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16 Investigation of Binary Liquid Mixtures via the Study of Infinitely Dilute Solutions

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D. A. JONAH University of Sierra Leone, Department of Mathematics, Fourah Bay College, Freetown, Sierra Leone, West Africa

The prediction and correlation of activity coefficients and their derivatives for binary liquid mixtures at infinite dilution have been considered on both the thermodynamic and molecular levels. The related considerations of gas solubilities in liquids and the solubilities of solids in supercritical fluids have also been studied. Based on a method of numerical differentiation using intercepts rather than slopes, it has been demonstrated that the above limiting thermodynamic functions can be estimated with sufficient accuracy from either excess Gibbs free energy or vapor pressure data available near both ends of the composition range. Limiting activity coefficients and their derivatives so estimated have been used in three empirical expressions for excess Gibbs free energy, to extrapolate from the dilute portion to the other parts of the composition range; such extrapolations have been found to be in good agreement with experimental measurements. Using rigorous statistical mechanics, combined with certain semiempirical arguments, a number of tested correlations of the limiting residual chemical potential (and also solubilities) with pure solvent properties are presented; these are found to be in good agreement with experimental data.

T

H E STUDY OF VERY DILUTE SOLUTIONS deserves serious attention for

at least three reasons: 1. Such a study has important practical applications in extractive and azeotropic distillations where important components often occur in very low concentrations.

0065-2393/83/0204-0395$08.00/0 © 1983 American Chemical Society


396

MOLECULAR-BASED STUDY OF FLUIDS

2. The constants in certain empirical equations for excess functions are related to thermodynamic functions at infinite dilution (e.g., the limiting activity coefficients and their first derivatives); further, these constants are needed for the prediction of multicomponent vapor-liquid equilibria. 3. Very dilute solutions are of considerable interest to the theoretician for at least two reasons. First, the absence of solute-solute interactions helps to isolate the unlike solutesolvent interactions against a background of solvent-solvent interactions, so that thermodynamic properties at infinite dilution provide a convenient source of information about these unlike pair interactions so necessary in any attempt to relate these to the like interactions. Second, the dominance of the solvent-solvent interactions makes the contribution from the solute-solvent interactions to the total energy a veritable small perturbation. Therefore, an infinitely dilute system is ideal for the application of a perturbation about the pure solvent, as one possible route in the prediction of mixture properties from those of pure components. It is therefore surprising that more attention has not been devoted to this important aspect of mixture theory. As is evident from Reason 2, the study of infinitely dilute solutions can form the basis for the study of binary mixtures at finite concentrations. This appears to be a viable alternative to the usual approach, which focuses attention on the mixture at finite concentration. Then, at some stage in the statistical thermodynamical development, information about the composition-dependence of the principal thermodynamic functions is injected into the theory, through assumptions expressing the composition-dependence of mixture molecular parameters in their relationship to corresponding parameters of the pure components. In this alternative approach, however, the form of compositiondependence of a basic thermodynamic function, such as the excess Gibbs free energy, is assumed beforehand through the use of one of the empirical equations with constants related to limiting thermodynamic quantities. Attention is then focused on the evaluation or prediction of these constants with the help of molecular thermodynamics. In this chapter, we shall apply this alternative approach to the study of binary nonelectrolyte mixtures. Infinitely dilute solutions will, of course, also be studied in their own right in attempts to predict such properties as Henry constants, solubilities of liquids in gases, and the solubilities of liquids and solids in supercritical gases. There will be two broad sections, one based on purely thermodynamic considerations, and the other on molecular thermodynamics. The "Thermodynamic Considerations" section briefly reviews some


16.

JONAH

397

Binary Liquid Mixtures

empirical equations whose constants are related to limiting quantities; special attention is given to a modified version of the two-constant Van Laar equation, called the Conic equation (J, 2). Examination of this equation as a correlating and extrapolative device—that is, a means for the extrapolation of excess functions measured for very dilute solutions into the rest of the composition range—is next considered; then follows an examination of the evaluation of the constants in this equation from vapor pressure-composition curves. Comparison is made with experimental data. The "Molecular Thermodynamics" section addresses the problem of predicting the constants in the empirical equations or correlating them with pure component properties. My approach combines rigorous statistical thermodynamics with empirical and semiempirical arguments. Also considered are the solubilities of liquids and solids in supercritical gases. Comparison is made between my equations and experimental data. Thermodynamic

