- Email: [email protected]
Partial molar volumes of gases dissolved in liquids. Part I. Selected literature data and some estimation techniques
Fluid Phase Equilibria, 8 ( 1982) 161- 180
161
Elsevier Scientific Publishing Company, Amsterdam-Printed
in The Netherlands
PARTIAL MOLAR VOLUMES OF GASES DISSOLVED IN LIQUIDS. PARTI. SELECTED LITERATURE DATA AND SOME ESTIMATION TECHNIQUEiS *
YASH PAUL HANDA ** and GEORGE C. BENSON Division of Chemistry, (Canada)
National
Research
Council of Canada,
Ottawq
Ontario
KIA
OR6
(Received July ISth, 1981; accepted in revised form October 16th, 1981)
ABSTRACT Handa, Y.P. and Benson, G:C., 1982. Partial molar volumes of gases dissolved in liquids. Part I. Selected literature data and some estimation techniques. Fluid Phase Equilibria, 8: 161-180 Experimental data, of precision better than 3%, available in the literature for the partial molar volumes of gases at infinite dilution in nonaqueous polar and nonpolar solvents are reviewed. Various empirical and semiempirical techniques for estimating this property are summarized and their usefulness is examined. Several graphical correlations provide useful rough estimates (within 10%). Values derived from corresponding-states and scaled-particle treatments vary considerably, and more-accurate experimental results are needed to establish the utility of these approaches.
INTRODUCTION
The partial molar volume of a gas in a solution is an important thermodynamic property for understanding the liquid-gas equilibrium, since it appears in various equations correlating gas solubility with temperature and pressure (Battino and Clever, 1966, Prausnitz, 1969, Chaps. 8 and 10). Although a large body of data on the solubihties of gases in liquids has been reported in the literature, there have been relatively few determinations of their partial molar volumes. The partial-molar-volume data available, however, do cover a variety of gases in liquids ranging from nonpolar to polar to fluorochemicals. * Issued as NRCC No. 19931. ** National Research Council of Canada Research Associate. Present address: Allied Chemical Corporation, 20 Peabody St., Buffalo, NY 14210, U.S.A. 0378-3812/82/0000-0000/$02.75
Q 1982 Elsevier Scientific Publishing Company
162 In this paper we present a brief review of the literature data on partial molar volumes and the various empirical and semiempirical techniques which have been proposed for their estimation. Some of these will be further tested in future papers, using the results of precise partial-molar-volume measurements currently being carried out in our laboratory. LITFRATURE DATA
Summaries of the experimental methods used to determine partial molar volumes have been given by Battino and Clever ( 1966) and Clever and Battino (1975). In the former article, the authors also give references to papers published prior to 1965 dealing with partial-molar-volume data for dissolved gases. Partial molar volumes V, have usually been obtained by one of three techniques: by measuring the dilation of the liquid due to dissolution of the gas, by determining the densities of the degassed and gas-saturated liquids, or by studying the pressure dependence of the gas solubility. Of these, the first technique usually gives the greatest precision.
TABLE I Partial molar volumes Vim (cm’ mol-‘) of various gases at infinite dilution in nonpolar solvents at 298 K (see Table 3 for references) Liquid
Gas H2
D2
He
Ar
n-Hexane n-Heptane
43.2 b
41.2 b
42.3 a 43.5 a
51.5 a 48.3 ’
2,2,4_Trimethylpentane Benzene
46.2 b 35.5 b.c.d.e
43.1 b 33.1b.c.c
47.8 a
49.8 b.c 44.6 b
Methylbenzene
35.8 =S
Cyclohexane Methylcyclohexane Chlorobenzeoe Tribromomethane Tetrachloromethane Carbon disulfide 1,I ,2-Trichloro- 1,2,2-trifluoroethane Perfluoro-n-heptane Perfluorotributylamine Perfluoromethylcyclohexane
41.0b
35 c 32.4 ’ 39.4 b
45 c 47.6 b 44c
33.9 d 37.7 d
54.2 =J
53.5 cX
44c 44= 45 c 51.8’ 549 51 c
163 Tables l-3 give the partial molar volumes for a number of gases in different solvents. The sources from which the data were taken are indicated by single-letter superscripts which refer to the footnotes to Table 3. A multilettered superscript means that the value thus denoted in the tables is the average of the values reported in the several papers cited. This was done only when the partial molar volumes from different sources agreed to within 2 cm3 mol- ‘. Where the difference was greater, values from independent measurements are reported separately. Since for most of the liquid-gas systems reported in Tables l-3 the solubility of the gas is small (the mole fraction of the gas is usually of the order of 10-3) and since the measurements were generally made at room temperature and at pressures close to atmospheric, the values in Tables l-3 can be regarded essentially as Vzm, the partial molar volumes at infinite dilution, 1 atm., and 298 K. Exceptions to this are the data from reference m which reports values at 300 K, from reference n where averages of results over the temperature range 253-293 K are given, and from reference o which gives data at 3 atm. The data from reference n are of doubtful precision in
N2
0,
62.8 a
55.7 a
52.6 d
46.1 d
co
51.7d
co,
Cl,
W’
46.9’
47.6 *
so2
46.8 d
SF,
102.6 105.5 103.8 97.1 105.5
b Ir b,k b k
101.2 bJ 49.6 d
43.0 d
46.1 d
44.4 d
53.1 d
45.2 d
52.4 d
47.9 d
66.0 f 66.1 b
47.5 i
44.6 d
47.0 d
46.5 *
53.3d
104k 126’ 101.4 Ir
52.7 j 93.7 f
164 TABLE
2
Partial molar volumes Vsm (cm3 mol-‘) of various solvents at 298 K (see Table 3 for references) Liquid
gases
at infinite
dilution
in nonpolar
Gas
CH, n-Hexane n-Heptane n-Octane 2,2,4-Trimetbylpentane Benzene Methylbenzene 1,3_Dimethylbenzene Cyclohexane Chlorobenzene Trichloromethane Tetrachloromethane Carbon disulfide
60.1 56.8 59.6 55.4
a*h m b m 45.9 n
56.6 b.m 56.1 a 52.6 b*d
50.4 d.”
60.5 d
72.0 ’ 66.3 d
57.8 d
63.5 d
60.9 d
66.1 d 61.4 ’
91.2”
47.2 D 47.4 n 48.7 d 52.3 d 56.1 ’
49.3 d 44.5 n 53.5 d
1,1,2-Trichloro1,2,2&ifluoroethane Perfluoro-n-heptane
68.4 h
82.9 h
view of the poorly controlled temperature. For solutions of SO, in Ccl,, (CH,),CO, and CH,COOCH,, and of Cl, in Ccl,, where the solubilities of the gases are fairly high (mole fractions of the order of 0.2 or higher), we fitted the volume of the solution to a polynomial in the mole fraction of the dissolved gas and derived the partial molar volume at infinite dilution. The footnotes also indicate where partial molar volumes have been reported as a function of temperature. Most of the data in Tables l-3 were obtained by direct dilatometric measurements. A few of the partial molar volumes were derived from density measurements; these also are indicated in the footnotes. In most cases, a precision of l-3% has been claimed for the partial-molar-volume data. For the gases CO,, N,O, SO,, C,H,, C,H,, and C,H,, Horiuti (1931) (reference d) reported a precision of 0.5%. We have not included in Tables l-3 any data derived from the pressure dependence of solubility. The scarcity of directly determined partial molar volumes in organic solvents is evident from the many gaps in Tables l-3 and from the small number of solvents for which such information is available. Krestov and Nedel’ko (1974) reviewed the literature data but did not
165
~-W-ho
V-W,0
CF,
C,F,
G’S
c-C,F,
CClF,
111 1
140’
144’
77’
104.9 a 88.6 b 86.4 m
105.0 a
67.0 d
86.0 b,m 82.8 b.m
86.7 b*’ 65.2 d 66.6 d
79.7 m
87.7 ’
cover as many systems as reported here, and did not report the sources of data for individual systems. These authors mistakenly used the molar volumes of gases at 298 K and 1 atm. instead of at 273 K and 1 atm. (STP) in calculating partial molar volumes from Horiuti’s (193 1) results (reference d) for the coefficient of dilation (defined as the change in volume of the solution divided by the standard volume of the gas dissolved). A similar error was made by Linford and Thornhill (1977). As a result, the partial molar volumes quoted by these authors are 4-7 cm3 mol - ’ higher than those reported here. Krestov and Nedel’ko also mistakenly reported the values calculated theoretically by Smith and Walkley (1962) as experimental partial molar volumes. ESTIMATION
OF PARTIAL
MOLAR VOLUMES
In the following discussion, subscripts 1 and 2 refer to the solvent and the gas (or solute) respectively. For most of the liquid-gas systems considered, T;4 298 K c < T;,where T? is the critical temperature of component i.
166 TABLE 3 Partial molar volumes Vzm (cm3 mol - ‘) of various gases at infinite dilution in polar solvents at 298 K Liquid
Gas
H2 Diethylether Dihexylether 2-Propanone Methylethanoate Dimethylformamide Dimethylsulfoxide Methanol
He
Ar
Kr
N2
02
co
50.5 d
65.6 d
56.0 d
61.8d
38.4 d 38.1 d
55.1 d 54.2 d
47.7 d 41.7 d
53.1 d 52.6 d
52 p
45 p
51 r
35p
35.3 s
48.6 a
58.5 s
a Ng and Walkley (1969); data at 288 and 308 K are also reported. b Walkley and Jenkins (1968). C Jolley and Hildebrand (1958). d Horiuti (1931); data at 283 and 313 K are also reported. e Walkley and Hildebrand (1959). r Hiraoka and Hildebrand (1964). a Krestov and Nedel’ko (I 974). h Gjaldbaek and Hildebrand (1950b); data at 273 K are also reported. i Hien et al. (1978); Vr derived from density measurements; data at 285, 288, 293, 303 and 308 K are also reported.
