Nonclassical Description of (Dilute) Near ... - American Chemical Society

5 Nonclassical Description of (Dilute) Near-Critical Mixtures J. M. H. Levelt Sengers, R. F. Chang, and G. Morrison

Downloaded by UNIV OF ROCHESTER on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

Thermophysics Division, Center for Chemical Engineering, National Bureau of Standards, Gaithersburg, MD 20899

The organizers of this symposium have asked us to review the nonclassical equations of state and their applications in near-critical fluids and fluid mixtures. In this contribution, the emphasis will be on fluid mixtures. The behavior of fluid mixtures near the gas-liquid critical line has undergone a revival of interest for a variety of reasons. One is the appearance of a number of reports of large anomalies in properties such as apparent heats of dilution (1), apparent molar volumes (2) and apparent molar specific heats (3) of dilute salt solutions in near-critical steam, and of extraordinarily large enthalpies of mixing at supercritical pressures in mixtures with components of very different critical temperature (4). Another reason is the strong push for exploration of the supercritical regime in separation processes, a promising alternative to liquid extraction (5). Supercritical solubility is governed, in part, by the partial molar properties of the solute, which have been reported to behave anomalously. It seems therefore useful at this point to take stock and review: (1) what are the differences in prediction between classical and nonclassical equations for fluids and fluid mixtures; (2) how the nonclassical behavior is going to be meshed with classical behavior further away and (3) what types of nonclassical equations are available for fluid mixtures and what is their range of validity. The anomalous properties of dilute near-critical mixtures, which are of interest in custody transfer and in supercritical solubility, are discussed in the last part of this contribution. C r i t i c a l Behavior of Pure Fluids This section defines the terminology and summarizes the differences between c l a s s i c a l and nonclassical behavior of pure f l u i d s i n the This chapter not subject to U.S. copyright. Published 1986, American Chemical Society

In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5.

LEVELT SENGERS ET A L .

Near-Critical Mixtures

111

b r i e f e s t of terms. The material presented here i s in standard notation and has been reviewed elsewhere ( 6 ^ 7j_ 8 ) . C r i t i c a l behavior i s called c l a s s i c a l (mean-field, or van der Waals-like) i f the Helmholtz free energy a ( V , T) can be expanded at the c r i t i c a l point i n terms of 6V = V - V and δΤ = Τ - T V being q

c>


the volume, Τ the temperature and the subscript c denoting the c r i t i c a l value. If such an expansion exists, thermodynamic properties vary i n accordance with simple power laws along given paths asymptotically near the c r i t i c a l point. The power laws and exponent values are summarized in Table 1 and the paths are indicated in Figure 1. Real f l u i d s do not behave c l a s s i c a l l y . Their coexistence curves and c r i t i c a l P - V isotherms are f l a t t e r than predicted by c l a s s i c a l equations, and their s p e c i f i c heat Cy shows a weak divergence (Table 1 ) . Renormalization-group theory explains how nonclassical exponent values result from the correlation of the c r i t i c a l fluctuations, and the theoretical exponent values for the universality class of 3D Ising-like systems have been confirmed in a number of delicate experiments, see, for instance ( 9 ) , ( 1_0). The scaling laws (6 - 8 , U_) that describe the thermodynamics of n e a r - c r i t i c a l f l u i d s can be viewed as a compact "packaging" of the power laws of Table 1. They imply the exponent equalities 2 - α - 0(6

+ 1)

; Ύ - β(δ - 1)

(1)

so that only two exponents are independent; these are, however, the same for a l l systems in the 3D Ising-like universality class. Likewise, only two amplitudes in the power-law expressions can be chosen independently; these are not universal. We w i l l c a l l the c r i t i c a l behavior shown by real f l u i d s and predicted by renormalization-group theory nonclassical. It i s important to note the difference between strongly (Ύ-like) and weakly (α-like) diverging properties i n the nonclassical case (j_2). Properties which are the same i n coexisting phases are called " f i e l d s " . Examples are pressure P, chemical potential μ and temperature T. Thus Ρ (μ, Τ ) , with Ρ = Ρ/Τ, μ = μ/Τ and Τ = 1/Τ i s an example of a thermodynamic potential i n which a l l variables are f i e l d s . First derivatives, the density ρ = (3Ρ/9μ) (3P/9T)

τ

and the energy density Up =

, are generally different in coexisting phases.

Such f i r s t

derivatives are called "densities". Other examples of densities include volume V , enthalpy H, entropy S, and composition x. The strong divergences are obtained ( c l a s s i c a l l y and nonclassically) i f a derivative i s taken of a density with respect to a f i e l d while another f i e l d i s kept constant, that i s , i n a d i r e c t i o n intersecting the coexistence curve in Figure 1 b . I f , however, a density i s kept constant at d i f f e r e n t i a t i o n , one obtains the nonclassical weak divergences; a constant - density direction i s p a r a l l e l to the coexistence curve in Figure 1 b . (J_2). Examples of strong and weak divergences in one-component f l u i d s are l i s t e d in Table I I .



