THERMODYNAMICS OF SOLID CARBON DIOXIDE SOLUBILITY IN LIQUID SOLVENTS A T LOW TEMPERATURES A.
L. M Y E R S 1 A N D J .
M . P R A U S N I T Z
Cryogenic Engineering Laboratory, National Bureau of Standards, Boulder, Colo., and Department of Chemical Engineering, University o j California? Berkeley, Calif.
A molecular thermodynamic method i s described for correlating solubility data for solid carbon dioxide at temperatures below its triple point, 2 16.56' K. The activity coefficient for carbon dioxide in liquid solution i s referred to pure subcooled liquid carbon dioxide at the tempeiature of the solution and i s related to pertinent intermolecular forces by a generalization of Scatchard's equation. The significant quadrupole moment of carbon dioxide as well as acid-base complex formation with unsaturated hydrocarbons i s taken into account. The correlation should be useful for predicting solid carbon dioxide solubility in various solvents as well as in solvent mixtures. HE solubility of solid carbon dioxide in liquid solvents at T l ~ w temperatures is of interest in the design of cryogenic processes. Reliable experimental data are not plentiful and cover only a limited range of conditions; it therefore appeared desirable to interpret and correlate the available data within a molecular-thermodynamic framework in order to facilitate the rational prediction of the solubility of solid carbon dioxide under conditions which have not been studied experimentally. Such a correlation is presented in this paper.
Basic Thermodynamic Equations
We consider the equilibrium between pure solid carbon dioxide and a saturated solution of carbon dioxide in a liquid solvent. Let subscript 1 refer to the solvent and subscript 2 to carbon dioxide. T h e equation of equilibrium is JZS
=
y:!x:!f2=
(11
where f s is the fugacity of the pure solid, f " is the fugacity of the pure subcooled liquid, x:! is the solubility (mole fraction) of carbon dioxide in the solvent, and y:!is the activity coefficient of carbon dioxide in the liquid solution (referred to the pure subcooled liquid), all a t the system temperature, T . T h e ratio of fugacities f 2 S and f?" can be evaluated from the known properties of pure carbon dioxide by the relation
Thermodynamic properties of carbon dioxide are summarized by Din (5). T h e data suggest that ACp is not constant but depends weakly on temperature; this variation with temperature was taken into account by using measured heats of sublimination and the generalized tables of Pitzer (70) for the heat of vaporization of nonpolar liquids. T h e fugacity ratio f&f2" for carbon dioxide from 140' K. to the triple point is shown in Figure 1. For an ideal solution ( 7 2 = 1) solubility x z can be calculated readily by using Equation l and the results shown in Figure l ; in the ideal case, the solubility falls from complete solubility a t the triple point temperature to x 2 = 0.09 a t 140' K. Activity Coefficients
In nonpolar solvents the activity coefficient of carbon dioxide can be described to a good approximation by Scatchard's equation (8)
RT In y:! = A 1 2 v 2 where oxide,
(3)
is the molar volume of subcooled liquid carbon di@1 is the volume fraction of the solvent, and , 4 1 2 is the UP
f at
TO
where To is the triple point temperature (216.56' K.), and ACp is the molar specific heat.of liquid carbon dioxide minus that of solid carbon dioxide. A derivation of Equation 2 is given in many' physical chemistry texts and in particular by Hildebrand and Scott ( 8 ) .
1
Present address, University of Pennsylvania, Philadelphia, Pa.
TEMPERATURE,
O K
Figure 1. Fugacity ratio for solid and subcooled liquid carbon dioxide VOL. 4
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209
exchange energy density. (Here we consider single solvents only. Solvent mixtures are considered later.) T h e exchange energy density is related to the cohesive energy densities by
Aiz = Cii
+ Czz - 2C12
(4)
where CI1 is the cohesive energy density of the solvent, CZZis that of subcooled carbon dioxide, and CIZ is the cohesive energy density corresponding to the intermolecular forces acting between solute and solvent. Frequently it is assumed that Clz is given by the geometric mean between CII and CZZ, in which case Scatchard's equation reduces to the familiar Scatchard-Hildebrand equation. This assumption. however, is reasonable only if all the intermolecular forces (solutesolute, solute-solvent, solvent-solvent) are limited to dispersion (London) forces; for solutions of carbon dioxide this assumption does not hold, since carbon dioxide has a significantly large quadrupole moment.