Considerations

Some Empirical Equations. Although several useful empirical and semiempirical equations express the composition-dependence of the excess Gibbs free energy, G , few of these have the useful and desirable property of their constants being simply related to the limiting derivatives of the excess functions. The more familiar equations in this desirable class are the well-known two-constant Van Laar and Margules equations and the one-constant regular solution equation. In each of these the constants are related to the limiting activity coefficients (or the limiting first derivatives of the excess function, G ). However, these equations seldom do well when the constants are identified as just stated; invariably, these constants have to be evaluated by some method (e.g., by least squares) using data over the whole concentration range. However, in not identifying the constants with limiting properties, we are sacrificing a very valuable feature of these equations that makes them useful not only as correlating equations, but also as extrapolative equations for deducing the excess function over the rest of the composition range, given a few measurements near either end of this range. Hence, these equations need to be modified to make it possible to use limiting thermodynamic quantities for the constants and achieve, at the same time, a close fit to the experimental data. Such modifications of the Van Laar and Margules equations have been suggested by Jonah and Ellis (J, 2) and by Abbott and Van Ness (3). The modified Van Laar form, the Conic equation, is given by E

E

q

2

where q =

G /RT. E

+ a xx l2

Y

2

+ qia^ + a x ) = 0 2

2

(1)

398


The modified Margules forms are Mi.

q = x x {A x 1

2

2l

l

- (\ i*i +

+ Ax l2

^12^1^2}

2

2

(2)

and q — X]X I A i%i + A x

M : 2

2

1 2

2

\

X xx

1 2

21

^12^-1


1

2

2

(3)

^"21^2,

The constants in Equations 1, 2, and 3 are all related to the limiting activity coefficients and their derivatives as follows X

—A

12

1 (d ln 7j

2A 12

2 1

dx

(4)

l

2A

X-21 — ^ 1 2

1 / d ln 7 _

91

2

dXo

6 ln 7x

+ (5)

*12

d In 7 ax ah

and

= ln 7?

1 2

+

9

^21

A

2

and

do = A

2 1

= ln 7

2

To see that Equation 1 is a modified form of the Van Laar equation, we rewrite it in the form

b q l2

2

+ x,x - q\j± 2

+ f-J

= 0

(6)

where b = a^ . When b = 0, Equation 1 reduces to the two-constant Van Laar equation; if A = A , the regular solution equation is recovered. In these modifications of the two-constant Van Laar and Margules equations the additional constants merely take account of the curvature l2

1

12

1 2

2 1


16.

JONAH

399


of the excess Gibbs free energy curves at the ends of the composition range. If the curvature is negligible, then we expect to find the twoconstant equations adequate for correlation purposes. Also, whereas the modified Margules equations, Equations 2 and 3, require limiting derivatives at both ends of the composition range, the Conic equation, Equation 1, requires just one of these same quantities. As will become evident later, this is a very useful feature. Often, in practice, both limiting derivatives cannot be evaluated with the same degree of certainty; in such cases we choose the more reliable of the two limiting derivatives. The introduction of curvature parameters into the two-constant Van Laar and Margules equations has the effect of constraining the fitted excess curve to pass through the maximum (or minimum) point. It is therefore not surprising to find that a in the Conic equation can be expressed in terms of the maximum (or minimum) excess function 12

_,

{1 - e(A V - Af, )} - 4 e A ^ 1

2

„

2

where e = G ^ . Using Equation 5 in Equation 7, we have the alternative expressions for the limiting derivatives of the activity coefficients:

aJnTiV _ dx

A? {1 2

e(A

l

2l

/

x

- Af, )} - 4eA 1

2

2

l2

2e

2A?

2

A

2

21

and

(7a) (9 In y Y 2

dx

2

J

_

A&1 - e(A^

- A )} 1

2l

2

- 4eA

2

21

2A

2

21

2e

2

For correlation purposes, when we have data over the whole composition range, Equation 7a is obviously preferable to a direct method of evaluating the limiting derivatives for data satisfying the Conic equation. On the other hand, Equation 7a is not useful for extrapolation of data from the very dilute regions of the concentration range into other regions of this range. However, for data satisfying the Conic equation, Equation 7a provides a standard by which to judge the effectiveness of a direct method, such as the method of intercepts, for evaluating derivatives, and it can serve as a check on the estimates of the derivatives used to evaluate this same constant by Equation 5. The Conic equation does surprisingly well in correlating the excess functions of a wide class of binary systems, in spite of its unattractiveness in that it gives q as an implicit function of composition. The criterion for applicability is simple: the excess function has to satisfy a "rule of rec-