112

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I C r i t i c a l e x p o n e n t d e f i n i t i o n s and v a l u e s The p a t h s a r e i n d i c a t e d i n F i g u r e 1 Property

Specific

D e f i n i t i o n of c r i t i c a l exponent

heat

Coexistence

Curve

c

v

- |T-T f

a

c

|p-p l " I T - T ι 0

Compressibility

κ

Critical

|P-P I

isotherms

τ

β

Value for real fluids

α =0

α =0 1

3 - 0.5

3 = 0 33

Ύ = 1

Ύ = 1

g

- |T-T f

Y

c

C

Classical Value

"

|P-P I

6

δ

=

3

C


δ

21

- 4 .8

5.

LEVELT SENGERS ET AL.


113

For d e t a i l s of the scaled form of the potential Ρ (μ, Τ) we refer to the l i t e r a t u r e (6 - 8). Conceptually, t h i s form i s construed as follows. The potential i s s p l i t into a part that i s analytic i n μ and Τ and a part that i s scaled.

The scaling

variables are chosen as a reduced temperature difference τ = Δ Τ i n the weak direction along the coexistence curve and a


reduced chemical potential difference h = Δμ i n the strong d i r e c t i o n intersecting this curve (Figure 2). The scaled part i s written i n algebraically closed form by means of Schofield's parametric variables, a distance-from-critical variable r and a contour variable θ. Two refinements are made. One goes by the name of "revised s c a l i n g " : the gas-liquid asymmetry c h a r a c t e r i s t i c of r e a l f l u i d s i s obtained by replacing the weak scaling variable by a linear combination of ΔΤ and Δμ (7, 1_3) which i s equivalent with " t i l t i n g " the h axis i n Figure 2. The other refinement i s c a l l e d "corrections to s c a l i n g " or "extended s c a l i n g " . These corrections, introduced by Wegner, account for the difference between the c r i t i c a l Hamiltonian of the r e a l f l u i d and the Hamiltonian characterizing the fixed point for 3D I s i n g - l i k e systems [7]. PVT and thermal data for a number of pure f l u i d s , including steam, ethylene and isobutane have been f i t t e d accurately i n the range of -0.01 to +0.1 i n reduced temperature around the c r i t i c a l point by means of revised and extended scaling with one Wegner correction term (8). Crossover from C l a s s i c a l to Scaled Behavior Universal scaling behavior r e s u l t s when the behavior of a system i s dominated by one length scale, that of the c o r r e l a t i o n length, which diverges at the c r i t i c a l point. When the system moves away from the c r i t i c a l point, other length scales begin to become important and the system gradually assumes the nonuniversal species-specific behavior that i s described by c l a s s i c a l equations. Only very recently, progress has been made with devising functions that scale near the c r i t i c a l point and are c l a s s i c a l at some distance from i t . There have been a number of attempts at empirically connecting " c r i t i c a l " and c l a s s i c a l " regimes. Switch functions have been introduced to smoothly connect the two regimes (14, 15)Î great d i f f i c u l t y i s , however, experienced with derived properties, because of anomalous contributions from higher derivatives of the switch function. Only i f c l a s s i c a l and scaled free energy are exceedingly close to each other i n the range of the switch can these anomalous variations be suppressed. H i l l reports good success with a switch function for steam (16). A true crossover function i s a function that behaves c l a s s i c a l l y far from and i s scaled close to the c r i t i c a l point. Fox transformed the variables i n a c l a s s i c a l thermodynamic potential i n such a way that nonclassical behavior was obtained near the c r i t i c a l point (j_7). His method gives good r e s u l t s for PVT and coexistence curve data but has not been t r i e d for higher derivatives such as C... White (18) has reported success with a


114

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I I .

Strong and Weak Divergences

Strong (Ύ-type)

weak (a-type)

other (nondivergent)

second derivative in direction intersecting coexistence surface in field space

second derivative taken in coexistence, but not in critical surface in field space

second derivative along critical surface in field space

one-component


τ

\dp)

V

K T

s --7

(lf)

N.A.

s

(")P

(")v

two-component, Δ=μ ~μ 1

τΔ

v \spy, ι

Tx

-

1

2

(*A

¥1

/

are f i n i t e at the c r i t i c a l point.

