in (,

and the functions H(^,t) or H(C,t) could be similarly expanded. Apart from certain

details (see App. C), this is the strategy employed within the "original" (as opposed to our

"modified") Leung-Griffiths formalism [2,27,28].

The problem with such a strategy is the extreme redundancy in the resulting

mathematical description. As previously noted, K can be any constant or, within limits,

any temperature—dependent function. Thus there is in principle an infinite set of

Leung—Griffiths models with different explicit parameter values but which are equivalent

in describing a particular coexistence surface. Such redundancy can be expected to make

convergence of a calculational algorithm extremely problematical. Furthermore, it is hard

to develop an intuition as to what effect the change of a particular parameter has on the

symbolted VLE surface, or to develop guidelines for the points of truncation of the

polynomials. Therefore, it is appropriate to search for hypotheses which further narrow down the model, even if such hypotheses have only empirical justifications.

The first hypothesis, central to the reformulation of the Leung and Griffiths model, was originally introduced by Moldover and Gallagher [3,4] and states that the functions

Ci(0 are linear in (, i.e.

Ci(C) = Ci'UcCi (62)

26
' (1) (2)

Ci = Ci -Ci (63)

In our most general model to date we shall retain these restrictions for i = 3, ... 6 but not

for i = 1 or 2. The above hypothesis represents a highly idealized assumption of the smoothness of

the coexistence surface in P—T—^ space. As explained earlier, if the principle of

corresponding states were obeyed exactly, the Ci would be universal constants for all pure

fluids. The linear interpolation of Eqs. (62) — (63) thus implies that Ci is typically close

to constant for all (^, < ^ < 1. In fact, our essential phenomenological concept is that the

thermodynamic behavior of the mixture along lines of constant ^ closely resembles that of a pure fluid, as was somewhat implied in the previous section. Loci of constant C, in our model, on a P—T graph, have essentially the same shape as, and are parallel to, the

pure—fluid vapor pressure curves. Similarly, lines of constant t have essentially the same

shape as, and are parallel to, the critical line. This is illustrated explicitly in Fig. 5. It has

recently been demonstrated that such an idea is useful for the prediction of interfacial

tension of mixtures [29], as well as for phase equilibrium.

We can anticipate some instances where our assumptions of ideal smoothness in

P—T—( space will clearly break down. For "Type 2" mixtures in the van Konynenberg—

Scott classification scheme [9], the phase diagram includes liquid—liquid equilibrium (LLE)

in addition to VLB, and an LLE critical locus or locus of "consolute points" distinct from

the VLE critical locus of "plait points". This second critical locus joins the VLE surface at a critical end point, and the LLE coexistence surface joins the VLE coexistence surface at a three—phase locus [11].

From the "180° rule" [30], the VLE coexistence surface, uj (C, t, h = 0), must have a

"crease" or discontinuity in dujjdC, at the three-phase locus. Such a discontinuity will end

at the critical end point, but will likely propagate beyond as a continuous but highly

curved ridge in the h = surface, and could greatly distort the shape of the surface

compared to its form in the absence of LLE.

Henceforth we shall assume that the mixture under consideration either displays no LLE, or has a locus of consolute points at temperatures well below the critical temperature

of the more volatile component. Specific guidelines for the temperature interval have yet

to be established. While liquid—liquid immiscibility has not been observed in carbon

dioxide + methane, that mixture has a freezing surface which comes very close to the locus

of plait points [31], and we have found evidence that the mixture, if it did not freeze, would

separate into two immiscible liquid phases not far below the critical temperature of

27
Figure 5. Loci of constant ( in the P—T plane (schematic). The broken curves are the pure—fluid vapor pressure curves, the dashed curve is the critical locus,

and the solid curves are loci of constant ( at, for example, ( = 0.2, 0.4, 0.6. and 0.8 from left to right. The dots denote even intervals of, for example,

0,02 in t along loci of constant ( .

28
methane. This probably explains why our model has failed to correlate carbon dioxide +

methane, as first noted by Al—Sahhaf et al. [32] and as explained more fully in Part II of

this series.

The change in density across the phase boundary is, from Eq. (60)

^P = Pl-P, = 2C^ (C) (-t)%^ (0 (64)

where the subscripts i and v refer to liquid and vapor, respectively. The change in

composition across the phase boundary, from Eq. (33), is

Ax = x« -X

t V = c(i-a Q (C,t)

Onuki [33] has introduced the two functions

{p~^-Pl) (65)

Qi = R-i As/(Ap/pc)

Q2 = Ax/(Ap/pc)

(66)

(67)

where As is the molar entropy difference between liquid and vapor. While these functions

could have meaning at any two—phase point, we are concerned here with their limiting

values at the critical locus, i.e. t = 0. Since As and Ax asymptotically approach the

critical locus as (—t) , the limits are finite. Also, the limits are the same for any

thermodynamic path not parallel to the critical locus [12].

Obviously, for normal mixtures 0:2 is zero at the two pure—fluid boundaries, and

has a single extremum typically near x = 0.5. The sign of 0.2 depends on the arbitrary

convention employed for composition; for ours, it is negative for normal mixtures. In the

case of azeotropic mixtures, a2 is zero at the azeotrope and has two extrema, one between

each of the pure fluids and the azeotrope.

We shall refer to 0:2 as the "distance from azeotropy". Onuki [33] calls it the

"degree of azeotropy", but his terminology is in our view less satisfactory, first because in

customary usage a given mixture either is or is not azeotropic, and second because a larger

29
absolute value of a2 denotes a state further from, not closer to, azeotropic behavior. The

concept is somewhat similar to "relative volatility", except that in customary usage

relative volatility is reserved for the function

"12 = ^i(l-\)/[\(l-^f,)

which approaches unity at the mixture critical point.

Another useful concept is a2m , which we call the "maximum distance from

azeotropy",

asm = max \a2\ (68)

Whereas relative volatility is a property of a particular two—phase state or (P,T) point of a

given mixture, distance from azeotropy, or 0:2 ^it critical, is a property of a particular

mixture and composition, and maximum distance from azeotropy is a property of the

mixture as a whole, as well as a measure of dissimilarity of the two pure fluids.

We alluded in the introduction to a quantitative measure for the difficulty of fitting a particular mixture. In fact, a-^m is precisely this quantitative measure. Within our model

most parameters will be determined from fits to experimental pure fluid coexistence

surfaces or the critical locus, but a limited number of additional parameters will be needed.

Based on our analysis of the highest quality VLE data, we have developed guidelines

within which only certain of the parameters may be used for a given mixture according to

the value of a2m- In only the most ideal cases (very small Q2m), there are no adjustable

parameters and our model reduces to that of Moldover and Gallagher [3,4], in which the

coexistence surface is determined from the pure fluid coexistence properties and the critical

locus. As Q2m iucrcases, more parameters become necessary.

From Eqs. (32), (64), (65) and (67), at the critical locus,

«2 = Pc^C(l-C)Q{C,t = 0)

(R^c)-^C(i-C) —d

dC Tc. +

d 1

.Tc. Tc

(69)

-P,

CtzzO

30
= (RPc)-^C{i-C)- d

dx

P.

+

d

dx

1

Tc.

Tc

dP

5T

-p

C,t=o

Equation (69) is a thermodynamic identity, not based on any approximations. Also, both

Q!2 and the above composition derivatives of critical properties are experimentally

measurable. For example, from isothermal VLE data Q2 = Pc i9x./dp) and hence can be

determined from the slope at the critical point of isotherms on diagrams such as Fig. 4. One can also determine 02 from VLE data along isopleths. With an argument

similar to those used by Wheeler et al. [34] to resolve an apparent paradox concerning

mixture compressibility, and by Levelt Sengers [35] to derive the slope of the so—called

"bird's beak" isotherm at T = Tc2, we may write

5T.

CRL

\dp]

c +

x,CXS

\dp]

c

T,CXS

r^xi

(70)

CRL

where, following the notation of Ref. 34, CRL denotes the critical line and CXS the

coexistence surface, and the superscript c denotes evaluation at the critical point. The

first factor of the second term is, from above, equal to Pc/ci2 > a^nd therefore

Pcdxc/dT(

0:2 = d^_

dTc

d£

(71)

5T

x,CXS

where all quantities on the right—hand side are measurable and the second term in the

denominator is the inverse slope of the isopleth at the critical point, cf. Fig. 2. Note that

for a pure fluid or an azeotrope the second term in the denominator becomes infinite, so a2-»0.

As shown in Sec. 3, if Tc(x) is monotonic there exists a K(T) such that x= 1 — ( on critical. In this case, the only expression not ordinarily measurable in Eq. (69) is {dP/dT)^ r\, the slope of the constant—^ locus at the critical point, and within our model

"2 = -(Rpc)-^x(l-x) d

dx

Pc

Tc. + C4PC —

dx

(72)

31
Empirical principle: If for a given Type 1 mixture of nonpolar fluids a2 is calculated from

Eq. (72) with a linearly interpolated C4(x), and a2m < 0-25, then Eqs. (62) — (63) are valid

for i = 3, ... 6 and the (unique) choice of K(T) such that x = 1 — ( along critical.

We do not assert that the above principle is in any sense exact, or that an

experiment specifically designed to observe deviations from it might not locate some. However, from extensive experience in VLE correlations we do claim that such a principle

leads to quantitatively successful fits of the highest quality VLE data presently available.

We have therefore found a very useful and important empirical relationship between

derivatives of the mixture critical properties and the ratio of amplitudes of composition

and density difference at the critical point. Only the ratio is thereby fixed; we do not know

the value of the amplitudes which are each proportional to Ci(^). If we consider the special case of exact corresponding states where Ci<^' = Ci^ > , and assume Eq. (62) is applicable

so Ci(0 = Ci<^' = Ci'2)^ then Ap/pdC) is the same function of t for the pure fluids

(C = or 1) as for the mixture, < ( <1. However, Ax -» for the pure fluid. Thus the

overall measure of phase change (difference in the properties of the coexisting fluids), as a

function of distance from the critical line would appear to be larger for the mixture than for

the pure fluid in this example.

