Utlonal Bureau of Standards

Library, ».W- BWb

AUG 11 1965

^ecknlcuL ^T,ot& 92o. 3/6

SOLUBILITY OF SOLIDS IN DENSE GASES

J. M. PRAUSNITZ

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^^ote 3/6

ISSUED July 1965

SOLUBILITY OF SOLIDS IN DENSE GASES

J. M. Prausnitz

Institute for Materials Research

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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Price: 35(f-

CONTENTS

ABSTRACT - 1

1. Introduction * 2

2. Basic Thermodynamic Equations 3

3. Previously Proposed Methods for Predicting

Solubility of Solids in Compressed Gases 5

4. Calculation of Fugacity Coefficients Using

Redlich's Latest Equation 7

Table 1 - 11

Table 2 - 14

Table 3 - - - 17

Table 4 - 19

5. Calculations Based on Regular Solution Theory 22

6. Fundamental Thermodynamic Equations 22

7. Evaluation of Parameters 25

Table 5 - - 28

8. The Carbon Dioxide -Nitrogen System -- 29

9. Correlation of Solubility Data 30

Table 6 31

10. Conclusion 32

111
11. Acknowledgment 34

12. References 35

Figure 1 37

Figure 2 38

Figure 3 - - -- 39

Figure 4 40

Figure 5 41

Figure 6 42

Figure 7 43

Figure 8 44

Figure 9 45

IV
SOLUBILITY OF SOLIDS IN DENSE GASES

J. M. Prausnitz

Institute for Materials Research

National Bureau of Standards

Boulder, Colorado

and

Department of Chemical Engineering

University of California, Berkeley

ABSTRACT

The thermodynamics of solid- dense gas equilibria is discussed, and two techniques are described for calculating the solubility of a solid component in a gas at high pressure. The first

one is based on the recent empirical equation of state of Redlich

which, in turn, is derived from Pitzer's generalized tables of

fluid-phase volumetric properties. The second one is based on

the Hildebrand-Scatchard theory of solutions. Both methods give

good semiiquantitative results but cannot accurately predict solubilities from pure -component data alone.
1. Introduction

High-pressure processes are becoming increasingly important

in the chemical and related industries; it is therefore of practical interest to develop techniques for the prediction of phase equilibria at advanced

pressures. In this report we consider the equilibrium between a solid

and a dense gas, i. e. , a gas at pressures around 100-300 atm. In particular, we attempt to calculate the solubility of the heavy (solid) component in the light (gaseous) component. Experiinental studies have

repeatedly shown that such solubilities are very much larger than those

computed by a simple calculation, which assumes ideal-gas behavior.

Indeed, the ratio, of observed solubility to solubility calculated on the

3

basis of ideality, is often of the order of 10 ; in some cases it is as high

as 10 tl].

We discuss two techniques for calculating the solubility of a

solid in a dense gas. One is based on a new empirical equation of state

recently developed by Redlich and co-workers; the other one is based on

the regular solution concept of Hildebrand and Scatchard [2]. Unfortunately neither of these techniques is capable of giving very accurate

quantitative predictions of the desired solubility. Both techniques however show the correct trends and give reasonable results. Furthermore,

the application of regular solution theory to this problem offers a method
of correlating and extending experimental data in a rational manner.

Before describing these two techniques we review briefly the basic thermodynamics and previously presented attempts to deal with the problem

of solubility of solids in compressed gases.

2. Basic Thermodynamic Equations

Let subscript 2 refer to the heavy (solid) component and let subscript 1 refer to the light (gaseous) component. The equation of equilibrium IS

h =V' ("

where f stands for fugacity, superscript s stands for solid, and g stands

for gas. The fugacity of the solid at temperature T and total pressure

P is easily calculated from solid vapor pressure and density data. We

assume that the solid is pure, i.e. , the gaseous component 1 does not

dissolve appreciably in the solid phase:

S S S I v^^dP

RT (2)

where s = saturation.

P = saturation (vapor) pressure of the solid at T,

g V = molar volume of the solid at T, and

s s

cp = fucacity coefficient of saturated vapor at T and P •
In almost all cases of practical interest, P is small ajid thus

Q

cp_ ?»1. Further, the solid may be considered to be incompressible and

thus (2) becomes

h =^2 ^^'P RT <^^>

The fugacity of component 2 in the gas phase is related to y ,

its mole fraction (i. e. , the solubility) in that phase, by

h - \'>z^ - <3)

where cp is the fugacity coefficient of 2 in the gas mixture at temperature T, composition y and total pressure P. The entire problem of

calculating the desired solubility lies in the calculation of cp .