Thus, these mixture properties are not analogous to their strongly diverging counterparts in one-component f l u i d s . Mixtures do show strong divergences; and example i s the osmotic s u s c e p t i b i l i t y (3χ/3Δ)ρ , with Δ = μ which diverges because of the Τ

1

c r i t i c a l i t y conditions: (3Δ/3χ)

2

ρτ

2

= (3 G/3x )

2

p T

2

- 0 ; 0 Δ/3χ )

3

ρ τ

3

= (3 G/8x )

p T

= 0

(2)

In Figure 3b we show the equivalent of Figure 1b for a mixture of constant composition; also indicated are the isobar that passes through the maximum on the phase boundary i n Figure 3a and b, and the isotherm tangential to the phases boundary. In this example, the c r i t i c a l point i s located between the two extrema i n Figure 3b. This i s the most common, but not the only case. In Figure 3b we also draw a few isochores. Note that the c r i t i c a l isochore changes slope at the phase boundary while the isochore passing through the temperature extremum does not change slope (22). By means of the law of corresponding states, van der Waals was able to predict



116


almost a l l of the often-complex phase behavior of binary mixtures that has l a t e r been found experimentally. Van Konijnenburg and Scott ( 2 3 ) generated by computer a l l binary-mixture phase diagrams that follow from the van der Waals equation through the law of corresponding states. Nonclassical C r i t i c a l Behavior of F l u i d Mixtures The defects of a c l a s s i c a l description are as obvious i n f l u i d mixtures as i n pure f l u i d s . C l a s s i c a l models for the coexistence curves of mixtures are never f l a t enough at the top, since binary gas-liquid and l i q u i d - l i q u i d coexistence curves are cubic (£4, 25). G r i f f i t h s and Wheeler (J_2) generalized the scaling laws for pure f l u i d s to f l u i d mixtures by means of the concept of c r i t i c a l - p o i n t u n i v e r s a l i t y . Although, as we have seen, a one-component f l u i d and a mixture of constant composition do not have the same two-phase behavior, G r i f f i t h s and Wheeler postulated that they do i f the mixture i s considered at constant f i e l d . This p r i n c i p l e i s indicated schematically i n Figure 4. In this figure the independent f i e l d s , f o r the scaled potential developed by Leung and G r i f f i t h s (26), are the weighted average a c t i v i t y a =

+ a

2

and the normalized a c t i v i t y of the second component: ζ = c Here c^, c

2

2

a

2

/ (c

a

1

+ c

2

a ) 2

(3)

are two constants related to the choice of zeropoints

of chemical potentials, a^^ - exp (yVRT), and ζ runs from 0 (pure component 1) to 1 (pure component 2). The variable ζ, rather than the customary composition variable x, connects the a-T diagrams of the two pure components. Whereas i f χ i s the variable, a coexistence surface consists of two sheets, a dew and a bubble surface, i t i s single valued i f ζ i s the variable, since ζ i s a f i e l d . A point on the c r i t i c a l l i n e can now be approached i n one of three ways. If two f i e l d s are kept constant, f o r instance ζ and the temperature, a path intersecting the coexistence curve i s obtained. We expect this d i r e c t i o n , indicated by h i n Figure 4, to be "strong" just as i n the pure f l u i d , Figure 2, and corresponding second derivatives to be strongly divergent. Several examples are given i n Table I I . Note that the osmotic susceptibility Ο χ / 3 Δ ) and the s u p e r c r i t i c a l s o l u b i l i t y ρ τ

Ox/3P)

T

^

constant μ

i n the presence of a s o l i d or l i q u i d phase of (almost) 2

are i n this category of strong, Ύ-like divergences.

If one density i s kept constant i n the d i f f e r e n t i a t i o n , f o r instance the volume or the entropy, then the c r i t i c a l point i s approached along the coexistence surface, the weak direction indicated by τ i n Figure 4, and second derivatives diverge weakly. Note, i n Table 2, that Κ , C and α are i n this category. Τ χ

p x

ρ χ

F i n a l l y , when two densities are held constant, the path to a point on the c r i t i c a l l i n e becomes asymptotically p a r a l l e l to this l i n e . Since f i r s t derivatives such as volumes and entropies vary




(b)

P

1 x.

/

m

V^ A

ψ /

Ρ /

ι T

!m '''''À


T/T

c

Figure 3 a, b. They are the equivalents of Figure 1a, b for a mixture of constant composition x, by virtue of the law of corresponding states. · i s the pure-fluid c r i t i c a l point, a pseudocritical point for the mixture. The mixture phase-separates on the dew-bubble curve and does not attain the states — ~ inside this curve. Ο i s the real c r i t i c a l point of the mixture. Τ i s the maximum temperature, Ρ the m m maximum pressure, f o r the dew-bubble curve at x^ . In (b), several isochores are indicated. The c r i t i c a l isochore changes slope at the dew-bubble curve while the isochore through Τ does not. m

I

Figure 4. of

Coexistence surface and c r i t i c a l

independent

l i n e i n the space

variables a (average a c t i v i t y ) , c(=a /a) and T. 2

The scaling variables h (strong) and τ (weak) are chosen i n the plane of constant ζ. ζ i s the third variable chosen along the c r i t i c a l l i n e .


