WINGY. NG AND JOHNWALKLEY
2274
The Partial Molal Volume of Gases Dissolved in Nonpolar Solvents by Wing Y. Ng and John Walkley Chemistry Department, Simon Fraser University, Burnaby W , British Columbia
(Received November 8, 1968)
It is argued that a relationship should exist between the partial molal volume (&) and the saturation solubility (XZ)of a dissolved gas of the form (v2/vo) - CY = A(R In xz/v,) where A is a constant for any given solvent and vc is the critical volume of a given gas. This relationship is used to examine the partial molal volume data of a range of gases in common nonpolar solvents at 25'. It is shown that the reference volume of the gas (02' = avo)may be found and that together with measured entropy of solution data the entropy term associated with the change in solvent structure around the solute molecule can be determined. This latter term is shown to be of a magnitude equal to the usual entropy of volume expansion and as such could not be ignored in any formal theory for the thermodynamic properties of gases in solution. We then make use of the expression given above to examine the partial molal volumes of argon, xenon, and methane in n-hexane solvent measured at 15, 25, and 35'.
Introduction
(& - v ~ ~ ) ( b p / b= T )(A
The experimental study of the volume change associated with the formation of a binary mixture from its pure components is of considerable importance to the thermodynamic interpretation of the mixing process. I n general, this quantity is needed in the correction of constant pressure experimental data to constant volume data, e.g., via the excess free-energy function
+
Av'(T,v) = G,'(T,p) - ( 2 ( ~ / 3 ) , ) - ~ ( V ' ) .~. . with the usual notation, V' the excess volume of mixing and p the coefficient of isothermal compressibility. In regular solution theory the volume change associated with the mixing process is used directly to correct the entropy of mixing through the relationship (A&'
- A&')
-
= ( b s / b ~ ) r , ~ , ( f iDZz')
Again the standard notation is used with the superscript 0 referring to the assumed reference state. Vzis the measured partial molar volume. Replacing the derivative (bs/dv),,N, by its thermodynamic equivalent, (bP/dT),,~,,o, this term has been shown quite adequate in correcting As, data to the As, relationship expected of regular solution theory in the R(d In xz/d In 5") us. -R In x z plot.' One notes that the use of this term requires the total volume of expansion to be a property of the solute and that the reference volume VZO is known. From regular solution theory, for a range of gases dissolved in any one solvent, it would appear that the entropy of expansion may be found in terms of the slope, A, of the experimental R(b In xz/b In T ) us. -R In xz plota2 In fact As(expansion) = (A - l ) R In x z Using this assumption we argued, in a recent paper,3 that it might be possible to establish the validity of reported experimental %-data through the relationship The Journal of Physical Chemistry
- l ) R In xz
For any gas in a range of solvents some constancy in
vzo might be expected. This was shown to be only partially true and certainly failed to discriminate between reported L% measurements even though discrepancies of up to 4% (in cubic centimeters mole-I) existed. Our presently reported partial molal volume data, at 25', when used in the above equation again fail to give a constant reference volume value. I n this paper we examine the entropy of volume expansion relationship in a more fundamental way and suggest that a relationship between gZand R In xz can be found which suggests a constancy in vzo does exist. Such a constancy, in opposition to the earlier calculations, does, however, throw an interesting light upon the assumption that (A - l ) R In xz is a simple entropy of expansion term. In the experimental investigation of the thermodynamic properties of gases dissolved in nonpolar solvents only the saturation solubility has received temperature-dependent study. For the most part even this property is reported over only a small temperature range. From these studies the resulting entropy of solution data have, however, been shown to be surprisingly well behaved, in the sense of regular solution theory.
Experimental Section The apparatus was essentially the same as the one devised by H o r i ~ t i . ~ The dilatometer bulb was 300 ml in capacity, and the two capillary side arms were 0.5(1) J. H . Hildebrand and R. L. Scott, "Regular Solutions," PrenticeHall Inc., Englewood Cliffs, N. J., 1962. (2) J. H. Hildebrand, Proc. N u t . Acad. Sci. U.S., 57, 642 (1967). (3) J. Walkley and W. I. Jenkins, Trans. Faraday SOC.,64, 19
(1968).
(4) J. Horiuti, Sci. Papers Inst. Phys. Chem. Res. (Tokyo), 17, 126 (1931).