400


tilinear diameters" (see Figure 1). I have yet to come across excess Gibbs free energy values for binary systems that depart significantly from this rule. The slope of the straight line joining the midpoints of the diameters can be simply related to the limiting first derivatives of the excess function. Evaluation of Limiting Derivatives. The usefulness of Equations 1-3 as extrapolating equations depends very much on the accurate evaluation of the limiting first and second derivatives of the excess functions. A method for the evaluation, by the method of intercepts, of the first and second derivatives of experimental data has already been developed and described elsewhere (7); the method has been tested against several sets of experimental data, and found to yield reliable estimates. I shall merely quote the principal equations here and give the results of applying them to the excess Gibbs free energy data for three binary systems. Let c|>(x) be some function of x whose first and second derivatives are required at x = b. Then these are given by

where

Figure 1. Rectilinear diameter criterion that must be satisfied by an excess function in order to be closely fitted by Equation 1. The slope s of the locus AB of the midpoints of diameters is given by s = 2A A /(A - A ). i2

M

2i

2i

16.

JONAH

401


and

^ax /

^ \ ax /

x = f c

x = b


where L?) -

lim * * (x - a)(x - b)

(9a)

q i

M

$ =

4>)/( - )(x - b) T

X

(10)

a

(x) and qr(x) are certain trial functions, which may be linear, quadratic, or otherwise. The linear forms (which we employ in this chapter) are given by x

T

I I / \ • Cj)(fc) ~ (fl) . . c() = 4>(fl) + — (x - a) — a

, V (11)

r

r(D _ Q l

= L

( l) a

+

r(i) 2_

(

x

_

(12)

a )

The value of a in the above equations is chosen according to convenience. The important feature in the above formulas is that they make use of intercepts, rather than slopes, for the evaluation of derivatives. They have been applied in the next section in evaluating the first and second limiting pressure derivatives (Equations 15 and 16), and also the limiting activity coefficients and their derivatives from excess Gibbs free energycomposition curves, for use in Equations 1-3. Three binary systems have been considered: ethanol-n-heptane (30 °C) (5) acetone-carbon tetrachloride (45 °C) (6) nitromethane-carbon tetrachloride (45 °C) (6) The results are summarized in Table I. In Table I, "g (l) calc" indicates excess free energies obtained from the Conic equation and the modified versions of Margules equations, M and M , given in Equations 1-3. The constants A and A have been chosen to obtain the best possible fit with the Conic equation, using the values obtained by the method of intercepts as first approximations. The limiting derivatives of the activity coefficients are evaluated according to Equation 7a and should be close to the true values, considering the very E

1

2

1 2

2 1

402


Table I. Excess Gibbs Free Energy From Equations 1-3 with Constants Identified with Limiting Properties Evaluated by Equations 8-12, 13a, and 14a g (From P-x Data)

g (2) Calc.

E

E

g (l) Calc. E

Expt.

M,

M

9


Ethanol (1) and n-Heptane (2) (30 °C) (19) 0.1 0.3 0.5 0.7 0.9

0.262 0.508 0.570 0.478 0.219

0.262 0.511 0.568 0.476 0.219

0.242 0.365 0.380 0.378 0.211

0. 246 0. 407 0. 440 0.,409 0,.214

0.262 0.511 0.566 0.475 0.219

Jonah

Constants 3.52 2.65

d ln 7

0.266 0.508 0.559 0.469 0.218

2

dx

3.55 2.65

0.275 0.449 0.444 0.393 0.210

Klaus et al.