118


smoothly

along

derivatives

critical

taken

in

the

In

to

construct

P(a, of

order

the

at

constant

pure

a regular

is

function

a

ζ

is

components

into

of

the

surface

and with

The

of

and the

the

pure

algebraically with

limited

adjustable for yet

been

with

corrections

simple

to

scaling

to

It

is

extent.

describe

one

of

v

mixtures

modifications, azeotropic Moldover, equation

it

He

order

region

describing

at

at

an angle

phenomenogical of

the

used

their

and

of

available It

is

and has

not

and

so-called

departure

e m p i r i c a l l y the

lines,

lines.

with

form,

scaling

nature,

number

fluids

the

variable.

in

equation

critical

ζ,

to

between

critical

large

scaled

near

split

parametric

interpolation

continuous

by

is

constant

interpolated

the

the that

Ρ(μ, Τ),

potential

l i q u i d - v a p o r asymmetry

(26,

from

the

corrections

properties,

27,

applied

as

to

such

have

the

true

to form

as

has

PVT a n d

of

minor

two-phase

modified been

and at is

of (29).

2

the

Leung-Griffiths

very

pressures

presently

in

successful

hydrocarbon mixtures,

model

states

and C 0 ~ e t h y l e n e

l i q u i d - v a p o r asymmetry

They have

curves

With

(28)

2

3]_).

Their

22). one-and

C0 ~ethane

in volatility,

pressure.

as

two-phase

describe

dew-bubble

differences

critical

been

(3£,

Τ ) , or

part,

one-component

and c o w o r k e r s

to

Pta^ the

for

postulated

The n o n u n i v e r s a l

therefore

a n d He

such

Rainwater

later.

il

J

has

mixtures

in

two-phase large

of

second of

The o r i g i n a l L e u n g - G r i f f i t h s

and

Q

C ,

(26)

Schofield s

only

which,

accommodate

a certain to

in

include

exponents

values,

to

used

ζ to

the

if

discussed

f

in

because

yet

result

exception

and h ,

fluid.

although

scaling

scaling.

critical

asymptotic

along,

with

power is

as

Again,

g a s - l i q u i d mixtures

generalized

apparent

been

of

τ,

restricted

parameters,

modeling

analogous

tedious,

be

background are

equation,

predictive

to

The s c a l e d

pure

components,

Leung-Griffiths

2).

written

analytic

1,

the

thermodynamic p o t e n t i a l

same f o r m

part.

the

with

and G r i f f i t h s

= 1 or

is

divergences

0 or

the

variables

analogy

those

-

Leung

of

(i

in

amplitudes

ζ

no

line,

a scaled

and a s c a l e d

coexistence complete

this

limits

gas-liquid mixture,

T)

line,

along

x-derivatives binary


the

are

even

down t o

limited

to

the in with

half

the

two-phase

states. The fluid

only

application in

other

mixtures

is

has

been

binary liquids,

Bartis

and H a l l

One o f liquid pure

us to

mostly

by

those

with

scaling

amplitudes

the

such

36)

model

that

lattice the

of

to

constructed

a

scaled the

Recently,

enthalpies

at

method,

analytic

of

potential

Johnson has

two

of

used

(3j0. (35).

for

variables

constant

the

boundaries

curves

equilibria

scaling

in

The

phase

binary of

pressure.

the In

nonuniversal

b a c k g r o u n d were

and volumes

applied

33).

coexistence gas-gas

a binary mixture

and the

been

(32,

description

a model

Leung-Griffiths

has

gas

closed-loop

a transformation of

pressure-dependent. excess

to

some w i t h

(L.S.,

analogy

correlate

decorated

applied

mixtures

fluid

nonclassical

the

made

this

potential

a binary liquid

to

mixture

(37). As

of

crossover importance urgent

this

date,

problem for of

no

attempt

fluid

has

mixtures.

supercritical fluid

been

made

to

solve

In

view

of

the

mixtures

this

the

growing

appears

to

be

task.


an

5.



119

E x p e r i m e n t a l E v i d e n c e , o r Lack o f I t , For N o n c l a s s i c a l Behavior i n Mixtures For a l l g r o s s f e a t u r e s o f phase b e h a v i o r o f m i x t u r e s , c l a s s i c a l e q u a t i o n s do a r e a s o n a b l e j o b . There a r e n o t many m i x t u r e s f o r which we have t h e w e a l t h o f a c c u r a t e e x p e r i m e n t a l d e t a i l t h a t we have f o r some o f t h e pure f l u i d s , so t h e shortcomings o f c l a s s i c a l e q u a t i o n s a r e not as a c u t e l y f e l t . As we a l r e a d y mentioned, t h e most s t r i k i n g shortcoming o f c l a s s i c a l e q u a t i o n s i s i n t h e r e p r e s e n t a t i o n o f t h e c o e x i s t i n g phases near t h e c r i t i c a l p o i n t (24, 25) where n o n c l a s s i c a l r e p r e s e n t a t i o n s make a major improvement (V7, J_8, 25 - 3£ ). A l s o , t h e s t r o n g anomaly o f t h e osmotic s u s c e p t i b i l i t y ( 3 χ / 3 Δ ) has been determined from t h e


ρτ

i n t e n s i t y o f s c a t t e r e d l i g h t i n l i q u i d - l i q u i d (38) and i n g a s - l i q u i d (39) m i x t u r e s and t h e n o n c l a s s i c a l exponent v a l u e was found f o r Ύ. The s i t u a t i o n w i t h r e s p e c t t o t h e weak anomalies i s f a r l e s s c l e a r - c u t . Only t h e weak d i v e r g e n c e p r e d i c t e d f o r t h e s p e c i f i c heat C has been c o n v i n c i n g l y demonstrated i n l i q u i d - l i q u i d (40) p x

mixtures.

