3230
J. Phys. Chem. lQ81,85,3230-3237
-r
0
e
P 0 0 I
I
7.8 7.67
I
I
-1
1
I
X
I
1
7.4 7.2 7.0 6.8 6.6 6.4 6.2 6.0 HYDRATION NUMBER
5
Flgure 2. Relatlonshlp between composltion and pressure of gas for hydrates of C103F, CHCIF,, CI,, and H2S.
at 0.316 atm; for H2S, n = 6.12 at 0.918 atm. (2) For each gas it was assumed that eq 5 was obeyed and that Ahw = 265 cal mol-l. (3) A simultaneous equation for chlorine was n = 7.47 = 46/(619~+ 2&) and for H2S, n = 6.12 = 46/(602 + (4) These pairs of simultaneous equations were solved to obtain and 4 for each gas at its quadruple point pressure. (5) These values were then substituted into eq 2 to obtain values for K 2 and K1 for each gas. (6) It was then possible to use eq 2 to calculate and 61 at various pressures and to obtain corresponding values
of n by use of eq 6. The lines drawn in Figure 2 pass through these calculated values of n. When considering the quality of agreement between theory and experiment for C12 and H2S, one should not take into account the location of the line. That is arbitrarily established in step 1of the calculation. The item to consider is the slope and shape of the curve, because this is determined through use of theoretical eq 2 and 5. In slope and shape, there is qualitative but not quantitative agreement in the case of chlorine. For H2S it appears that the agreement is nearly quantitative within the limits of experimental precision and that 61 and 62 are substantially equal. The assumed value of Ahwplays a part in the shape of the theoretical curves. A value higher than 265 cal molw1 would make the curves rise more steeply with pressure and agree less well with experiment for C12 and H2S but better for CHClF2and C103F. A value lower than 265 cal mol-I would improve the situation for Clz and H2S but make it worse for CHClF2 and C103F. The lowest value of Ab,,, which can give a hydration number of 6.12 (that estimated for H2S) at the quadruple point is 265 cal mol-l. It is, of course, possible that the theory is strictly correct and that the experimental compositions for hydrates of C103F, CHC1F2,and C12 are systematically in error. The author feels, however, that the experimental data given here are close to correct. He is continuing to work with other systems and hopes that the added information will be helpful. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for the partial support of this research. The author thanks D. W. Davidson for his interest and helpful advice.
Solubility of Gases in Liquids. 13. High-Precision Determination of Henry's Constants for Methane and Ethane in Liquid Water at 275 to 328 K Tlmothy R. Rettlch, Y. Paul Hands,+ Rubin Battino,' and Emmerlch Wllhelmt Departmen?of Chemistry, Wright State University, Dayion, Ohio 45435 (Received:April 30, 1981; In Final Form: June 26, 1981)
A high-precision apparatus of the Benson-Krause type has been used to measure the solubilitiesof methane and ethane in pure liquid water in the pressure range 50-100 kPa and temperature range 275-328 K. From these data, Henry's constant Hz,l is derived. Its temperature dependence is accounted for by both a ClarkeGlew-type fitting equation, and a power series in 1/T. The imprecision of our measurements is characterized by average deviations of Hz,l from these smoothing equations of ca. &0.06% for H20 + CHI, and ca. f0.08% for HzO + CzHG.From the temperature variation of H2,1, partial molar quantities pertaining to the solution process, Le., standard changes in enthalpy, entropy, and heat capacity, have been obtained. In addition, several thermodynamic quantities used in discussing hydrophobic interaction are presented.
Introduction In recent years there has been a resurgence of interest in the solubility of gases in liquids in general,l-s and in liquid water in particular?Sa The need for high-precision experimental data on very dilute aqueous solutions can be 'Division of Chemistry, The National Research Council of Canada, Ottawa, Canada KIA OR6. *Visiting Associate Professor of Chemistry from Institut fur Physikalische Chemie, Universitat Wien, Wahringerstrasse 42,A1090 Wien, Austria.
traced to research activities in a variety of areas.12p26-2e Measurements covering sufficiently large ranges of tem(1) R. Battino and H. L. Clever, Chern. Reu., 66, 395 (1966). (2) E. Wilhelm and R. Battino, Chern. Rev., 73, 1 (1973).
(3) H.L. Clever and R. Battino in "Weissberger: Techniques of Chemistry", Vol. 8, Part 1,M.R.J. Dack, Ed., Wiley-Interscience, New York, 1975, p 379. (4) R. A. Pierotti, Chern. Reu., 76, 717 (1976). (5)W. Gerrard, "Solubility of Gases in Liquids. A Graphic Approach", Plenum Press, New York, 1976. (6) E. Wilhelm, Fortschr. Verfahrenstechn.,Abt. A, 16, 21 (1977). (7) S. Goldman, Acc. Chem. Res., 12, 409 (1979).