Schumm and Brown (1953) found that for a single gas in different nonpolar solvents, values of Vz~/V,o fall on parabolic curves when plotted against a,, where I$” is the infinite-dilution partial molar volume of the dissolved gas, and V,” and S, are respectively the molar volume and solubility parameter of the solvent. They constructed plots for N,, CH,, C,H,, and CF, at T- 298 K in solvents with solubility parameters ranging from 12 to 20 MPa’/* with 4-7 data points for each gas. We have tested this correlation for the gases Ar, N,, CH,, C,H,, CF,, and SF, using the VT data for nonpolar and polar solvents, including methanol. Values of VP and 6, were taken from Barton (1975) and Hildebrand et al. (1970). The plots are given in Fig. 1. In order to show the correlation for each gas more clearly, the ordinates for the gases N,, CH,, C,H,, CF,, and SF, have been displaced upwards by 0.2, 0.4, 0.6, 0.8, and 1.0 respectively. The correlation is linear provided that the points for the solvents C1,FC - CClF,, C,H,Cl, CS,, and (CZH&O, which deviate considerably from the linear plots in Fig. 1, are rejected. The parabolic curves reported by Schumm and Brown (1953) also
167
CO,
N,O
CH,
C,H,
C,H,
C,H,
(CH,),O
SO2
61.9d 61.6 d
68.3 d 68.1 d
67.6 d 67.9 d
43.1 d 44.3 d
59.0 d 44.4 d 44.2 d
43 p
45.8 d 45.5 d
54.8 d 52.8 d
52P
45.9 ” 47.9 d.0 48.2 d 46.1 n 42.8 n 43.4” 49.3 a
J Gjaldback and Hildebrand (195Oa); Vj” derived from density measurements; data at 273 K are also reported. ’ Hiraoka and Hildebrand (I 963). ’ Linford and Hildebrand (1969). m Schumm and Brown (1953). ” Ryutani (1 %I); data for C, H, in 1,5-dimethylpyrrolidone, 3-methyl-5-ethylpyridine, and a number of polyhydroxyethers are also reported. o Kiyama and Hiraoka (1956); data at 283 and 313 K are also reported. P Kritchevsky and Iliinskaya (1945); data at 273 and 323 K are also reported.
reduce to linear correlations if the points for the solvent CS, in their plots are rejected. From VT data for many gases in the same solvent, Prausnitz (1958) constructed plots of Vzm/l$ versus T/T; at T = 298 K, where V; is the molar critical volume of the gas. Three smooth curves each corresponding to a certain value of 6, were reported. The curves extend from T/T” values of - 0.7 to 10.0. Prausnitz did not specify the VT data used in constructing the plots. If the gases H,, Ar, and He are ignored, then at 298 K, T/T,C for the other gases in Tables l-3 is less than.2.5. This means that the positions of the curves in the range 2.5 < T/q c 10.0 are based on three points at the most. In Fig. 2 we have plotted Vzw/V- versus T/T; using the Vzm data for the solvents n-C6H,.,, C6HIZ, C,H,, Ccl,, C,H,Cl, CH,COOCH,, (CH,),CO, and CH,OH. These solvents have solubility parameters ranging from 14.9 MPa’12 for r&,H,, to 29.7 MPa’i2 for CH,OH. The v data for the gases H,, Ar, and He were not used. The critical parameters for the gases were taken from Kudchadker et al. ( 1968) and Reid et al. (1977). Because of
Fig. I. Plot of the volume ratio VF/Vp for liquid (I)+gas (2) systems at 298 K versus the solubility parameter 6, (MPa’/*) of the liquid, for the gases: 0. Ar; A, N,; 0, CH,: V, C,H,; 0, CF,; X, SF,. Values of v;“/Vp for gases after Ar in this list have been increased by successive multiples of 0.2. Broken lines are least-squares fits of results for the same gas in various liquids.
Fig. 2. Plot of the volume ratio v;P/ V; versus the temperature ratio T/c solvents at 298 K. The broken line is a least-squares fit of ah results.
for gases in various
169
-L
$ G
C&T;
__--
loo-
-____o,-"-"vi----G-0
_,__---
__P---
_____
CH, ___---
ml-
5
__--
o ~ _“,__-
0
__--
N2
0 e-m-
fJ--ip,‘8
,“_“-_o_9--o
-
-
,---b’
Ar 40-
0
6-o-
-
CL-
V~/cm2mol-~~
for liquid (l)+gas(2) Fig. 3. Plot of the volume difference [V2’-V20(Tb)] (cm’ mol-‘) systems at 298 K versus the molar volume VP (cm3 mol-‘) of the liquid, for the gases: +, H,; 0, Ar; A. N,; Cl, CH,; V, C,H,; 0, CF,; X, SF,. Values of V2- - Vp(Tb) for gases after H, in this list have been increased by successive multiples of 20 cm3 mol-‘. Broken lines are least-squares fits of results for the same gas in various liquids.
the scatter of the points for each individual solvent, Fig.2 resembles a universal reduced plot, and it is difficult to construct with any reliability a series of smooth curves each corresponding to a specific value of S,, as was done by Prausnitx (1958). Consequently, the use of Fig. 2 would predict the same value of Vm for a given gas in different solvents at a constant temperature. However, Tables l-3 show that Vzm of a gas in different solvents does not change by more than lo!%, except in the case of fluorochemicals or expanded solvents such as (C,H,),O, n-C,H,,, etc. Krestov and Nedel’ko (1974) correlated VT - V;( Tb) with Q“( Tb) and with VP, where Vzo(Tb) is the molar volume of the gas in the liquid state at its normal boiling point. We tested these two correlations using the Vrmdata from Tables l-3 and values of Vro(Tb) taken from HiIdebrand et al. (1970) and Reid et al. (1977). Plotting Vm - Vt( Tb) versus l$‘( Tb) for the solvents C,H,, Ccl,, and (CH,),CO gave a scatter of points without any indication of the nearly smooth correlation reported by Krestov and Nedel’ko. In Fig. 3, Km - &“(Tb) is plotted versus V,” for the gases H,, Ar, N,, CH,, C,H,, CF,, and SF,. For clarity, the ordinates for each successive gas have
170 been displaced upwards by a further multiple of 20 cm3 mall’ with respect to H,. For a highly dilute solution, Hildebrand et al. (1970, pp. 79-81) derived a relation between the partial molar volume of expansion and the partial molar entropy of expansion, which can be written as
v, =
v; + s;xp/(asp/w)
T.P
(1)
where Vzois the molar volume of the pure solute in the liquid state at the temperature and pressure of the solution, (EGp/aV)r,, is equal to the thermal pressure coefficient of the solvent, (V, - VP) is called the partial molar volume of expansion, and SFP is the partial molar entropy of expansion. For nonpolar solvents at 298 K, SyP is given (Hildebrand and Lamoreaux, 1974) by S,‘“r(JK-‘mol-1)=R(~lnx,/~lnT),+Rlnx,+87.9
(2)
where x2 is the saturation mole-fraction solubility of the gas in a solution at a partial pressure of 1 atm. The first two terms on the right-hand side of eqn. (2) correspond to the standard partial molar entropy of solution, AS’, as defined by Wilhelm and Battino (1973). AS0 values for a large number of aqueous and nonaqueous solutions of gases have been tabulated by Wilhelm and Battino (1973) and by Wilhelm et al. (1977). The last term in eqn. (2) is the entropy of condensation of a gas in its own hypothetical liquid boiling at 298 K, and is thus taken as a reference-state value. For solutions of gases in nonpolar solvents, the validity of eqn. (1) has been tested by Hildebrand (1967), Walkley and Jenkins (1968), Ng and Walkley (1969) and Linford and Hildebrand ( 1969). One problem associated with the use of eqn. (1) is the choice of a value for VP, as this term refers to the molar volume of the gaseous solute in its hypothetical liquid state at the temperature and pressure of the solution. Linford and Hildebrand (1969) used Vr”(Tb) in place of VP and obtained excellent, agreement between the VF values calculated for a number of fluorine-containing gases. in C,H,, and those measured experimentally. In a similar manner, Linford and Thornhill (1977) applied eqns. (1) and (2) to predict Vzwof 17 gases in 11 solvents. For 56 liquid-gas systems, their calculated values can be compared with the experimental data recorded in Tables 1 and 2. In 12 cases, involving H,, D,, and He, the discrepancy is usually greater than 75% and cannot be ascribed entirely to the translational quantum effects associated with these gases. In 24 cases, involving Ar, N,, O,, CO, and CH,, the agreement, on the average, is of the order of 30%. In the remaining 20 cases, involving C,H,, C,H,, C,H,, CF,, and SF,, the agreement, on the average, is of the order of 10%. In general, the agreement improves as the boiling point of the gas increases. One feature of eqn. (1) is
171 that, in all but five cases, it predicts values which are higher than the experimental results. The problems associated with eqn. (1) are obvious: the choice of Vzo, the limited precision and availability of data for thermal pressure coefficients, and the large error associated with SyP since it is obtained by differentiation of solubilities which usually have a precision of 1% or worse. Smith and Walkley (1962) proposed’ that the partial molar volume of a dissolved gas can be approximated by the volume occupied by one mole of the gas compressed to the internal pressure of the solvent. In other words, the dissolution process can be divided into two steps: the gas is first compressed to a liquid-like volume at the internal pressure of the solvent, and is then mixed with the solvent. No volume change is associated with the mixing process. Assuming the gas to be a collection of hard spheres, they equated the expression for the pressure generated by the change in the free volume of the gas during compression to the internal pressure of the solvent and obtained the relation v,m/v,o(o)
=[I
- (R/V;“)/(aP/az-)v]-3
(3)
where Q’(O) = yNu3 is the molar volume of the gas at absolute zero, N is Avogadro’s number, and u is the collision diameter of the gas molecule. The constant y depends on the geometry of the lattice and is usually assumed to be equal to unity. Smith and Walkley used eqn. (3) to predict 1/2mfor 11 gases in 7 nonpolar solvents, including the fluorocarbons c-C,F,, and n-C,F,,. Comparison of the predicted values with the data in Tables 1 and 2 shows that eqn. (3) works well. The values of Vzmare predicted to within 10% for H,, D,, Ar, N,, O,, CH,, C,H,, CF,, and N,O, and to within 20% for co,. Lyckman et al. (1965) argued that since, to a good approximation, the thermal pressure coefficient is related to the solubility parameter of a solvent by the equation (aP/aT)v=s:/~
(4)
eqn. (3) can be written in the following functional Vp;/RT;:
=
f( TP;/T#
where (RTT/P,C) is used For T< G, they plotted nonpolar as well as polar correlation was obtained and Gubbins, 1972). For solvent, Lyckman et al. values which were much
form: (5)
as a reducing parameter for volumes instead of VP. VzmPi/Rc versus TPic/QSf using V,oOdata for solvents, including water. An approximately linear which led to reasonable estimates of I/’ (Tiepel temperatures close to the critical temperature of the (1965) found that their correlation predicted Vzm less than the experimental values.