Those i n t h e expansion c o e f f i c i e n t

compressibility Κ

Τ χ

have not been seen y e t .

m i x t u r e s , t h e reason i s o b v i o u s . relations α

/ Κ

Ρχ Τχ "

d P / d T

l RL

:

C

C

α

ρ χ

and t h e

In binary l i q u i d

Since, according t o the Pippard

Px

/ V T a

Px *

d P / d T

l RL

where CRL denotes t h e c r i t i c a l l i n e , and s i n c e d P / d T | l a r g e i n b i n a r y m i x t u r e s , t h e anomaly i n 0 b e i n g v i s i b l e , f o l l o w e d by those i n α

ρ χ

ρ χ

W

C

C R L

i s very

has t h e b e s t chance o f

and i n Κ

Τ χ

, i n this

order.

I n g a s - l i q u i d m i x t u r e s , t h e L e u n g - G r i f f i t h s model has been used t o f i n d t h e d i s t a n c e from t h e c r i t i c a l l i n e where t h e weak 3 4 divergence of Κ can be seen. For He - He , t h i s d i s t a n c e was Τ χ

1 0

e s t i m a t e d t o be about 10 Κ ( 2 6 ) , which i s beyond e x p e r i m e n t a l accessibility. The p r e d i c t i o n o f a weakly d i v e r g i n g c o m p r e s s i b i l i t y l e a d s t o an apparent paradox (£l_) i n t h e case d e p i c t e d i n F i g u r e 3* because the c r i t i c a l i s o t h e r m has t o c r o s s t h e phase boundary a t t h e c r i t i c a l p o i n t w i t h a h o r i z o n t a l s l o p e ( F i g u r e 5 ) . From t h e relation d V / d P

l ,x a

=

(

9

v

/

9

p

)

+ T

x

(3V/3T)

p x

dT/dP|

(5)

o > x

where σ i s t h e phase boundary, we note t h a t i f ( 3 V / 3 P )

T x

i s to

d i v e r g e t o -«. w h i l e (dV/dP) and dT/dP| a r e f i n i t e and ° σχ σχ n e g a t i v e , then (3V/3T) has t o d i v e r g e t o -« . T h i s means t h a t i n 1


120

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

the one-phase region near the c r i t i c a l point, the expansion c o e f f i c i e n t must pass through zero and diverge to The negative expansion c o e f f i c i e n t makes i t appear that the isotherm enters the two-phase region while i t i s actually s t i l l i n one phase, and vice versa on the other side of the c r i t i c a l point. Some indications of negative expansion c o e f f i c i e n t s have been seen i n l i q u i d - l i q u i d mixtures near consolute points (42, 43) but never near gas-liquid c r i t i c a l l i n e s . By using the Leung-Griffiths model, Wheeler and two of us (RFC and GM) have found that i n 0.28 mole f r a c t i o n C0 i n 2


-22 ethane, the expansion c o e f f i c i e n t changes sign at 10 i n reduced temperature from the c r i t i c a l l i n e ! (41). While several of the weak anomalies thus escape detection, some of the non-divergent properties actually appear to diverge.

Q

4 Thus the s p e c i f i c heat C of He - He behaves l i k e the weakly-divergent s p e c i f i c heat Cy of pure f l u i d s , while theory predicts i t to remain f i n i t e (j_2). The Leung-Griffiths model explains that the same mechanism, the small size of a weakly-diverging property, i s the cause of the apparent f i n i t e value of Κ« and the apparent divergence of C (26). A related theoretical prediction i s that the isochore-isopleth, a curve on which two densities, V and x are constant, should become confluent with the c r i t i c a l l i n e and therefore with the constant - χ phase boundary (Figure 5). The most detailed measurement of (9Ρ/3Τ) i s that of Doiron et a l . i n 3 4 He - He (22) i n which the c r i t i c a l point was approached to 1 mK but O P / 3 T ) remained f i n i t e . The Leung-Griffiths V x

Γχ

V x

t

νχ

Vx

model predicts 10~ approach dP/dT|

CRL

.

11

Κ for the region i n which (3Ρ/3Τ)

νχ

begins to

In the C0 - C Hg system, the isochoric slope 2

2

has to become negative (Figure 5) - according to the -22 Leung-Griffiths model, t h i s happens at 10 i n reduced temperature, the same point where the expansion c o e f f i c i e n t turns negative (41). It i s obvious that for a l l p r a c t i c a l purposes most of the predictions of nonclassical theory for the weak anomalies can be safely ignored. An exception i s the weak divergence i n 0 . The departures from c l a s s i c a l behavior f o r the strong divergences, however, are quite v i s i b l e and need to be taken into account i f accuracy i s desired. ρ χ

Dilute N e a r - C r i t i c a l Mixtures: C l a s s i c a l Analysis Dilute n e a r - c r i t i c a l mixtures show a p e c u l i a r i t y that at a f i r s t glance i s offensive to the thermodynamic1st; p a r t i a l molar properties of the solvent do not necessarily approach pure-solvent properties i n the l i m i t x-0 at the solvents c r i t i c a l point. This behavior was f i r s t experimentally studied by Krichevskii and coworkers (4_4, 45), and modeled by a c l a s s i c a l equation by Rozen (46). Wheeler (47) gave a nonclassical treatment based on the decorated l a t t i c e gas.