0022-3654/81/2085-3230$01.25/0 0 1981 American Chemical Soclety
Solubility of Gases in Liquids
perature constitute, with very few ex~eption,~’-~~ the only source of information for the changes of enthalpy and heat capacity associated with solution processes. These quantities are usually significantly more sensitive to approximations introduced in model theories than Henry’s constant itself, and are thus better suited for testing the limitations of such theories. Despite the substantial body of data on the solubilities of gases in water, very few extend above 310 K and, when critically reviewed,12appreciable differences between solubilities reported by different authors become apparent. Water + methane and water + ethane were selected primarily because thermodynamic data on both systems have been used extensively (a) in the theory of hydrophobic (b) in the development of empirical group contribution methods?l* and (c) in the quantitative assessment of the extent of hydrocarbon pollution in natural waters.34 In recent years, the most significant advance in experimental techniques for measuring the solubility of gases in liquids has been the apparatus constructed by Benson and Krause.l’~~~ Its precision (and accuracy) is considerably better than that of any previous design, even sur~ ~ on Benson passing that due to Cook and H a n ~ o n .Based and Krause’s work (and their assistance), we have constructed a similar apparatus which is capable of routinely yielding results with small enough imprecisions to allow evaluation of enthalpy and heat capacity changes upon solution with confidence in the range 273.2-335 K. In this article we detail some differences between our apparatus and that of Benson and K r a ~ s e .Our ~ ~ method of data reduction will be treated in considerable detail, including a comparison of two methods for presenting our results (8) E. Wilhelm, B o g . Chem. Eng. A , Fundam. Chem. Eng., 18, 21 (1980). (9) K. W. Miller and J. H. Hildebrand, J. Am. Chem. Soc., 90, 3001 (1968). (10) F. Franks and D. S. Reid in “Water. A Comprehensive Treatise”, Vol. 2, F. Franks, Ed., Plenum Press, New York, 1973, p 323. (11) M. H. Klapper, Adu. Chem. Phys., 23, 55 (1973). (12) E. Wilhelm, R. Battino, and R. J. Wilcock, Chem. Reu., 77, 219 (1977). (13) R. A. Pierotti, J.Phys. Chem., 69, 281 (1965). (14) A. Ben-Naim, J. Wilf, and M. Yaacobi, J. Phys. Chem., 77, 95 (1973). (15) A. Ben-Naim and M. Yaacobi. J. Phvs. Chem.. 78. 170 (1974). (16) S. Yamamoto, J. B. Alcauskas,’and T:E. Crozier, J.’Chem. Eng. Data, 21, 78 (1976). (17) B. B. Benson and D. Krause, J. Chem. Phys., 64, 689 (1976). (18) L. I. Gordon, Y. Cohen, and D. R. Standley, Deep-Sea Res., 24, 937 (1977). (19) J.’Muccitelli and W.-Y. Wen, J. Solution Chem., 7, 257 (1978). (20) R. W. Potter, 11, and M. A. Clynne, J. Solution Chem., 7, 837 (1978). (21) W.-Y. Wen and J. A. Muccitelli, J.Solution Chem., 8,225 (1979). (22) A. Douabul and J. Riley, J. Chem. Eng. Data, 24, 274 (1979). (23) B. B. Benson, D. Krause, and M. A. Peterson, J. Solution Chem., 8, 655 (1979). (24) J. Muccitelli and W.-Y. Wen, J.Solution Chem., 9, 141 (1980). (25) F. Franks in “Water. A Comprehensive Treatise”, Vol. 1, F. Franks, Ed., Plenum Press, New York, 1972, p 1. (26) A. Ben-Naim, “Water and Aqueous Solutions”, Plenum Press, New York, 1974. (27) D. M. Alexander, J. Phys. Chem., 63, 994 (1959). (28) R. Jadot, J. Chim. Phys., 70, 352 (1973). (29) J. Cone, L. E. S. Smith, and W. A. Van Hook, J. Chem. Thermodyn., 11, 277 (1979). (30) R. Battino and K. N. Marsh, Austr. J. Chem., 33, 1997 (1980). (31) N. Nichols, R. Skold, C. Spink, J. Suurkuusk, and I. Wadso, J. Chem. Thermodyn., 8, 1081 (1976). (32) J. P. Guthrie, Can. J. Chem., 55, 3700 (1977). (33) G. Perron and J. E. Desnoyers, Fluid Phase Equilib., 2, 239 (1979). (34) J. W. Swinnerton and R. A. Lamontagne, Enuiron. Sci. Technol., 8, 657 (1974). (35) M. W. Cook and D. N. Hanson, Reu. Sci. Instrum., 28,370 (1957).
The Journal of Physical Chemistry, Vol. 85,No. 22, 198 1 3231
as a function of temperature. The high-precision Henry’s constants for methane and ethane, respectively, dissolved in pure liquid water have been obtained between ca. 275 and 328 K in steps of about 5 K. Smoothed values a t rounded temperatures are given in tabular form, together with derived thermodynamic quantities pertaining to the solution process. Experimental results are then compared with calculated values by means of scaled particle theory. Finally, some quantities of relevance in the discussion of hydrophobic interaction will be presented.
Experimental Section Materials. Precision and accuracy of the experimental method, and data reduction, were verified by determining Henry’s constant of oxygen dissolved in water between 275 and 328 K. The oxygen used was Matheson ultrahighpurity quality, i.e., 99.95 mol % minimum purity. The methane used for most of the experiments was Airco CP grade, 99.0 mol % minimum purity. The ethane used for most of the experiments was Matheson CP grade, 99.0 mol % minimum purity. Airco ultrahigh-purity methane, 99.99 mol % minimum purity, and Matheson ultrahighpurity ethane, 99.96 mol% minimum purity, were used for comparison. Reverse osmosis “house-distilled’’ water was further purified by continuous pumping through a series of 111co-Way research model ion exchangers and through a 1.2-wm pore size Millipore filter. The purity of the water was regularly monitored by a flow conductivity cell using a Yellow Springs Instrument Co. resistance meter (Model 31). The water prepared in this fashion had a resistivity greater than 5 X lo4 Q m. Apparatus and Operation. The technique used to degas liquids has been previously d e ~ c r i b e d . ~The ~ maximum residual of dissolved gas in water after degassing is estimated to be 0.001% or less of the saturation value at room temperature and atmospheric pressure. The equilibrator used to assure saturation of water by gas was essentially that described by Benson and Krause,23 as were the procedures followed. Studies by Benson and Krause indicated that 4-20-h equilibration times are sufficient for this apparatus for all except the highest temperatures. To attain reproducible results at the higher temperatures we used equilibration times of up to 60 h, although the usual times were from 16 to 48 h. Temperatures were determined with Leeds and Northrup knife blade platinum resistance thermometers. These were checked weekly in triple-point-of-water cells, and twice each year in an NBS-certified benzoic acid cell. The 2-3-h drift of the main thermostat for equilibration was f0.002 K, and the overnight drift was f0.004 K. Gas-phase samples (GPS) and liquid-phase samples (LPS) of precisely known volumes (respectively, uG and uL) were analyzed separately to determine the amount of dry gas in each. The extracted gas in each case was confined over mercury in a series of calibrated volumes37 (*0.003%) which were connected via capillary tubing to a Ruska quartz Bourdon tube manometer (Model XR38). Although the resolution of this manometer is &0.0003% of full scale (0-106 kPa), the accuracy with calibration is *0.01% . This manometer was periodically (ca. every 2 months) checked against a Ruska gas-lubricated piston pressure gauge (Model 2465) with traceability of calibration to the National Bureau of Standards. The manometer, itself, the connecting lines, and the water bath in which the calibrated volumes reside were all thermostatted to (36) R. Battino, M. Banzhof, M. Bogan, and E. Wilhelm, Anal. Chem., 43, 806 (1971).