PARTIAL M O L A L VOLUME OF GASES Table I : The Intercept
a
2275
for Various Gas Solvent Systems
-
7
Solvent n-Heptane
.
1 1 1 -
CCl4
CeHa
i-CeHls
CClzFCCIFz
a (nonfluorocarbon
0.41 i 0.02
0.41 f 0.03
0.43 f 0.02
0,43& 0.03
gases a (fluorocarbon gases)
0.43 i 0.02
...
...
0.43f0.02
n-Hexane
0.42 f 0.02
0.42 & 0.02 (15') 0.40 f 0.02 (25') 0.42 & 0.02 (35')
...
Table 11: The Partial Molal Volume of Dissolved Gases a t 25'
-------------
Gas
7 -
Nz
-
Solvent
CHI
CzHe
CsHs
C4HlO
Ar
CnHn n-C& n-C,Hln .i-CsHis
56.1 60.2
72.0 67.9
91.2 93.0
105.0 104.9
...
...
...
...
51.5
62.8
55.7
42.3 43.5 47.8
mm diameter precision tubing. The arms were calibrated along their whole length with mercury. The dilatometer was filled with degassed solvent and then immersed in a thermostat, the temperature of which was maintained to within =kO.OOl". After the dissolution of a known volume of gas the resulting increase in volume of the solution was measured using the constant pressure technique devised by Walkley and Hildebranda6 Over the low concentration range of these experiments (x2 < the partial molal volume can be defined as the total volume of expansion per mole of dissolved gas. The linear relationship expected between the number of moles of gas dissolved and the measured volume increase was considered as a strict requirement for any run. The solvents n-hexane and benzene were spectrograde reagents (Matheson Coleman and Bell) and were first dried over calcium chloride. All gases were supplied by Matheson and their minimum purity was quoted as 99.9%.
Results The experimental data a t 25" are presented in Table I1 and the data for argon, xenon, and methane at 15 and 35" are presented in Table 111. The data are considered accurate to *l cc mol-'.
g2
- vZ0 =
OB
(A - 1)(R In x z ) / ( b p / b T ) ,
He
(1)
or
the latter relationship suggesting that for different gases in the same solvent at the same temperature a linear relationship should exist between fiz/v2O and (R In x2)/v20a We shall make the assumption that the reference volume v20 should for all (classical) gases have some proportionality to either the critical volume of the gas (v,) or its boiling point volume (ob). Assuming the first proportionality, viz., v20 = av,,then we can rewrite eq 2 as
--
i72
VC
- 1)
R In x2 a = (W b T k - ) (A
(3)
The assumption made above would suggest that a linear plot between i72/vc and (R In x2)/vc should now exist of slope presumably verifiable in terms of experimental properties, and, more important, of intercept at x2 = 1 of value a. Since eq 3 relates to the interpretation of different gases in the same solvent, the linear relationships for different solvents should give a family of lines of identical intercept a.
Discussion Table I11 : The Partial Molal of Gases in n-C6H14 (cc mol-') 15'
Ar CH4 Xe
46.1 55.7 58.8
350
53.5 62.9 63.7
Theory We rewrite the expression relating the partial molar volume of the saturation solubility and entropy of solution as
All available data were fitted to eq 3 by the leastsquares method, equal weight being assigned to each datum point. The data could only be reconciled to eq 3 if it was assumed that for any solvent data for the fluorocarbon gases fell on a line of considerably different slope from that for the nonfluorocarbon gases. This behavior pattern was previously noted in the plot of the entropy of solution data for these two types of gases in the same solvent, that is, the R(d In x2/d In T ) us. -R (5) J. H. Hildebrand and J. Walkley, J . Amer. Chem. SOC.,81, 4439 (1959).