We first addressed this problem in Ref. 36, where we conjectured that there exists

some total "amount of phase change" Ao(t), a combination of Ax and Ap, which "obeys

corresponding states" in the sense that it is nearly independent of ( in the asymptotic

limit of small 1 1

1 . We initially chose a quadratic combination for Ao(t),

\ (t) = (Ap)2 + 2Cj [p^iOY {Ax)'

1/2

IpAO

*v " '

Pc(C)

1 + C, IpU)]'

-.2 Ax

Ap\

(73)

to leading order in Ax , where C„ is an adjustable parameter set equal to 1 in Ref. 36.

This quadratic combination was used by Al—Sahhaf et al. [32] to study VLE of several

fluids in the critical region. However in subsequent work Rainwater and Moldover [6] replaced Eq. (73) by a linear combination, i.e.

A(t) _A2_

PciO

+ [1 + 0^1^2(01] (74)

32
The switch to a linear combination was motivated in part by a conjectured analogy

with revised scaling [23] as discussed below Eq. (61). In revised scaling, t is altered in

Eq. (45) by a small linear admixture with h , and there may be some empirical or mathematical connection with the idea of linearly mixing Ax with A/? . An explicit

connection, however, is as yet undemonstrated and must await a future study of the

mixture one—phase region.

From Eqs. (64) and (65), Ao(t) is asymptotically proportional to (—t) . If

Ci<^) = Ci^2)^ the constant of proportionality for Ao(t) may be made independent of (

by the choice

Ci(0 = ^^^. (75)

1 + CjC (1-C) IQ(C,o)|/pjC)

It should be emphasized that Ao(t) is independent of ( only in the asymptotic limit of

small 1 1 1 , not over the entire VLE critical region in {(,i) space. For the latter condition to

be satisfied, Cj would need to be a function of t as well as ( . However, it is explicit in

the development of Sec. 2 that, whereas any function Ci(0 obeying the boundary

conditions leads to a thermodynamically consistent model, Cj clearly cannot depend on t

or h.

For Cj^ ^' ^ Ci< 2)

J the natural generalization of Eq. (75) is to replace the numerator

by a linear interpolation in ( . Furthermore, we have found that for some mixtures

sufficiently far from azeotropy, the fit is substantially improved by replacing C„ with

C (1 + C C)) where C is a second adjustable parameter that, in effect, makes C

linearly (—dependent. The final expression for Ci(() is thus

( 2) ( 1) _( 2) Ci + c c , - c

Ci(C) = "> ^ ^ ' "l )±^^^^ (76)

1 + C, (1 + CM |a2(C)

X ^ ' Y

The above equation may be straightforwardly generalized. C„ was first introduced

in Ref. 6, whereas C is new except for brief mention in Ref. 29. As with our subsequent

parameters, setting C„ or C^ equal to zero (equivalent to not introducing them in the

first place) reduces the model to that of Moldover and Gallagher [3,4]. Our guidelines are

that C is necessary only if a2m > 0-10 and C only if Q;2m > 017.

33
On intuitive grounds, one would expect that C_ should always be positive, since

Ao(t), whatever its precise functional form, should be a monotonically increasing function

of A/j and Ax . However, our correlation of the carbon dioxide + hydrogen sulfide

system [37] appears to require a negative C_ . This is an anomalous result; more than 30

other normal mixtures we have successfully correlated require C„ to be either positive or

zero, as will be discussed in Part II of this series.

As pointed out by Griffiths and Wheeler [12], there are three independent density

variables for the binary fluid mixture, which may be chosen as the density p , the

composition x , and the entropy per mole s . It thus might be argued that Ao(t) should

include a contribution from As, the change in entropy per mole across the phase boundary,

e.g.

^ Pc(C)

1 + a \0L. + c. \Q..

PciO

+ Cj Ax + R~^Cg As (77)

and thus incorporation of As into a new definition of Ao(t) requires the evaluation of ai

.

Unfortunately, to do this we need explicitly the value of K in the definition of C , but in our model K is "indeterminate". Specifically, for h = 0, Onuki [33] shows that

a I = -p-^ ^(To;)

dT = -P-' a; + T

T dT

+ T

A C

dT

dT A

(78)

where A = p,2 — (ii . However, even for the simple case of a constant K,

K

dT RT2

C(l-C) = -C(l-C)in iK

1-C.

/T (79)

and thus we cannot employ a revised Ao(t) as redefined by Eq. (77), even though it may be

fundamentally more appropriate.

Onuki calculates explicitly both Oi and a2 along the critical line for the helium 3

+ helium 4 mixture according to the fit of Leung and Griffiths [2]. The calculation of Qi was possible only because Leung and Griffiths explicitly evaluated K for the mixture by

means of specific heat data from absolute zero to the temperature for VLE. Subsequent

applications [3^,6-6,27—29] of the Leung—Griffiths method to other mixtures have left K

indeterminate.

34
It is, in fact, interesting and convenient that ease of experimental measurement

correlates inversely with the need for knowledge about K . For correlation of usual VLE

data in P, T, X and /?, K drops out of the equations and nothing must be known about it. For the less frequently measured enthalpy, knowledge of dK/dT is necessary. Finally, for

such normally unmeasured quantities as the entropy and Gibbs or Helmholtz free energies, an explicit value of K is required.

A negative value of C„ is not entirely unreasonable in that a mixture may have, in

relative terms, a much larger entropy change than its constituent pure fluids. In this case, a negative C_ might be necessary to make Ao(t) as redefined by Eq. (76) independent of

( . Nevertheless, we should expect that for a substantial majority of mixtures C is nonnegative.

It would be equivalent, and perhaps more to the taste of some, to write Ci(C) as a

linear interpolation plus a polynomial series, e.g. a truncated sum of terms of the form

C° (1 — 0) ^ ^^ integer, with adjustable coefficients. However, we find the explicit tie

with Q2 intuitively appealing, and our formalism leads to a somewhat regular pattern in

C„ with increasing a2m In fact, according to our experience with binary mixture VLE

data, C„ is nearly linear in a2m , which might lend support to a return to the quadratic

coupling of Eq. (73) and Ref. 36 with a nearly universal C_. But while such inter—mixture

comparisons support a quadratic model, intra—mixture analysis or fits of different

dew—bubble curves for the same mixture tend to favor a linear model.

For C2(C)) the slope of the rectilinear diameter, we use the frankly empirical

expression

c,{Q = c;%cc; + c^xjc) x,(0

c (80)

where C2 is defined by Eq. (63) and the adjustable parameter C_, again a deviation from

a, linear interpolation, is new here except for brief mention in Ref. 29. Here Xc(C) is the

critical composition as a function of C , and, as previously stated, equals (1-0 for

Q2m < 0.25. Thus for nearly azeotropic mixtures the last term in Eq. (80) is C, C(1~C)> but

K, we have found for mixtures far from azeotropy that use of Xc(0 provides a superior fit. Our guidelines are that C„ is necessary only if Q2m > 0-17. For all mixtures studied

a. to date, C„ , if needed, is positive and increases fairly rapidly with a2m

a.

35
5. Specification of the Function H((,t)

At this point we have completed the specification of the thermodynamic potential

a; (C, t, h), but not the model as a whole. In particular, specification of the function S((,t)

is still necessary for determination of the coexisting compositions.

From Eq. (33), we see that on the critical locus

Xc(C)-i-C + C(i-C)S(C,t = o) (81)

As previously stated, for mixtures not too far from azeotropy we make the

restriction that xdC) = I — C, which implies the condition H(C, t = 0) = 0. Furthermore,

as there is no reason to suppose that H is nonanalytic, we can expand H as a power

series in t with ^—dependent coefficients. For the range of applicability of the model, it turns out that a linear term in t is sufficient.

However, as was also noted in the previous section, for a2m > 0-25 we cannot

simultaneously set Xc = 1 — C ^^^ assume linear interpolation for Ci(0 in Eq. (61). The

objectives of this section are therefore twofold, first to extend the model to mixtures far

from azeotropy with a nonzero H at t = 0, and second to model the contribution to H

which is linear in t and is present for almost all mixtures.

As motivation for our strategy, we first prove that in the dilute limits (small x or small xj = 1 — x), the distance from azeotropy Q2 , which largely governs the shape of the

coexistence surface, is to leading order independent of dxc/d^. Consider Eq. (69) for a

dilute solution of fluid 2 in fluid 1, i.e.

a2 = (Rpc)-^C(l-C)-

dC

d

dx .Tc. +

d

_dx

1

Tc.

Tc

dP

dT

-Pc+0(x)

Ic

(82)

where (dP/dT)ic is the slope of the vapor pressure curve of fluid 1 at its critical point.

Because of the boundary conditions on the chemical potential, Eqs. (8)—(9), in the dilute

limit loci of constant ^ must be parallel to the pure vapor pressure curves. Now consider an arbitrary variation of Xc with ( subject to boundary conditions,

i.e.

1 - C = e X + 0(x2) (83)

36
where e does not necessarily equal one. Expanded in a Taylor series in x , Eq. (82)

becomes

0C2 = (Rpci)-^ex(-l/e)

.dx[Tc.