3232 The Journal of Physical Chemktry, Vol. 85, No. 22, 108 1
Rettich et al.
within fO.O1 K of each other at approximately 313 K. The use of a quartz Bourdon tube manometer for determining pressures is the main difference between our procedures and those of Benson and K r a ~ s who e ~ ~used a mercury manometer. Real gas corrections were used throughout, the amounts of dry gas being determined by direct PVT measurements. Using auxiliary information on the second virial coefficients, the vapor presswe and molar volume of water, and the partial molar volume at infinite dilution of dissolved gas, we determined the fugacity and the mole fraction of dissolved gas and hence Henry's constant at the equilibrium temperature and pressure. The temperature of the cold traps was maintained at roughly 193 K (dry ice/ acetone) at which temperature the pressure of water vapor over ices is about 0.05 Pa, thus contributing negligibly to the measured gas pressures. All molar quantities are based on the relative atomic mass table of IUPAC, 1975.39 For the gas constant a value of R = 8.31441 J K-' mol-l was used.40
Henry's constants were derived from the raw data by a method differing in several aspects from that advanced by Benson and K r a ~ s e .We ~ ~evaluated the various gasphase correction terms theoretically by utilizing experimental results on the second virial coefficient, while Benson and K r a ~ s used e ~ ~an empirical approach. Furthermore, the pressure dependence of Henry's constant was also accounted for explicitly via eq 2. The amounts of dry gas in the GPS bulb, nZG,and the LPS bulb, nZL,were separately determined by measuring pressure, temperature, and volume v of the dry gas contained in the calibrated manometric system (MS) described above:
Results Data Reduction. The solubility of a gas (often a supercritical component) in a liquid is determined by the equations of phase e q ~ i l i b r i u m . Introducing ~~ the auxiliary functions vapor-phase fugacity coefficient 6 and liquid-phase activity coefficient y (for which the asymmetric convention is adopted), the relevant relation for compound i at experimental pressure P and thermodynamic temperature T is
where B22is the second virial ~ o e f f i c i e n t . ~The ~ imprecision associated with these PVT measurements never exceeded k0.025 5%. The total amount of substance in the LPS bulb was calculated from
n2 = (RZ2)-1[PV/mMS
with appropriate superscripts L or G attached to nP The compressibility factor Z2of the pure gas was computed from a truncated virial series
nL = nZL + ul0/V,O
= 6i(P,T)yip = yi(P,T)xifiO(P,T)
(1)
Here, yi and xi indicate, respectively, the vapor-phase and the liquid-phase mole fractions, and f/' is an appropriate liquid-phase standard-state fugacity. For the gaseous component
where denotes Henry's constant for solute 2 dissolved in solvent 1. At saturation pressure (indicated by a subscript a) of the solvent, Pu,l,Henry's constant is accessible from experimentally determined ratios of fugacity over mole fraction, that is to say (3)
V2mis the partial molar volume of dissolved gas at infinite dilution. The standard-state fugactiy for the solvent is the Poynting-corrected orthobaric fugacity of pure solvent
(7)
where to an excellent approximation u10
fi
(5)
= nlLV,ON
uL
- n2LV2""
(8)
Hence, the liquid-phase mole fraction is
x 2 = nzLv,O/(u,O+ nzLVlo)
(9)
The last term in eq 8 is considerably smaller than the bulb volume. As a consequence, uncertainties associated with the partial molar volumes at infinite d i l ~ t i o n have ~~-~~ negligible influence on vl0 and hence on x 2 (less than 15 ppm for a 10% uncertainty in V29. V2mfor methane dissolved in water was expressed as a function of temperature (t = T/K - 273.15) by In (V2"/(cm3 mol-l)) = 3.541
+ 1.23 X
(10)
+ 1.22 X
(11)
and for ethane dissolved in water In (V2m/(cm3mol-l)) = 3.914
In order to calculate the corresponding fugacity, we obtained the equilibrium mole fraction of gas in the vapor phase from y 2 = nzG/nG= nzGRTZ/(PvG)
(12)
+
where 410= 41(Pu,17T) is the fugacity coefficient of pure solvent vapor, and V,O is the molar volume of pure liquid solvent; for the pressure range considered, it is essentially the orthobaric molar volume.42 (37)P. H. Bigg, Br. J.Appl. Phys.,15, 1111 (1964). (38)E. W. Washburn in "International Critical Tables", Vol. 111, E. W. Washburn, Ed., McGraw-Hill, New York, 1928. (39)IUPAC, Pure Appl. Chem., 47,75 (1976). (40)IUPAC, Pure Appl. Chem., 51, 1 (1979). (41)J. M. Prausnitz, "Molecular Dyanmics of Fluid-Phase Equilibria", Prentice-Hall, Englewood Cliffs, NJ, 1969. (42)G. S. Kell, J. Chem. Eng. Data, 20,97 (1975).
where nG = nlG n2Gis the total amount of substance, and 2 is the compressibility factor of the vapor phase. Thus, combining eq 9 and 12 with the defining relation for Henry's constant and the equilibrium condition, Le., eq 1 and 3 (43)(a) J. H.Dymond and E. B. Smith, "The Virial Coefficients of Gases. A Critical Compilation", Clarendon Press, Oxford, 1969. (b) A. Pompe and T. H. Spurling, "Virial Coefficients of Gaseous Hydrocarbons'', CSIRO Australian Division of Applied Organic Chemistry Technical Paper No. 1, 1974,p 1. (44)R. Kobayashi and D. L. Katz, Ind. Eng. Chem., 45,440 (1953). (45)W.L.Masterton, J. Chem. Phys.,22, 1830 (1954). (46)D. N.Glew, J. Phys. Chem., 66,606 (1962). (47)T.D. OSullivan and N. 0. Smith, J.Phys.Chem.,74,1460(1970). (48)E. W.Tiepel and K. E. Gubbins, J.Phys.Chem., 76,3044(1972).