The fugacity coefficient cp can be found from the rigorous thermodynamic relation [ 3] ,

dp \ RT

an ; V

T,V,n^

dV - in z , (4)

where V is the volume of the gas nnixture at T and P having n moles of

1 and n moles of 2, and where z is the compressibility factor of the
mixture:

PV

^ = (Hj + n^) RT • <^^

In order to utilize (4) it is necessary to have an equation of state

for the gas mixture, i.e. , volumetric data for the mixture as a function

of pressure and composition at the temperature of interest.

3. Previously Proposed Methods for Predicting Solubility

of Solids in Compressed Gases

During the past dozen years there has been a variety of publications dealing with the calculation of cp as given by (4). In each case a

particular equation of state is used and a particular set of mixing rules

is employed to predict the volumetric properties of the gas mixture

using only experimental data on the pure components. Equations of state

which have been used for this purpose include those proposed by Benedict,

Webb, and Rubin; by Beattie and Bridgeman; by Redlich and Kwong, and

by Martin and Hou [4, 5] . In a similar but slightly different approach,

attempts have been made to calculate cp using a pseudocritical method

and generalized thermodynamic charts as described by Joffe [6]. In

some cases these methods have produced good results; in others the

results have been only fair, aind in a few cases the results have been

poor. The absence of consistently good results is due to the inadequacy
of the equations of state and, more important, to the very rough nature

of the empirical mixing rules which are used to relate mixture constants

to those of the pure components. In some cases such rules are reliable

but in others they are not, and it is very difficult in the absence of pertinent experimental information to decide a priori whether or not a

given mixing rule is likely to be appropriate for a given mixture.

The most reliable method for calculating cp uses the theoretically valid equation of state known as the virial equation. Although this

equation was originally proposed about sixty years ago on purely empirical grounds, it has been shown to have a sound statistical mechanical

basis. The compressibility factor for the gas mixture is given by

B C

mix mix ... z = 1 + + + . . . , (6)

V

where v is the molar volume, i.e. , V/(n + n ), and the virial coefficients for the mixture are given rigorously as a function of composition by

Smix = yi'^ll + ^V2^12 + y2S2 (^'

^mix = yi'^lU + ^^1 Vll2 + ^yiy2'^122 + ^2^^222 ' '8>
The fugacity coefficient is determined by substituting (6), (7), and (8)

into (4), yielding

^" *2 = I '^1^12 * y2»22> + -TZ '^I'^IU + 271720,22 + 72'c222)

2v

+ ..." in z. (9)

The various virial coefficients B.. and C.., are functions only of temperature. Various techniques have been proposed for calculating the

necessary virial coefficients [7,8], In general it is possible to make

good estimates of the second virial coefficients B., but, at present, it

is not possible to estimate accurately the third virial coefficients C. .

As a result, use of the virial equation for calculating cp is limited to

moderate densities or pressures (typically up to 20-60 atm depending on

the temperature cind on the nature of the components), and it is not at

all useful for equilibria at very high pressure.

4. Calculation of Fugacity Coefficients Using

Redlich's Latest Equation

During the past few years Dr. Otto Redlich and co-workers have

developed an equation of state which is an empirical modification of the

equation proposed by Redlich and Kwong [9]. The modification is based

on the generalized compressibility factor tables of Pitzer et al. [2].

Redlich's equation is of the form
where z is the compressibility factor as given by the Redlich-Kwong RK

equation and f (T , P , cj) is a generalized analytical deviation function R R

whose independent variables are reduced temperature T , reduced pres- R

sure P , and acentric factor co. The Redlich-Kwong equation is R

RK V - b „^RT^7^77)-

(U)

The two constants a and b are related to the critical temperature

and pressure by

a = 0.4278 R —^ ' (12)

c ^^ RT

b = 0.0867 -^-^ . (13)

c

For a binary mixture of components 1 and 2,

and

^mix = ^1^^ + ^-^l^Z'i^Z + y2'^2 ' <^^^

b . = y,b, +y,b_ . (15) mix ' 1 1 ' 2 2

The function f (T , P ,w) is a very complicated but analytical expression R R

obtained by forcing agreement between (10) and Pitzer's generalized

tables. To apply f (T , P , co) to a mixture, a pseudocritical hypothesis R R

is used to calculate the reduced temperature, the reduced pressure, and

the acentric factor for the mixture. The simplest procedure for doing
so is to accept Kay's rule [4], but it was found by Redlich, et al that an.

alternate empirical procedure gives a much more accurate representation of volumetric data for gaseous mixtures. The details of this alternate procedure are described in a forthcoming publication [ 10] . Redlich

ajid co-workers found that their equation reproduced with high accuracy

volumetric data for a variety of nonpolar mixtures at moderate and high

pressures. However, there were strong indications that at conditions

near the critical of the mixture, the equation is not reliable.

