THERMODYNAMICS OF SOLID SOLUBILITY IN CRYOGENIC SOLVENTS G .
T.
P R E S T O N
A N D
J.
M. P R A U S N I T Z
Department of Chemical Engineering, University of California, Berkeley, Calif. 94720 A thermodynamic framework is given for estimating the solubilities of solids in cryogenic liquid solvents. The activity coefficient of the solid in solution is referred to the pure subcooled liquid; activity coefficients are calculated from Scatchard's equation with a correction to Hildebrand's geometric-mean assumption for cohesive energy density. Alternatively, activity coefficients may be calculated from a cell-theory partition function. The methods discussed are applicable to single and mixed solvents.
IN
THE design of cryogenic processes, it is frequently necessary to estimate the solubility of solids in liquid solvents at low temperatures. Reliable experimental studies of such solubilities are not plentiful and therefore, in order to make predictions with confidence, it is desirable to interpret and correlate existing data within a thermodynamic framework. In this work we discuss such a framework and present results for typical cryogenic systems, including multicomponent systems. The results of our work suggest that, if good judgment is used, it is possible to make reasonable estimates of solid solubilities as required for many typical low-temperature processes. Our work is somewhat similar to but more general than that of Cheung and Zander (1968). We are concerned with solid-liquid equilibria for which the solid phase is pure solute and the liquid phase is a saturated solution of the solute in the solvent. For such systems the equation of equilibrium is flS
=
7 2
x?f?i
temperature, T,; second, the entropy of fusion a t T,, Sf,is negligibly different from that a t T,; and third, since the two terms containing I C p are opposite in sign, they tend to cancel each other for temperatures not too far removed from the triple point. With these simplifications, Equation 2 becomes
Figure 1 shows a plot of Equation 3. Values of AS,/R and T , for several typical substances are presented in Table I. Solubility in an ideal mixture can be calculated from Equations 1 and 3 upon assuming that y2 = 1. However,
(1)
where subscript 2 refers to the solute, f s is the fugacity of pure solid, f " is the fugacity of pure subcooled liquid, x is the mole fraction in solution, and y is the activity coefficient in solution referred to the pure subcooled liquid, all at the temperature of the system, T . The fugacity ratio fLs/fk is evaluated from properties of the pure solute by the relation
where T , is the triple-point temperature, L3ft is the molar entropy of fusion at T,, and I C p is the molar specific heat of the liquid solute minus that of the solid solute. Equation 2 is derived in many texts-see Hildebrand and Scott (1964). Equation 2 includes the assumption that I C ~is independent of temperature. Three further simplifications are commonly made. First, a t normal pressures T , is nearly equal to the normal melting 264
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
\
-7 I
'
1
I
I
I
I
2
3
4
5
6
Tm/ T
Figure 1. Activities of solid solutes referred to their pure subcooled liquids
Table I. Pure Component Properties"
Cc Mole
p,
L,,
I
T,, OK.
'S, R
83.8 63.2 68.1 54.3
1.69 1.37 1.47 0.99
Methane Acetylene Ethylene Ethane Propylene Propane Butane Pentane Isopentane (2-methylbutane) 1-Hexene Hexane Benzene Toluene p-Xylene Cyclohexane Methylcyclohexane Carbon dioxide Ammonia Sulfur dioxide
90.7 192.4 104.0 89.9 87.9 85.5 134.8 143.4 113.2
1.25 2.55 3.88 3.82 4.11 4.96 4.16 7.04 5.47
133.3 177.8 278.7 178.2 286.4 279.7 146.6 216.5 195.4 197.7
8.43 8.81 4.25 4.47 7.19 1.15 5.54 4.86 3.48 4.50
Hydrogen sulfide
187.6
1.52
Component Argon Nitrogen Carbon monoxide Oxygen
'trans
'
'K
'trans
35.5 61.5 43.8 23.8 20.4
0.78 1.24 2.04 0.47 0.45
107.6
2.31
186.1
4.36
103.5 126.2
1.79 0.48
T,, OK.
Atm.