38
37 -
E
2 36V
p35W
5
5
0 1
34-
a 33-
Q:
-1
TEMPERATURE, Figure 2.
O K
Molar volume of subcooled liquid carbon dioxide
Effect of Quadrupole Forces
T h e cohesive energy density of a liquid is defined as the energy of complete vaporization divided by the volume :
where Aut = u t (vapor a t zero pressure and T ) - ut (saturated liquid a t T ) , and ut is the molar liquid volume. For carbon dioxide solutions, where forces due to quadrupoles are not negligible, it is necessary to split the cohesive energy density into two parts :
Cft =
CtfD
+ CtfQ
(6)
where the superscripts refer to the contributions from dispersion forces and quadrupolar forces. If we retain the use of a geometric mean for that part of the cohesive energy density which is due to dispersion forces, we have :
TEMPERATURE,
O K
Figure 3. Cohesive energy density for subcooled liquid carbon dioxide
where C12Qis the cohesive energy density corresponding to quadrupole-quadrupole interactions between solute and solvent. Thus, for solutions where quadrupolar forces may not be neglected, the interchange energy density is given by
Al2
=
Cil
+ CZZ- 2 [ d C 1 1 D C ~ 2+DC I Z ~ ]
(8)
I n any solvent whose quadrupole moment is negligibly small,
CliQ = C12Q = 0 ; thus CiiD = C11. Evaluation of Cohesive Energy Densities
T h e potential energy for the interaction of two identical, ideal quadrupole moments has been calculated by Buckingham (7).
3 QI"
UftQ = - ytt5
4
F(6')
(9)
where Q t is the quadrupole moment of molecule z , r t f is the distance between the two quadrupole moments, and F(6') is a function of their mutual orientation. When the quadrupole energy is small relative to the thermal energy, a statistical average over all possible mutual orientations of the two quadrupole moments (for a fixed value of rtf) is of the form
I
I40
I
I
I
I60 '80, TEMPERATURE, K
I
200
Figure 4. Dispersion and quadrupole cohesive energy densities for subcooled liquid carbon dioxide 210
I&EC FUNDAMENTALS
where cy is a numerical constant and k is Boltzmann's constant. We now assume additivity of intermolecular potentials and that r I I is proportional to the cube root of the liquid volume. We then obtain for the quadrupole cohesive energy density:
considering this contribution is considerable. For example, consider the solubility of solid carbon dioxide in a nonpolar (and nonquadrupolar) solvent having the (typical) value of C l l = 60 cal. per cc. for its cohesive energy density a t 160' K. If we d o not take proper account of the presence of quadrupolar forces, we calculate '4 according to the Hildebrand formula A12 =
where p is a numerical (dimensionless) constant. For the potential energy between two different, ideal quadrupole moments separated by distance rij, Buckingham has shown that
We therefore find that
where P is the same numerical constant as in Equation 11. For u I j we assume the relation
To evaluate p, we uiilize carbon dioxide solubility data in liquid ethane (.3) and in liquid cyclopropane (6). Both of these solvents have negligible quadrupole moments (thus C12Q= 0) and as saturated hydrocarbons they d o not form acid-base complexes with carbon dioxide. For a given temperature the activity coefficient, y2, was calculated from Equation 1 using the experimental solubility, x p , and the fugacity ratio shown in Figure 1. T h e exchange energy density, A12, was then calculated from Equation 3 using the molar volumes for subcooled liquid carbon dioxide sho\vn in Figure 2 ; these volumes were calculated by Lydersen's method ( 9 ) . T h e cohesive energies, CI1, for each solvent were calculated from known thermodynamic properties of the solvents using Equation 5 and the total cohesive energy, ( 2 2 2 , for subcooled liquid carbon dioxide was also calculated from Equation 5 using liquid heats of vaporization as determined from Pitzer's tables and liquid volumes from Figure 2 ; the total cohesive energy density, C22, for carbon dioxide is shown in Figure 3. Finally, /3 was found by substituting Equations 6 and 11 into Equation 8. resulting in a value of 1.92 i 0.25. T h e most recent value of the quadrupole moment of carbon e.s.u. (2). dioxide was used; it is 4 1 X Cohesive energy densities for subcooled liquid carbon dioxide are given in Figure 4 ; the contributions of dispersion forces and quadrupole forces are indicated separately. As expected, the contribution from dispersion forces is the larger one but the quadrupole term is by no means negligible, especially a t lower temperature, as indicated by the theoretical Equation 11. Equations 1. 