-'x=0

+

d 1 To

dP

V -Pc + 0(x2)

dx T. dT ^ "^'x=U - ^ ^ ic

-(RpciTci)-ix dP,

dx — dPdT

a dT,

dx

L ^X::0 '- ic x=oJ

(84)

+ 0(x2)

Thus to leading order a2 is independent of e and depends on quantities that are experimentally measurable. The ratio of the amplitudes of Ax and Ap is then fixed by

the critical properties and vapor pressure curve of the solvent and the initial slope of the

critical line. Since Ci(() is presumably analytic in ( and therefore x , the leading—order

amplitudes of Ax and Ap separately are also determined by these measurable quantities.

Similar arguments apply to a dilute solution of fluid 1 in fluid 2. The importance of the combination of derivatives in brackets in Eq. (84) was first noted by Krichevskii [38] and later by Rosen [39], who defined a quantity A as

A =

dPg

.dx.

x=0

5P

_5T.

Ic

dT,

.dx.

(85)

-'x=0

SO that Q2 = - (RPci Tci)'^ Ax + 0(x2).

Levelt Sengers and co-workers [34,40-42] have shown that a wide range of

thermodynamic properties, particularly excess molar volumes, are dependent on A for

dilute mixtures. While many of these properties involve thermodynamic paths off the

coexistence surface (h = 0), or derivatives in directions away from the coexistence surface,

Levelt Sengers [35] has shown that on the VLE surface the slope dP/dx of the so—called

"bird's beak" isotherm T = Tc2 (the critical temperature of the more volatile component)

is equal to A .

37
Rizvi et al. [43] have suggested that a theory of dilute solutions based on A may be

useful in understanding supercritical solubility, and may be a preferred alternative to our methods which normally require extensive and high—quality VLE data. Supercritical

solubility is quantitatively described by the behavior of the dew curves of a mixture with a

large a2m , eg- carbon dioxide + toluene, near the critical point of the more volatile

component. Unfortunately, as shown by the 0.96 propane + 0.04 n—octane mixture in

Part II of this series, the formal expressions for the dilute model break down much more

rapidly for this case than for mixtures rich in the less volatile component or for mixtures

with smaller a 2m • As explained in Sec. 2, the degrees of freedom embodied in K(T) allow the

construction of many different but equivalent modified Leung—Griffiths models, with

different functions Xc{C) but the same P—T—x—p VLE surface. If Tc(x) is monotonic, a

choice of K(T) has been proven to exist such that Xc = 1 — C- We expect that another

choice of K(T) exists such that, at least in some best—fit sense, loci of constant (^ are

parallel to the pure vapor pressure curves, or more precisely Eqs. (62)—(63) hold for

3 < i < 6. The empirical principle of Sec. 4 states that, for small a 2m > these choices of

K(T) are essentially identical.

Since the empirical principle does not lead to accurate VLE fits for a2m > 0.25, we

must assume that the above two choices of K(T) begin to diverge at this point. We then

have our choice whether to retain the condition Xc = 1 — C or to retain the condition of

parallel constant—^ curves. We elect to retain the latter condition, for two reasons. First, it allows us to retain a physically appealing, nonadjustable relationship between a (P,T) data point and a point on the critical locus. Equations (29) and (61) define a mapping from (^,t) to (P,T) and

vice versa, which is one—to—one for normal mixtures and two—to-one in some regions for

azeotropic mixtures. If the parallel structure of constant—^ lines is unaltered, a given

(P,T) point is always associated with the same point on the critical locus as a function of x

(if not Q, and t for that datum is unaltered.

Second, we can exploit the observation that the model is insensitive to dxg/d^ at the

dilute limits. For maximum adjustability, as seen from Eq. (69), we need the ability to

vary dxc/d^ for the equimolar mixture. Our choice for Xc(C) to achieve such adjustability

is

Xc(C) = l-C + HiC(l-0(l-2C) (86)

38
^ = -1 + Hi(l-6C + 6C2) (87)

and, from Eq. (80)

H(C,t = 0) = Hi (1-20 (88)

The second term in Eq. (87) has an extremum at x = 72 » so variation of Hj has the

greatest effect on the equimolar mixture, as desired.

We have introduced for the first time (except for brief mention in Ref. 29) a fourth

adjustable parameter Hj , with guidelines that it be used only if a2m > 0.25. There are obvious outer bounds for this parameter; Hi < 1 ensures that < Xc < 1 and Hi > —

2

ensures that dxc/d^ is always negative. The bounds to ensure thermodynamic stability over the entire surface are not established but are probably much more restrictive. In

practice, we have kept | Hi| < 0.65 for all correlations to date.

Before considering the t—dependence of H, it is appropriate to establish a range of

applicability for the present model. Since it is specifically designed for critical conditions,

it should be expected to work only within a certain distance from the critical line. The

variable which measures such a distance is (—t), and since the thermodynamic potential is expressed as a nonanalytic power series in this expansion parameter, the model should be

quantitatively correct only when the expansion parameter is an order of magnitude less

than unity, say —0.1 < t < 0. Moreover, for pure fluids, and consequently for our mixture

model, for t < —0.1 Eq. (60) on the vapor side enters the regime when the virial equation of

state is applicable, and begins to deviate significantly from that virial equation.

However, along a pure vapor pressure curve or a line of constant C

lim

T-.Tc(0

a tn P

a in T = 1 + C4(C) (89)

and, typically, C4 « 6. Because of the steepness of the vapor pressure curve near critical, a

ten percent decrease in temperature corresponds approximately to a 50 percent decrease in

pressure. On a P—T graph, therefore, the "extended critical region" for VLE is bounded by

the critical line, the two pure vapor pressure curves, and the locus t = —0.1 which is approximately the locus P = Pc(C)/2, as shown in Fig. 5. It is within this region that

conventional, analytic equations of state typically break down, so, as emphasized in

App. A, our model and analytic equations of state complement each other.

39
Within this extended critical region, we find that only a linear dependence of H on

t is needed, in contrast to o^an oi Eq. (42) for which the truncation is at the cubic term.

We again follow the approach taken by Moldover and Gallagher [4] in their Appendix B.

As a function of C and r , according to Eq. (34), H when expanded in a power series in r becomes

H(C, r) = S(C, 0) + r

52 H

Mdr_ T=0 dC

dm

5r2 ^7=0

+ 0(r2) (90)

In the limit of pure fluid 2 (C -» 0), from Eqs. (9) and (12)

H(0, r) = 1/2 + ^n K2 = ^^ + tn Kj

RT

(91)

where the superscript a again denotes the value on the saturation curve. The first term

in the brackets of Eq. (90) is difficult to interpret physically, but the partial derivative in

the second term for pure fluid 2, with Eq. (11), may be written

dm

5r2 J C=T=0

dT2

(92)

where the possible T—dependence of K2 has been neglected. In terms of t instead of r

,

to linear order the contribution to S is

dBc(C)

dC

dm

dr^

T=0 = _^dT^C t Tc(0 dC "

(93)

where

C„ = lim T d2/X2

T-»TpR dT2

(94)

40
Moldover and Gallagher [4] estimate, from consideration of consequences of the van der Waals equation and fits to some pure fluid data, that C„ « —12 for pure propane.

In their model for VLE of propane + n—octane they chose C„ = —25, although a reexamination of that mixture, as will be shown in Part II of this series, suggests that

C„ « —15 at the pure propane end.

n In our model, we treat C„ as an adjustable parameter, neglect the first term in

brackets of Eq. (90), and in effect assume that Eq. (93) is applicable for all ( , not just

(^ = or (I" = 1. In fact, it has been found that minus twelve is a typical value for C„ . For mixtures, sufficiently far from azeotropy, it has also been found that the fit can be

improved by allowing a linear dependence of C„ on ( , i.e. C„ -* C (1 4- C„Q where C„

is also an adjustable parameter. This is analogous to the replacement of C„ by

C (1 + C () in Eqs. (75)—(76). In summary, our model for H is

fi(C,t) = Hi (1-20 ^^3:^^c (l + CC)t (95)

Te(C) dC "

^

where C„ and C„ have appeared in several previous papers [6,8,29].

Our guidelines are that C„ may be used for all mixtures, whereas C„ is used only

if Q2ni > 0.17. The only mixture for which we have set C = is helium 3 + helium 4, the

subject of the original Leung—Griffiths work [2]. As with Ci(C), it would be essentially equivalent, and perhaps more to the taste of

some, to replace the second term of Eq. (95) by (t) times a simple polynomial in C with

adjustable coefficients, rather than by tying this term to dTc/d^. However, we have found

the above form to be convenient in that C„ lies in a fairly restricted interval (usually

—15 < C„ < — 6 ) over a range of mixtures for which dTc/d(^ can differ by an order of

magnitude or more.

The modeling of H((,t) rather than H(C,t) is convenient for the calculation of the

P—T—X—p coexistence surface (or one—phase region), but is done at some expense to the

calculation of other thermophysical properties. For example, it would be of interest to

correlate the chemical potentials //i and H2 as functions of P and T . If H(C,t) were

modeled, as well as a;(C,t), then the equations for H and the definition of ( , Eq. (7),

provide two algebraic equations for two "unknowns", fi^ and /Z2 , in terms of "known"

quantities ( and t . These can be solved for the chemical potentials, after which the

Gibbs free energy can be calculated from Eq. (2) and other properties such as entropy from

41
the usual thermodynamic relations; see Eqs. (2.37) — (2.40) of Leung and Griffiths [2]. We

see that, as with the entropy difference of Eqs. (77) — (78), calculation of the total entropy

requires explicit knowledge of K .

In the present formalism where H(^,t) is modeled instead, the "unknown" /tj and

fi2 are represented by one algebraic equation and one partial differential equation,

Eq. (34), so their calculation would be much more difficult. Also the enthalpy, as calculated from Eqs. (6) and (28), requires {dR/dr)^ by itself, as well as in combination

with {dR/dQ^ according to Eq. (34). Consequently, in future correlations of the enthalpy

it probably will be necessary, after correlating the P—T—x—p surface and obtaining a model

H((^,t), to solve Eq. (34) as a partial differential equation, subject to the boundary

condition of Eq. (91), and thereby obtain a self—consistent H((,t). In summary, the

simplicity of calculating the P—T—x—p surface comes at the price of complexity for other

thermodynamic functions.