The Journal of Physical Chemistry, Vol. 85, No. 22, 1981 3233
Solubility of Gases In Liquids
3.96
For the sake of a more compact notation we set OL = :v + nzLV? N vL - n2L(V2m- Vlo), and
298.15 K --
318.1 5 K
a in
n2G OL R T h2,l(P,T) = - - n2L v G Vf whence eq 13 becomes H2,1(Pu,1,T) =
lim lh2,1(P,T)Z42(P,T)J (15)
-. r” 4.55
x2+
The “uncorrected Henry’s constant” hz,l(P,T)consists entirely of experimentally accessible quantities, and their combination minimizes the significance of almost all systematic analytical errors.23 We note that the total equilibrium pressure P does not appear explicitly in eq 15; it has to be known, however, for the evaluation of the correction term Z+,(P,T) (which is nearly unity). For gaseous mixtures a t low pressure, it is convenient to use the virial equation in its volume-explicit form49(see eq 6). Let B11, Bz2, and B12 denote the second virial coefficient of pure components 1 and 2, and the second virial cross coefficient, respectively, and 812 = 2B12 - B11 - B22 (16) The compressibility factor is then given by z = 1 + PO’iBii + ~ $ 2 2 + Y I Y ~ ~ ~ Z I / R(17) T and the fugacity coefficient by
4i = exP(P(Bij + Yj2812)/Rfl
i
# j
(18)
where, for conciseness, the parenthetical mention of the functional dependence on P and T has been omitted. ’ The total pressure P a t any experimental temperature and the vapor-phase mole fraction y2 = 1- y 1 were calculated through an iterative procedure using the equilibrium condition in the form
- - - - - - -00 318.15 K
0
40
80
0
12c
PIkPa Figure 1. Henry’s constant H2,1(Pu,l,T) for methane and ethane in water plotted against the pressure Pat two temperatures. The plots give essentially zero slope.
of pure water vapor were taken from O’Connell;60the cross coefficients were extracted from the experimental work of Rigby and Prausnitzsl (H20+ CHI) and Coan and King2 (H20 C2H,). For the vapor pressure of water we used the Chebyshev polynomial representation of Ambrose and Lawrens~n.~~ Since the extrapolative procedure according to eq 15 is rather tedious, a slightly modified approach was adopted. The uncorrected Henry’s constants h,,(P,T)were determined as a function of pressure at two selected temperatures only, appropriate corrections due to vapor-phase nonideality and the Poynting correction were applied, and the resulting quantity plotted against pressure as shown in Figure 1. Within our experimental precision, the essentially zero slope corroborates the internal consistency of our approach12and thus permits evaluation of Henry’s constant at any desired temperature and at the corresponding saturation pressure of pure water from data obtained at pressure P via
+
y1 = (1 - x2)(Pu,1/P)(+?/4~) exp{Vt(P- P , , l ) / R T )
%,l(Pu,l,T) = h2,1(P,T)Z+z exp(-V2”(P - P,J)/RTJ (21)
(19) in conjunction with eq 12. When the mole fraction of dissolved gas is as low as in this work, the liquid-state activity coefficient of the solvent, yl, can be taken to be unity without introducing any appreciable error.1z Further, in the pressure range under consideration, liquid water is essentially incompressible, and hence the Poynting correction of eq 4 simplifies to the exponential in eq 19. In the first approximation (indicated by a superscript 1 in parentheses) the total pressure is calculated as the sum of the dry gas pressure plus Raoult’s law pressure for water. The corresponding approximation for the vapor-phase mole fraction is yl(’) = (1- X2)Pu,,/P(’) (20)
The average combined random error (imprecision) in H2,1(P,,1,T) is estimated to be less than *0.06% for both water + methane and water + ethane. The maximum systematic error is believed to be less than f0.03% for the former, and less than f0.05% for the latter mixture. Random errors were estimated from repetitive measurements of the number of moles of hydrocarbon via PVT measurements for the gas- and liquid-phase samples; systematic errors were estimated from uncertainties in calibrated volumes, temperature measurements, virial coefficients, etc. To assess the performance of the apparatus, we determined Henry’s constants for oxygen in water. The results of these measurements and a comparison with the most reliable literature valuesz3are given in Table I. For this table, our values were calculated by using the Benson and KrauseB empirical approach to permit a direct comparison with their results. Since there is a difference of about 0.1 % , our values being a bit lower on the average, there may be a small systematic error. However, we consider
leading to 41(1) via eq 18. P1) and 41(1) are then used to via eq generate an improved y1(2)(eq 19), leading to SZ) 17, which in turn yield a second approximation to the pressure pC2) (from eq 12) and hence a new 41(2),and so forth. This iterative scheme converges rapidly, resulting in consistent equilibrium values for total pressure, vapor phase mole fraction, fugacity coefficient, and compressibility factor. Auxiliary data on the second virial coefficient (49)J. G.Hayden and J. P. OConnell, Ind. Eng. Chern.,Process Des. Deu., 14,209 (1975).
(50)J. P. O’Connell, Ph.D. Thesis, University of California, Berkeley, 1967. (51)M. Rigby and J. M. Prausnitz, J. Phys. Chern., 72, 330 (1968). (52)C. R. Coan and A. D. King, Jr., J. Am. Chem. Soc., 93, 1857 (1971). (53) D. Ambrose and I. J. Lawrenson, J. Chern. Thermodyn., 4,755 (1972).