In view of the mathematical complexity of Redlich' s equation, it

is essential to use an electronic computer for calculating cp and for

finding the trial-and-error solution to the equation of equilibrium

(Equation 1). Dr. Redlich kindly supplied the author with the necessary

computer programs.

Calculations with Redlich' s equation were performed for four

systems for which high-pressure data (beyond the range of the second

virial coefficient) were available; in each system the solubility of the

heavy (solid) component in the gas phase was calculated. The four systems are: carbon dioxide -nitrogen, carbon dioxide-air, oxygen-hydrogen,

and naphthalene -ethylene. The calculated results are compared with

experimental solubilities in tables 1, 2, 3, and 4.

In table 1 we compare calculated and experimental results for

the carbon dioxide -nitrogen system. The results are expressed in terms
of enhancement factors E defined by

y (actual)

E = ,., ,. , (16)

y^ (ideal)

where by y- (ideal) we mean the value calculated by assuming 9=1

and f^ = P^. Thus

y^ (ideal) =^- . (17)

Results are shown for the temperature range 140° to 190° K and

for pressures up to 100 atm. For all temperatures except the lowest,

agreement between calculated and experimental results is very good.

At 140° K the agreement is somewhat worse, although still not bad.

Deviations between experimental and calculated results at high pressure

at this low temperature are probably due to the fact that the mixture is

not far removed from its critical state, where Redlich's equation (as

all other equations of state) is not very reliable.

Table 2 gives calculated and experimental solubilities for the

carbon dioxide-air system up to 200 atm for the temperature range

133° to 163° K. Agreement is not good especially at the lower temperatures. Again, the poor agreement at the higher pressures is probably

due to the fact that the mixture is not sufficiently removed from its

critical conditions.

10
Table 1. Solid-vapor equilibria for carbon dioxide -nitrogen.

Comparison of predicted and experimental results.

^°2 T = c 304. 19 ° K P = c 72. 9 atm 0) = 0. 225

^2 T = c 126. 2° K P = c

33. 5 atm (ju = 0. 04

E = ^2 P/P s 2

Pressure, atm Calculated Experimental

Enhcincement factor at 140° K (P

s 2 = 0. 00184 atm)

10 1.36 1. 3043

20 1.96 1.9565

30 3. 18 3.0978

40 5.44 5. 4348

50 10.86 11.685

60 21.80 25.435

70 35.40 42. 609

80 54.40 60.435

90 65.10 77. 283

Enhanc ement factor at 150° K (P

s 2 = 0. 00834 atm)

10 1.32 1. 2723

20 1.80 1.6409

30 2. 52 2. 2830

40 3.62 3.3769

50 5. 70 5. 5886

60 9.15 9. 1320

70 14.02 14.316

80 20. 30 21.118

90 27. 00 28.894

100 35.00 36.147

11
Table 1 (continued)

E = y^P/p/

Pressure, atm Calculated Experimental

Enhancement factor at 160° K (P^ = 0. 03114 atm)

10 1.26 1.2580

20 1.62 1.5987

30 2.12 2.0732

40 2.84 2.7643

50 3.87 3.8694

60 5.55 5.3503

70 7.69 7.4013

80 10.52 10.013

90 13.68 13.213

100 17.42 17.070

Enhancement factor at 170° K (P^ = 0. 0993 atm)

10 1.22 1.2185

20 1.509 1.4804

30 1.885 1.8369

40 2.34 2.3041

50 3.02 2.9204

60 3.93 3.7341

70 4.83 4.8288

80 6.65 6.1229

90 8.44 7.7946

100 10.59 9.7482

12
Table 1 (continued)

s

Pressure, atm Calculated Experimental

Enhancement factor at 180° K (P = 0. 274 atm)

10 1.193 1.1861

20 1.430 1.3869

30 1.696 1.6533

40 2. 04 2. 0000

50 2.54 2.4270

60 3.06 2.9562

70 3.81 3.6022

80 4.61 4.3796

90 5.35 5.2883

100 6.75 6.4234

Enhancement factor at 190° K (P^^ = 0. 697 atm)

10 1.160 1.1458

20 1.349 1.3402

30 1.589 1.5641

40 1.849 1.8321

50 2.17 2.1576

60 2.58 2.5626

70 3. 02 3. 0206

80 3. 67 3. 5700

90 4.30 4.2415

100 5.16 4.0074

13
Table 2. Solid-vapor equilibria for carbon dioxide -air.

uu = 0. 225

UD = 0. 03

uu = 0. 04

^°2 T = c 304. 19° K P = 72. 9 atm

c °2 T = c 154. 35° K P = 49. 7 atm

c ^2 T = c 126. 2° K P = 33. 5 atm

c

essure

^2

Pr

4

atm (gauge) Calc. X 10 Expt. X 10^ '^^J

Mole fraction of CO in air at 133° K (P^ = 0. 000567 atm)