172
006-
’ 0.01
0
I 0.W
I 0.03
, 0.04
I 0.06
1 0.06
0.07
TPC/T,CP: *
Fig. 4. Plot of V,OP,C/Rq versus TPiC/P/Tt for gases in various solvents broken line, calculated from eqn. (a), corresponds to a correlation coefficient
at 298 K. The of 85%.
Although Sf is approximately equal to the internal pressure Pf for nonpolar solvents, this is not so for polar solvents and especially for hydrogen-bonded solvents. Thus we have used P{ instead of 6: in eqn. (5). A plot of VzmPf/Rq versus TP,‘/P:T,’ for Vzmdata from Tables l-3 is shown in Fig. 4. The internal pressures were obtained from Dack (1975a,b), from Hildebrand et al. (1970), or from sources cited in those references. Although there is considerable scatter in the plot, a linear least-squares fit gives the following equation: VTP;/Rq
= 0.092 + 2.674TP;/cP;
(6)
with a correlation
coefficient of 85%. O’Connell (197 1) derived equations relating the solution thermodynamic properties to direct correlation functions. For an infinitely dilute binary solution, these can be written as I/P/RTsr,, O =1-c,,
(7)
and JY/RT where
O = 1 -c,, kr.1 V,’ and
KF,
(8) are
the molar
volume
and coefficient
of isothermal
173 compressibility for the solvent, and the quantities C, are spatial integrals of the molecular direct-correlation functions. Brelvi and O’Connell (1972) obtained corresponding-states formulations for eqns. (7) and (8). Using experimental compressibility data for the liquids Ar, N,, 02, CH,, Ccl,, n-C&,,, n-C,J&, n-C,J%,, and n-Clb&,, they derived the following relation from eqn. (7): ln( 1 + VF/RTK:,,)
= - 0.42704& -0.42367(
- 1) + 2.089(&
- 1)2
p, - 1)3
(9)
where PI = VT/VI0 is the reduced density and VT is a characteristic molar volume for each substance. For nonpolar liquids V: = VF. For polar liquids, the characteristic volume is normally less than the critical volume. Equation (9) is applicable over the reduced density range 1.5 G p, a 3.7, and can be used to predict RF,, to within 6-8%. Alternatively, the characteristic volumes for polar liquids can be calculated from eqn. (9) if K:,~ is known. Using eqn. (9) and VF data for a number of liquid-gas systems, the following expressions were derived for C ,2: ln[ -C,,WW”~62] = -2.4467
for2.0Gfi,G2.785
+ 2.12074&
= 3.02214 - 1.87085& + 0.71955&
(10)
for 2.785 4 p, G 3.2
Brelvi and O’Connell (1972) used eqns. (8)-( 10) to predict V2= for a number of liquid-gas systems at various temperatures. They compared their results with experimental values and with those calculated from the correlation given in eqn. (5). For TC c, their predictions are of the same accuracy as those from eqn. (5), and in most cases are within 5% of the experimental values. The superiority of their method lies in the fact that it can be used to predict VZmto within 15% at temperatures close to TF. Based on the formalism of the scaled-particle theory, Lebowitz et al. (1965) derived the following equation for the pressure of a binary mixture of hard spheres: P/kT=
(6/n)[&,(l
-t,)-’
where k is the Boltzmann
+ 35,&il constant
- &-2
(11)
+ 35:(1 - t3)-3]
and the variables
&, are defined
by
(12) where pi = n, N/V is the number density, ni is the number of moles of component i in volume V of the mixture, and ai is the hard-sphere diameter. Differentiating eqn. (11) with respect to the number of moles n, of solute,
174 solving for the differential gives Vr;o= vp[ 1 +y,a,(3
(aV,&,),,,,,
and taking
the limit as nz 4 0
+ 3u, + ur’) + 3Y$+(3 + zu,) + 9Y$3]
x(1 +7y* + 15Y,z +9y;3)-’
(13)
where (14)
U, = UJU, Yr =Y/(l
-y)
(15)
and
Although eqn. (13) is derived for a reference system of hard spheres, Klapper (1971) suggested that it could be applied semiempirically to real liquid-gas systems by equating Vrs (the molar volume of the solvent consisting of hard spheres) to the experimental molar volume of the real solvent, and ai to the collision diameter. Thus the deviations of the real system from the reference system are indirectly taken into account. Using eqn. (13) in this way, Klapper predicted Pm for a variety of gases in Ccl, and H,O, and obtained good agreement with the experimental values. Pierotti (1963) combined the scaled-particle theory with the cavity model first proposed by Sisskind and Kasarnowsky (1933) and later developed by Uhlig (1937) and Eley (1939) to formulate a solubility theory for liquid-gas systems. According to this theory, the partial molar free energy of the solute, G2, associated with the solution process is given by G, = GcBy,+Gint, +RTln(
RT/V,‘)
(17)
where G,,, is the partial molar free energy for creation of a cavity in the solvent large enough to accommodate the gas molecule, and Gint, is the partial molar free energy for the interaction of the gas molecule in the cavity with the surrounding solvent molecules. The last term in eqn. (17) arises because of a different choice for the solute standard state in the liquid and gas phases. Differentiating eqn. (17) with respect to pressure at constant temperature and composition gives vz” = L.
+ vi”,. $-RW,,
Pierotti expression
(1976) used the scaled-particle for Vcav,:
(18) theory
to obtain
the following
v,,,,=R?-K$'(l-JJ)-3[(1-J')Z+3(1-Y)U,+3(1+2Y)U;] + (rN/6)
u,”
(19)
175 Assuming a Lennard-Jones (6,12) pairwise additive potential, the interaction term Gint, for a nonpolar gas in a polar solvent can be written (Wilhelm and Battino, 1971) as Gint, = - (N/V,“)[04444R(o, + 10.664~N&~(u,
+a,)3(r,,/k)“2 + c,)-‘1
(~~/‘k)“~ (20)
where ei, and u, are the characteristic parameters in the Lennard-Jones potential (note that ci is identified with the hard-sphere diameter), CL,is the dipole moment of the solvent, and CQis the polarizability of the solute. For a polar gas in a nonpolar solvent, Gin*, is obtained by interchanging the subscripts for p and (r in the last term of eqn. (20), and for a nonpolar gas in a nonpolar solvent this term is zero. If the gas and solvent molecules are both polar, eqn. (20) must be augmented by a dipole-dipole term (Pierotti, 1976). It follows from eqn. (20) that V,“,. = kF,rGint.
(21)
Equation (18) was used by Patyi et al. (1978), Pierotti (1963, 1965, 1976), and Wilhelm et al. (1977) to predict V2mfor nonpolar and polar gases in both nonpolar and polar solvents. The agreement with experimental values of VZm was good in most cases. Some of these calculations ignored the interaction terms, on the assumption that 1I$,,, ( was small compared to V,,, However, we find that 1Vi,,, 1 often amounts to - 30% of I$,,, and generally makes a significant contribution to VZm. Pierotti’s approach is also semiempirical, since the terms V,,, and V,,,, are calculated only approximately. The calculation of V,,. is based on scaledparticle theory, and like Klapper, Pierotti assumed that using experimental values of the solvent molar volume and coefficient of isothermal compressibility in eqns. (17)-(21) would account for the deviations of the real system from the reference hard-sphere system. In calculating the interaction term function g,, is unity. Gin,. 3 Pierotti assumed that the radial distribution Recently developed perturbation theories calculate these terms in a more fundamental way by using only molecular properties (Boublik and Lu, 1978; Goldman, 1979; Gubbins, 1973). The superiority of perturbation techniques lies in the possibility of providing better estimates of the cavity and interaction terms. However, this does not mean that the predictions from perturbation theory will necessarily be better than those from eqn. (18), since they very much depend upon the form of the terms in the perturbation expansion. Thus Goldman (1979) compared the predictions for VT of Ne in Ar (liquid) obtained from perturbation treatments due to Leonard-Henderson-Barker, Mansoori-Canfield, and Weeks-Chandler-Anderson, and found that they differed from each other by as much as 7 cm3 mol --I.
176 DISCUSSION
The graphical correlations shown in Figs. l-3 should be useful for rough estimates of Vzm. In each case the result is obtained readily with little information about the components. We do not recommend eqn. (1). It not only requires, a priori, the solubility data as a function of temperature, but the predictions from it are consistently poor, as already noted. Calculations of VW according to eqns. (3), (8), (13) and (18) were carried out for various gases in the solvents &C,H,,, n-C,H,,, 2,2,4-C,H,,, C,H,, c-C,H,,, C,H,Cl, Ccl,, PPC,F,~, (CH,),CO, and CH,OH. In all, 93 systems were considered, comprising most of the cases for which experimental results are given for these solvents in Tables l-3 . The main sources of the characteristic parameters a, and sir were the papers by Wilhelm and Battino (197 l), Pierotti (1976), and references cited therein. Isothermal compressibilities of the solvents were taken from various literature sources. A limited set of VW values, calculated for Ar, N,, 02, CH,, and CO, in C,H, and in Ccl,, is summarized in Table4. These systems were chosen so that the predictions could also be compared with those from the perturbation theory of convex molecules as reported by Boublik and Lu (1978), as well as with the experimental results from Tables 1 and 2. It should be noted, however, that the perturbation treatment used different values of a, and cii. If we use the same values of these parameters as Boublik and Lu, the predictions from
TABLE 4 Comparison of estimates of the partial molar volume Vzm (cm’ mol-‘) of various gases at infinite dilution in GH, and Ccl, at 298 K with experimental results from Tables 1 and 2 System
C,H,
+Ar +CH, +N, +0, + co,
Ccl,
+Ar +CH, +Nz +4 +co,
VT (cd
mol-
Expt.