3234
The Journal of Physical Chemistry, Vol. 85, No. 22, 1981
TABLE I: Comparison of Values of Henry’s Constant for Oxygen Dissolved in Water with Literature Valuesas b
275.45 288.15 298.15 298.15 298.15 298.15 298.16 303.14 318.14 318.16 318.16 318.17 318.17 328.14
2.7199 3.6697 4.4058 4.3980 4.4068 4.3955 4.4070 4.7521 5.6985 5.6908 5.6895 5.6979 5.6956 6.1995
2.7258 3.6691 4.4039 4.4039 4.4039 4.4039 4.4045 4.7539 5.6995 5.7006 5.7006 5.7012 5.7012 6.2140
-0.22 0.02 0.05 -0.13 0.07 -0.19 0.06 - 0.84 -0.02 -0.17 - 0.19 - 0.05 - 0.10 -0.23
Rettich et al.
CGW and a four-term BK equation for water + methane; for water + ethane, the same criterion resulted in selecting five-term equations for both the CGW and the BK expansion. In all cases, the actual significance level for the number of terms chosen was well below 1% , Table IV contains the relevant coefficients Ai and ai for eq 22 and 23. The last two columns each in Tables I1 and I11 show the percent deviations of the data from their fit to either the CGW- or BK-type smoothing functions, i.e. 6 = 100[H2,1(expt)- H2,1(calcd)]/H2,1(calcd) (24) as indicated by the appropriate subscript. One measure of the dispersion of experimental results is the average percent deviation a = N-1c16il N
(25)
where N is the number of data points (each with Si). For water + methane a = 0.06%, in satisfactory accord with the estimated experimental uncertainty. In the case of Reference 23. Tis the thermodynamic temperature. water + ethane, however, for no obvious reason the scatter is slightly larger than our estimate, Le., a = 0.08%. an interlab correspondence of 0.1% to be excellent. Although comparison of our data with previous results Tables I1 and I11 show Henry’s constants H2,1(Pu,1,T)for is of limited significance because of the high precision of water + methane, and for water + ethane, respectively, in our measurements,a brief review of earlier work is in order. the temperature range 275-328 K. The possible influence In fact, a is about five to ten times smaller as compared of impurities was investigated by performing measureto that of any older work. ments with ultrahigh-purity gases. Within our experiWater + Methane. Henry’s constants, recalculated from mental error we found no difference that was a function the original data of Winkler:’ Morrison and Billett,% and of gas purity. All the results were well within the experShoor et are all considerably higher (up to about 4 % ) imental error found for measurements made with the than ours. The data of Wen and Hungsoand Ben-Naim slightly less pure gas samples. et al.14J5 me all somewhat higher, but usually within about Variation of H2,1with Temperature. For representing 1% of the present data. Yamamoto16 reported results the temperature dependence of Henry’s constant, many which are in good agreement with our values, and a set of methods have been reported in the literature. The most solubilities recently determined by Muccitelli and Wen24 useful approaches to date are those advocated by Clarke is more or less equally scattered about our fit with an and Glew,Mand by Benson and Krause.l7S These authors average deviation of ca. &0.6%. Claussen and Polglase’ssl assume that In (H2,1/Pa) and the associated standard work shows pronounced deviations toward smaller values thermodynamic function changes are well-behaved, conof Henry’s constant. tinuous, and derivable functions of either the thermodyWater + Ethane: Again, earlier measurements exhibit namic temperature T, or the inverse temperature T1. wide disparity. Claussen and Polglase’ssl values are again Subsequently, the enthalpy of solution is expanded in a significantly lower (by about 3%). Morrison and Billet’sm Taylor series about either (a) a suitably chosen reference data are rather high, as are WinklerV7 and Wen and temperature or (b) an inverse reference temperature. Hung’sso (deviations of several percent). The data deThus, when adopting the former procedure, our data were termined by Ben-Naim et al.l49l5are closest to ours, though fitted by unweighted least-squares analysis to a Clarkeslightly above by about 170, Glew-type equation modified by we is^^^ (CGW) Discussion In [H2,1(Pu,1,T)/Pa1= Thermodynamic Functions. Partial molar quantities A. A17-l A2 In T A ~ + T A ~ +T ...~ (22) pertaining to the solution process, i.e., the standard where T = 10-2T/K. When the method of Benson and changes in enthalpy AHz”, entropy AS2’, heat capacity Krause (BK) was used, the results were fitted by a power ACPo2etc. are derived by appropriate differentiation of series in T-l either eq 22 or 23. We emphasize, however, that these fitting equations represent the temperature dependence n of Henry’s constant under saturation conditions, that is In [H2,1(P,,1,T)/Pal = CaAT/K)-’ (23) i=O to say both temperature and pressure vary. Thus, for example, the enthalpy of solution at (T,P,,) is given by The CGW equation (22) has been used extensively by the exact relation Wilhelm et al.12in their recent critical review of gas solubilities in water. As a guide in deciding whether the inclusion of an additional term in eq 22 or 23 was justified, the F, test%was used. Generally, a percentage probability of 4 5 % is conwhere the first term on the right-hand side is obtained sidered acceptable, leading to the selection of a four-term
+
+
+
(54)E.C. W. Clarke and D. N. Glew, Trans. Faraday Soc., 62, 539 (1966).For a condensed treatment see P. D. Bolton, J. Chem. Educ., 47, 638 (1970). (55)R. F. Weiss, Deep-sea Res., 17,721 (1970). (56)P. R. Bevington, “Data Reduction and Error Analysis for the Physical Sciences”, McGraw-Hill, New York, 1969.
(57)L. W.Winkler, Berichte, 34, 1408 (1901). (58)T.J. Morrison and F. Billett, J. Chem. Soc., 3819 (1952). (59)S. K.Shoor, R. D. Walker, and K. E. Gubbins, J. Phys. Chem., 73,312 (1969). (60)W.-Y. Wen and J. H. Hung, J. Phys. Chem., 74,170 (1970). (61)W. F.Claussen and M. F. Polglase, J. Am. Chem. Soc., 74,4817 (1952).