10 0.79 0.77

20 0.689 0.62

30 23.2 0.82

40

c^ liquid air

Mole fraction of CO in air at 143° K (P^ = 0. 00304 atm)

10 4.31 4.19

20 3.06 2.92

30 3.32 3.54

40 32.1 5.19

50 39.6 7.39

60 45.0 13.1

70 49.4 17.1

80 53.7 18.8

100 59.6 22.9

150 71.5 27.3

200 83.1 30.2

14
Table 2 (continued)

^2

Pressure

atm (gauge) Calc. >x 10' Expt. x

4 - 1^4 [12]

Mole fraction of CO in air at 152° K (P^ = 0. 01129 atm)

10 13.92 12.8

20 9.97 8.65

30 9. 68 8. 33

40 11.62 99.33

50 21.20 12.20

60 54.70 15.1

76 71.90 22.2

80 84.10 28.0

90 94.40 34.7

100 97.10 39.0

150 123.2 52.0

200 142.0 58.1

Mole fraction of CO in air at 163° K (P^ = 0. 0456 atm)

10 53.6 49.2

20 36. 1 33. 6

30 32. 2 28. 1 40 33. 2 28. 5

50 38.4 30,6

60 50. 1 36. 5

70 72. 43. 3

80 98.4 52.9

90 121.0 61.0

100 141. 69. 5

150 203.0 94.9

200 240.0 123.0

15
Table 3 gives calculated and experimental solubilities for the

oxygen-hydrogen system up to 50 atm for the temperature range 40-53°K,

At normal temperatures 50 atm is not considered to be a very high pressure, but in this case it is relatively very high since the corresponding

molar density is large due to the very low temperature. Since hydrogen

is a quantum fluid which does not obey the classical theorem of corresponding states, effective critical properties rather than actual critical

constants were used. The effective values were determined by obtaining optimum agreement between calculated and observed volumetric

properties of gaseous hydrogen.

The agreement between calculated and experimental solubilities

is surprisingly good. The lowest temperature (40°K) is only about 7°K

above the (true) critical temperature of hydrogen and one might expect

that the calculations would be poor because of proximity to critical conditions. However, at this low temperature level, 7°K is a large amount.

Furthermore, at 40° K the mole fraction of oxygen in the gas phase is of

- f> the order of 10 and therefore naust have a very small effect on the

critical properties of the gaseous mixture.

Finally, table 4 gives results for the solubility of naphthalene

in ethylene up to 270 atm for the temperature range 285-323°K. The

critical temperature of ethylene is 282° K. The results are only fair;

at the higher pressures the calculated and experimental values differ by

16
Table 3. Solid-vapor equilibria for oxygen-hydrogen.

Comparison of predicted and experimental results.

O, T =154. 35° K P =49.7atm uu = 0. 03

2 c c H (pseudo) T = 40° K P = 28 atm uu =

^2

Pressure, atm Calculated Expt.

[13, 14]

s -6

Mole fraction of O in H at 40° K (P = 1. 585 x 10 atm)

5 5.72x10"^ 1.2x10"^

10 5. 53 X lO"^ 1.4 X 10"

15 7.94 X

10"^ 1. 1 X

10"^

20 1.56x10"^ 3.0x10"^

30 2. 82 X

10"^

40 2.82 X

10"^

50 4.28x10"^

s -4

Mole fraction of O in H at 48° K (P = 1. 286 x 10 atm)

5 3. 78 X lO"^ 3.73 x lO"^

10 2.84 X

10"^ 2.81X lO"^

15 2.9 X

10"^ 2.8 X10'^

20 3.43 X lO"^ 3. 7 X lO"^

30 6.14x 10"^ 7.0 X lO"^

40 1.4x lO""^ 1. 65 X lO"'*

50 3.78 X lO""^ 2.70 x lO""^

17
Table 3 (continued)

^2

Pressure, atm Calculat ed Expt.

s -4

Mole fraction of O in H at 53° K (P = 8. 95 x 10 atm)

5 2. 38 X lO" 2. 2 X lO"^

10 1. 61 X

10""^

1. 62 X

10""^

15 1.46 X lO"'^ 1. 3 X

10"^

20 1. 515 X lO"'^ 1. 66 X

10"^

30 1.99 X

10"^ 2. 17 X 10"^

40 3. 11 X

10"^ 3. 15 X lO"'^

50 5. 31 X

10"^ 4. 6 X lO"^

as much as a factor of two. This system provides a particularly sensitive test of the calculational method; not only is the system rather near

its critical conditions (notice the relatively large mole fraction of

naphthalene in the vapor at the higher temperature and pressure), but,

in addition, this system is a highly asymmetric one where the solute

molecule is a very large one, much larger than the solvent molecule,

and already on the borderline of the corresponding states correlation.