150.7 126.2 132.9 154.8
48.0 33.5 34.5 50.1
75.2 90.1 93.1 74.4
-0.002 0.040 0.041 0.021
191.1 308.7 283.1 305.6 365.1 370.0 425.2 469.8 461.0
45.8 61.7 50.5 48.3 45.4 42.0 37.5 33.3 32.9
98.7 112.9 124 148 181 200 255 311 308
0.013 0.186 0.087' 0.105 0.143 0.152 0.201 0.252 0.206
504.0 507.9 562.6 594.0 618.8 553.2 572.1 304.2 405.6 430.7
31.1 29.9 48.6 40.0 33.9 40.0 34.3 72.8 111.5 77.7
356 368 260 320 378 308 344 94.1 72.5 122
0.283" 0.290 0.215 0.233' 0.293b 0.211' 0.237' 0.225 0.250 0.246'
373.6
88.9
97.7
(*
0.100
a With few exceptions, hydrocarbon data are from A P I Project 44 (19641, inorganic data are from Din (1961), and values of Pitzer's acentric factor and some critical constants are from Reid and Sherwood (1966). Other references are Hoge, 1950: Cryogenic Engineering Neus, 1968: Lercis and Randall, 1961; Hildebrand and Scott, 1962; Hodgman, 1959; Perry et al., 1963; and Large, 1961.'Calculated from vapor pressure data.
for many systems of industrial interest, y 2 = 1 is a bad assumption-for example, y~ for hydrogen sulfide in methane is of the order of 100. The effect of neglecting the two AC, terms is not serious at moderate values of T,/T. For example, for a system at T,/T = 1.2 with methane as the solute, inclusion of the AC, terms increases the calculated solubility by 2%. With ammonia as the solute, the correction is 6 % , and for carbon dioxide the correction is 3.5%. Some substances undergo solid-phase transitions from one crystal structure to another. If the solute. of interest is known to have such a transition, and if the transition temperature, Ttrans,is greater than T , the temperature of interest, another term must be added to Equation 3. The appropriate relation is then
The exchange energy density is related to the cohesive energy densities, C,, by where C11 is the cohesive energy density of the solvent, CJZis that of the subcooled liquid solute, and Clr is a cohesive energy density describing the intermolecular forces between solvent and solute. For Clr we use
CI? =
(1 - I I ? )
(cl!cL2)1
where L I Z is a constant of the order of of a given solute-solvent pair, and, to tion, independent of temperature and introduce the solubility parameter 6 6, and 7, we obtain
(7)
lo-*, characteristic a close approximacomposition. If we into Equations 5 ,
Temperatures and entropies of transition are given in Table I for several substances known to exhibit solidphase transitions.
where 61 = (Cl1)' ', 6 2 = (CZ2)'', and R = 1.987 cal./moleOK. When 112 = 0, Equation 8 reduces to the familiar Scatchard-Hildebrand equation.
Activity Coefficient
Subcooled Liquid Volume and Solubility Parameter
For nonpolar systems, an approximate method for obtaining liquid-phase activity coefficients is provided by Scatchard's equation (Hildebrand and Scott, 1964) for a binary system:
(5) where uz is the molar volume of the subcooled liquid solute, *I is the volume fraction of solvent in the solution, and A12 is the exchange energy density.
T o estimate liquid molar volumes and solubility parameters for the solute and the solvent, we used the correlations of Lyckman et al. (1965) extrapolated to low reduced temperatures in order to include the subcooled liquid region. For liquid volumes we used Lyckman's tabulated reduced volumes a t moderate reduced temperatures as well as an empirical correlation of Prausnitz and Chueh (1968) a t lower reduced temperatures. The results are shown in Figures 2 and 3 and in the Appendix. To use Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
265
0.45
0.40 v "C
0.35
0.3C I 0.2
I
I 0.4
0.3
1 0.5
T/Tc
I 0.6
I
0.7
Figure 2. Reduced molar volumes for stable and subcooled liquids See A p p e n d i x for t a b u l a t e d results
E
& .