The Journal of Physical Chemistry, Vol. 85, No. 22, 1981 3235
Solubility of Gases in Liquids T A B L E 11: Henry's Constant Temperature T u
T/K
P/atm
275.46 278.14 283.95 288.10 288.15 293.16 298.14 298.16 298.16 303.15 308.15 313.16 318.15 318.16 323.16 328.15
0.80626 0.84553 0.85559 0.84911 0.90323 0.90510 0.50789 0.85312 0.94704 0.97011 1.03092 1.03439 0.50108 1.09037 1.14714 1.16763
H,,l(Pu,l,T) for Methane Dissolved in Water as a F u n c t i o n of the T h e r m o d y n a m i c
h,,,/atrn
23 099.2 25 004.6 29 142.0 32 138.1 32 168.9 35 786.1 39 264.9 39 380.5 39 369.8 42 783.4 46 089.2 49 217.9 5 1 796.8 5 1 910.5 54 400.4 56 737.9
0.998 116 0.998 100 0,998 220 0,998 328 0.998 219 0.998 338 0.999 198 0.998 559 0.998 390 0.998479 0,998 517 0.998650 0.999 854 0.998752 0.998 889 0.999 122
(??$
0.998093 0.998 072 0.988 178 0.998 266 0.998 159 0.998 247 0.998 983 0.998 414 0.998 255 0.998 280 0.998 230 0.998 247 0.998 673 0.998149 0.998044 0.997 904
Hz4-
POPth cor
(PU,lT)/
atm
23 011.7 24 909.0 29 037.1 32 028.7 32 052.5 35 663.9 39 193.5 39 261.4 39 237.8 42 644.9 45 939.4 49 065.3 51 720.5 51 749.8 54 233.6 56 569.3
1.001 22 1.001 27 1.001 27 1.001 24 1.001 32 1.001 30 1.00069 1.001 20 1.001 33 1.00134 1.001 39 1.001 36 1.00057 1.001 39 1.001 42 1.001 39
22 983.6 24 877.3 29 000.4 3 1 989.1 32 010.3 35 617.7 39 166.3 39 214.4 39 185.6 42 587.9 45 875.6 48 998.7 5 1 691.2 5 1 677.8 54 156.7 56 490.8
z
@Z
atm
6
H24-
'P&/
2.328 8 1 2.520 69 2.938 46 3.241 30 3.243 44 3.608 96 3.968 52 3.973 40 3.97048 4.315 22 4.648 34 4.964 80 5.237 6 1 5.236 25 5.487 42 5.723 93
BK
CGW
-0.049 0,076 0.023 -0.017 -0.041 -0.058 -0.016 0.066 0.005 -0.044 0.026 0.166 -0.036 -0.065 -0.130 0.088
-0.050 0.077 0.025 -0.012 -0.041 -0.059 -0.017 0.065 0.004 -0.046 0.026 0.168 -0.034 -0.062 -0.130 0.085
a 6 is t h e p e r c e n t deviation f r o m either s m o o t h i n g e q 22 (CGW) or 23 (BK). Average p e r c e n t deviation, e q 25: +0.056; CYBK=k0.056.
T A B L E 111: Henry's Constant
C Y C G=~
H,.l(Pu.l,T) for E t h a n e Dissolved in Water as a F u n c t i o n of T h e r m o d y n a m i c Temperature T u 6
275.45 275.44 278.15 283.16 283.14 283.77 288.15 293.15 298.15 298.15 298.14 298.16 303.15 308.16 313.14 318.16 318.14 318.16 318.16 318.16 318.16 323.14 323.15
0.696 37 0.719 50 0.840 25 0.689 00 0.551 08 0.852 22 0.756 75 0.886 26 0.573 27 0.944 3 1 0.714 05 0.500 80 0.911 36 0.818 89 1.061 32 1.067 45 1.08542 1.091 25 0.669 4 1 0.906 50 0.598 89 0.730 96 0.958 29
13 981.1 13 992.5 15 654.0 1 8 766.1 1 8 714.1 19 240.6 22 271.4 26 020.3 29 639.3 29 860.4 29 739.3 29 669.1 33 750.7 37 686.9 4 1 850.4 45 500.8 45 531.7 45468.1 45 303.3 45 440.4 45 180.3 48 635.6 48 831.9
0.993 342 0.993 1 2 1 0.992 195 0.993 953 0.995 166 0.992 536 0.993 705 0.993 018 0.995 775 0.992 970 0.994 706 0.996 329 0.993 579 0.994 586 0.993 326 0.993 741 0.993 627 0.993 591 0.996 324 0.994 768 0.996 804 0.996 328 0.994 865
0.993 349 0.993 127 0.992 202 0.993 976 0.995 189 0.992 554 0.993 806 0.993 060 0.995 777 0.993 017 0.994 734 0.996 631 0.993 615 0.994 561 0.993 336 0.993 646 0.993 539 0.993 505 0.995 928 0.994 585 0.996 314 0.995 696 0.994 508
13 795.7 13 800.7 15410.7 18 540.3 18 534.1 18 954.8 21 994.1 25 659.3 29 389.4 29 443.4 29 426.0 29 460.6 33 319.9 37 279.1 4 1 294.0 44 928.7 44 949.2 44 883.2 44 953.0 44 957.9 44 869.9 48 248.4 48 314.4
1.001 53 1.001 59 1.001 84 1.001 48 1.001 18 1.001 83 1.001 60 1.001 85 1,001 15 1.001 93 1.001 44 1.00099 1.001 82 1.001 58 1.002 03 1.00198 1.002 0 1 1.002 02 1.001 17 1.001 65 1.00102 1.001 22 1.001 68
6 is t h e p e r c e n t deviation f r o m either s m o o t h i n g e q 22 (CGW) o r 23 (BK). ~ 0 . 0 7 8 CYBK ; = ~0.078.