5.

121



The reason for this abnormal behavior i s that the solvent c r i t i c a l point i s the intersection of the locus of i n f i n i t e dilution, (3Δ/3χ) + and of the c r i t i c a l l i n e , ( 3 Δ / 3 χ ) ρτ

0;

ρτ

as a consequence the l i m i t i n g values assumed by c e r t a i n p a r t i a l properties depend on the path of approach. This anomalous behavior i s r e a d i l y elucidated for the p a r t i a l molar volume V

- V - χ (3V/3x)

1

(6)

pT

since (3V/3x)


In the l i m i t x+0,

Κ

pT

- VK

Tx

(3P/3x)

approaches the compressibility of the pure

Τ χ

f l u i d which i s strongly divergent. by (44,

(7)

VT

The l i m i t of (3P/3x)

VT

i s given

48)

[Lim (3P/3x) ]° - dP/dx| VT

where dP/dx|°

are

dT/dx|^L

T

L/HL

- dP/dT|£

CRL

dT/dx|£

xc

(8)

RL

are l i m i t i n g slopes of the

critical

LKL

l i n e i n P-x and T-x space, respectively; the l i m i t i n (8), i n general, i s f i n i t e . Thus (3V/3x) diverges at the solvent's c r i t i c a l point. If the l i m i t x-0 i s taken f i r s t , i t diverges strongly, as the compressibility. As a consequence, the p a r t i a l molar volume of the solute pT

V

2

= V + (1 - x ) ( 3 V / 3 x )

(9)

pT

diverges strongly, with a sign equal to that of (3P/3x)

VT

in (8).

These diverging p a r t i a l molar volumes of the solute have been reported by Khazanova and Sominskaya for C0 (45), by Benson 2

et a l .

for NaCl i n steam (2) and by Eckert et a l .

i n ethylene

(49).

for naphthalene

Close to the solvent c r i t i c a l point,

and

V

2

can be predicted from the solvent properties and the i n i t i a l slope of the c r i t i c a l l i n e (48, 50, 51). In the c l a s s i c a l treatment of d i l u t e mixtures, the divergence of (3V/3x) along a path of choice i s obtained by expanding pT

(VK^)""

1

equals 0.

around the solvent's c r i t i c a l point, at which point i t We obtain (48);

V

K

ix

-

a

VVx

X

+

3

VVT

(

δ

Τ

)

+

a

VVVV

(

5

V

)

2

/

2


0

0

)

122


where subscripts V, χ, Τ indicate repeated p a r t i a l d i f f e r e n t i a t i o n of the Helmholtz free energy a(V, T, x) with respect to the variable indicated, and where the superscript c indicated that the derivatives are evaluated at the solvent's c r i t i c a l point. Note that (VK,^)

1

i s quadratic i n volume because the c r i t i c a l i t y c

conditions imply that a ^

c = 0, a^yy

v

• 0.

A t y p i c a l path i s the c r i t i c a l l i n e (Figure 5). ν κ

Here δΤ and

1

δν are linear i n x, so ( ) i s linear i n χ and (3V/3x) diverges as χ . The product x(3V/3x) i n (6) i s a f i n i t e Τ χ

pT

1

pT


constant and

does not approach V »

A second path i s the

c

2 coexistence curve at T=T (Figure 6). Here (δν) ~ χ and (3V/3x)p s t i l l diverges as 1/x, but with a different amplitude. On any 2 isothermal path on which (δν) varies more slowly than x, however, 2 the term i n (δν) w i l l dominate the c r i t i c a l behavior. One such path i s the c r i t i c a l isotherm-isobar (Figure 6) on which (δν) - χ, so that V- diverges as x*~ . On t h i s path, x(3V/3x) c

3

T

2/3

2

pT

approaches 0 and V^ •» V .