18
Table 4. Solid-vapor equilibrium for naphthalene -ethylene.

Comparison of predicted and experimental results.

C ^H„ T =751.69°K P = 40. 6 atm o) = 0. 306

10 8 c c C^H, T =283. 09° K P = 50. 5 atm m = 0. 085

2 4c c

^2

4 4 r 1 51 Pressure, atm Calc. x 1 Expt . x 1

Mole fraction of C H in C H at 285° K (P^ = 3. 1 x lO"^ atm)

140.2 185 106

169.6 226 119

189.8 250 133

237.4 310.5 157

s -4

Mole fraction at 298° K (P = 1.15x10 atm)

111.4 72.9 115

126.3 88.8 141

150.6 115 172

189.8 152.2 209

255.9 208 253

s -4

Mole fraction at 308° K (P = 2. 92 x 10 atm)

126.3 96.2 170

142.2 121.0 204

145.7 127.9 217

189.8 181.0 301

237.4 234.0 366

270.3 268.0 401

19
Table 4 (continued)

^2

4 4 [ 1 5l Pressure, atm Calc. x 10 Expt. x 10 "

s -4

Mole fraction at 318° K (P = 7. 05 x 10 atm)

40.6

45.4

52.0

61. 2

70.4

79.8

98.5

117.4

121. 2

137. 2

142. 2

156.4

185. 1 208.8

237.4

270. 3

46.6

65. 2

75.1

84. 7

94. 2

103.9

113. 6

123.

132.4

142. 2

151.8

161.0

Mole fraction at 323

1.33 4.8

1.59 3.4

2.13 7.

3.52 8.3

6.51 9.8

13.2 23.9

41.4 85.4

78. 2 175.5

87.9 191. 5

125 257

139 286

168 351

220 471

262 575

306 676

356 766

J23° K (P/ = 1. 070 X 10" atm)

2. 27 13.9

5. 37 15.6

9.84 18.8

18.31 33.9

32. 3 61.4

49. 5 98.0

70. 2 157. 5

91.6 220

116.0 294

134. 378

161.0 461

185.5 563

20
Table 4 (continued)

^2

4 4 [ 1 51 Pressure, atm Calc. x 10 Expt. x 10

s -3

Mole fraction at 323° K (P = 1. 070 x 10 atm) (cont'd)

175.5 218 739

189.8 246 1075

204.2 256 1534

218.3 273 1654

237.4 298 1737

270.3 362 1798

In summary, it appears that Redlich's equation of state is capable of giving fairly good predictions of the solubility of a solid in a

compressed gas. Predicted values are probably good at least within a

factor of 2, and at conditions remote from critical they are undoubtedly

much better than that. Like all equations of state^ Redlich's is least

reliable in the critical region. The great advantage of doing solubility

calculations based on Redlich's equation is that the calculations may

easily be performed with an electronic computer and require no more

than the critical temperature, critical pressure and acentric factor of

each component.

21
5. Calculations Based on Regular Solution Theory

A dense gas is usually considered as an extreme case of a very

nonideal gas, and by using the ideal gas as a reference state, fugacity

coefficients can be calculated from volumetric properties; this approach

is the common one of dealing with dense gas mixtures, and this was the

one used in the previous section. An aulternate approach, however, is

suggested by the fact that a dense gas may be considered as an expanded

liquid. When this view is adopted the thermodynamic properties of the

mixture are calculated not from volumetric properties but from a suitable theory of liquid solutions. For nonpolar components a simple and

reasonable description of liquid mixtures is given by the HildebrandScatchard theory of regular solutions.

6. Fundamental Thermodynamic Equations

We consider the equilibrium between a solid designated by subscript 2 and a dense fluid (subscript 1) and we wish to find the solubility

X in the fluid phase. The thernaodynamic relations appropriate for this

situation are well known and are briefly summarized below.

The overall equation of equilibrium in terms of fugacities is

f/ = f/, (18)

where the superscripts s and f designate the solid and fluid phases. For

22
convenience we rewrite the equation of equilibrium:

f (pure) f RT in +RTin—^- =0 , (19)

f^ f^ (pure)

where f (pure) is the fugacity of pure subcooled liquid 2 at T, the temperature of interest. The first term in (19) is the Gibbs energy change

of going, at temperature T and pressure P, from pure solid 2 to pure

subcooled liquid 2. The second term in (19) is the partial Gibbs energy

change experienced by subcooled liquid 2 as it goes, at temperature T,

from the pure state to the solution in the fluid where its mole fraction is

X . The first term in (19) is given in terms of pure -component properties of component 2;

f (pure) ,- _-i/^m ^nriAc

RT in = Ah . s fusion

2 at T

m -> ^T ^ ^'T

m

+ V AvdP , (20)

where Ah, . , ^ is the molar enthalpy of fusion at the normal meltfusion at T

^^ m

ing temperature T and,

. _ liquid solid (21) Ac =c - c

23
. _ liquid solid ,,^. Av = V ^ - V , (22)

where c is the molar specific heat at (low) constant pressure, v is the

P

Q molar volume at temperature T, and P is the saturation (vapor) pressure of the solid at T.