T A B L E I V : Values of t h e Parameters Ai of E q 22 (CGW j a n d ai of E q 23 (BK) CGW coefficients
BK coefficients
Water t Methane A , = 127.173804 a, = 9.881539 a , = 9.864392 X l o 3 A ,= -155.575631 a , = - 2.150246 X l o 6 A , = -65.2552591 A , = 6.16975729 a , = 8.810241 X 10' Water A , = 1340.027128 A , = -2216.17099 A, = -2158.42179 A , = 718.779402 A, = -40.5011924
+ Ethane a , = -2.056101 X 10' a , = 2.624901 x l o 5 a , -1.128303 X 10' a , = 2.159875 X 10" a4 = -1.569604 X 10"
__
from the fitting equations, and (dP/d!7'), from vapor pressure data of water.53 For temperatures sufficiently below the critical, the effect of pressure on properties of condensed phases is generally very small. Even at the
13 774.5 13 778.9 15 382.4 18 512.9 1 8 512.3 1 8 920.1 21 959.0 25 612.1 29 355.8 29 386.7 29 383.6 29 431.4 33 259.4 37 220.2 4 1 210.5 44 840.1 44 859.0 44 792.6 44 900.6 44 883.9 44 824.0 48 189.4 48 233.2
1.395 71 1.396 15 1.558 62 1.875 82 1.875 76 1.917 08 2.224 99 2.595 14 2.974 48 2.977 60 2.977 29 2.982 13 3.370 01 3.771 34 4.175 65 4.543 42 4.545 33 4.538 61 4.549 55 4.547 86 4.541 79 4.882 79 4.887 23
BK
CGW
-0,032 0.024 0.082 -0.092 -0.039 -0.086 0.093 0.147 -0.102 0.000 0.006 0.131 - 0.151 -0.136 0.163 -0.023 0.060 -0.124 0.118 0.070 -0.059 -0,.065
-0.034 0.022 0.092 -0.091 -0.038 -0.086 0.082 0.139
0.017
-0.101 0.001 0.007 0.131 -0.141 -0.124 0.166 -0.030 0.054 -0.130 0.112 0.064 -0.065 -0.056 0.027
Average percent deviation, e q 25: C Y C G W =
highest temperatures investigated, the last term of eq 26 contributes insignificantly (cf. Table V); it amounts to about 14 J mol-l for HzO + C2H6at 328.15 K. The same comment applies for the other thermodynamic properties. Some conventional standard thermodynamic quantities based on the ideal gas state at 1atm (= 101.325 kPa) are given in Table V at selected temperatures. Note, that the four-term CGW equation predicts an essentially linear temperature dependence of the heat capacity change upon solution ACpo2, in contrast to the more complex behavior suggested by the corresponding BK equation. While the actual numerical differences between values generated from either the CGW or BK equation are exceedingly small within the temperatue range covered by our experiments, care must be exercised when extrapolation to higher temperatures is desired20because of increasing divergence of the smoothing functions. It is gratifying to note that Glew's careful treatment46 of older experimental data57,58,61@ yielded ACPo2 for water
3236
The Journal of Physical Chemistry, Vol. 85, No. 22, 1981
TABLE V: Conventionala Partial Molar Gibbs Energy of Solution AG;, Partial Molar Enthalpy of Solution AH,", Partial Molar Entropy of Solution AS;, and Partial Molar Heat Capacity of Solution ACp', of Methane and Ethane Dissolved in Waterb As,'/
T/K
ACp",/
(J mol-' (J mol-' (kJ mol-') (kJ mol-') K-l) K-') AH,'/
AG,"/
273.15 283.15 293.15 298.16 303.15 313.15 323.15 333.15
Water t Methane 22.648 -19.43 -154.0 -16.85 -144.8 24.142 25.546 -14.38 -136.2 26.217 -13.19 -132.2 26.868 -12.02 -128.2 28.113 -9.75 -120.9 -7.59 -114.1 29.288 30.398 -5.54 -107.9
262 252 242 237 231 221 211 201
273.15 283.15 293.15 298.15 303.15 313.16 323.15 333.15
Water t Ethane 21.423 -27.02 -23.68 23.134 24.737 -20.85 -19.50 25.503 -18.13 26.246 27.662 -15.10 -11.39 28.972 30.151 -6.55
373 302 271 270 280 330 420 551
-177.3 -165.3 -155.5 -151.0 -146.4 -136.6 -124.9 -110.2
a Based on the ideal gas state at 1 atm = 101.325 kPa. Values are based on the CGW smoothing equation (see Table IV).
I \
I
I
I
I
I (
I
Rettich et at.
many gases in a variety of s o l v e n ~ , 3 ~ 4 ~ 1It2considers ~13~~~~ the dissolution process as consisting of two steps: (a) creation of a cavity in the solvent large enough to accomodate the solute molecule, and (b) insertion into the cavity of a solute moelcule which interacts with the solvent in some specified way. Associated with step (a) is the partial molar Gibbs energy of cavity formation, kea",and with step (b) the partial molar Gibbs energy of interaction, pint. Henry's constant is thus given by In (H2,1/Pa) = pCav/RT+ pint/RT
+ In (plokT/Pa) (28)
where pio = N*/Vl0 is the molecular density of solvent, N Ais Avogadro's constant, and k is Boltzmann's constant. SPT provides a reasonable approximation for p,4,86*88 in that it yields an asymptotic expansion in the radius of the cavity to be created, retaining terms up to the cubic. In semiempirical applications of SPT, this expansion is usually used with the tacit assumption that the molecules behave as hard bodies but that volume and pressure at a given temperature are determined by the real intermolecular interactions in the fluid phase. Thus, retaining the actual experimental pressure and volume preserves information as to the attractive interactions in the solvent!" The various approximations involved in calculating the interactional term in eq 28 have been amply discussed in the l i t e r a t ~ r e . ~In J ~particular, the assumption of zero interactional entropy is expected to result in calculated entropies of solution which are too positive. Experimental and calculated Henry's constants and entropies of solution are compared in Table VI. Hydrophobic Interaction (HI). Following the approach of Ben-Naim,26when discussing HI between two solute particles we refer to the indirect part of the Gibbs energy required to bring the solute particles from change, GGHT(r), fixed positions at infinite separation to some close distance r (at constant temperature and pressure). The link with experimentally accessible quantities is established by noting that an indication of the strength of HI is provided by the approximate relation
GG"(r=d) = A ~ O C -~ ~AP'CH, H~
+
Flgure 2. Temperature dependence of ACpo2for water methane: this work, upper line; Glew's results, lower line (see text).