3, and 8 are the key equations for calculating the solubility of solid carbon dioxide in nonpolar solvents. The necessary parameters for carbon dioxide are given in Figures 1. 2, and 4. (-The cohesive energy density shown in Figure 3 is the xim of those given in Figure 4 . ) Splitting the cohesive energy density into a dispersion part and a quadrupole part has a n important effect on the calculated solubility. Even though the contribution from quadrupole forces is not large. the error incurred by not separately
[Cii'/' - C2z1/']'
Assuming = .te: we then obtain for the solubility x2 = 0.067. However, if we account properly for quadrupolar 0.016. forces (Equation 8),we obtain for the solubility 1 2 : Effect of Acid-Base Complex Formation
Parameter A 1 2in the Scatchard equation is a function of the intermolecular forces which exist in the liquid solution. Equation 8 takes into account dispersion forces and forces due to quadrupole-quadrupole interactions, but in some cases allowance must also be made for specific (or chemical) forces which lead to the formation of a n acid-base (or charge-transfer) complex. Carbon dioxide has acidic properties and therefore it has a tendency to form complexes in basic solvents. 'This tendency, which is also called solvation, results in increased solubility. Unsaturated hydrocarbons are basic, because of the high polarizability of r-electrons; this property accounts, for example, for the abnormally high solubility of carbon dioxide gas in benzene and toluene noted by Hildebrand and Scott (7). T h e solubility of solid carbon dioxide is also larger in unsaturated hydrocarbons than in saturated hydrocarbons and for the same reason. Unfortunately there is no suitable theoretical framework for predicting or even correlating the strength of acid-base interactions. However, since experimental data are available for the solubility of carbon dioxide in three unsaturated hydrocarbon solvents (3!4, 6 ) : an empirical correlation is possible. To account for acid-base interactions, we write, as before, A12
=
CII
+
"22
- 2C12
(1 5)
However, the term C12 must now take into account (chemical) complex formation in addition to the other forces discussed previously. For C12 we have e 1 2
+ ClZQ+
= dCIl~C22D
(16)
C12comp1ex
T h e parameter Cl~complex was found from experimental solubility data in the solvents ethylene, propylene, and acetylene. T h e first two of these have negligible quadrupole moments and thus, for these solvents, C1zQis zero. However, the quadrupole moment for acetylene is estimated to be 3.0 X 10-26 e.s.u. (7), and therefore solute-solvent interactions due to quadrupole-quadrupole forces may not be neglected. For solutions of carbon dioxide in acetylene ClzQ was found from Equations 13 and 14 using the previously determined value of
P. Values of c ~ ~ for ~ carbon ~ ~ dioxide P ' ~and ~ unsaturated hydrocarbons are shown in Figure 5. Since the enthalpy of complex formation is inevitably exothermic, it is not surprising that C12comp1exfalls with rising temperature. Solutions of carbon dioxide in paraffinic and olefinic solvents exhibit positive deviations from Raoult's law, but in acetylene slightly negative deviations are observed due to the contributions of both ClsQand C,2comPlex. Mixed Solvents
T h e extension of Scatchard's equation to multicomponent solutions has been discussed in various references [most notably VOL. 4
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211
’
IO
depends on r g ; as a first trial calculate xg with -yn = 1 and then use this initial value for the next iteration. Convergence in these calculations is very rapid.