Eqn. (3)
Eqn. (8)
Eqn. (13)
Eqn. (18)
PTCM a
44.6 54.4 52.6 46.1 41.6
41 48 48 42 55
45 53 50 44 51
34 40 40 35 46
42 48 52 44 53
40 44 48 37 43
44 52.3 53.1 45.2 47.9
42 49 49 44 56
43 50 48 42 49
36 42 42 37 48
41 47 51 43 52
41 44 48 38 43
‘)
’ Perturbation theory of convex molecules (Boublik and Lu, 1978).
177 eqns. (3), (13), and (18) differ greatly from the experimental results as well as from the calculations of Boublik and Lu. In general, the absolute deviations between the calculated and experimental results vary widely on a percentage basis and for each of the four equations amount to lo-20% on the average over the 93 systems. There is very little to recommend one equation over another. It seems clear that some of the discrepancies are due to the low accuracy of the experimental results. On the other hand, the predictions from eqns. (3), (13), and (18) are quite sensitive to the choice of a, values for the solute and/or solvent. For example, changing a, by 0.1 X IO-* cm in these equations changes vzm by 2-3 cm3 mol-‘; a similar change of u, leads to changes of - 10 cm3 mol-’ in the estimates of 1/2mby eqn. (18). Values of ui are available in the literature for a very limited number of molecules and values from different sources frequently show considerable disparity. Consequently, precise experimental values of vzrn are needed to test both the reliability of the a, values (especially CT,)and the assumptions underlying the theory. LIST OF SYMBOLS
c G k
lb P Pi R
s T V X Y Y, a Y
6 AX0 c KT
i” P
spatial integral of molecular direct-correlation function Gibbs free energy Boltzmann constant number of moles Avogadro’s number pressure internal pressure gas constant entropy temperature volume mole fraction compactness factor defined in eqn. (16) ratio defined in eqn. (15) molecular polarizability geometric packing factor solubility parameter standard partial molar change of X for solution process Len&d-Jones interaction-energy parameter isothermal compressibility molecular dipole moment sum defined in eqn. (12) density or number density
178 CT
0,
collision diameter ratio of collision diameters,
see eqn. (14)
Superscripts b normal boiling point C critical point value for expansion exp pure-component value 0 infinite-dilution partial molar value co * characteristic value reduced value Subscripts cav. int. 1 2
partial molar value for cavity formation partial molar value for interaction component 1, solvent liquid component 2, solute gas
REFERENCES Barton, A.F.M., 1975. Solubility parameters, Chem. Rev., 75: 731-753.. Battino, R. and Clever, H.L., 1966. The solubility of gases in liquids. Chem. Rev., 66: 395-463. Boublik, T. and Lu, B.C.-Y., 1978. Solubility of gases in liquids from the first-order perturbation theory of convex molecules. J. Phys. Chem., 82: 2801-2805. Brelvi, S.W. and O’Connell, J.P., 1972. Corresponding states correlations for liquid compressibility and partial molal volumes of gases at infinite dilution in liquids. Am. Inst. Chem. Eng. J., 18: 1239-1243. Clever, H.L. and Battino, R., 1975. The solubility of gases in liquids. In: M.R.J. Dack (Ed.), Solutions and Solubilities, Part I. Techniques of Chemistry, Vol. VIII, Wiley, New York, pp. 379-441. Dack, M.R.J., 197Sa. The importance of solvent internal pressure and cohesion to solution phenomena. Chem. Sot. Rev., 4: 21 I-229. Dack, M.R.J., 1975b. Solvent structure. The use of internal pressure and cohesive energy density IO examine contributions to solvent-solvent interactions. Aust. J. Chem., 28: 1643-1648. Eley, D.D., 1939. On the solubility of gases. Part 1. The inert gases in water. Trans. Faraday Sot., 35: 1281-1293. Gjaldbaek, J. Chr. and Hildebrand, J.H., 1950a. The solubility of chlorine in normal perfluoro-heptane and other liquids. J. Am. Chem. Sot., 72: 609-611. Gjaldbaek, J. Chr. and Hildebrand, J.H., 1950b. On some partial molal volumes of gases in solution. J. Am. Chem. Sot., 72: 1077- 1078. Goldman, S., 1979. A modem approach to nonelectrolyte solubility theory. Act. Chem. Res., 12: 409-415. Gubbins, K.E., 1973. Perturbation methods for calculating properties of liquid mixtures. Am. Inst. Chem. Eng. J., 19: 684-698.
179 Hien, N.Q., Ponter, A.B. and Peier, W., 1978. Density and viscosity of carbon tetrachloride solutions containing chlorine. J. Chem. Eng. Data, 23: 54-55. Hildebrand, J.H., 1967. Solutions of gases in normal liquids. Proc. Natl. Acad. Sci. U.S., 57: 542-547. Hildebrand, J.H. and Lamoreaux, R.H., 1974. Solubility of gases in liquids: fact and theory. Ind. Eng. Chem. Fundam., 13: 110-l IS. Hildebrand, J.H., Prausnitz, J.M. and Scott, R.L., 1970. Regular and Related Solutions, Van Nostrand-Reinhold, New York. Hiraoka, H. and Hildebrand, J.H., 1963. Partial molal volumes of sulfur hexafluoride. J. Phys. Chem., 67: 1919. Hiraoka, H. and Hildebrand, J.H., 1964. The solubility and entropy of solution of certain gases in (C,F,),N, CCI,F.CClF,, and 2,2,4-(CH,),C,H,. J. Phys. Chem., 68: 213-214. Horiuti, J., 1931. On the solubility of gas and coefficient of dilatation by absorption. Sci. Pap., Inst. Phys. Chem. Res. (Tokyo), 17: 125-264. Jolley, J.E. and Hildebrand, J.H., 1958. Solubility, entropy and partial molal volumes in solutions of gases in non-polar solvents. J. Am. Chem. Sot., 80: 1050-1054. Kiyama, R. and Hiraoka, H., 1956. Partial molal volumes of compressed acetylene gas in liquids. Rev. Phys. Chem. Jpn., 26: 56-69. Klapper, M.H., 1971. On the nature of the protein interior. Biochim. Biophys. Acta, 229: 557-566. Krestov, G.A. and Nedel’ko, B.E., 1974. Partial molar volumes of dissolved gases. Termodinam. Stroen. Rastvordv, 2: 112-120 (in Russian). Kritchevsky, I. and Iliinskaya, A., 1945. Partial molal volumes of gases dissolved in liquids. (A contribution to the thermodynamics of dilute solutions of non-electrolytes). Acta Physicochim. U.R.S.S., 20: 327-348. Kudchadker, A.P., Alani, G.H. and Zwolinski, B.J., 1968. The critical constants of organic substances. Chem. Rev., 68: 659-735. Lebowitz, J.L., Helfand, E. and Praestgaard, E., 1965. Scaled particle theory of fluid mixtures. J. Chem. Phys., 43: 774-779. Linford, R.G. and Hildebrand, J.H., 1969. Partial molal volumes of fluorochemical gases in cyclohexane. Trans. Faraday Sot., 65: 1470-1472. Linford, R.G. and Thomhill, D.G.T., 1977. Use of Hildebrand and Lamoreaux’s theory to predict solubility and related properties of gas-liquid systems. J. Appl. Chem. Biotechnol., 27: 479-497 Lyckman, E.W., Eckert, CA. and Prausnitz, J.M., 1965. Generalized reference fugacities for phase equilibrium thermodynamics. Chem. Eng. Sci., 20: 685-69 I. Ng, W.Y. and Walkley, J., 1969. The partial molal volume of gases dissolved in nonpolar solvents. J. Phys. Chem., 73: 2274-2278. O’Connell, J.P., 197 1.Thermodynamic properties of solutions based on correlation functions. Mol. Phys., 20: 27-33. Patyi, L., Furmer, I.E., Makranczy, J., Sadilenko, A.S., Stepanova, Z.G. and Berengarten, M.G,, 1978. Solubilities of gases in certain organic liquids. Zh. Prikl. Khim. (Leningrad), 5 I : 1296- 1300 (in Russian). Pierotti, R.A., 1963. The solubility of gases in liquids. J. Phys. Chem., 67: 1840- 1845. Pierotti, R.A., 1965. Aqueous solutions of nonpolar gases. J. Phys. Chem., 69: 281-288. Pierotti, R.A., 1976. A scaled particle theory of aqueous and nonaqueous solutions. Chem. Rev., 76: 717-726. Prausnitz, J.M., 1958. Regular solution theory for gas-liquid solutions. Am. Inst. Chem. Eng. J., 4: 269-272.
180 Prausnitz, J.M., 1969. Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, Englewood Cliffs, NJ. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York. Ryutani, B., 1961. Solubility of acetylene in various organic solvents at low temperature under high pressure. V. Pressure and temperature dependence of the solubility of acetylene and the resulting expansion of the solution. Nippon Kagaku Zasshi, 82: 5 13-5 17 (in Japanese). Schumm, R.H. and Brown, O.L.I., 1953. On some partial molal volumes of gases in solution. J. Am. Chem. Sot., 75: 2520-2521. Sisskind, B. and Kasarnowsky, I., 1933. Untersuchungen tiber die Loslichkeit der Gase. 2. Loslichkeit des Argons. 2. Anorg. Allg. Chem., 214: 385-395. Smith, E.B. and Walkley, J., 1962. The thermodynamic properties of gases in solution. I. The partial molal volume. J. Phys. Chem., 66: 597-599. Tiepel, E.W. and Gubbins, K-E., 1972. Partial molal volumes of gases dissolved in electrolyte solutions. J. Phys. Chem., 76: 3044-3049. Uhlig, H.H., 1937. The solubilities of gases and surface tension. J. Phys. Chem., 41: 12151225. Walkley, J. and Hildebrand, J.H., 1959. Partial molal volumes of hydrogen and deuterium. J. Am. Chem. Sot., 81: 4439. Walkley, J. and Jenkins, W.I., 1968. Entropy of volume expansion of gases dissolved in non-polar solvents. Trans. Faraday Sot., 64: 19-22. Wilhelm, E. and Bat&o, R., 1971. Estimation of Lennard-Jones (6,12) pair potential parameters from gas solubility data. J. Chem. Phys., 55: 4012-4017. Wilhelm, E. and Battino, R., 1973. Thermodynamic functions of the solubilities of gases in liquids at 25°C. Chem. Rev., 73: 1-9. Wilhelm, E., Battino, R. and Wilcock, R.J., 1977. Low-pressure solubility of gases in liquid water. Chem. Rev., 77: 219-262.