The second term in (19) is given by

RT in— = RT in y + RT in x^ , (23)

f^ (pure)

where the activity coefficient y refers to pure subcooled liquid 2 at

temperature T and pressure P as the standard state. This activity

coefficient is given by the regular solution theory [2] :

RT in Y2 = v^^ $^^ (6^ - 6^)^ , (24)

where v is the molar volume and 6 the solubility parameter of pure

subcooled liquid 2, and 6 is the solubility parameter of the (fluid) solvent, all at temperature T and pressure P. Under the conditions of

interest here, x <<< 1 and therefore $ , the volume fraction of solvent,

is for all practical purposes equal to unity. When (20), (23), and (24)

are substituted into (19) we obtain, for the desired solubility x ,

24
RT in— = Ah. . X fusion

at Tm

1- ^

T T .

+ J

^AvdP + v^^$^^6^-6^)2 . (25)

^2

In (25) the important terms on the right-hand side are the first

and the last. The terms involving Ac and Av are of secondary importance.

7. Evaluation of Parameters

The heat of fusion, the normal melting point, and the molar volume of a typical nonpolar solid are usually available from the literature.

The specific heats of the pure liquid and solid are also known, at least

for the more common substances; frequently, however, they are not

known as a function of temperature in which case Ac may be treated as P

s L

a constant with little error. Figures 1 and 2 give the f /f ratios for

subcooled liquid carbon dioxide and for subcooled liquid naphthalene.

The molar volume of the subcooled liquid must be obtained by

extrapolation; since the density of a liquid at temperatures near the

melting point does not vary appreciably, this extrapolation is not a sensitive factor in applying (25). Molar volumes of subcooled liquid carbon

dioxide ajid subcooled liquid naphthalene are shown in figures 3 and 4.

Much more important is the evaluation of the two solubility

25
parameters 6 and 6 The solubility parameter is defined by

Au 1/2

where v. is the molar volume of component i at the temperature T and

pressure P where its solubility parameter is desired, and where

Au. = u.{T, «) - u. (T,v.) , (27)

u.(T, °°) is the internal energy of i at temperature T and at infinite volume (i.e. , ideal gas) and u.(T,v.) is the internal energy of i at temiperature T and molar volume v..

1 First we consider the heavy component 2. Assuming that subcooled liquid 2 is incompressible and that its saturation (vapor) pressure

is small (say, less than a few atmospheres).

/Ah ^ ^ - RT . 1/2

^2

The enthalpy of vaporization Ah must be evaluated at temperature T; since T is below the melting temperature T this enthalpy of

vaporization must be found by extrapolation using the equation

26
T'

Ah

vap at T vap at T'

f , L V) dT

\ (c - c Jn. P P

(29)

L V

where c and c are the molar specific heats of the saturated liquid

P P

and vapor, respectively. The (?.rbitrary) temperature T' is some convenient temperature in the real liquid region where the heat of vaporization is known accurately. Table 5 gives solubility parameters for

subcooled liquid carbon dioxide.

The solubility parameter 6 is for the fluid solvent, i.e. ,

the dense gas. Since the volume of the gas is highly dependent on the

pressure it follows that 6 (unlike 6 ) is a strong function of pressure.

The volume of the gas is given by

RT

v' = z , 1 P • (30)

and the compressibility factor z is given by Pitzer and Brewer [ 2] in the

form

z = z^°^ (P„, T„) +ojz^^^ (P„, T^) , ^ R R' ^ R R'

(31)

where z and z are generalized, tabulated functions of reduced pressure P„ and reduced temperature T„ and oo is the acentric factor defined R

^ R

by

27
11. 71

11. 42

11. 19

10. 89

10. 58

Table 5. Solubility parameters for subcooled liquid

carbon dioxide from pure -component data.

1/2 Temperature, °K 6, (cal/cc)

140

150

160

170

180

-^-logCl^') -1.000. <^2^ ^ c '^T_ = 0. 70 R

g where P is the critical pressure and P is the saturation (vapor) pressure at T = 0. 70.

The change in internal energy given by (27) can also be found

from Pitzer's generalized tables for the isothermal effect of pressure

on enthalpy by using the relation between internal energy and enthalpy:

Au = Ah - A(Pv) , (33)

where

Ah = h(T,P = 0) - h(T,P) . (34)

For A(Pv) we write

A(Pv) = RT Az (35)

28
with

Az = z(ideal gas) - z(P, T) . (36)

Figures 5 and 6 show reduced solubility parameters as a function

of reduced temperature and reduced pressure for two values of acentric

factor. Figure 7 gives the solubility parameter of liquid and gaseous

nitrogen over a wide range of pressure and temperature including the

critical region where the solubility parameter is very sensitive to small

changes in temperature or pressure. Data for the preparation of figure

7 were taken from Strobridge [ 16]

.