6. Representation of the Critical Locus

As a final item, we require representations of the mixture critical pressures,

temperatures, and densities as functions of ( or, equivalently, as functions of x . Typically

the experimentalist provides a table of these critical properties for the composition that were measured. We first fit the equations below to the experimentalist's table, but the

coefficients are understood to have a limited adjustability, in the sense that a slightly

different critical locus from that reported by the experimentalist may, within our model,

provide a better fit to the coexistence surface. As functions of x , the representations we

choose are — = i-::^ + —^

+ x(1-x)[Ti + (1-2x)T2

RTc(x) RTci RTc2 L (96)

+ (l-2x)2T3 + (l-2x)3T4]

Pc(x) ^ (l-x)Pci

I xP,2

RTc(x) RTc 1 RTc2

(97)

+ X (1 - x) [Pi + (1 - 2x) P2 + (1 - 2x)2 P3 + (1 - 2x)3 P4]

42
^c(x) = (1 -x) Pel + x/?c2 + X (1 -x) [pi + (1 - 2x) P2 + (1 - 2x)2 P3] (98)

where Pd , T^ and pd are the critical parameters for pure fluid i . Since experimental

sources typically provide dew—bubble curves for five or six mixtures, from the critical point

tables alone the introduction of more than three or four degrees of freedom in

Eqs. (85)—(87) would lead to overfitting. For most mixtures, only the terms through T3

and P3 are actually used.

Rainwater and Moldover [6] have developed an alternative to Eqs. (97)—(98) which

we employ in the present model for nonazeotropic mixtures. They introduced a new

variable x,_ ,

^

^ 1/T,,- 1/T,(x)

1/Tci - 1/Tc2

(99)

which has the property that x = when x = and x = 1 when x = 1, and for the

mixture is only a function of Tc(x). If Tc(x) is monotonic in x , x varies smoothly form

to 1 as X varies over the same interval. In the alternative formulation, the polynomials

in X of Eqs. (96)—(97) are replaced by polynomials in x,^ , i.e. P^^ (1-x^^x^

RT (x)

c ' RT

cl

RT

c2

+ x^(l-x^) P^ + (l-2x^) P2 + (l-2x^)^ P3 + (l-2x^)^ P4 (100)

Pjx) = (1-x^) p^^ + x^ p^2 + ^T ^^~^T^ ^1 "• (^~2x^) ^2 "^ (1~2^T^ ^3

v2 - (101)

Equations (100)—(101) have a practical advantage for fitting over Eqs. (97)—(98), in

that the former determine the critical line in pressure versus temperature or temperature

versus density respectively. Thus adjustments made in Eq. (96), with the parameters of

Eqs. (100)—(101) held constant, will not affect the shape of the critical line on a P—T graph

or a T—p graph. This would not be the case if Eqs. (97)—(98) were used instead. However,

for most azeotropic mixtures Tc(x) has a minimum, and since x no longer is singlevalued, a correlation therefore must be made in x .

43
In principle, the fitting of the critical line is a separate problem from use of the

modified Leung—Griffiths or any other thermodynamic model, and the number of

parameters needed for a quality fit depends on the length and curvature of the critical line

for the mixture under consideration. Usually, only three degrees of freedom in Eqs. (96)

and (100) or (97) are needed, but there are counterexamples. For carbon dioxide with

n—butane [29] or a heavier alkane, the P—T critical locus is nearly linear near the alkane

end and curves abruptly near carbon dioxide. We have found that a fourth degree of

freedom in Eqs. (96) and (100) is absolutely essential to reproduce the critical locus and

dew—bubble isotherm in the high temperature range.

The specification of the thermodynamic model is now complete. The coexistence

surface in P—T—p—x space is given by Eq. (60) and Eq. (33), where the explicit

representation of the function Q(C, t), from Eqs. (32), (41), (42), (45), (47) and (63), with

6"^ = 1, a = 0.1 and where dx/d( is given by Eq. (87) on the critical locus, is

Q(^,t) = dx^T,M.A[p^(x)/Te(x)]

dC RTPc(x) dx

+ C3(-t)^-^ + Clt + c'5t^ + C6t^

-1.9C3(C)(-t)°-^ + C4(C)

RTJx)

(102)

dC R dx

+ 2C5(C)t + 3C6(C)t^

For many mixtures, Hj = so dx/d^ = —1.

Excluding the pure fluid critical points, the model can have as many as 29

parameters. However, all but six are fixed by the pure—fluid coexistence properties and the

critical line. The first set of parameters, which characterizes the pure fluid vapor pressure

and temperature—density coexistence curves, is Cj^ i', i = 1, 2 and j = 1, ... 6 of Eqs. (37)

and (39) (although frequently Ce^^' is not used). The second set of parameters, which

characterizes the critical line, is T. and P. or P. , i = 1, 2, 3, 4 and p. or p. ,i = 1,

2, 3of Eqs. (96)-(101). Finally, the parameters C„ and C„ ofEq. (76), C. ofEq. (80),

H. ofEq. (88) and C„ and C„ ofEq. (95) are properties of the entire coexistence surface

1 a L of the mixture.

44
Our model has been and continues to be under development. As 0:210 increases,

more parameters will be necessary, and we will develop guidelines as to what parameters

are necessary and permissible for a given value of a2m • A previous edition of the

model [6,7], with only C , C„ and C , was found to be satisfactory for the P—T—

x

surface if a2m < 0.25 and for coexisting densities if a2m < 0.17. The present edition extends

the range of P—T—x correlation to (at least) a2m = 0.3 and the range of density correlation

to Q!2m = 0.25. With our guidelines, we believe the model can fit quality VLE data to

within their precision while not overfitting, and can be used as a data evaluation technique

to identify erroneous data.

While a systematic study has yet to be undertaken, it appears that 0:201 has an

upper bound of about 0.7 to 0.8 for Type I binary mixtures. If the pure fluids are too

dissimilar the critical locus "breaks" and becomes discontinuous. For example, methane

(Tc = 190.55 K) forms a continuous critical locus with 2,2 dimethylbutane, the hexane

isomer with the lowest critical temperature (Tc = 488.7 K), but a discontinuous critical

locus with the four other hexane isomers [44]. We are working towards the addition of new

features onto the model to extend its range, and the present sub—project will be considered

complete when the model incorporates sufficient features to yield an accurate P—T—x—

p

coexistence surface up to 02ni ~ 0.8.

7. Summary

This report has provided a detailed and self—contained presentation of a thermodynamic model to describe VLE of Class I binary mixtures over an extended critical

region. The model is based on that of Leung and Griffiths [2] as modified by Moldover and

Gallagher [3,4], but with considerable subsequent development. Insights are presented

based on our experience and hindsight from successfully correlating 35 binary mixtures to

date. Conjectures of possible ways to improve and to extend the range of the model are included.

We have attempted here to examine carefully the logical structure of the model.

The formalism is in part rigorous thermodynamics, in part incorporation of a simplified

modern understanding of critical phenomena, and in part empiricism, although motivated

as far as possible by heuristic principles such as corresponding states as well as rigorous

boundary conditions.

45
The new results and insights of this report, not included in previous publications,

are as follows:

A. The "constant" K (or Kj or K2) in the definition of ( , Eq. (7), is generalized to a temperature—dependent function. In particular, it has been shown that the formal

expressions for the P—T—x—p surface are left unchanged.

B. With a generalized K , new terms proportional to dK/dT are derived for the energy

density u , Eq. (28), and consequently the (measurable) coexisting enthalpy.

C. It is demonstrated that the temperature dependence of K includes and extends the

degrees of freedom available from the choice of zeroes of chemical potential. Also, it

is explicitly shown that chemical potential shifts of /xio and //20 lead to a shift in

energy density of pi //jo + P2 A*20 > a-s required.

D. From Eq. (36), it is demonstrated that for monotonic Tc(x) a K(T) exists such that

Xc(0 = 1 — C) or within limits Xc equals any other function such that Xc = 1 for

^ = and vice versa, exactly.

E. A relationship is derived, Eq. (71), between Q2 , the reduced ratio of composition

and density amplitudes, and the slope of isopleths on a T—p diagram at critical.

F. An empirical principle is affirmed that, for 0210 = max |q!2| sufficiently small, the

model with Xc = 1 — C on critical coincides with the model with loci of constant (

linearly interpolated between the pure vapor pressure curves, according to

Eqs. (62)-(63).

G. A new parameter C^ is introduced in Eq. (76). It was briefly mentioned (as were a andHJinRef. 29.

a. 1

H. It is shown that calculation of Onuki's [33] ai , the reduced ratio of entropy and

density amplitudes, requires explicit knowledge of K , unavailable from usual VLE

correlations with the model.

I. A new parameter C,, that adjusts the slope of the rectilinear diameter, is introduced

in Eq. (80).

46
J. It is proven that, in the leading-order dilute limit, a2 is independent of dxc/d(,

and depends only on quantities which are experimentally measurable, the slopes of

the solvent vapor pressure curves and the critical locus.

K. With motivation from the previous result, a new parameter H is introduced in

Eq. (95).

Furthermore, the modified Leung—Griffiths model is compared and contrasted with

conventional phase equilibrium calculation techniques in App. A and with the "original"

Leung—Griffiths model [2,28,29] in App. C. Nevertheless, the primary justification for this

work is its success in correlating VLE for a wide variety of binary mixtures, as is demonstrated in Part II of this series.