161
Elsevier Scientific Publishing Company, Amsterdam-Printed
in The Netherlands
PARTIAL MOLAR VOLUMES OF GASES DISSOLVED IN LIQUIDS. PARTI. SELECTED LITERATURE DATA AND SOME ESTIMATION TECHNIQUEiS *
YASH PAUL HANDA ** and GEORGE C. BENSON Division of Chemistry, (Canada)
National
Research
Council of Canada,
Ottawq
Ontario
KIA
OR6
(Received July ISth, 1981; accepted in revised form October 16th, 1981)
ABSTRACT Handa, Y.P. and Benson, G:C., 1982. Partial molar volumes of gases dissolved in liquids. Part I. Selected literature data and some estimation techniques. Fluid Phase Equilibria, 8: 161-180 Experimental data, of precision better than 3%, available in the literature for the partial molar volumes of gases at infinite dilution in nonaqueous polar and nonpolar solvents are reviewed. Various empirical and semiempirical techniques for estimating this property are summarized and their usefulness is examined. Several graphical correlations provide useful rough estimates (within 10%). Values derived from corresponding-states and scaled-particle treatments vary considerably, and more-accurate experimental results are needed to establish the utility of these approaches.
INTRODUCTION
The partial molar volume of a gas in a solution is an important thermodynamic property for understanding the liquid-gas equilibrium, since it appears in various equations correlating gas solubility with temperature and pressure (Battino and Clever, 1966, Prausnitz, 1969, Chaps. 8 and 10). Although a large body of data on the solubihties of gases in liquids has been reported in the literature, there have been relatively few determinations of their partial molar volumes. The partial-molar-volume data available, however, do cover a variety of gases in liquids ranging from nonpolar to polar to fluorochemicals. * Issued as NRCC No. 19931. ** National Research Council of Canada Research Associate. Present address: Allied Chemical Corporation, 20 Peabody St., Buffalo, NY 14210, U.S.A. 0378-3812/82/0000-0000/$02.75
Q 1982 Elsevier Scientific Publishing Company
162 In this paper we present a brief review of the literature data on partial molar volumes and the various empirical and semiempirical techniques which have been proposed for their estimation. Some of these will be further tested in future papers, using the results of precise partial-molar-volume measurements currently being carried out in our laboratory. LITFRATURE DATA
Summaries of the experimental methods used to determine partial molar volumes have been given by Battino and Clever ( 1966) and Clever and Battino (1975). In the former article, the authors also give references to papers published prior to 1965 dealing with partial-molar-volume data for dissolved gases. Partial molar volumes V, have usually been obtained by one of three techniques: by measuring the dilation of the liquid due to dissolution of the gas, by determining the densities of the degassed and gas-saturated liquids, or by studying the pressure dependence of the gas solubility. Of these, the first technique usually gives the greatest precision.
TABLE I Partial molar volumes Vim (cm’ mol-‘) of various gases at infinite dilution in nonpolar solvents at 298 K (see Table 3 for references) Liquid
Gas H2
D2
He
Ar
n-Hexane n-Heptane
43.2 b
41.2 b
42.3 a 43.5 a
51.5 a 48.3 ’
2,2,4_Trimethylpentane Benzene
46.2 b 35.5 b.c.d.e
43.1 b 33.1b.c.c
47.8 a
49.8 b.c 44.6 b
Methylbenzene
35.8 =S
Cyclohexane Methylcyclohexane Chlorobenzeoe Tribromomethane Tetrachloromethane Carbon disulfide 1,I ,2-Trichloro- 1,2,2-trifluoroethane Perfluoro-n-heptane Perfluorotributylamine Perfluoromethylcyclohexane
41.0b
35 c 32.4 ’ 39.4 b
45 c 47.6 b 44c
33.9 d 37.7 d
54.2 =J
53.5 cX
44c 44= 45 c 51.8’ 549 51 c
163 Tables l-3 give the partial molar volumes for a number of gases in different solvents. The sources from which the data were taken are indicated by single-letter superscripts which refer to the footnotes to Table 3. A multilettered superscript means that the value thus denoted in the tables is the average of the values reported in the several papers cited. This was done only when the partial molar volumes from different sources agreed to within 2 cm3 mol- ‘. Where the difference was greater, values from independent measurements are reported separately. Since for most of the liquid-gas systems reported in Tables l-3 the solubility of the gas is small (the mole fraction of the gas is usually of the order of 10-3) and since the measurements were generally made at room temperature and at pressures close to atmospheric, the values in Tables l-3 can be regarded essentially as Vzm, the partial molar volumes at infinite dilution, 1 atm., and 298 K. Exceptions to this are the data from reference m which reports values at 300 K, from reference n where averages of results over the temperature range 253-293 K are given, and from reference o which gives data at 3 atm. The data from reference n are of doubtful precision in
N2
0,
62.8 a
55.7 a
52.6 d
46.1 d
co
51.7d
co,
Cl,
W’
46.9’
47.6 *
so2
46.8 d
SF,
102.6 105.5 103.8 97.1 105.5
b Ir b,k b k
101.2 bJ 49.6 d
43.0 d
46.1 d
44.4 d
53.1 d
45.2 d
52.4 d
47.9 d
66.0 f 66.1 b
47.5 i
44.6 d
47.0 d
46.5 *
53.3d
104k 126’ 101.4 Ir
52.7 j 93.7 f
164 TABLE
2
Partial molar volumes Vsm (cm3 mol-‘) of various solvents at 298 K (see Table 3 for references) Liquid
gases
at infinite
dilution
in nonpolar
Gas
CH, n-Hexane n-Heptane n-Octane 2,2,4-Trimetbylpentane Benzene Methylbenzene 1,3_Dimethylbenzene Cyclohexane Chlorobenzene Trichloromethane Tetrachloromethane Carbon disulfide
60.1 56.8 59.6 55.4
a*h m b m 45.9 n
56.6 b.m 56.1 a 52.6 b*d
50.4 d.”
60.5 d
72.0 ’ 66.3 d
57.8 d
63.5 d
60.9 d
66.1 d 61.4 ’
91.2”
47.2 D 47.4 n 48.7 d 52.3 d 56.1 ’
49.3 d 44.5 n 53.5 d
1,1,2-Trichloro1,2,2&ifluoroethane Perfluoro-n-heptane
68.4 h
82.9 h
view of the poorly controlled temperature. For solutions of SO, in Ccl,, (CH,),CO, and CH,COOCH,, and of Cl, in Ccl,, where the solubilities of the gases are fairly high (mole fractions of the order of 0.2 or higher), we fitted the volume of the solution to a polynomial in the mole fraction of the dissolved gas and derived the partial molar volume at infinite dilution. The footnotes also indicate where partial molar volumes have been reported as a function of temperature. Most of the data in Tables l-3 were obtained by direct dilatometric measurements. A few of the partial molar volumes were derived from density measurements; these also are indicated in the footnotes. In most cases, a precision of l-3% has been claimed for the partial-molar-volume data. For the gases CO,, N,O, SO,, C,H,, C,H,, and C,H,, Horiuti (1931) (reference d) reported a precision of 0.5%. We have not included in Tables l-3 any data derived from the pressure dependence of solubility. The scarcity of directly determined partial molar volumes in organic solvents is evident from the many gaps in Tables l-3 and from the small number of solvents for which such information is available. Krestov and Nedel’ko (1974) reviewed the literature data but did not
165
~-W-ho
V-W,0
CF,
C,F,
G’S
c-C,F,
CClF,
111 1
140’
144’
77’
104.9 a 88.6 b 86.4 m
105.0 a
67.0 d
86.0 b,m 82.8 b.m
86.7 b*’ 65.2 d 66.6 d
79.7 m
87.7 ’
cover as many systems as reported here, and did not report the sources of data for individual systems. These authors mistakenly used the molar volumes of gases at 298 K and 1 atm. instead of at 273 K and 1 atm. (STP) in calculating partial molar volumes from Horiuti’s (193 1) results (reference d) for the coefficient of dilation (defined as the change in volume of the solution divided by the standard volume of the gas dissolved). A similar error was made by Linford and Thornhill (1977). As a result, the partial molar volumes quoted by these authors are 4-7 cm3 mol - ’ higher than those reported here. Krestov and Nedel’ko also mistakenly reported the values calculated theoretically by Smith and Walkley (1962) as experimental partial molar volumes. ESTIMATION
OF PARTIAL
MOLAR VOLUMES
In the following discussion, subscripts 1 and 2 refer to the solvent and the gas (or solute) respectively. For most of the liquid-gas systems considered, T;4 298 K c < T;,where T? is the critical temperature of component i.
166 TABLE 3 Partial molar volumes Vzm (cm3 mol - ‘) of various gases at infinite dilution in polar solvents at 298 K Liquid
Gas
H2 Diethylether Dihexylether 2-Propanone Methylethanoate Dimethylformamide Dimethylsulfoxide Methanol
He
Ar
Kr
N2
02
co
50.5 d
65.6 d
56.0 d
61.8d
38.4 d 38.1 d
55.1 d 54.2 d
47.7 d 41.7 d
53.1 d 52.6 d
52 p
45 p
51 r
35p
35.3 s
48.6 a
58.5 s
a Ng and Walkley (1969); data at 288 and 308 K are also reported. b Walkley and Jenkins (1968). C Jolley and Hildebrand (1958). d Horiuti (1931); data at 283 and 313 K are also reported. e Walkley and Hildebrand (1959). r Hiraoka and Hildebrand (1964). a Krestov and Nedel’ko (I 974). h Gjaldbaek and Hildebrand (1950b); data at 273 K are also reported. i Hien et al. (1978); Vr derived from density measurements; data at 285, 288, 293, 303 and 308 K are also reported.