0.4
1.0
0.9 I
0.2
1
I 0.3
1
I
I
I
I
0.5
04
/
0.6
I
I 07
T / 5,
Figure 3. Reduced solubility parameters for stable and subcooled liquids See A p p e n d i x for t a b u l a t e d results
these results, critical constants and acentric factors are required. Some critical temperatures, pressures, and volumes, and acentric factors are shown in Table I.
gas mixtures. Nevertheless, for simple, noncomplexing molecules of similar size, k12 and L12 are nearly the same, and for other binary pairs it is often possible, using good judgment, to predict roughly how Ll2 differs from h12. Prausnitz (1969) reports k12's for a large number of binary systems. When no binary data are available, it is still possible in many cases to estimate a reasonable llr by analogy with results for chemically similar systems. For cryogenic systems 11, is of the order of l o - * and usually is positive. If solute and solvent are similar chemically and of nearly the same molecular size, L12 is near zero. I n general, L12 increases with differences in molecular size and chemical nature. However, if specific chemical forces can act between solute and solvent-e.g., hydrogenbonding or acid-base complexing- they tend to lower l l L . Solubility Calculations
The solubility of solute 2 in solvent 1 is determined by combining Equations 1, 3 or 4, and 8. Since x2 and (P2 are unknown, the calculation is trial and error but converges rapidly. We calculated the solubility in three ways: first, assuming llr = 0; second, using llr = h12, the geometric-mean correction obtained from gas mixture data; and third, using an empirical I l l which best reproduces the experimental solubility data. Typical results are shown in Figures 4 and 5 . The experimental data and the best value for 112 over the indicated temperature range are summarized in Table 11. In many cases the best 112 obtained from solid-liquid data is close to the gas-mixture h12. Notable exceptions are systems which form complexes (carbon dioxide or hydrogen sulfide in an unsaturated hydrocarbon), where the recommended l l ? is significantly less than h12. For binary systems which form solid solutions, Equation 1 is not valid, since the solid phase is not pure solute. For the systems methane-argon, argon-nitrogen, and methane-nitrogen, solid-mixture data are available; for these systems we calculated the solubility from
x,sf?F= y ? x 2 f 2 L
(14
Geometric-Mean Correction
Activity coefficients calculated from Equation 8 are sensitive to L12, which is a measure of deviation from the geometric-mean assumption. Since Llr is a binary constant, it cannot be estimated from pure-component data alone. When possible, Ilr should be determined empirically from solid-liquid equilibrium data for the system of interest, as we discuss below. However, other kinds of mixture data can sometimes be used to obtain a useful estimate of l12. Liquid-phase activity coefficients, obtained from vaporliquid equilibrium data, can give lI2 by direct application of Equation 8--for example, the methane-oxygen vaporliquid data of Hodges and Burch (1967) yield L12 = 0.05, in agreement with our results obtained from solid-liquid data. If no liquid-phase information is available, we can turn to gas-phase data. I n their study of gas-phase mixtures, Chueh and Prausnitz (1967) suggested that the characteristic temperature T , is given by the geometric mean corrected by the factor (1 - hlz):
T,
180 I
170 I
160 I 0 0
O K
150
140
130
I
I
I
DATA OF C H E U N G 8 Z A N D E R DATA OF C L A R K 8 DIN C A L C U L ATED
0.01 0.008 0.006 0.0040.002 -
I
I
I
I
I
0.55
0.60
0.65
0.70
0.75
102/T,
The binary parameters hlr and 112 are not identical, since II? reflects several liquid-phase effects that are absent from 266
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
1 0.E
OK-'
Figure 4. Solubility of carbon dioxide in ethane
Table II. Summary of Literature Data
Man. Error.
Best l,?
T , K."
Methane-argon
0.02
72-90
2
Argon-nitrogen
0.00
70-83
1
Methane-nitrogen
-0.02
70-90
5
Ethane-nitrogen Ethylene-nitrogen Acetylene-nitrogen Propylene-nitrogen Carbon dioxide-nitrogen Argon-oxygen
0.01 0.06 0.07 -0.01 -0.18 0.06
77 70-85 77 65-80 77 63-83
...