8. Acknowledgments

The author thanks M.R. Moldover for his valuable advice and guidance throughout

this project and thanks D.G. Friend, J.J. Lynch, T.S. Storvick and L.J. Van Poolen for

careful readings of the manuscript. Although the opinions expressed in this report,

particularly Apps. A and C, are the author's own, and he does not mean to imply that the

following individuals necessarily agree with them, he thanks R.F. Chang, J.F. Ely, J.R.

Fox, A.J. Kidnay, J.M.H. Levelt Sengers, G. Morrison, E.D. Sloan and J. A. Zollweg for

illuminating discussions. This research has been supported in part by the U.S. Department

of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences.

47
9. References

[1] J.S. Rowlinson and F.L. Swinton, Liquids and Liquid Mixtures (Butterworth,

London, 1982). Chapter 6 is an excellent introduction to vapor—liquid equilibria.

[2] S.S. Leung and R.B. Griffiths, Phys. Rev. A 8, 2670 (1973).

[3] M.R. Moldover and J.S. Gallagher, In Phase Eouilibria and Fluid Properties in the

Chemical Industry . ACS Symposium Series No. 60, S.I. Sandler and T.J. Storvick,

Eds., American Chemical Society, Washington, 1977, p. 498.

[4] M.R. Moldover and J.S. GaUagher, A.I.ChE. J. 24, 267 (1978).

[5] H.P. Clegg and J.S. Rowlinson, Trans. Far. Soc. 51, 1333 (1955).

[6] J.C. Rainwater and M.R. Moldover, in Chemical Engineering at Supercritical Fluid

Conditions . M.E. Paulaitis et al., Eds., Ann Arbor Science (1983). pp. 199.

[7] J.C. Rainwater and F.R. Williamson, Int. J. Thermophys. 7, 65 (1986).

[8] J.C. Rainwater and R.T. Jacobsen, Cryogenics, 28, 22 (1988).

[9] P.H. Van Konynenberg and R.L. Scott, Phil. Trans. Roy. Soc. London 298, 495

(1980).

[10] In Ref. 9, a distinction was made between the "class" of a mixture, i.e. whether its critical line is continuous (Class I) or discontinuous (Class II), and the six "types"

of overall phase behavior. There is some confusion in that certain sources, e.g. Ref.

11, use "class" in the sense that Ref. 9 uses "type". We find the distinction between

type and class useful and will retain the original terminology in the present work.

[11] W.B. Streett, in Chemical Engineering at Supercritical Fluid Conditions M.E.

Paulaitis et al., Eds., Ann Arbor Science (1983), p. 3.

[12] R.B. Griffiths and J.C. Wheeler, Phys. Rev. A 2, 1047 (1970).

[13] See S.M. Khodeeva, L.A. Subbotina and I.V. Gubochinka, Russ. J. Phys. Chem. 51,

266 (1977) and references therein. These papers cite direct fugacity mixtures along

a small initial interval of the critical line for seven dilute mixtures. They find the

fugacity varies as xTM with m < 1, while our model, and, as they acknowledge, a decorated lattice gas model due to Wheeler (Ref. 14) has m = 1. They admit that a true asymptotic limit may not have been reached. In any case, the measurements

are far too limited to serve as a basis for a complete VLE surface correlation.

[14] J.C. Wheeler, Ber. Bunsen. Phys. Chem. 76, 308 (1972).

[15] L.J. Van Poolen and J.C. Rainwater, Int. J. Therm. 8, 695 (1987).

[16] Thermophvsical Properties of Working Fluids for Binarv Geothermal Cycles . D.E.

Diller, J.S. Gallagher, B. Kamgar—Parsi, G. Morrison, J.C. Rainwater, J.M.H.

Levelt Sengers, J.V. Sengers, L.J. Van Poolen and M. Waxman, Nat. Bur. Stand.

(U.S.) Interagency Report NBSIR 85-3124 (May 1985).

48
[17] See, for example, F. Mohling, Statistical Mechanics: Methods and Applications

(Wiley, New York, 1982) p. 309-15.

[18] T.J. Edwards, Specific—Heat Measurements Near the Critical Point of Carbon

Dioxide, Ph.D. Thesis, Univ. of Western Australia, 1984.

[19] S.C. Greer, Phys. Rev. A 14, 1770 (1976); D.T. Jacobs, D.J. Anthony, R.C. Mockler

and W.J. O'Sullivan, Chem. Phys. 20, 219 (1977).

[20] B. Widom, J. Chem. Phys. 43, 3898 (1965).

[21] Ref. 1, p. 20.

[22] See, for example, R.D. Goodwin, Adv. Cryo. Eng. 23, 611 (1978).

[23] F.W. Balfour, J.V. Sengers, M.R. Moldover and J.M.H. Levelt Sengers, Proc. 7th

Symp. Thermophysical Prop. (A. Cezairliyan, Ed.) 1977 p. 86.

[24] F.J. Wegner, Phys. Rev. B5, 4529 (1972); F.J. Wegner, Phys. Rev. B6, 1891 (1972).

[25] P. Schofield, Phys. Rev. Lett. 22, 606 (1969).

[26] J.M.H. Levelt Sengers, W.L. Greer, and J.V. Sengers, J. Phys. Chem. Reference

Data 5, 1 (1976).

[27] G. D'Arrigo, L. Mistura and P. Tartaglia, Phys. Rev. A12, 2587 (1975).

[28] R.F. Chang and T. Doiron, Int. J. Thermophys. 4, 337 (1983).

[29] M.R. Moldover and J.C. Rainwater, J. Chem. Phys. 88, 7772 (1988).

[30] J.C. Wheeler, J. Chem. Phys. 61, 4474 (1974); Phys. Rev. A 12, 267 (1975).

[31] H.G. Donnelly and D.L. Katz, Ind. Eng. Chem. 46, 511 (1954).

[32] T.A. Al-Sahhaf, E.D. Sloan and A.J. Kidnay, A.LCh.E. J. 30, 867 (1984).

[33] A. Onuki, J. Low Temp. Phys. 61, 101 (1985).

[34] J.C. Wheeler, G. Morrison and R.F. Chang, J. Chem. Phys. 83, 5837 (1985).

[35] J.M.H. Levelt Sengers, Fluid Phase Equil. 30, 31 (1986).

[36] J.C. Rainwater, Nat. Bur. Stand. (U.S.) Technical Note 1061, p. 83 (1983).

[37] J. A. Bierlein and W.B. Kay, Ind. Eng. Chem. 45, 618 (1953).

[38] I.R. Krichevskii, Russ. J. Phys. Chem. 41, 1332 (1967).

[39] A.M. Rozen, Russ. J. Phys. Chem. 50, 837 (1976).

[40] R.F. Chang, G. Morrison and J.M.H. Levelt Sengers, J. Phys. Chem. 88, 3389

(1984).

49
[41] J.M.H. Levelt Sengers, R.F. Chang and G. Morrison, ACS Symposium Series 300 . 110 (1986).

[42] R.F. Chang and J.M.H. Levelt Sengers, J. Phys. Chem. 90, 5921 (1986).

[43] S.S.H. Rizvi, A.L. Benado, J. A. Zollweg and J. A. Daniels, Food Technology 40 (6),

55 (1986).

[44] A.J. Davenport and J.S. Rowlinson, Trans. Far. Soc. 59, 78 (1963).

[45] See, for example, S.M. Walas, Phase Equilibria In Chemical Engineering

(Butterworth, London, 1985).

[46] 0. Redlich and J.N.S. Kwong, Chem. Rev. 44, 233 (1949).

[47] D.Y. Peng and D.B. Robinson, Ind. Eng. Chem. Fund. 15, 59 (1976).

[48] M. Benedict, G.B. Webb and L.C. Rubin, J. Chem. Phys. 8, 334 (1940).

[49] J.F. Ely, Nat. Bur. Stand. (U.S.) Technical Note 1061, p. 1 (1983).

[50] H. Knapp, R. Doring, L. Oellrich, U. Plocker and J.M. Prausnitz, Vapor—Liquid

Equilibria for Mixtures of Low Boiling Substances . DECHEMA Chemistry Data

Series Vol. VI (DECHEMA, Frankfurt, 1982).

[51] C.Y. Tsang and W.B. Streett, J. Chem. Eng. Data 26, 155 (1981).

[52] E. Chang, J.C.G. Calado and W.B. Streett, J. Chem. Eng. Data 27, 293 (1982).

[53] J.C.G. Calado, E. Chang and W.B. Streett, Physica 117A, 127 (1983).

[54] R. Schmidt and W. Wagner, Fluid Phase Equil. 19, 175 (1985).

[55] D.G. Friend, J.F. Ely and H. Ingham, Nat. Inst. Stand. Tech. TN 1325 (1989).

[56] J.J.—C. Hsu, N. Nagarajan and R.L. Robinson, Jr., J. Chem. Eng. Data 30, 485

(1985).

[57] B.E. Eaton, J. Stecki, P. Wielopolski and H.J.M. Hanley, J. Res. NBS 86, 419

(1981).

[58] J.R. Hastings, J.M.H. Levelt Sengers and F.W. Balfour, J. Chem. Therm. 12, 1009

(1980).

[59] J.M.H. Levelt Sengers, G. Morrison and R.F. Chang, Fluid Phase Equil. 14, 19

(1983).

[60] J.V. Sengers and J.M.H. Levelt Sengers, Ann. Rev. Phys. Chem. 37, 189 (1986).

[61] Ref. 2, p. 311-12.

[62] J.R. Fox, Fluid Phase Equil. 37, 123 (1987).

[63] J.R. Fox, and T.S. Storvick, Int. J. Thermophys., in press.

50
[64] J.R. Fox, Fluid Phase Equil. 14, 45 (1983).

[65] D. Stauffer, M. Ferer and M. Wortis, Phys. Rev. Lett. 29, 345 (1972).