8. The Carbon Dioxide -Nitrogen System

Equation (25) was used to calculate the solubility of solid carbon

dioxide in nitrogen in the region 140-170°K and 90-200 atm. When the

calculated results are compared with experimental solubilities [ 11] the

comparison shows that the calculated results are too low by one order

of magnitude. This poor result is undoubtedly due to the fact that the

solubility x is very sensitive to the values of 6 and 6 • even a slight

change in 5 - 6 can seriously affect the solubility. This sensitivity is

necessarily always present whenever 6 and 4 are very different from

one another, as they are in this ca^e and as one may expect them to be

in any solid-dense gas equilibrium situation. It therefore appears likely

that the regular solution approach described above is not useful for

29
accurately predicting solubilities from pure-compon ent data alone.

However, (25) is a very useful equation for correlation and extrapolation

of solubility data provided at least some data for the binary system are

available.

9. Correlation of Solubility Data

Equation (25) may be considered as a semi-empirical correlating

equation if 6 , the solubility parameter of the solute, is treated as an

empirical parameter, dependent on temperature but not on pressure.

For example, table 6 gives the solubility parameter of naphthalene as

calculated from the regular solution equation and from solubility data in

saturated liquid carbon dioxide [ 17]; figure 8 gives the solubility parameters of carbon dioxide which were used. The solubility parameter of

naphthalene calculated in this way is close to, but not identical with, the

solubility parameter calculated from pure -component properties as

given by (28) and (29). Figure 9 shows the solubility parameter of naphthalene as calculated from solubility data in compressed ethylene at

several temperatures and pressures. These solubility parameters are

somewhat lower than those determined from solubility data in carbon

dioxide. Figure 9 shows that 5- is essentially invariant with pressure

which is consistent with the derivation of (25). The pressure -sensitive

term in (25) is 6 , the solubility parameter of the gaseous solvent, as

indicated by figures 5 and 6.

30
Table 6. Solubility parameters for subcooled liquid

naphthalene from solubility measurements

in liquid carbon dioxide.

1/2 Temperature, °C 6, (cal/cc)

-20

-10

10

20

25

12. 20

11. 85

11. 46

10. 97

10. 33

9. 90

Equation (25) with figures 5 and 6 correctly predicts that the

solubility of a solid in" a dense fluid changes very rapidly with temperature and pressure in the vicinity of the critical temperature and pressure of the solvent; near the critical point, 5 is very sensitive to small

changes in temperature and pressure as shown by figures 5, 6, and 7.

Equation (25) indicates that near the critical point of the solvent a small

increase in pressure or a small decrease in temperature can increase

the solubility of a solid by one or more orders of magnitude. Such

effects have been observed in the nitrogen-hydrogen system [ 13] and in

the carbon dioxide-air system [ 18].

Application of (25) for the correlation of solid- dense gas solubility data was mentioned by Omar and Dokoupil [ 19] in their study of

31
the solid nitrogen-liquid hydrogen systern.

10. Conclusion

This study shows once again the validity of a remark made by

Professor Pitzer some years ago when he said that while our understanding of the liquid state is small, our understanding of the dense-gas

state is still smaller. In both the liquid state and in the dense -gas state

we are held back by our lack of knowledge concerning intermolecular

forces in multi-body assemblies. However, in the liquid state the geometry of the assembly is at least reasonably regular and the density of

a liquid (well below the critical point) is only a mild function of temperature and a very weak function of pressure. As a result it is not surprising that various lattice -type theories of liquids and liquid mixtures

are reasonably successful, at least for approximation purposes; the

simplification that a liquid is a "loose" solid is a fairly good one for

describing thernaodynamic properties of liquid solutions. However, in

a dense gas, this simplification is no longer applicable; the density is a

very strong function of temperature, pressure, and composition- -especially at conditions not far removed from the critical.

This study has briefly considered two approaches to the problem

of solid-dense fluid equilibria. The first approach considers the dense

fluid as a highly nonideal gas and starting out from the ideal- gas limit,

attempts to correct for the nonideality by pure -component empirical

32
constants, generalized by the theorem of corresponding states. This

approach is only moderately successful primarily because the extension

to mixtures of corresponding states correlations for pure components is

based on fixed but essentially arbitx*ary mixing rules. Unfortunately,

these rules are not invariant with the system under consideration nor

with the conditions of temperature, pressure, and composition.