Methane-oxygen
0.05
70-90
3
Ethane-oxygen
0.03
60-75
20
Ethylene-oxygen Acet ylene-oxygen
0.06 0.18
75-90 90
...
Propane-oxygen Propylene-oxygen
0.02 0.05
60-70 67-75
115 30
Butane-oxygen Carbon dioxdp-oxygen Argon-methane
0.08 0.03 0.00
90 90 72-83
... ...
0.10 -0.02
105-135 110-140
40 10
Hydrogen sulfidemethane Acetylene-ethane Carbon dioxide-ethane
0.04 0.11 0.08
120-150 150-170 110-170
40 30 15
Hydrogen sulfide-ethane Acetylene-ethylene
0.07 0.02
120- 160 100-1 60
30 25
Carbon dioxideethylene Hydrogen sulfide-ethylene Carbon dioxide-acetylene Carbon dioxide-propane
0.00 -0.01 -0.02 0.08
140-170 120-185 175-190 120-200
10 15 3 10
Hydrogen sulfide-propane Carbon dioxide-propylene
0.06 0.01
140- 170 130-210
45 20
-0.005 0.09
130-180 140-200
5 20
0.05 0.04 0 0 0.01' 0.01' 0.01'
140-170 170-188 26-33
25
Solute-Soluent
Acetylene-methane Carbon dioxide-methane
Hydrogen sulfide-propylene Carbon dioxide-butane Hydrogen sulfide-butane Acetylene-carbon dioxide Nitrogen-hydrogen Nitrogen-oxygen Methane-ethane Methane-ethylene E thane-ethylene
35
... 75
... 25
15
2
3
Reference Van't Zelfde et ai. (19681, Fedorova (1938) Long and DiPaolo (1963). Din et al. (1955). Fedorova (1938) Omar et a/.!1962), Fastovskii and Krestinskf (1941) Cox and DeVries (1950) Tsin (19401 Fedorova (1940) Tsin (1940) Fedorova (1940) Din et a/. (1955), Fedorova (1938) McKinley and Himmelberger (1957), Fastovskii and Krestinskii (1941) McKinley and Wang (1958), McKinley and Himmelberger (1957) McKinley and Wang (1958) Kanvat (19581, McKinley and Himmelberger (19571, Fedorova (1940) McKinley and U'ang (1958) McKinley and U'ang (19583. Tsin (1940) Karwat (1958) Fedorova (1940) Van't Zelfde et a/.(1968), Fedorova (1938) Seumann and Mann (1969) Cheung and Zander (1968), Davis et al. (1962), Sterner (19601, Brewer and Kurata (1958), Donnelly and Katz (1954) Cheung and Zander (1968) Clark and Din (1950) Jensen and Kurata (1969), Cheung and Zander (1968). Clark and Din (1953) Cheung and Zander (1968) Neumann and Mann (1969), Clark and Din (1950) Clark and Din (1953) Cheung and Zander (1968) Clark and Din (1950) Jensen and Kurata (1969), Cheung and Zander (1968) Cheung and Zander (1968) Cheung and Zander (19681, Haselden and Snowden (1962) Cheung and Zander (1968) Jensen and Kurata (1969) Cheung and Zander (1968) Cheung and Zander (1968) Clark and Din (1950) Omar and Dokoupil (1962)
Temperature range recommended for use of tabulated lln.
bh'an. error, % = man
{ 1%.