[66] M.R. Moldover, Phys. Rev. A 31, 1022 (1985); H. Chaar, M.R. Moldover and J.W.

Schmidt, J. Chem. Phys. 85, 418 (1986).

[67] B. Wallace, Jr. and H. Meyer, Phys. Rev. A 5, 953 (1972).

[68] B. Wallace, Jr. and H. Meyer, Tabulation of the Original PressureVolume—Temperature Data for He^—He'^ Mixtures and for He'^ . Technical Report,

Department of Physics, Duke University, May 1971, 93pp.

[69] G.G. Haselden, D.M. Newitt and S.M. Shah, Proc. Roy. Soc. A 209, 1 (1951).

[70] G.G. Haselden, F.A. Holland, M.B. King and R.F. Strickland - Constable, Proc.

Roy. Soc. A 240, 1 (1957).

[71] N.E. Khazanova, L.S. Lesnevskaya and A.V. Zakharova, Khim. Prom. 44, 364

(1966).

[72] N.E. Khazanova, E.E. Sominskaya, A.V. Zakhorova, M.B. Rozovskii and N.L.

Nechitailo, Russ. J. Phys. Chem. 53, 902 (1979).

[73] R.F. Chang, Nat. Inst. Stand. Tech., Gaithersburg, private communication.

[74] M.R. Moldover, Nat. Inst. Stand. Tech., Gaithersburg, private communication.

51
Appendix A

Comparison with Conventional Equations of State

In this appendix we give a brief overview of conventional VLE calculation

techniques and their difficulties in the critical region, and contrast them with the present

model. Phase equilibrium is of course extremely important to the chemical and petroleum

industries, and many different equations of state and computer packages have been

developed [45]. For low pressures, in particular atmospheric pressure, these equations and

packages largely fulfill industrial needs.

However, such conventional methods in principle have deficiencies at critical

pressures. Our model, being essentially an expansion about critical conditions with correct

critical exponents, usually yields accurate VLE calculations from the critical pressure down

to about half that pressure, below which conventional methods are generally reliable. Thus

our model is intended to supplement, not to supersede, classical methods.

By conventional definition, an equation of state for a binary mixture takes the form

P = P{T,p,x) (Al)

and is therefore a mixed representation of a field variable (pressure) as a function of one

field variable (temperature) and two density variables (density and composition). The

analogous expression in our model is a; ((, r, h), Eqs. (41), (42) and (45)—(47), which

involves only field variables.

While modern liquid—state theories have made progress in predicting an equation of

state microscopically [1], in practice empirical forms are usually employed. Of course, it is considerably easier to correlate the VLE surface alone than to correlate the entire

thermodynamic volume (liquid and gas), so simple cubic equations or variants of the van

der Waals equation, e.g. Redlich—Kwong [46], Peng—Robinson[47], etc., are frequently

utilized for VLE correlations. For a more global thermodynamic description, manyparameter equations of state such as the Benedict—Webb—Rubin (BWR) equation [48] are

typically used.

Equations of state are usually developed first for pure fluids. To generalize them to

mixtures, the parameters of the equation are made composition—dependent by means of

various mixing and combining rules. Our expressions for Ci(0 in Sec. 4 can be viewed as the analog of mixing rules. An important difference is that in conventional methods a

liquid state of composition x« is matched with a coexisting vapor state of a different

52
composition Xy , and thus the equation—of—state parameters across the phase boundary are

different for liquid and vapor. However, our "mixing rules" use ( rather than x , and (

is the same for liquid and vapor. The latter picture may be the more natural.

The specific equations of state cited above have the property that the chemical

potentials fii and fi2 iiiay be calculated by analytical integration. The VLE surface is mapped out by various algorithms, e.g. "flash calculations," which search numerically for

liquid and vapor state points at the same pressure and temperature such that chemical

potentials, or equivalently fugacities, are equal. The most commonly used algorithm is described in detail by Ely [49].

There are two problems associated with conventional VLE calculations in the

extended critical region, in our formalism —0.1 < t < and particularly for very small |t|

.

The first problem is that many algorithms converge slowly or not at all near the critical

locus. Knapp et al. [50] present a large number of P—x diagrams with isothermal VLE data

and calculated dew—bubble curves similar to Fig. 3. The calculated curves, in most cases based on the Peng—Robinson equation [47], frequently end at pressures below critical and

in some cases are not presented at all for the highest isotherms.

However, any reasonably well—behaved equation of state does contain within it a

prediction of the critical line, which is the locus of points satisfying the condition

d^Gr

6bc2

P,T

d'G^

5x3 = . (A2)

P,T

where G^ is the Gibbs free energy per mole. Furthermore, the equation of state also

contains a coexistence surface that extends up to the critical line. Thus, the failure of a

particular algorithm to converge near the critical line indicates a deficiency in the

calculational algorithm, not in the equation of state. A second and fundamentally distinct problem is that conventional equations of state

tend to predict an incorrect critical locus and coexistence surface near critical. The

compilation of Knapp et al. [50] shows several dew—bubble curves for which the

calculations begin to diverge significantly from experiment at high pressure before the limit

of convergence is reached. This problem is generally more severe for wide dew—bubble

curves or, in our terminology, large |q;2|- Streett and co-workers [51—53] have

demonstrated for mixtures with relatively small | a2 \ that the data and predictions of cubic

equations diverge near critical.

53
In our view, this second problem is a manifestation of a fundamental shortcoming of

classical equations of state. If, in Eq. (Al), P is an analytic function oiT, p, and x , the

critical exponents miLst in the asymptotic limit reach certain classical (and incorrect)

values; a , of Eq. (38), classically is zero compared to the established "nonclassical" value

of 0.110; /? of Eq. (39), classically is one—half rather than 0.325; 7 , of Eq. (49) classically

is unity rather than 1.24. The "effective" values a = 0.1 and = 0.355 used in our model

do not coincide exactly with the true asymptotic values, but are much closer to them than

the classical critical exponents.

For practical purposes, the classical value 0= 72 in out view is a much more

important cause of the failure of equations of state in the critical region, at least for VLE,

than the discrepancies in other exponents. This problem is, of course, present for pure

fluids as well. A classical equation of state cannot truly mathematically represent the

temperature—density coexistence curve of a pure fluid, but can mimic it to any desired

accuracy given enough terms. Schmidt and Wagner [54] have introduced a manyparameter equation of state that describes the near—critical thermodynamics of pure fluids,

e.g. methane [55], quite accurately. Its generalization to mixtures requires a choice of

mixing rules, etc., and is not straightforward. To our knowledge, no classical equation has

yielded critical region mixture VLE correlations of the accuracy shown in Part II of this

series and our previously published correlations [6—8,29].

It has been suggested that our fit quality is largely due to use of the experimental

critical locus, not to the value of P that we use, and that if the parameters of a classical

equation were varied to obtain the best fit to the critical locus, the VLE predictions would

be comparable. We do not agree, but a resolution of this question must await an algorithm

for classical equations that converges very near critical. The fit of Moldover and

Rainwater [29] to the carbon dioxide + n—butane dew—bubble isotherms of Hsu, Nagarajan

and Robinson [56] shows an agreement of shape with their data near critical which, as the

experimentalists themselves point out, is characterized by ^a YsIn fact, the insights of the present work may be useful in developing an algorithm

for VLE from classical equations that converges near critical. First, the critical line could

be determined from Eq. (A2) from techniques such as those of Eaton, et al. [57] Then, our

results such as Eq. (72) could be used to construct approximate asymptotic forms for

dew—bubble curves near critical as functions of derivatives of the critical locus. These

forms would not be exact, but as with any numerical root—finding algorithm, a good first

guess should lead to rapid convergence. Here we would not be injecting the Leung—Griffths

approach into classical equations, but rather would be more efficiently calculating the

54
critical locus and near—critical VLE surface contained within the classical equation, using

insights from Leung—Griffiths theory. With such an algorithm, a true and fair quantitative

test of the relative merits of the two approaches could be conducted.

Various hybrid techniques, in part incorporating conventional approaches and in

part incorporating more modern concepts of field variables and nonclassical critical

exponents, have been proposed. One idea is to start with a scaling—law equation of state

for a reference pure fluid, and then to construct a mixture equation of state using classical

corresponding states [58]. Unfortunately, this approach does not work [59]. Contrary to

experiment, the P—T critical locus in the dilute limits is co—linear with the pure vapor

pressure curves [39,60]. Furthermore, the correct critical thermodynamic behavior of the

mixtures is not predicted [59,61].

Fox [62] has proposed a method in which an equation of state is rewritten in terms

of field variables, and a "reference" mixture is formed from two identical fluids, thus being

in reality a pure fluid with arbitrarily different labels "one" and "two" for the molecules.

A transformation is then made to a "target" mixture by changing a background function,

or analytic part of a thermodynamic potential, of one label only, thus yielding a critical

locus and coexistence surface. In Fox's initial paper [62], the resulting critical line had

Pc(x) variable but Tc(x) a constant, though in subsequent work [63] both Pg and Tc can

be varied with a reasonable amount of freedom. Whether this approach will lead to

accurate P—T—x—p correlations of VLE for a wide variety of mixtures remains to be seen. Admittedly, classical equations of state still do retain certain advantages over the

modified Leung—Griffiths model in its present stage of development. The difficulties

described at the end of Sec. 5 for calculating quantities other than P, T, p and x, such as enthalpy or entropy, are not present to the same degree within classical approaches. Also,

at least qualitatively as van Konynenberg and Scott [9] have shown, even the van der

Waals equation can predict five of the six known types of binary mixture phase equilibria.

The present modified Leung—Griffiths model is restricted to Type 1 mixtures, although a generalization to Type 2 is in progress.