The second approach considers the dense fluid as a liquid mixture and the solid solute as a hypothetical pure liquid whose properties

are estimated by extrapolation. This extrapolation necessarily becomes

increasingly uncertain as the temperature under consideration departs

more and more from the melting temperature of the solid. The process

of dissolving the hypothetical liquid in the "liquid" solvent is treated as

a constant volume process, and again, somewhat arbitrarily, the intermolecular forces between solute and solvent are calculated in a fixed

manner from those of the pure components by a geometric -me aui assumption. These various simplifications, especially the last one, place a

heavy strain on the difference between theory and reality. What clearly

is needed is a better theory of fluids in the vicinity of the critical point;

such theory is now under active development by workers in statistical

mechanics and will, in time, be useful for increasing our ability to

describe solid-dense gas equilibria.

33
11. Acknowledgment

The author is grateful to Mr. Pi-teh Shen for extensive assistance in the calculations and in the preparation of the figures.

34
12, References

I] McKinley, C. , Cryogenic Technology, R. W. Vance, ed. , (John Wiley & Co. , New York, N. Y. ), 1963, p. 95.

2] Lewis, G. N. , and M. Randall, Thermodynamics (2d. ed. , revised by K. S. Pitzer and L. Brewer)(McGraw Hill Book

Co., New York, N. Y. ) 1961.

3] Beattie, J. A., Chem. Rev. 44, 141 (1949).

4] Comings, E, W. , High Pressure Technology, (McGraw Hill

Book Co. , New York, N. Y. ), 1956.

5] Martin, J. J., and Y. C. Hou, A. I. Ch. E. Journal 2., 142(1955).

6] Joffe, J., Ind. Eng. Chem. 40, 1738(1948).

7] Prausnitz, J. M. , A. I. Ch.E. Journal _5, 3(1959).

8] Prausnitz, J. M. , and A. L. Myers, ibid . , 9, 5(1963).

9] Redlich, O. , and J. N. S. Kwong, Chem. Rev. 44, 233(1949).

1 0] Redlich, O. , F. J. Ackerman, R. D. Gunn, M. Jacobson, and

S. Lau, to be published in J. Chem. Eng. Data.

II] Smith, G. E. , R. E. Sonntag, ajid G. J. Van Wylen, Advances

in Cryogenic Engineering, Vol. 8, K. D. Timmerhaus, ed. , (Plenum Press, Inc., New York, N. Y. ), 1963, p. 162.

12] Webster, J. J., Proc. Roy. Soc. (London), A214, 61 (1952).

13] Dokoupil, Z. , Progress in Low Temperature Physics, Vol. Ill,

C. Gorter, ed. , (Interscience Publishers, Inc., New York, N. Y. ) 1961, p. 454.

[ 14] McKinley, C. , J. Brewer, and E. S. J. Wajig, Advances in

Cryogenic Engineering, Vol. 7, K. D. Timmerhaus, ed. , (Plenum Press, Inc., New York, N. Y. ), 1962, p. 114.

35
[15] Diepen, G. A. M, , and F. E. C, Scheffer, J. Phys. Chem.

^, 575 (1953).

[16] Strobridge, T. R. , NBS Technical Note No. 129, P. B. 161630

(Jan. 1962).

[17] Quinn, E. L. , J. Am. Chem. Soc. ^, 672(1928).

[18] Geist, J. M. , personal commiinication.

[19] Omar, M. H. , and Z. Dokoupil, Physic a 28, 461 (1962).

36
140 160 180 200

TEMPERATURE, ^K

220

Fig. 1. Fugacity ratio for solid and (subcooled) liquid carbon dioxide.

k

37
-30 -20 -10 10 20 30 40 50 60 70

T. ^'C

Fig. 2. Fugacity ratio for solid and (subcooled) liquid naphthalene,

38
00 r^ CD lO sr ro CJ

ro ro ro ro ro ro ro

o

ou

a

u

T3

0) I—

I

O

o

Xid

CO

O

CO

ci

•1-1

ro 9iouj/oo *2a 3wnnoA avnow

39
C.C.

124 h-

-30 -20 -10 10 20 30 40 50

T, *»C

Fig. 4. Molar volume of (subcooled) liquid naphthalene.

40
23456789 10

Fig. 5. Solubility parameters for dense gases with an acentric factor of zero. 41
Fig. 6. Solubility parameters for dense gases with an acentric factor of 0. 075.

42
~——\ 81 O QQOOQO

o

^1o

+J

Oh

1—

(

o

in

2/,

(33/ 1 DO) "S

43
8

Fig. 8. Solubility parameter for saturated liquid carbon dioxide.

44
200 220 240

PRESSURE, atm.

280

i| Fig. 9. Solubility parameter of naphthalene calculated from solubility data in

gaseous ethylene.

45 GPO 861 -348


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