3.75 2.68

4.09 2.69

-26.2

-21.8

-23.11

-40.62

-11.18

-11.3

-12.81

-12.42

2

6.45 3.90

8.71 3.81

6.74 4.80

Nitromethane (1) and Carbon Tetrachloride (2) (45 °C) (23) 0.0459 0.0918 0.1954 0.2829 0.3656 0.4659 0.5366 0.6065 0.6835 0.8043 0.9039 0.9488

0.110 0.199 0.350 0.432 0.482 0.504 0.501 0.476 0.432 0.317 0.176 0.099

0.108 0.198 0.350 0.434 0.482 0.504 0.498 0.475 0.429 0.313 0.176 0.099

0.108 0.196 0.340 0.415 0.456 0.476 0.471 0.452 0.412 0.307 0.175 0.099

0.108 0.197 0.343 0.421 0.465 0.485 0.480 0.459 0.418 0.309 0.175 0.099

0.108 0.199 0.354 0.441 0.491 0.514 0.507 0.483 0.435 0.317 0.176 0.099

Constants 2.578 2.072

A12 (d In In 7 i V dx d ln 7 dx

2

0.276 0.456 0.4.54 0.399 0.211

-10.59 -5.50

2

2.211 1.184

2.547 2.056 -9.37

16.

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403


Table I. Excess Gibbs Free Energy From Equations 1-3 with Constants Identified with Limiting Properties Evaluated by Equations 8-12, 13a, and 14a (Continued) g (From P-x Data)

g (2) Calc. E

g (l)Calc. E

g Expt. C M M C C M Acetone (1) and Carbon Tetrachloride (2) (45°C) (6) x

Xl


E

0.0556 0.0903 0.2152 0.2929 0.3970 0.4769 0.5300 0.6047 0.7128 0.8088 0.9090 0.9636

0.057 0.087 0.161 0.188 0.209 0.208 0.203 0.191 0.158 0.117 0.061 0.026

0.057 0.086 0.161 0.189 0.207 0.208 0.204 0.191 0.158 0.117 0.061 0.026

0..057 0,.086 0,.157 0..182 0,.198 0..199 0,.195 0..183 0,.154 0,.115 0,.061 0,.026

2

0.057 0.086 0.161 0.188 0.206 0.207 0.203 0.190 0.158 0.117 0.061 0.026

0. 057 0. 086 0. 162 0. 189 0.,208 0. 210 0. 205 0. 192 0. 160 0. 118 0. 062 0. 026

0.055 0.084 0.160 0.188 0.207 0.208 0.204 0.191 0.158 0.116 0.061 0.025

,

0.055 0.084 0.159 0.187 0.206 0.209 0.205 0.193 0.161 0.119 0.062 0.026

M

2

0.056 0.086 0.168 0.199 0.222 0.224 0.219 0.204 0.167 0.121 0.062 0.026

Constants A A (d In 12

21

\

7 l

/a In

7

2

dX

=

-4.869

-4.70

=

- 1.258

—

-0.756

— —

0.584 0.044

N

J

2

\

1.147 0.730

N

f /

tei

= =

f

= —

^12 ^21

1.102 0.718

1.146 0.739

0.8705 0.316

-4.14

Note: g = G /RT E

E

close fits by the Conic equations. The modified Margules equations M and M do not give as close fits as the Conic equations, except for the system acetone-carbon tetrachloride, for which M does as well as the Conic equation. It is, however, probable that M and M may give closer fits, by not identifying the constants with limiting thermodynamic properties. The close agreement between g (l) and experimental values demonstrates the effectiveness of the above equations for purposes of correlation. In Table I, "g (2) calc" indicates values of the excess free energy as obtained from the Conic equation using constants that have been calculated entirely from limiting activity coefficients and their derivatives, as estimated by method of intercepts. Corresponding values for M and M are not shown, as these equations require both limiting derivatives, x

2

2

l

2

£

£

x

2

404


which in general are not available. The good agreement between g (2) and the experimental values demonstrates the potentiality of extrapolating excess free energy from the dilute ends of the concentration range to other parts of the range. We next examine the possibility of calculating excess free energies over the whole composition range, given a few total vapor pressure measurements near both ends of the range. Excess Gibbs Free Energy from Vapor Pressure Data. The limiting activity coefficients and their derivatives can be expressed in terms of the limiting first and second pressure derivatives with respect to composition; corresponding expressions in terms of limiting first and second temperature derivatives with respect to composition for isobaric data are also available. Gautreax and Coates (8) were the first to derive such relationships but only for the limiting activity coefficients within the symmetric standard states convention. Jonah has rederived these relations for both the symmetric and unsymmetric choice of standard states; analogous relations for the limiting first derivatives of the activity coefficients within the symmetric standard states convention, have also been derived (9). Only the isothermal relations, which have been applied to two binary systems, acetone-carbon tetrachloride (45 °C), and ethanoln-heptane (30 °C) are given here:


£

S-

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