Additionally, of course, conventional equations of state yield superior results at temperatures and pressures well below critical, in our formalism t < —0.1. We believe the

ultimate solution to an accurate global description of VLE will be to merge the two

approaches. Hopefully, for most simple mixtures there will be an interval in t, say

—0.1 < t < —0.07, where both approaches work. In this case, a global model could transfer

55
from one to the other with a switching function, or an alternative method along the lines

proposed by Fox [64]. Such a merging of the modified Leung—Griffiths model with classical

equations of state is planned, but is deferred until a comprehensive study of binary mixture

VLE near critical has been completed.

56
Appendix B

Critical Divergence of the Specific Heat at Constant Volume

The objective of this appendix is to derive the relationship between the term

proportional to (—t)^'*^ in the pure fluid vapor pressure curve, Eq. (37) and the critical

divergence of the specific heat according to Eq. (38). This is basically a small facet of the

thermodynamics as predicted by scaling— law equations, which is presented in much more

detail elsewhere, e.g. by Levelt Sengers et al. [26] Here we extract only the pieces from

that lengthy formalism that are needed for assigning a value to C3

.

The internal energy per unit volume is given by Eq. (28). Since H is analytic in

the present formalism and there is certainly no need to use nonanalytic forms for K^ or K2 , the singular part of Eq. (28) is

using

dujsing

dT

C,h

dujsong

di

i+V

Ch

(Bl)

where Wging is given by Eq. (47) parametrically in terms of r and 9

The specific heat per unit volume Cy for pure fluid 2 is

Cv =

5u

^T

/j,C=0

1 TcdM

p,(=0

(B2)

and similarly for pure fluid 1 where ( = I. (We shall confine our discussion to pure fluid 2,

and henceforth in this appendix it is understood that ( = 0.) From Eqs. (45)—(47), (Bl)

and the relation

dusing

di ^ ^(^sing,h)/a(r, 6)

^ d{t, h)/d (r, 9)

(B3)

we find that

^smg = RTc(l + t)2

P C.

_E ^

fl-a n =E P,n ^2n

In RT a^

m =E Po. ^2m

(B4)

57
where Pm and P2m are constant coefficients. (The only coefficients that survive in the

final result are Pio = (2-a) ao and P20 = (b^ — 1)"^).

To obtain the leading—order singularity in the specific heat we replace the factor

(1 + t)2 by unity in Eq. (B4). We thus define

^sing_ 1 L-v —

d uS^ng

di

(B5)

where Using = Using (1 + t^)'^ Once again the partial derivative must be evaluated

parametrically, i.e.

sing

I ^(Uslng, P)/d{T,9)

Cv = —

Te d{t , p)ld{l,e)

(B6)

According to Eqs. (14), (41), (42), (45)-(47) and (59),

p{i,e) = pc l + rC2

1 _ h2 ^2 ^„P3n 0"^^ ± ? ? OrT n = _^r n^iL

b2 - 1 E Pon. ^2m

m =

(B7)

where Psn are additional constant coefficients.

s in£^ Evaluation of Eq. (B6) yields Cy as a function of r and 6 . However, we are

s in^

specifically interested in the behavior of Cy along the critical isochore in the one—phase

region as a function of t , since experiments along the critical isochore produce a divergence in the specific heat according to Eq. (38) and a measure of the amplitude of

divergence. From Eq. (B7), the critical isochore in (r,^) space according to our model is given by

E Psn ^2n

9 E^^ = C2 r 5—^

E Po. ^2m b2 - 1 (B8)

m = 2m

58
For the special case C2 = 0, i.e. an infinite slope for the rectilinear diameter, the

critical isochore is simply the locus ^ = (for ( = 0). With a nonzero C2 , the critical

isochore is a more complicated function of r and 9 . sinfiT However, our objective is to determine the leading singularity in Cy as a function

of t as the critical point is approached, i.e. as r -» 0, along the critical isochore. From

Eqs. (45) and (B8), asymptotically

b2 - 1 1- C2 b^(b2- 1)2

r^% (B9)

and, since /? < 1, the leading—order singularity along the critical isochore is equivalent to

that along the locus ^ = 0. Upon evaluating the Jacobians of Eq. (B6), setting ^ = 0, and identifying the

leading singularity, we find that

p r

,sing _ ^c 3

'V — r-c'(l-a)(b2-l) 10

20

^c^3 ^0 .^,.2

(BIO)

2-a

Tc S— t (b2-l) (2-a)(l-Q)

which establishes the result that Cy diverges along the critical isochore according to

Eq. (38) and provides the constant of proportionality between C3 and the amplitude of

the divergence.

The thermodynamic properties of a variety of fluids in the critical region have been

correlated by Levelt Sengers, et al. [26] From their Eqs. (1.2), (1.3) and (2.4a) we may

write

sing

f-£ f^— t Tc a

(Bll)

59
where A is their reduced amplitude. Noting from their Eq. (3.14a) that ao = — fo/ak and

using their Eqs. (3.8) and (3.13), we find that

sing p -a 2 0-6

Cy ^ = ilfi t (2-Q)(l-a)(b -1) ao(axo ) (B12)

where a and Xq are critical region parameters tabulated for a variety of fluids in their

Table 34. Comparison of Eqs. (A12) and ( A14) yields

3+a-2

C3 = (b2-l)^ a.

r -P' ^0 = 0.7222 ax.

-p- (B13)

As explained in the following report, C3 must be assigned a value before a fit of the

pure vapor pressure curve is undertaken. For the fluids listed in Ref. 26, Table 34, we

determine C3 from Eq. (B13). Since C3 « 30 for all of these fluids (except the helium

isotopes), when we require a fit to a pure fluid not correlated in Ref. 26 we simply set C3

equal to 30.

VLE correlations are not particularly sensitive to the choice of C3, since, in

Eqs. (37) and (61), the terms in (—t) = (—t)

' and t are highly correlated, and

changes in C3 can be compensated by alterations in C5 . However, C3 is very important

for analysis of interfacial tension (IFT), for which the amplitude varies as C3 ^. Moldover

and Rainwater [29] have analyzed the carbon dioxide 4- n—butane IFT data of Hsu,

Nagarajan and Robinson [56] by means of the present model and concepts from the two—

scale-factor universality theory of Stauffer, et al. [65], as verified by Moldover [66] for IFT

data from pure fluids and binary mixture LLE. Except for one isotherm very near critical,

the Nagarajan—Robinson data are predicted to within ten percent, comparable to their

scatter. One could also invert the procedure and determine C3 from IFT data.

60
Appendix C

Comparison of the Original and "Modified" Leung—Griffiths Models

Since the publication of the original Leung—Griffiths paper in 1973, there have been

two rather distinct approaches to its use for VLE calculations. First, some researchers, in

particular D'Arrigo et al. [27] and Chang and Doiron [28], have adhered very closely to the

original Leung—Griffiths nomenclature and methodology, and have made changes only in

the truncation points of certain polynomial fitting functions. We refer to this line of effort

as the "original" Leung—Griffiths model. Second, Moldover and Gallagher [3,4] introduced

rather substantial changes into the structure and philosophy of the model, including the

introduction of the variable t , the use of corresponding states ideas, the possibility of a

sloping rectilinear diameter, and the hypothesis, given further justification in this work,

that X = 1 — ^ on critical. The present work has built onto this "modified" LeungGriffiths model by adding features as necessary to accommodate mixtures farther from

azeotropy.

The various modifications introduced by Moldover and Gallagher are largely

independent of each other. Therefore, one could construct various hybrid models that

include some, but not all, of their modifications. Nevertheless, it is our view that all of

their modifications help to make the Leung—Griffiths model a more efficient and useful

technique for VLE (and perhaps for one—phase thermodynamic) correlations.

The original Leung—Griffiths formalism follows the development of Sec. 2, except for

the temperature—dependent K , up to Eq. (29). The variable t is not introduced in the

original model; in contrast, r is retained and largely plays the subsequent roles of t . For

example in the construction of a;sing by means of the Schofield linear model, our

Eqs. (45)—(47), the original model uses in place of Eq. (45) the relations

r = HOr (CI)

T = r(l-b2^2)/RTci (C2)

where i {Q is a linear function, but Eqs. (46) and (47) are retained apart from some

notational difference.

For the initial theoretical development, i.e. Eqs. (11)—(28), r is useful in that the

necessary partial derivatives are kept relatively simple. But for purposes of correlation, t is

clearly preferred as it is dimensionless whereas r has dimension. Furthermore, by

61
corresponding states Eqs. (37) and (39) will have nearly the same coefficients for the pure

fluids in terms of t, but not in terms of r , particularly if the critical temperature ratio is

large.

The presence of i {() in Eq. (CI), therefore, may be viewed as a compensation

factor for an artificial distortion of the model by use of r instead of t . Such a distortion

also adversely affects correlation of the regular part of u . We see utterly no advantage in

the use of r in place of t , and strongly advocate that t be employed in future

calculations.

The original formalism then expands Bc(C), Eq. (11), H(^,t), Eq. (12), and

^an (C) ''"> h) Eq. (41), in straightforward polyomial series in the respective independent

field variables. The coefficients of these polynomials constitute the parameters of the

model, and such polynomial fits are, respectively, analogous to our Eq. (96) for Bc(x) and

Eq. (86) for Xc(C), Eq. (95) for H((, t), and Eq. (42) for t^an (C, t, h), with €3(0 given by

Eq. (80) and Ci(0, i > 2, by Eq. (62). The original model does not introduce the functions

Q of Eq. (32) or S of Eq. (34), and models H(C,r) rather than S(C, t). From Eq. (34), a truncation of H linear in t is equivalent to a truncation of H quadratic in r .