Since the derivative (3H/3x)

pT

is

asymptotically proportional to (3V/3x) : pT

Lim (H /V ) = Τ (dP/dT|£ x-»0 2

2

)

(11)

we expect the same anomalies i n p a r t i a l enthalpies as in p a r t i a l molar volumes (48, 50, 51_) as was also noted by Wheeler (47). Strongly increasing slopes i n the excess enthalpy curves of binary gas-liquid mixtures were reported by Christensen on isotherm-isobars near the c r i t i c a l points of each of the components (4, 50). The above analysis i s readily extended to higher-ordered derivatives such as the p a r t i a l molar s p e c i f i c heat C

p 2

- (3C /3x) . p

pT

This quantity i s predicted to diverge much more 2

strongly than the p a r t i a l molar volume, namely as (K ) T

i n the

2 l i m i t x+Q and as (1/x) strong divergence of C

along the c r i t i c a l l i n e . p 2

This extremely

was experimentally found by Wood et a l .

for NaCl i n steam (3). The c l a s s i c a l results are summarized i n Table 3. Dilute N e a r - C r i t i c a l Mixtures; Nonclassical Analysis We have investigated the nonclassical c r i t i c a l behavior of d i l u t e mixtures (48) by writing




X

X

1

1

1 Ρ /

2

/

\\ \\


\ (a)

(b)

Figure 5. Nonclassical behavior near a mixture c r i t i c a l point. The compressibility i s weakly divergent, so the c r i t i c a l isotherm i s horizontal (a). The isochore-isopleth i s confluent with the phase boundary and the c r i t i c a l l i n e (b). If the slope of the phase boundary i s i s negative at the c r i t i c a l point, the expansion c o e f f i c i e n t diverges to -°° and the slope of the isochore-isopleth i s negative at the mixture's c r i t i c a l point.

Figure 6. The paths in V-x space at the c r i t i c a l temperature of the solvent along which we have calculated the divergence of V « 2

The c r i t i c a l l i n e i s shown i n projection.


124


(aV/3x)

= θΥ/3Δ) /Οχ/3Δ)

pT

ρ τ

(12)

ρ τ

Both numerator and denominator, being second derivatives [of the potential μ (Δ, Ρ, Τ), (_1_2)] i n a d i r e c t i o n intersecting the 2

coexistence surface, diverge strongly at the c r i t i c a l l i n e . As a consequence, (3V73x) i s f i n i t e at the c r i t i c a l l i n e j u s t as i n pT


the c l a s s i c a l case, and not weakly divergent as stated i n r e f . (52). The osmotic s u s c e p t i b i l i t y i n the denominator, however, faces a dilemma when x+0 because i t i s zero at the pure-fluid axis and i n f i n i t e at the c r i t i c a l l i n e . By means of the Leung-Griffiths model we have been able to obtain the structure of this term. Schematically, i t can be given, to the leading order i n ζ and r , as Οχ/3Δ)

* c(flnlte) + c (strong)

(13)

2

ρ τ

While the structure of the numerator i s θν/3Δ)

~ c(strong)

ρ τ

(14)

where ( f i n i t e ) denotes a thermodynamic derivative that remains f i n i t e , (strong) one that diverges strongly at the c r i t i c a l l i n e . Thus the structure of (3V73x) i s pT

(3V/3x) ax;

* .„ , (strong) ( f i n i t e ) + c(strong)

pT

(

}

Schofield's r denotes the distance from the c r i t i c a l l i n e at constant ζ, and the strong term can be written as proportional to -Ύ r . A path to the solvent c r i t i c a l point can be characterized by stating how r goes to zero as ζ shrinks, or by the r e l a t i o n r _

(16)

ε ζ

If ε i s small, ζ decreases much faster than r and the path i s close to the pure-fluid axis. I f ε i s large, ζ decreases slower than r and the path i s close to the c r i t i c a l l i n e . Intermediate paths are obtained by varying the value of ε. Thus we may indicate the behavior of (3V/3x) schematically by pT

(3V/3X)

~

il

-y

( f i n i t e ) + cr

-

^

£

Ύ

.

e

T

^

(

(finite) + ζ

Since for small ζ, ζ ~ χ, because the RTxfcnx term dominates the free energy, we obtain


1

7

)

5.



(3V/3x)

"

125

L

— (finite)x

(18) e T

+ χ

When ζ goes to 0 at f i n i t e r, we retrieve, from (15), the strong divergence of (3V/3x) . If ζ and r both go to zero, there i s a pT

competition between the term i n χ ε i s large,

ε Ύ

and the term i n χ i n (18).

ε > 1/Ύ, εΎ > 1, ε > 0.81 then (3V/3x)


V

1

pT

(19)

behaves l i k e 1/x and so does V

does not approach V .

2

(9).

In t h i s case

If ε i s small

ε < 1/Ύ, εΎ < 1, ε < 0.81 then (3V/3x)

pT

behaves l i k e 1/χ

does approach V .

εΎ

(20)

and so does \? . 2

On the c r i t i c a l l i n e (ε large) V

q

If

In t h i s case 2

diverges as

1/x. On the isothermal or isobaric coexistence curve h equals 0 and τ i s linear i n ζ to lowest order. Since τ i s linear i n r , we have ε=1 and (3V/3x) goes at 1/x. pT

On the c r i t i c a l isoterms-isobar, the pressure and temperature are to be kept constant and equal to those at the solvent's c r i t i c a l point. In this case both τ and h vary l i n e a r l y with ζ for ft Λ

small ζ leading to r (3V/3x)

and so does V « 2

pT

~ ζ or ε = 1/36. . 1/χ

Ύ / Β δ

1

« 1/χ "

The value of 1-1/6

I t follows that 1 / δ

= 0.79.