Schumm and Brown (1953) found that for a single gas in different nonpolar solvents, values of Vz~/V,o fall on parabolic curves when plotted against a,, where I$” is the infinite-dilution partial molar volume of the dissolved gas, and V,” and S, are respectively the molar volume and solubility parameter of the solvent. They constructed plots for N,, CH,, C,H,, and CF, at T- 298 K in solvents with solubility parameters ranging from 12 to 20 MPa’/* with 4-7 data points for each gas. We have tested this correlation for the gases Ar, N,, CH,, C,H,, CF,, and SF, using the VT data for nonpolar and polar solvents, including methanol. Values of VP and 6, were taken from Barton (1975) and Hildebrand et al. (1970). The plots are given in Fig. 1. In order to show the correlation for each gas more clearly, the ordinates for the gases N,, CH,, C,H,, CF,, and SF, have been displaced upwards by 0.2, 0.4, 0.6, 0.8, and 1.0 respectively. The correlation is linear provided that the points for the solvents C1,FC - CClF,, C,H,Cl, CS,, and (CZH&O, which deviate considerably from the linear plots in Fig. 1, are rejected. The parabolic curves reported by Schumm and Brown (1953) also
167
CO,
N,O
CH,
C,H,
C,H,
C,H,
(CH,),O
SO2
61.9d 61.6 d
68.3 d 68.1 d
67.6 d 67.9 d
43.1 d 44.3 d
59.0 d 44.4 d 44.2 d
43 p
45.8 d 45.5 d
54.8 d 52.8 d
52P
45.9 ” 47.9 d.0 48.2 d 46.1 n 42.8 n 43.4” 49.3 a
J Gjaldback and Hildebrand (195Oa); Vj” derived from density measurements; data at 273 K are also reported. ’ Hiraoka and Hildebrand (I 963). ’ Linford and Hildebrand (1969). m Schumm and Brown (1953). ” Ryutani (1 %I); data for C, H, in 1,5-dimethylpyrrolidone, 3-methyl-5-ethylpyridine, and a number of polyhydroxyethers are also reported. o Kiyama and Hiraoka (1956); data at 283 and 313 K are also reported. P Kritchevsky and Iliinskaya (1945); data at 273 and 323 K are also reported.
reduce to linear correlations if the points for the solvent CS, in their plots are rejected. From VT data for many gases in the same solvent, Prausnitz (1958) constructed plots of Vzm/l$ versus T/T; at T = 298 K, where V; is the molar critical volume of the gas. Three smooth curves each corresponding to a certain value of 6, were reported. The curves extend from T/T” values of - 0.7 to 10.0. Prausnitz did not specify the VT data used in constructing the plots. If the gases H,, Ar, and He are ignored, then at 298 K, T/T,C for the other gases in Tables l-3 is less than.2.5. This means that the positions of the curves in the range 2.5 < T/q c 10.0 are based on three points at the most. In Fig. 2 we have plotted Vzw/V- versus T/T; using the Vzm data for the solvents n-C6H,.,, C6HIZ, C,H,, Ccl,, C,H,Cl, CH,COOCH,, (CH,),CO, and CH,OH. These solvents have solubility parameters ranging from 14.9 MPa’12 for r&,H,, to 29.7 MPa’i2 for CH,OH. The v data for the gases H,, Ar, and He were not used. The critical parameters for the gases were taken from Kudchadker et al. ( 1968) and Reid et al. (1977). Because of
Fig. I. Plot of the volume ratio VF/Vp for liquid (I)+gas (2) systems at 298 K versus the solubility parameter 6, (MPa’/*) of the liquid, for the gases: 0. Ar; A, N,; 0, CH,: V, C,H,; 0, CF,; X, SF,. Values of v;“/Vp for gases after Ar in this list have been increased by successive multiples of 0.2. Broken lines are least-squares fits of results for the same gas in various liquids.
Fig. 2. Plot of the volume ratio v;P/ V; versus the temperature ratio T/c solvents at 298 K. The broken line is a least-squares fit of ah results.
for gases in various
169
-L
$ G
C&T;
__--
loo-
-____o,-"-"vi----G-0
_,__---
__P---
_____
CH, ___---
ml-
5
__--
o ~ _“,__-
0
__--
N2
0 e-m-
fJ--ip,‘8
,“_“-_o_9--o
-
-
,---b’
Ar 40-
0
6-o-
-
CL-
V~/cm2mol-~~
for liquid (l)+gas(2) Fig. 3. Plot of the volume difference [V2’-V20(Tb)] (cm’ mol-‘) systems at 298 K versus the molar volume VP (cm3 mol-‘) of the liquid, for the gases: +, H,; 0, Ar; A. N,; Cl, CH,; V, C,H,; 0, CF,; X, SF,. Values of V2- - Vp(Tb) for gases after H, in this list have been increased by successive multiples of 20 cm3 mol-‘. Broken lines are least-squares fits of results for the same gas in various liquids.
the scatter of the points for each individual solvent, Fig.2 resembles a universal reduced plot, and it is difficult to construct with any reliability a series of smooth curves each corresponding to a specific value of S,, as was done by Prausnitx (1958). Consequently, the use of Fig. 2 would predict the same value of Vm for a given gas in different solvents at a constant temperature. However, Tables l-3 show that Vzm of a gas in different solvents does not change by more than lo!%, except in the case of fluorochemicals or expanded solvents such as (C,H,),O, n-C,H,,, etc. Krestov and Nedel’ko (1974) correlated VT - V;( Tb) with Q“( Tb) and with VP, where Vzo(Tb) is the molar volume of the gas in the liquid state at its normal boiling point. We tested these two correlations using the Vrmdata from Tables l-3 and values of Vro(Tb) taken from HiIdebrand et al. (1970) and Reid et al. (1977). Plotting Vm - Vt( Tb) versus l$‘( Tb) for the solvents C,H,, Ccl,, and (CH,),CO gave a scatter of points without any indication of the nearly smooth correlation reported by Krestov and Nedel’ko. In Fig. 3, Km - &“(Tb) is plotted versus V,” for the gases H,, Ar, N,, CH,, C,H,, CF,, and SF,. For clarity, the ordinates for each successive gas have
170 been displaced upwards by a further multiple of 20 cm3 mall’ with respect to H,. For a highly dilute solution, Hildebrand et al. (1970, pp. 79-81) derived a relation between the partial molar volume of expansion and the partial molar entropy of expansion, which can be written as
v, =
v; + s;xp/(asp/w)
T.P
(1)
where Vzois the molar volume of the pure solute in the liquid state at the temperature and pressure of the solution, (EGp/aV)r,, is equal to the thermal pressure coefficient of the solvent, (V, - VP) is called the partial molar volume of expansion, and SFP is the partial molar entropy of expansion. For nonpolar solvents at 298 K, SyP is given (Hildebrand and Lamoreaux, 1974) by S,‘“r(JK-‘mol-1)=R(~lnx,/~lnT),+Rlnx,+87.9
(2)
where x2 is the saturation mole-fraction solubility of the gas in a solution at a partial pressure of 1 atm. The first two terms on the right-hand side of eqn. (2) correspond to the standard partial molar entropy of solution, AS’, as defined by Wilhelm and Battino (1973). AS0 values for a large number of aqueous and nonaqueous solutions of gases have been tabulated by Wilhelm and Battino (1973) and by Wilhelm et al. (1977). The last term in eqn. (2) is the entropy of condensation of a gas in its own hypothetical liquid boiling at 298 K, and is thus taken as a reference-state value. For solutions of gases in nonpolar solvents, the validity of eqn. (1) has been tested by Hildebrand (1967), Walkley and Jenkins (1968), Ng and Walkley (1969) and Linford and Hildebrand ( 1969). One problem associated with the use of eqn. (1) is the choice of a value for VP, as this term refers to the molar volume of the gaseous solute in its hypothetical liquid state at the temperature and pressure of the solution. Linford and Hildebrand (1969) used Vr”(Tb) in place of VP and obtained excellent, agreement between the VF values calculated for a number of fluorine-containing gases. in C,H,, and those measured experimentally. In a similar manner, Linford and Thornhill (1977) applied eqns. (1) and (2) to predict Vzwof 17 gases in 11 solvents. For 56 liquid-gas systems, their calculated values can be compared with the experimental data recorded in Tables 1 and 2. In 12 cases, involving H,, D,, and He, the discrepancy is usually greater than 75% and cannot be ascribed entirely to the translational quantum effects associated with these gases. In 24 cases, involving Ar, N,, O,, CO, and CH,, the agreement, on the average, is of the order of 30%. In the remaining 20 cases, involving C,H,, C,H,, C,H,, CF,, and SF,, the agreement, on the average, is of the order of 10%. In general, the agreement improves as the boiling point of the gas increases. One feature of eqn. (1) is
171 that, in all but five cases, it predicts values which are higher than the experimental results. The problems associated with eqn. (1) are obvious: the choice of Vzo, the limited precision and availability of data for thermal pressure coefficients, and the large error associated with SyP since it is obtained by differentiation of solubilities which usually have a precision of 1% or worse. Smith and Walkley (1962) proposed’ that the partial molar volume of a dissolved gas can be approximated by the volume occupied by one mole of the gas compressed to the internal pressure of the solvent. In other words, the dissolution process can be divided into two steps: the gas is first compressed to a liquid-like volume at the internal pressure of the solvent, and is then mixed with the solvent. No volume change is associated with the mixing process. Assuming the gas to be a collection of hard spheres, they equated the expression for the pressure generated by the change in the free volume of the gas during compression to the internal pressure of the solvent and obtained the relation v,m/v,o(o)
=[I
- (R/V;“)/(aP/az-)v]-3
(3)
where Q’(O) = yNu3 is the molar volume of the gas at absolute zero, N is Avogadro’s number, and u is the collision diameter of the gas molecule. The constant y depends on the geometry of the lattice and is usually assumed to be equal to unity. Smith and Walkley used eqn. (3) to predict 1/2mfor 11 gases in 7 nonpolar solvents, including the fluorocarbons c-C,F,, and n-C,F,,. Comparison of the predicted values with the data in Tables 1 and 2 shows that eqn. (3) works well. The values of Vzmare predicted to within 10% for H,, D,, Ar, N,, O,, CH,, C,H,, CF,, and N,O, and to within 20% for co,. Lyckman et al. (1965) argued that since, to a good approximation, the thermal pressure coefficient is related to the solubility parameter of a solvent by the equation (aP/aT)v=s:/~
(4)
eqn. (3) can be written in the following functional Vp;/RT;:
=
f( TP;/T#
where (RTT/P,C) is used For T< G, they plotted nonpolar as well as polar correlation was obtained and Gubbins, 1972). For solvent, Lyckman et al. values which were much
form: (5)
as a reducing parameter for volumes instead of VP. VzmPi/Rc versus TPic/QSf using V,oOdata for solvents, including water. An approximately linear which led to reasonable estimates of I/’ (Tiepel temperatures close to the critical temperature of the (1965) found that their correlation predicted Vzm less than the experimental values.