I
FROM EXPERIMENTAL DATA FOR SOLUBILITY OF SOLID CO2 IN: 0 ETHYLENE A PROPYLENE ACETYLENE LINEAR LEAST-SQUARES FIT OF EXPERIMENTAL POINTS
Acknowledgment
T h e authors are grateful to the Xational Science Foundation for financial support. Nomenclature
A
=
C C,
= cohesive energy density
molar specific heat a t constant pressure a‘function of orientation f fugacity AHfusion = heat of fusion k = Bpltzmann’s constant = Avogadro‘s constant Q = quadrupole moment R = gas constant r = intermolecular distance 7 ’ = temperature To = triple point temperature x = mole fraction u = liquid molar volume l u = energy of complete vaporization u = potential energy f f = a numerical constant P = a numerical constant (dimensionless) Y = activity coefficient in liquid phase @ = volume fraction in liquid phase
F(0)
I60 I80 TEMPERATURE, “ K
I40
200
Figure 5. Contribution of complex formation to cohesive energy density
by Hildebrand and Scott ( 8 ) and Wohl ( 7 7 ) ] , since this equation is merely a special form of the van Laar equation. I n particular, for a mixed solvent containing liquid solvents 1 and 3, the activity coefficient for component 2 (carbon dioxide) is
RT In YZ
+
= UZ[AIZW
A23W
+ @ I @ ~ ( A4-~ z
A23
- AH)]
(17 ) where A t j is the exchange energy density for the ij binary. T h e method described in this paper should be used for A12 and A23. Excess free energy data for the carbon dioxide-free liquid mixture should be used to obtain ’413. Conclusions
A molecular thermodynamic method is presented for calculating the solubility of solid carbon dioxide in liquid solvents a t low temperatures. T h e required calculation steps are : For the temperature, T , of interest. obtain the value of f z s / f z ” from Figure 1: For temperature T,find C22 from Figure 3 and u z from Figure 1
L.
From the thermodynamic properties of the solvent, find a1 and calculate Cll using Equation 5. If the solvent has a significant quadrupole moment, calculate C l l D from Equations 6 and 11. Calculate C12 using Equation 16. If the solvent has a negligible quadrupole moment: set C12Q= 0; otherwise calculate C1zQ using Equations 13 and 14. For carbon dioxide find CzzD from Figure 4. If the solvent is a saturated hydrocarbon, or otherwise inert toward carbon dioxide, set C12comPlex = 0. If the solvent is a n unsaturated hydrocarbon, find ClzcomP1ex from Figure 5. Calculate solubility x z using Equations 1 and 3. T h e calculation is trial-and-error, since the volume fraction, @ I ,
212
I&EC FUNDAMENTALS
exchange energy density
= = =
SUBSCRIPTS 1: 3 = solvents 2 = carbon dioxide i, j = components SUPERSCRIPTS = dispersion forces = liquid phase S = solid phase Q = quadrupole moment
D L
literature Cited (1) Buckingham, A. D., Quart. Reo. (London) 13, 183 (1959). (2) Buckingham, A. D., Disch, R. L.. Proc. Roy. Soc. (London) A263, 275 (1963). (3) Clark, A. M.. Din, F., Discussions Faraday SOC.15, 202 (1953). (4) Clark, ’4. M., Din, F.; Trans. Faraday Soc. 46, 901 (1950). (5) Din, F.. “Thermodynamic Functions of Gases,” Vol. 1,
Butterworths, London. 1956. (6) Haselden, G. G.: Snowden. P., Trans. Faraday Soc. 5 8 , 1515 (1 9 62).
(7) Hildebrand, J. H., Scott. R. L., “Regular Solutions,’‘ Prentice-Hall, Yew York. (1962). (8) Hildebrand, J. H., Scott, R. L., .‘Solubility of NonElectrolytes” Reinhold: New York, 1950. (9) Lydersen, A. L.. in ”Properties of Gases and Liquids,” R. C. Reid and T. K. Sherwood, McGraw-Hill, New York, 1958. (10) Pitzer, K. S., et al.. quoted in “Thermodynamics.” G. N. Lewis and M. Randall. 2nd ed., .4ppendix I, McGraw-Hill, New York. 1961. (11) Wohl, K., Trans. Am. Inst. Chem. Engrs. 42, 215 (1946). RECEIVED for review June 1; 1964 ACCEPTED September 10. 1964