Volume 73, Number 7 July 1069
2276
WINGY. Na AND JOHNWALKLEY
In xz plot. The values obtained for the intercept a are given in Table I, and a remarkable constancy in this value is seen to exist. I n Figures 1, 2, and 3 we present the fjz/v0 vs. (-log XZ)/% plots and for each plot we make use of an average a value of 0.42. For the solvents CCld and C&, the entropy of solution plots have the same slopes, 1.75 for the noduorocarbon gases and 1.6 for the gases CF4 and SFe. For these two solvents it is seen from Figure 1 that a well-defined linearity exists for the relationship predicted by eq 3 the two lines having slopes (which we will call A,) of value 1.3 for the common gases and 1.4 for the fluorocarbon gases. We note that both values are less than the A1n z1 values obtained from the slopes of the entropy of solution plot. The points for Hz and Dz depart widely from the line for the common gases. I n Figure 2 we give data for the solvents n-C6H14) n-C7H16j and iCsH1,.6 The data for the solvent n-C7H16 gives a A, value of 1.3 for the common gases and 1.5 for the fluorocarbon gases; these compare to the A i n z2 values of 1.55 and 1.91, respectively. Data for i-C6Hu is again sparse, but the three available points, for CH,, Ar, and He, certainly serve to define a line with the common intercept a value. The points for Hz and Dz do not fall upon this line. For the solvent n-hexane the available datae again defines a line of the expected intercept. gives points Though the data for both CH, and falling on this line the data for CZH6 does not. This
I
I
I
I
I
I
o H2 r 3 H2
@
D2
Figure 1. ~ 2 / v , us. ( - log xt)/v, relationship for various gases in: CCh solvent, 0;CeHa solvent, 0 . The Journal of Physical Chemistry
Oaf.
1
I
I
I
.01
-
Figure 2. u ~ / u , us. (-log xa)/u, relationship for various gases in: n-CTHla solvent, "; n-CeHla solvent, 0;i-CBHlB solvent, f ; n-CTHle solvent, 8.
could be experimental inaccuracy but could be symptomatic of the unusual behavior often found in the solubility thermodynamic properties of hydrocarbon gases in hydrocarbon solvents. For this solvent Agt = 1.4 and Ainz2 = 1.75. Noting the distinctly different line for fluorocarbon and nonfluorocarbon gases in the same solvent we plot in Figure 3 the %/v, vs. - (log z2)/v, relationship for the available range of gases in CC12FCCIFz and C7F16 solvents.6 For the first of these solvents the data for Nz and CH, certainly lie on a line with the correct intercept, but the point for argon falls well off this line. The points for SF6 and CF4 in this solvent are not well defined by a line passing through the common a value but certainly, if we accept this intercept, then within the limits of the theory it would seem that they lie on a line of slightly greater slope than that for Nz and CH,. For C7F16 solvent we are limited to data for Nz, CH4, and Hz and possibly we may accept SFe in (C4F&N solvent.e Accepting the common intercept a value the points for Nz, CH4, and SFe are seen to lie on a common line. The point for Hz (6) Experimental data from the following sources: H. L. Clever, R. Battino, J. H. Saylor, and P. N. Grass, J . Phys. Chem., 62, 375 (1958); J. C. Gjaldback, Acta Chem. Scand., 6, 623 (1952); A. Lannung and J. C. Gjaldback, ibid., 14, 1124 (1960); G. Archer and J. H. Hildebrand, J . Phys. Chem., 67, 1830 (1963); H. Hiraoka and J. H. Hildebrand, ibid., 68, 213 (1964).
PARTIAL MOLALVOLUMEOF GASES
I
I
2277 fluorocarbon gases in hydrocarbon solvents parallels the observed solubility relationships. The relationship given in eq 3 in that it requires the universal identity that v2o/v, = a cannot be argued as exact. Implying a corresponding states form of behavior, the identity required only exists exactly if for each gas T/T,is also a constant. This latter condition is certainly not true but if we recognize the definition of vzo as the reference volume of 1 mol of gas at 25" and as found as around 30 cc mol-1 this implies that at these effectively high pressures and within the accuracy of the available data the constancy of a is to be expected, both for all gases at a single temperature and for any gas over a limited temperature range. I n Figure 4 we plot O~/V,against (log xz)/vc for the argon, xenon, and methane data at 15, 25, and 35". Again we find that
I
I
-
0.8
v2
c
VC
0.6
I
I Ar
-
0.4
( log X i " , Figure 3. &/v, us. (-log z~)/v,relationship for various gases in: CClzFCClFz solvent, 8 ; C ~ F M solvent, 0 ; (CaFs)aN solvent, V.