Different truncation points of these polynomials are utilized in the fits by Leung and

Griffiths to the helium 3 + helium 4 data of Wallace and Meyer [67,68], by D'Arrigo et al.

to the carbon dioxide + ethylene data of Haselden et al., [69,70] and by Chang and Doiron

to the carbon dioxide + ethane data of Khazanova et al. [71]. No specific guidelines have

ever been given for these truncation points, and it is implied that they must be determined on a purely ad hoc basis for each separate mixture.

Leung and Griffiths [2] recognize the redundancy of the model due to the choice of

K and zeroes of chemical potential, whereas the other groups evidently do not. Thus Leung

and Griffiths specifically restrict Bc(C), and the inverse critical temperature, in their

Eq. (3.6) to be linear in ( . In their Appendix B, they point out that, with the above

degrees of freedom and a constant K , such a linear relation can be satisfied

approximately. The argument is similar to ours in Eqs. (35)—(36) and, in fact, for

monotonic Tc(x) a temperature dependent K may similarly be shown to exist such that

inverse critical temperature is strictly linear in ( . However, the other two cited mixtures do not exhibit a monotonic Tc(x). The

evidence for a minimum in Tc of carbon dioxide + ethylene was inconclusive from the

data of Haselden et al. [69,70], and D'Arrigo et al. [27] fit Bc(C) to a quadratic form so that a minimum occurs at pure ethylene. A subsequent experiment by Khazanova, et al. [72],

clearly showed a minimum in Tc at ten percent carbon dioxide. While it was a significant

accomplishment to show that the Leung—Griffiths model can describe azeotropy,

62
D'Arrigo et al., by removing any restriction associated with the degrees of freedom in K , made the formalism in principle highly redundant. Chang and Doiron went further in

using a cubic form for carbon dioxide + ethane, which has a crescent-shaped P—T critical

locus with a pronounced temperature minimum [71].

These three mixtures all have a relatively small Q2in and thus, according to this

report and the following one, are relatively easy to fit. For helium 3 + helium 4,

a2m = 0.074; for carbon dioxide 4- ethylene, a2m = 0.07; for carbon dioxide + ethane,

Q2m = 0.053. It would be expected that mixtures far from azeotropy, e.g. propane +

n—octane, would require many additional terms in the polynomial series, and in the absence

of guidelines quality fits would be quite difficult to achieve and a clear danger of overfitting

would be present.

For the boundary conditions that yield pure—fluid critical and coexistence

properties, the original model yields a cumbersome set of coupled equations, e.g. Eqs. (29)—(34) of Chang and Doiron [28], in place of our more direct representations of

Eqs. (61)—(63), (76) and (80). While perhaps a minor point, this does further demonstrate

the efficiency of the modified model over the original one. In the discussions of relative merits, key issues have been the slope of the rectilinear

diameter and the potential applicability of the model to the one—phase region. The sloping

diameter is described by the term proportional to C2 in Eq. (39) or (60), and originates

from the term (th) in Eq. (42). The original model could include such a sloping diameter

with a (rh) term in a;an (or Uj^ in the alternate notation). However, neither Leung and

Griffiths (their Eq. (3.2)), nor D'Arrigo et al. (their Eq. (5)), nor Chang and Doiron (their

Eq. (9)), choose to include such a term.

Again, for helium 3 + helium 4 this omission is justified, because C2 for these pure

fluids is exceptionally small [4]. For most ordinary fluids, C2 « —1 and its omission leads to deviations from experiment, as was recognized by Chang and Doiron [28]. From visual

inspection the p—x plot of their Fig. 8 presents a fit clearly inferior to the earlier fit of

Moldover and Gallagher [4] (their Fig. 4) of the same mixture and data.

The three correlations with the original Leung—Griffiths model cited above consider

the supercritical one—phase region as well as VLE. The research to date with the modified

Leung—Griffiths model has not, although this has been largely at our election. As

described in Part II of this series, while many VLE surfaces have been correlated, many

more have not, and problems remain for correlating azeotropic mixtures, normal mixtures

very near or very far from azeotropy, mixtures without coexisting density data, and VLE

in the presence of LLE. Related problems include the correlation of multicomponent

mixture VLE, interfacial tension, and coexisting enthalpy data, the investigation of formal

63
nonlinear least—squares fitting, and the merging of our model near critical with a classical

equation of state away from critical. While correlation of the one—phase region is certainly an important problem, we have seen no cause for assigning it priority over these other

important and interesting problems at present. In any case, it is quite premature to

conclude that the modified Leung—Griffiths theory will not work in the one—phase region.

Throughout this project, it has been recognized that a sloping rectilinear diameter,

or nonzero C2 , in principle leads to problems in the one—phase region. For pure fluids,

the h = locus in the supercritical region represents a symmetry locus, that of inflection

points of isotherms on a P—p diagram, that empirically is close to the critical isochore,

whereas the present model in the supercritical region would incorrectly give a sloping line on such a diagram for that symmetry locus.

The generalization of revised scaling to mixtures, as discussed below Eq. (61), offers

the most promising hope of escaping this dilemma. In unpublished work, Chang [73] has

incorporated revised scaling, as well as a (rh) term, into the original Leung—Griffiths

formalism. For carbon dioxide + ethane, an improved VLE fit is obtained comparable in

quality to that of Moldover and Gallagher, as well as an improved fit of the mixture

specific heat. Also in unpublished work, Moldover [74] has included revised scaling into the

pure—fluid limit of the modified Leung—Griffiths model and has found good agreement with

carbon—dioxide data and correlations in the supercritical region.

We recomm^end that a study of the one—phase region by means of the LeungGriffiths model, if undertaken, be conducted along the lines of the present project. First, an appropriate extended critical region should be defined, e.g. r < 0.1 where r is the

Schofield parameter of Eqs. (45)—(47). Second, an appropriate criterion for thoroughness of

measurement over this region should be estabfished. Third, F—T—p—x data from all sources should be collected, and, fourth, priorities should be assigned to the mixtures according to

the proximity to azeotropy (smallness of a2m) and thoroughness and reliability of the data.

Pressures from programmatic considerations to focus on a particular mixture or narrow set

of mixtures should initially be resisted.

The absence of a sloping rectilinear diameter in the original method may have given

it some initial advantage in the one—phase region. However, within either method, the

ultimate requirements will probably be both revised scaling and a (th) or (rh) term in

Wan • The more important modifications of the model, the use of t instead of r and the

incorporation of corresponding states, may well yield advantages for the one—phase region

as well as for VLE. Just as it is a useful first approximation to assume that a curve of

64
constant ( on the coexistence surface behaves as a pure fluid, it may be a similarly useful

hypothesis that a surface of constant ( behaves as a pure fluid in the near—critical

one—phase region.

None of the statements in this appendix should be construed as criticism of the

research efforts of Leung and Griffiths [2]. Clearly, the success of our project has depended

on their understanding of field and density variables, their clever construction of ( with

its many useful properties, and numerous other relationships and insights. The primary

motivation of the Leung—Griffiths paper was not to develop a general formalism for

correlation of all binary mixtures. Rather, it was to reassert the predictions by Griffiths

and Wheeler [12] of the behavior of specific heats at constant composition in view of some

apparently contradictory data on helium 3 + helium 4 due to Wallace and Meyer [65,66].

It is therefore not surprising, and no reflection on Leung and Griffiths, that subsequent

researchers have established a more efficient revision of the model for non—cryogenic binary

mixtures.

Nevertheless, except perhaps for the modeling of H instead of H (as discussed at the

conclusion of Sec. 5), we see no particular advantages in the original Leung—Griffiths model

over the present version. We strongly recommend that future research in critical

phenomena of binary mixtures be conducted starting from the present, modified formalism.

65

NBS-n4A IREV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NIST TN-1328

2. Performing Organ. Report No. 3. Publication Date

April 1989

4. TITLE AND SUBTITLE

Vapor-Liquid Equilibrium of Binary Mixtures in the Extended Critical Region

I. Thermodynamic Model

5. AUTHOR(S)

Jan;es C. Rainwater

6, PERFORMING ORGANIZATION (If joint or other tlian NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Supported in part by the

U.S. Department of Energy

Office of Basic Energy Sciences

Division of Chemical Sciences, Washington, DC 20585

10. SUPPLEMENTARY NOTES

[^J Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual sumnnary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

The thermodynamic model of Leung and Griffiths for binary mixture vapor—liquid

equilibrium near the critical locus, as modified by Moldover, Rainwater and co—workers, is extended to accommodate fluid pairs of greater dissimilarity. The maximum absolute

value of a dimensionless ratio of composition difference to density difference (0:201) is shown

to be a very useful measure of dissimilarity of components. New parameters are introduced

into the model with guidelines that the number of permissible parameters depends on, and

increases with, a^m The quantity K in the Leung—Griffiths fugacity ratio ( is generalized from a constant to a temperature—dependent function. Among other

consequences, the previously approximate relation that the mole fraction x = l—( on the

critical locus can for most mixtures be made exact. For a2m < 0.25, an empirical principle

is asserted which predicts a relationship between the critical slopes of

constant—composition curves on temperature-density plots or isotherms on

density—composition plots and derivatives of critical locus parameters. The coexistence

surface is shown to be insensitive to dx/dC on the critical locus at the dilute limits. The

range of the model is extended to Q2m ~ 0-3, e.g. propane + n—octane or carbon dioxide +

n—butane. The modified model is contrasted with conventional equation—of—state methods

and with the original Leung—Griffiths formalism.

12. KEY WORDS (IS/x to twelve en-tries; alohabe.tjcal order: capitalize only, proper namfis: and separate,key wprd-s^v semicolons)

binary mixtures; coexisting densities; corresponding states; critical region; dew—DubDle

curves; field variables; Leung—Griffiths model; scaling—law exponents; vapor—liquid

equilibrium

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