- 3,calcd I 3.obsd
obsd
100)
For mixed s o l m t calculations, values of 111 calculated from experimental results quoted by Prausnitz (1969)
Ind. Eng. Chem. Process Des. Develop., Vol. 9 , No. 2 , 1970
267
T
180
160
140
O K
I20 DATA OF CHEUNG 8 ZANDER CALCULATED
L
\
Figure 5. Solubility of carbon dioxide in propane
instead of Equation 1. The importance of taking the solid phase composition into account is illustrated in Figure 6. The lI2's given in Table I1 for these systems were obtained using Equation l a . Argon-oxygen probably forms solid solutions, but in the absence of solid-mixture data Equation 1 was used for this system. It is difficult to explain the l I 2 of -0.18 for carbon dioxide in nitrogen. Inclusion of the A C p terms in Equation 2 only raises lI2to -0.13. Taking into account the quadrupole moments of carbon dioxide and nitrogen could only lower Ll2 further. Although our solubility parameter correlation is uncertain at low reduced temperatures, our cal-
T, T
1.0
85
80
culated lI2 of 0.03 for carbon dioxide-oxygen a t 90°K. is not unreasonable. Fedorova's data for carbon dioxideoxygen and carbon dioxide-nitrogen a t the solvent's normal boiling point are likely to be correct, since they have been confirmed by McKinley (1969). Equation 8 indicates that the calculated solubility is sensitive to the values of u2, 61, 6 2 , and 1 1 2 . We estimate the accuracy of our liquid volume correlation to be 2% or better. The uncertainty in the solubility parameter from our correlation is about 10% a t the lowest reduced temperatures and about 2% a t high reduced temperatures; a t low reduced temperatures and for very low solubilities, in an extreme case, log x L could be in error by 30%. The effect of changes in l l L is illustrated in Figures 4 and 5 . For binary systems listed in Table 11, the uncertainties in u and 6 are absorbed in 1 1 2 , since Ill was determined from experimental solubility data using our generalized correlations for u and 6. I t is very important therefore that our lI2's always be used together with our u and 6 correlations. We have mentioned four methods for determining 1 1 2 : (1) from solid-liquid data, (2) from vapor-liquid data, (3) from gas-mixture data, and (4) an intelligent guess. Our results--for example, Figure 5-show that even an intelligent guess for L I Z usually improves the accuracy of a solubility prediction relative to that obtained with lI2 = 0. Equation 8 cannot be used to calculate solubilities near the critical temperature of the solvent where the excess volume of mixing is not negligible as assumed by regular solution theory. Nevertheless, solubility in the critical region can be estimated if we make the reasonable assumption that In y varies inversely with temperature a t constant composition. We first calculate In y 2 a t a lower temperature and then use the 1/ T temperature dependence to calculate in y? a t other temperatures. This procedure was used for calculating the solubility of nitrogen in liquid hydrogen in the range 26" to 33°K. (The critical temperature of hydrogen is 33.0"K.) In these calculations 112 = 0. Calculated and experimental solubilities are shown in Figure 7 . We also applied this technique to calculate
OK O K
70
75
65
I
T , "K
0.8
0
DATA OF O M A R 8 DOKOUPIL
- CALCULATED xir= 1
0.6
\
XAr
0.4 0 0 A
0.2
-
-
DATA OF FEDDRDVA (19381 DATA OF LONG 8 DIPADLO (19631 DATA OF DIN ET AL. (19551 CALCULATED, PI2 - 0 0
Tc li
5-
-
00
I 1.2
I I .3
I
I I .4 lO'/T,
1.5
I
I
I
I
I
OK-'
Figure 6. Solubility of argon in nitrogen Effect of solid-phase composition
268
3
1.6
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
Figure 7. Solubility of nitrogen in hydrogen near critical temperature of hydrogen
the solubility of carbon dioxide in methane in the range 110" to 190"K., using 1 1 , = -0.02; the critical temperature of methane is 191.1"K. The results are shown in Figure 8.
I 0
I
I
SMOOTHED DATA CALCULATED, h
b=.f ( S E E TEXT1
SCATCHARD-HILDEBRAND
Extension to Solvent Mixtures
For multicomponent mixtures, the general form of Equation 5 is (Hildebrand and Scott, 1964):
where A,, = 0 and A , = (6, - 6 ) 2 + 21,,6,6,. [Hildebrand and Scott (1964) erroneously omit the factor l / 2 with A , . ] For the case of a solute, 2, in a mixture of two solvents, 1 and 3, Equation 9 becomes
In
y2 = 1,
:-[AI?@;
RT
+ Ai?@