(0.691, 0.02105) is not given in the figure but lies well below this line. It would certainly appear from the available data plotted in Figures 1, 2, and 3 that the relationship proposed between v2/v, and ( R In xZ)/v, does in fact exist. The constancy of the a value allows the vZo value (= av,)given in Table IV to be determined. The difference in the A,, value for fluorocarbon and non-
Table IV: Critical Volume ( v , ) ~and Standard Reference Volume (0~0) (both in cc mol-')
Va v ~ Q
vo v20
Ar
CHI
C2Ha
C8Hs
H2
D2
75.2 31.6
99.0 41.6
148.0 62.2
200.4 84.2
65.0 27.3
60.3 25.3
CF4
SFs
N2
0 2
co
He
146.7 61.6
194.2 81.6
90.1 37.8
78.1 32.8
93.1 39.1
57.7 24.2
a Data from R. C. Reid and T. K. Sherwood, "The Properties of Gases and Liquids," 2nd ed, McGraw-Hill Book Co., Inc., New York, N. Y., 1958.
,O3
Figure 4. Temperature dependence of the partial molal volume of argon, methrtne, and xenon.
for any given temperature a linear relationship is observed and again the family of lines has a constant (a) intercept. Within the accuracy of the solubility data available the A1nx2 values at 15, 25, and 35" are 1.87, 1.75, and 1.66. The corresponding AC2values are 1.28, 1.35, and 1.37. The difference between A,, and AR In can be conveniently ascribed to "the loosening of the surrounding solvent structure due to the presence of the gas molecule." The magnitude of this entropy term, (AR l n xa A,JR In x 2 for CeHe solvent is given in Table IV and is seen to be quite large. From the plot of the data at 15, 25, and 35" we see that these two sets of values move, with respect to temperature, in opposite directions; the difference, then, becomes smaller as the temperature increases. Volume 73, Number 7 July 1969
WINGY. NG AND JOHNWALKLEY
2278 The value a, the intercept, corresponds to a hypothetical mole fraction x2 = 1. It is easily verified that using o b as the reduction parameter a family of lines is found to exist for the plot of 2'2/2)b us. (log 22)/ub. These lines are again of common intercept and lead to the same value of u t , for any given gas as given in Table IV. Neither avonor aBbcorrespond to the ratios U I / V ~or where VI is the molar volume of the pure solvent. This behavior pattern follows that recently noted by R/liller7for the relationship between RT In x g and E b u (the energy of vaporization of the pure solute), where for a range of gases in any one solvent a linear relationship is observed with fluorocarbon and nonfluorocarbon gas falling on different lines. For this plot a common intercept at x2 = 1 is not found (it is not, in fact, a corresponding states plot) and for neither sets of gases does the intercept correspond to the E b u for the pure solvent. The accuracy of s~/v,(log xz)/vc plot relies greatly upon the accuracy of the experimental data. The behavior of the gases Hz and Dz in comparison with other gases remains enigmatic and appears by no means obviously to be a quantum e f f e ~ t . ~The partial molar volume data for HZ and Dz appears to require further investigation particularly in view of the data for n-CVH16 and i-CsHls solvents. The departure of certain gases from the expected line can easily be suggested as due t o experimental error (thus one would suspect that for Ar in CClzFCCIFza value & = 57.5 cc
The Journal of Physical Chemistry
mol-l, which would place this gas on the expected line, is much more reasonable a value than the reported value of 51.8 cc mol-'). It would appear that both the more recent gas solubility data and the partial molal volume data conform Table V : Entropy of Solution Terms for Gases in Benzene Solvent a t 25" Gas
104~~ (ref 6 )
8.77 20.7 148 2.58 2.66 5.75 26.4 8.15 6.63
82
- szg
(V2 - v * 9 x
0.6 -1.2 -7.5 5.1 4.9 2.0 -2.7 1.2 1.8
HZand DZdo not fall on linear &/v0
(&nm
-
(dp/dT)
Aue)R In xz
3.862 4.308 2.911 2.406 2.198 6.149 4,605 4.041 4.427
5.157 4.528 3.086 6.053 (?)a 6.035 ( ? ) a 2.952 2,352 5.211 6.386
us.
-
(log xz)/v, plot.
to expected behavior patterns at least to the point of showing internal consistency. The detailed interpretation in terms of statistical mechanical theories must wait until a lot more is known about the intermolecular forces in bulk media. (7) K. W. Miller, J . Phys. Chem., 7 2 , 2248 (1968)