(21)

Therefore χ

(3V/3x)

0 and V\ V . The crossover from 1/x to slower-than 1/x 1 c behavior occurs at ε - 1/Ύ » 0.81 which i s between the isothermal or isobaric coexistence curve (ε*1) and the c r i t i c a l isotherm-isobar (ε=0.64). We have also analyzed the behavior of the p a r t i a l molar s p e c i f i c heat C . p2

The results of our analysis

are summarized i n Table I I I . The c l a s s i c a l and nonclassical treatments of d i l u t e mixtures show the same global features but d i f f e r i n d e t a i l . The differences are due to the different values assumed by the strong exponents 3, Ύ, δ and should therefore be experimentally detectable. Impure N e a r - C r i t i c a l Fluids In the custody transfer of f l u i d s , the density of the f l u i d i s often obtained from measurements of pure Ρ and temperature Τ and


pT

126


from the known equation of state. I t i s important to know what effect the presence of an impurity can have on the density at given Ρ and T. I f V indicates the molar volume, we may write V(P, T, x) = V(P, T, 0) + (3V/3x) x + . . . .

(22)

PT

Since (3V/8x)

pT

i s proportional to the solvent's compressibility,

impurity effects become larger the larger the compressibility. The expansion (23) i s not v a l i d at the solvent's c r i t i c a l point. At 1 /Λ


T

c*

P

w

c*

e

h

a

v

e

χ

( Ρ and two pressures Ρ , P„ at which the cm d cm m l actually induces a phase t r a n s i t i o n . 0

impurity


5.



129

where we have used a ^ - -dP/dT|£ , t h e l i m i t i n g s l o p e o f t h e T

xc

vapor p r e s s u r e c u r v e . I f t h e major component has been t r e a t e d n o n c l a s s i c a l l y , o r w i t h a very accurate c l a s s i c a l equation, the coefficient a ^ a

VVT *

9

(

ν

κ

χ

1

)

/

9

V T

Τ

w i l l be z e r o o r v e r y s m a l l because ~

Δ

τ

Ί

Μ

and Ύ > 1.

Thus, t h e i n i t i a l s l o p e o f


the c r i t i c a l l i n e i s f o r c e d t o be t h e e x t e n s i o n o f t h e vapor p r e s s u r e c u r v e , which c o n t r a d i c t s experiment. The combining o f a n o n c l a s s i c a l p u r e - f l u i d equation with corresponding s t a t e s f o r the m i x t u r e induces a s e r i o u s d i s t o r t i o n i n t h e phase diagram. Conclusions For one-component f l u i d s , a c c u r a t e n o n c l a s s i c a l f o r m u l a t i o n s o f t h e c r i t i c a l region are a v a i l a b l e , i n c o r p o r a t i n g the v a p o r - l i q u i d asymmetry as w e l l as c o r r e c t i o n s t o s c a l i n g . E m p i r i c a l t e c h n i q u e s f o r s w i t c h i n g from c l a s s i c a l t o c r i t i c a l b e h a v i o r have been worked o u t . The fundamental s o l u t i o n o f t h e c r o s s o v e r problem has advanced t o t h e p o i n t t h a t a p p l i c a t i o n t o one-component f l u i d s i s near. F o r b i n a r y m i x t u r e s a n o n c l a s s i c a l f o r m u l a t i o n i s a v a i l a b l e f o r t h e one-and-two-phase r e g i o n s m i x t u r e s w i t h c o n t i n u o u s c r i t i c a l l i n e s . A v a r i e t y o f n o n c l a s s i c a l models have been used t o r e p r e s e n t c o e x i s t i n g phases. No attempts have been made t o d e v i s e crossover functions. D i l u t e n e a r - c r i t i c a l m i x t u r e s p r e s e n t some s p e c i a l anomalies not o f t e n encountered i n thermodynamics. We d e s c r i b e these w i t h b o t h c l a s s i c a l and n o n c l a s s i c a l e q u a t i o n s . We argue t h a t a n o n c l a s s i c a l thermodynamic d e s c r i p t i o n o f t h e s o l v e n t can n o t be combined w i t h a c o r r e s p o n d i n g - s t a t e s treatment o f t h e d i l u t e mixture. For i n c o r p o r a t i o n of the n o n c l a s s i c a l e f f e c t s i n t o p r a c t i c a l thermodynamic s u r f a c e s o f f l u i d m i x t u r e s , t h e s o l u t i o n o f the c r o s s o v e r problem appears t o be t h e most urgent t a s k a t hand. Acknowledgment We have p r o f i t e d from d i s c u s s i o n s w i t h P.C. A l b r i g h t and J.C. Wheeler.

References 1. 2. 3.

4.

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5. 6. 7.


8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32. 33.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

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131

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November 8, 1985


Nonclassical Description of (Dilute) Near ... - American Chemical Society

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