+ methane which are in satisfactory agreement with our
results. For the latter quantity, a comparison is given in Figure 2. Evidently, the fitting equation reported earlier12 is inadequate to describe the temperature dependence of ACpo2. A similar comment applies to the heat capacity change for water + ethane. In engineering applications41 it is often convenient to define adjusted Henry's constants at some arbitrary reference pressure Pr,usually zero pressure. For the present systems, to an excellent approximation
(29)
where d = 0.1533 nm is the carbon-carbon distance in ethane, and
A p o / R T = - lim [In ( p z L / p z G ) ] = - lim [In L(P,T)] = PlL.0
Pa-
-In L" (30) Here, L denotes the Ostwald coefficient,2eand p2Land pzG denote the molecular density of the dissolved gas in the liquid and the vapor phase, respectively, at equilibrium. The Ostwald coefficient can be derived directly from our experimental results (cf. eq 5) via
L(P,T) = p2L/p2G =
n2LuG/(n2GuL)
(31)
and its value at infinite dilution, L" = L(P,,l,T), may be obtained by extrapolation, similar to the method outlined where the partial molar volumes are given by eq 10 and 11. Scaled Particle Theory (SPT). Scaled particle theory has been used with fair success to predict solubilities of (62)0.L. Culberson and J. J. McKetta, Jr., J. Petrol. Technol.,3,223 (1951).
(63)R. A. Pierotti, J. Phys. Chern., 67, 1840 (1963). (64)R.J. Wilcock, J. L. McHale, R. Battino, and E. Wilhelm, Fluid Phase Equil., 2, 225 (1978). (65)(a) H.Reiss and H. L. Frisch, J. Chern.Phys., 31,369(1959). (b) H.Reiss, H. L. Frisch, E. Helfand, and J. L. Lebowitz, J. Chern. Phys., 32, 119 (1960). (66)H.Reiss, Adu. Chern. Phys., 9,1 (1965).
The Journal of Physical Chem;stry, Vol. 85, No. 22, 1981 3237
Solubility of Gases in Liquids
TABLE VI: Comparison between Experimental and Calculated Quantities via Scaled Particle Theory of Henry's Constant H, ,(P, ,T). Partial Molar Enthalpy of Solution AH,". and Partial Molar Entrow of Solution AS," ln Hl~l[(~,,',T)/Pal
AH,"/(kJmol-')
2°K
expt
calcd
283.15 298.15 313.15 328.15
21.781 22.102 22.324 22.467
21.129 21.453 21.717 21.930
expt Water t Methane -16.85 -13.19 -9.75 -6.54
283.15 298.15 313.15 328.15
21.353 21.814 22.151 22.368
20.330 20.826 21.237 21.576
Water t Ethane - 23.68 -19.50 -15.10 -9.13
AS,"/(J mol-' K - l )
calcd
expt
calcd
-15.61 -13.30 -11.15 -9.05
- 144.8 -132.2 -120.9 -110.9
-135.0 -127.2 -120.4 -114.1
- 23.60 - 20.48
-165.3 -151.0 - 136.6 -118.0
-156.6 -146.0 -136.9 -128.5
-17.59 -14.74
TABLE VII: Ostwald Coefficients L at Selected Temperatures for Methane and Ethane Dissolved in Water Calculated from H,,l(Pu,l,T)Values by BK Fitting Equation 102L" T/K 273.15 278.15 283.15 288.1 5 293.15 298.15 303.15 308.15 313.15 318.15 323.15 328.15
water t methane 5.8052 5.0954 4.5372 4.0942 3.7400 3.4554 3.2261 3.0412 2.8927 2.7742 2.6809 2.6091
water t ethane 9.9553 8.2410 6.9586 5.9761 5.2104 4.6061 4.1258 3.7421 3.4373 3.1977 3.0129 2.8763
Figure 3. The Gibbs ener and enthalpy changes for hydrophobic interaction 6GH'(d)and dH'(d) as a function of temperature T .
TABLE VIII: Regression Coefficients of BK-Type Fitting Equations for In L- Calculated from Smoothed H,, ( P ,,TI Values water t ethane water t methane
l__l___--
1,
t 9.884592
11
-1.024474 X t 2.184432 x -9.176532 X
1, 13
14
lo4 lo6 lo'
x 10' -2.602551 X lo5 t 1.117022 x lo* -2.137281 X 10" t 1.552598 x lo', t 2.231684
above. However, a much more convenient approach exploits the limiting relation
which to an excellent approximation may be recast into nm
( ~ u , l / R T ) 2 ~ l l ( 2-~Bldl l z (33)
Here, C$~"(P~,~,T) is the fugacity coefficient of gas at infinite dilution in solvent vapor. Thus by using smoothed results for Henry's constant in conjunction with eq 33, we obtained Ostwald coefficients a t infinite dilution L" pertaining to selected temperatures (Table VII). Regression coefficients for BK-type fitting equations are given in Table VIII. Figure 3 shows GGH1(d)and BHH1(d)as a function of temperature as derived from our new solubility data. and heat Figure 4 presents the changes in entropy GSHT(d) capacity GCpH1(d)associated with HI, which may be obtained by appropriate differentiation of eq 29. As already pointed out by Ben-Naim, the strength of HI initially
273
31 3
293
333
T/K Figure 4. The entropy and heat capacity changes for hydrophobic interaction 6SH1(d)and 6C,"(d) as a function of temperature T . The extrapolation to higher temperatures for the heat capacity change shows a rapid increase. This and the minima in 6SH'(d)and 6HH1(d) have to be regarded with reservation pending further experiments at temperatures above the present range.
increases with increasing temperature, i.e., GGHT(d) becomes more negative, although the rate of increase becomes progressively smaller at higher temperatures as evidenced by Figures 3 and 4. We note, however, the existence of a minimum in GS"(d) (and GHH1(d))around 323 K. Since it is located just at the edge of our experimental range, further experiments at somewhat higher temperatures are desirable to firmly establish or disprove its existence and the corresponding trend inversion.
Acknowledgment. The authors acknowledge the support of the Public Health Service via the National Institutes of General Medical Sciences Grant GM 14710-13 for this work. We also acknowledge the generous assistance given us by Professor B. B. Benson and Dr. D. Krause, Jr., in building the apparatus.