172
006-
’ 0.01
0
I 0.W
I 0.03
, 0.04
I 0.06
1 0.06
0.07
TPC/T,CP: *
Fig. 4. Plot of V,OP,C/Rq versus TPiC/P/Tt for gases in various solvents broken line, calculated from eqn. (a), corresponds to a correlation coefficient
at 298 K. The of 85%.
Although Sf is approximately equal to the internal pressure Pf for nonpolar solvents, this is not so for polar solvents and especially for hydrogen-bonded solvents. Thus we have used P{ instead of 6: in eqn. (5). A plot of VzmPf/Rq versus TP,‘/P:T,’ for Vzmdata from Tables l-3 is shown in Fig. 4. The internal pressures were obtained from Dack (1975a,b), from Hildebrand et al. (1970), or from sources cited in those references. Although there is considerable scatter in the plot, a linear least-squares fit gives the following equation: VTP;/Rq
= 0.092 + 2.674TP;/cP;
(6)
with a correlation
coefficient of 85%. O’Connell (197 1) derived equations relating the solution thermodynamic properties to direct correlation functions. For an infinitely dilute binary solution, these can be written as I/P/RTsr,, O =1-c,,
(7)
and JY/RT where
O = 1 -c,, kr.1 V,’ and
KF,
(8) are
the molar
volume
and coefficient
of isothermal
173 compressibility for the solvent, and the quantities C, are spatial integrals of the molecular direct-correlation functions. Brelvi and O’Connell (1972) obtained corresponding-states formulations for eqns. (7) and (8). Using experimental compressibility data for the liquids Ar, N,, 02, CH,, Ccl,, n-C&,,, n-C,J&, n-C,J%,, and n-Clb&,, they derived the following relation from eqn. (7): ln( 1 + VF/RTK:,,)
= - 0.42704& -0.42367(
- 1) + 2.089(&
- 1)2
p, - 1)3
(9)
where PI = VT/VI0 is the reduced density and VT is a characteristic molar volume for each substance. For nonpolar liquids V: = VF. For polar liquids, the characteristic volume is normally less than the critical volume. Equation (9) is applicable over the reduced density range 1.5 G p, a 3.7, and can be used to predict RF,, to within 6-8%. Alternatively, the characteristic volumes for polar liquids can be calculated from eqn. (9) if K:,~ is known. Using eqn. (9) and VF data for a number of liquid-gas systems, the following expressions were derived for C ,2: ln[ -C,,WW”~62] = -2.4467
for2.0Gfi,G2.785
+ 2.12074&
= 3.02214 - 1.87085& + 0.71955&
(10)
for 2.785 4 p, G 3.2
Brelvi and O’Connell (1972) used eqns. (8)-( 10) to predict V2= for a number of liquid-gas systems at various temperatures. They compared their results with experimental values and with those calculated from the correlation given in eqn. (5). For TC c, their predictions are of the same accuracy as those from eqn. (5), and in most cases are within 5% of the experimental values. The superiority of their method lies in the fact that it can be used to predict VZmto within 15% at temperatures close to TF. Based on the formalism of the scaled-particle theory, Lebowitz et al. (1965) derived the following equation for the pressure of a binary mixture of hard spheres: P/kT=
(6/n)[&,(l
-t,)-’
where k is the Boltzmann
+ 35,&il constant
- &-2
(11)
+ 35:(1 - t3)-3]
and the variables
&, are defined
by
(12) where pi = n, N/V is the number density, ni is the number of moles of component i in volume V of the mixture, and ai is the hard-sphere diameter. Differentiating eqn. (11) with respect to the number of moles n, of solute,
174 solving for the differential gives Vr;o= vp[ 1 +y,a,(3
(aV,&,),,,,,
and taking
the limit as nz 4 0
+ 3u, + ur’) + 3Y$+(3 + zu,) + 9Y$3]
x(1 +7y* + 15Y,z +9y;3)-’
(13)
where (14)
U, = UJU, Yr =Y/(l
-y)
(15)
and
Although eqn. (13) is derived for a reference system of hard spheres, Klapper (1971) suggested that it could be applied semiempirically to real liquid-gas systems by equating Vrs (the molar volume of the solvent consisting of hard spheres) to the experimental molar volume of the real solvent, and ai to the collision diameter. Thus the deviations of the real system from the reference system are indirectly taken into account. Using eqn. (13) in this way, Klapper predicted Pm for a variety of gases in Ccl, and H,O, and obtained good agreement with the experimental values. Pierotti (1963) combined the scaled-particle theory with the cavity model first proposed by Sisskind and Kasarnowsky (1933) and later developed by Uhlig (1937) and Eley (1939) to formulate a solubility theory for liquid-gas systems. According to this theory, the partial molar free energy of the solute, G2, associated with the solution process is given by G, = GcBy,+Gint, +RTln(
RT/V,‘)
(17)
where G,,, is the partial molar free energy for creation of a cavity in the solvent large enough to accommodate the gas molecule, and Gint, is the partial molar free energy for the interaction of the gas molecule in the cavity with the surrounding solvent molecules. The last term in eqn. (17) arises because of a different choice for the solute standard state in the liquid and gas phases. Differentiating eqn. (17) with respect to pressure at constant temperature and composition gives vz” = L.
+ vi”,. $-RW,,
Pierotti expression
(1976) used the scaled-particle for Vcav,:
(18) theory
to obtain
the following
v,,,,=R?-K$'(l-JJ)-3[(1-J')Z+3(1-Y)U,+3(1+2Y)U;] + (rN/6)
u,”
(19)
175 Assuming a Lennard-Jones (6,12) pairwise additive potential, the interaction term Gint, for a nonpolar gas in a polar solvent can be written (Wilhelm and Battino, 1971) as Gint, = - (N/V,“)[04444R(o, + 10.664~N&~(u,
+a,)3(r,,/k)“2 + c,)-‘1
(~~/‘k)“~ (20)
where ei, and u, are the characteristic parameters in the Lennard-Jones potential (note that ci is identified with the hard-sphere diameter), CL,is the dipole moment of the solvent, and CQis the polarizability of the solute. For a polar gas in a nonpolar solvent, Gin*, is obtained by interchanging the subscripts for p and (r in the last term of eqn. (20), and for a nonpolar gas in a nonpolar solvent this term is zero. If the gas and solvent molecules are both polar, eqn. (20) must be augmented by a dipole-dipole term (Pierotti, 1976). It follows from eqn. (20) that V,“,. = kF,rGint.
(21)
Equation (18) was used by Patyi et al. (1978), Pierotti (1963, 1965, 1976), and Wilhelm et al. (1977) to predict V2mfor nonpolar and polar gases in both nonpolar and polar solvents. The agreement with experimental values of VZm was good in most cases. Some of these calculations ignored the interaction terms, on the assumption that 1I$,,, ( was small compared to V,,, However, we find that 1Vi,,, 1 often amounts to - 30% of I$,,, and generally makes a significant contribution to VZm. Pierotti’s approach is also semiempirical, since the terms V,,, and V,,,, are calculated only approximately. The calculation of V,,. is based on scaledparticle theory, and like Klapper, Pierotti assumed that using experimental values of the solvent molar volume and coefficient of isothermal compressibility in eqns. (17)-(21) would account for the deviations of the real system from the reference hard-sphere system. In calculating the interaction term function g,, is unity. Gin,. 3 Pierotti assumed that the radial distribution Recently developed perturbation theories calculate these terms in a more fundamental way by using only molecular properties (Boublik and Lu, 1978; Goldman, 1979; Gubbins, 1973). The superiority of perturbation techniques lies in the possibility of providing better estimates of the cavity and interaction terms. However, this does not mean that the predictions from perturbation theory will necessarily be better than those from eqn. (18), since they very much depend upon the form of the terms in the perturbation expansion. Thus Goldman (1979) compared the predictions for VT of Ne in Ar (liquid) obtained from perturbation treatments due to Leonard-Henderson-Barker, Mansoori-Canfield, and Weeks-Chandler-Anderson, and found that they differed from each other by as much as 7 cm3 mol --I.
176 DISCUSSION
The graphical correlations shown in Figs. l-3 should be useful for rough estimates of Vzm. In each case the result is obtained readily with little information about the components. We do not recommend eqn. (1). It not only requires, a priori, the solubility data as a function of temperature, but the predictions from it are consistently poor, as already noted. Calculations of VW according to eqns. (3), (8), (13) and (18) were carried out for various gases in the solvents &C,H,,, n-C,H,,, 2,2,4-C,H,,, C,H,, c-C,H,,, C,H,Cl, Ccl,, PPC,F,~, (CH,),CO, and CH,OH. In all, 93 systems were considered, comprising most of the cases for which experimental results are given for these solvents in Tables l-3 . The main sources of the characteristic parameters a, and sir were the papers by Wilhelm and Battino (197 l), Pierotti (1976), and references cited therein. Isothermal compressibilities of the solvents were taken from various literature sources. A limited set of VW values, calculated for Ar, N,, 02, CH,, and CO, in C,H, and in Ccl,, is summarized in Table4. These systems were chosen so that the predictions could also be compared with those from the perturbation theory of convex molecules as reported by Boublik and Lu (1978), as well as with the experimental results from Tables 1 and 2. It should be noted, however, that the perturbation treatment used different values of a, and cii. If we use the same values of these parameters as Boublik and Lu, the predictions from
TABLE 4 Comparison of estimates of the partial molar volume Vzm (cm’ mol-‘) of various gases at infinite dilution in GH, and Ccl, at 298 K with experimental results from Tables 1 and 2 System
C,H,
+Ar +CH, +N, +0, + co,
Ccl,
+Ar +CH, +Nz +4 +co,
VT (cd
mol-
Expt.
Eqn. (3)
Eqn. (8)
Eqn. (13)
Eqn. (18)
PTCM a
44.6 54.4 52.6 46.1 41.6
41 48 48 42 55
45 53 50 44 51
34 40 40 35 46
42 48 52 44 53
40 44 48 37 43
44 52.3 53.1 45.2 47.9
42 49 49 44 56
43 50 48 42 49
36 42 42 37 48
41 47 51 43 52
41 44 48 38 43
‘)
’ Perturbation theory of convex molecules (Boublik and Lu, 1978).
177 eqns. (3), (13), and (18) differ greatly from the experimental results as well as from the calculations of Boublik and Lu. In general, the absolute deviations between the calculated and experimental results vary widely on a percentage basis and for each of the four equations amount to lo-20% on the average over the 93 systems. There is very little to recommend one equation over another. It seems clear that some of the discrepancies are due to the low accuracy of the experimental results. On the other hand, the predictions from eqns. (3), (13), and (18) are quite sensitive to the choice of a, values for the solute and/or solvent. For example, changing a, by 0.1 X IO-* cm in these equations changes vzm by 2-3 cm3 mol-‘; a similar change of u, leads to changes of - 10 cm3 mol-’ in the estimates of 1/2mby eqn. (18). Values of ui are available in the literature for a very limited number of molecules and values from different sources frequently show considerable disparity. Consequently, precise experimental values of vzrn are needed to test both the reliability of the a, values (especially CT,)and the assumptions underlying the theory. LIST OF SYMBOLS
c G k
lb P Pi R
s T V X Y Y, a Y
6 AX0 c KT
i” P
spatial integral of molecular direct-correlation function Gibbs free energy Boltzmann constant number of moles Avogadro’s number pressure internal pressure gas constant entropy temperature volume mole fraction compactness factor defined in eqn. (16) ratio defined in eqn. (15) molecular polarizability geometric packing factor solubility parameter standard partial molar change of X for solution process Len&d-Jones interaction-energy parameter isothermal compressibility molecular dipole moment sum defined in eqn. (12) density or number density
178 CT
0,
collision diameter ratio of collision diameters,
see eqn. (14)
Superscripts b normal boiling point C critical point value for expansion exp pure-component value 0 infinite-dilution partial molar value co * characteristic value reduced value Subscripts cav. int. 1 2
partial molar value for cavity formation partial molar value for interaction component 1, solvent liquid component 2, solute gas
REFERENCES Barton, A.F.M., 1975. Solubility parameters, Chem. Rev., 75: 731-753.. Battino, R. and Clever, H.L., 1966. The solubility of gases in liquids. Chem. Rev., 66: 395-463. Boublik, T. and Lu, B.C.-Y., 1978. Solubility of gases in liquids from the first-order perturbation theory of convex molecules. J. Phys. Chem., 82: 2801-2805. Brelvi, S.W. and O’Connell, J.P., 1972. Corresponding states correlations for liquid compressibility and partial molal volumes of gases at infinite dilution in liquids. Am. Inst. Chem. Eng. J., 18: 1239-1243. Clever, H.L. and Battino, R., 1975. The solubility of gases in liquids. In: M.R.J. Dack (Ed.), Solutions and Solubilities, Part I. Techniques of Chemistry, Vol. VIII, Wiley, New York, pp. 379-441. Dack, M.R.J., 197Sa. The importance of solvent internal pressure and cohesion to solution phenomena. Chem. Sot. Rev., 4: 21 I-229. Dack, M.R.J., 1975b. Solvent structure. The use of internal pressure and cohesive energy density IO examine contributions to solvent-solvent interactions. Aust. J. Chem., 28: 1643-1648. Eley, D.D., 1939. On the solubility of gases. Part 1. The inert gases in water. Trans. Faraday Sot., 35: 1281-1293. Gjaldbaek, J. Chr. and Hildebrand, J.H., 1950a. The solubility of chlorine in normal perfluoro-heptane and other liquids. J. Am. Chem. Sot., 72: 609-611. Gjaldbaek, J. Chr. and Hildebrand, J.H., 1950b. On some partial molal volumes of gases in solution. J. Am. Chem. Sot., 72: 1077- 1078. Goldman, S., 1979. A modem approach to nonelectrolyte solubility theory. Act. Chem. Res., 12: 409-415. Gubbins, K.E., 1973. Perturbation methods for calculating properties of liquid mixtures. Am. Inst. Chem. Eng. J., 19: 684-698.
179 Hien, N.Q., Ponter, A.B. and Peier, W., 1978. Density and viscosity of carbon tetrachloride solutions containing chlorine. J. Chem. Eng. Data, 23: 54-55. Hildebrand, J.H., 1967. Solutions of gases in normal liquids. Proc. Natl. Acad. Sci. U.S., 57: 542-547. Hildebrand, J.H. and Lamoreaux, R.H., 1974. Solubility of gases in liquids: fact and theory. Ind. Eng. Chem. Fundam., 13: 110-l IS. Hildebrand, J.H., Prausnitz, J.M. and Scott, R.L., 1970. Regular and Related Solutions, Van Nostrand-Reinhold, New York. Hiraoka, H. and Hildebrand, J.H., 1963. Partial molal volumes of sulfur hexafluoride. J. Phys. Chem., 67: 1919. Hiraoka, H. and Hildebrand, J.H., 1964. The solubility and entropy of solution of certain gases in (C,F,),N, CCI,F.CClF,, and 2,2,4-(CH,),C,H,. J. Phys. Chem., 68: 213-214. Horiuti, J., 1931. On the solubility of gas and coefficient of dilatation by absorption. Sci. Pap., Inst. Phys. Chem. Res. (Tokyo), 17: 125-264. Jolley, J.E. and Hildebrand, J.H., 1958. Solubility, entropy and partial molal volumes in solutions of gases in non-polar solvents. J. Am. Chem. Sot., 80: 1050-1054. Kiyama, R. and Hiraoka, H., 1956. Partial molal volumes of compressed acetylene gas in liquids. Rev. Phys. Chem. Jpn., 26: 56-69. Klapper, M.H., 1971. On the nature of the protein interior. Biochim. Biophys. Acta, 229: 557-566. Krestov, G.A. and Nedel’ko, B.E., 1974. Partial molar volumes of dissolved gases. Termodinam. Stroen. Rastvordv, 2: 112-120 (in Russian). Kritchevsky, I. and Iliinskaya, A., 1945. Partial molal volumes of gases dissolved in liquids. (A contribution to the thermodynamics of dilute solutions of non-electrolytes). Acta Physicochim. U.R.S.S., 20: 327-348. Kudchadker, A.P., Alani, G.H. and Zwolinski, B.J., 1968. The critical constants of organic substances. Chem. Rev., 68: 659-735. Lebowitz, J.L., Helfand, E. and Praestgaard, E., 1965. Scaled particle theory of fluid mixtures. J. Chem. Phys., 43: 774-779. Linford, R.G. and Hildebrand, J.H., 1969. Partial molal volumes of fluorochemical gases in cyclohexane. Trans. Faraday Sot., 65: 1470-1472. Linford, R.G. and Thomhill, D.G.T., 1977. Use of Hildebrand and Lamoreaux’s theory to predict solubility and related properties of gas-liquid systems. J. Appl. Chem. Biotechnol., 27: 479-497 Lyckman, E.W., Eckert, CA. and Prausnitz, J.M., 1965. Generalized reference fugacities for phase equilibrium thermodynamics. Chem. Eng. Sci., 20: 685-69 I. Ng, W.Y. and Walkley, J., 1969. The partial molal volume of gases dissolved in nonpolar solvents. J. Phys. Chem., 73: 2274-2278. O’Connell, J.P., 197 1.Thermodynamic properties of solutions based on correlation functions. Mol. Phys., 20: 27-33. Patyi, L., Furmer, I.E., Makranczy, J., Sadilenko, A.S., Stepanova, Z.G. and Berengarten, M.G,, 1978. Solubilities of gases in certain organic liquids. Zh. Prikl. Khim. (Leningrad), 5 I : 1296- 1300 (in Russian). Pierotti, R.A., 1963. The solubility of gases in liquids. J. Phys. Chem., 67: 1840- 1845. Pierotti, R.A., 1965. Aqueous solutions of nonpolar gases. J. Phys. Chem., 69: 281-288. Pierotti, R.A., 1976. A scaled particle theory of aqueous and nonaqueous solutions. Chem. Rev., 76: 717-726. Prausnitz, J.M., 1958. Regular solution theory for gas-liquid solutions. Am. Inst. Chem. Eng. J., 4: 269-272.
180 Prausnitz, J.M., 1969. Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, Englewood Cliffs, NJ. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York. Ryutani, B., 1961. Solubility of acetylene in various organic solvents at low temperature under high pressure. V. Pressure and temperature dependence of the solubility of acetylene and the resulting expansion of the solution. Nippon Kagaku Zasshi, 82: 5 13-5 17 (in Japanese). Schumm, R.H. and Brown, O.L.I., 1953. On some partial molal volumes of gases in solution. J. Am. Chem. Sot., 75: 2520-2521. Sisskind, B. and Kasarnowsky, I., 1933. Untersuchungen tiber die Loslichkeit der Gase. 2. Loslichkeit des Argons. 2. Anorg. Allg. Chem., 214: 385-395. Smith, E.B. and Walkley, J., 1962. The thermodynamic properties of gases in solution. I. The partial molal volume. J. Phys. Chem., 66: 597-599. Tiepel, E.W. and Gubbins, K-E., 1972. Partial molal volumes of gases dissolved in electrolyte solutions. J. Phys. Chem., 76: 3044-3049. Uhlig, H.H., 1937. The solubilities of gases and surface tension. J. Phys. Chem., 41: 12151225. Walkley, J. and Hildebrand, J.H., 1959. Partial molal volumes of hydrogen and deuterium. J. Am. Chem. Sot., 81: 4439. Walkley, J. and Jenkins, W.I., 1968. Entropy of volume expansion of gases dissolved in non-polar solvents. Trans. Faraday Sot., 64: 19-22. Wilhelm, E. and Bat&o, R., 1971. Estimation of Lennard-Jones (6,12) pair potential parameters from gas solubility data. J. Chem. Phys., 55: 4012-4017. Wilhelm, E. and Battino, R., 1973. Thermodynamic functions of the solubilities of gases in liquids at 25°C. Chem. Rev., 73: 1-9. Wilhelm, E., Battino, R. and Wilcock, R.J., 1977. Low-pressure solubility of gases in liquid water. Chem. Rev., 77: 219-262.