J . Phys. Chem. 1988, 92, 1994-2000
1994
found for measurements closer to infinite dilution and closer to the critical point. The theoretical and experimental findings that V20 is proportional to K~ shows that any theory or correlation must include this factor in order to work well near the critical point. The correl~~ this sponding-states treatment of Brelvi and O ' C ~ n n e includes factor. The scaled particle theory of gas solubility3' and the perturbation theory of Neff and M c Q ~ a r r i egive ~ ~ reasonable results only if the experimental value of the solvent density and its temperature derivative are used in the calculation since this builds in the experimental value of K~ for water. Marshall and co-workers have correlated chemical equilibria as simple functions of the density of the solvent and the t e m p e r a t ~ r e . ~ This ~.~~ procedure gives terms proportional to K~ for AV, and these equations can fit data near the critical point.
TABLE 11: Comparison of Henry's Law Constants for Argon, kH, Calculated from Solubilities and Heat Capacities
T IK 298.15' 308.15' 318.15' 328.15e 338.15e 348.15' 309.9' 335.9' 365.3' 368.3' 397.3' 424.7f 453.7f 568.d
k,lMPa solubilitya solubilityb heat capacity' P,,, 17.2 MPa 17.2 MPa 4023 4671 5264 5777 6190 6495 4748 6093 6409 6601 6144 53 36 4354 1774
4927 5691 6384 6985 7462 7813 5788 7351 7689 7915 7358 6384 5216 1774
AkH/MPad 17.2 MPa
1846 5724 6417 6996 7443 7752 5631 7355 1967 7966 7492 6520 5283 1438
81 -33 -3 3 -1 I 19 61 157 -4 -278 -5 1 - 1 34 -1 36 -67 336
Acknowledgment. This work was supported by the National Science Foundation under G r a n t s CHE8009672 arid CHE8412592. W e also thank Dorothy E. White, William E. Davis, and Patricia S. Bunville for their help.
"Henry's law constants, kH.at saturation pressure derived from solubility data. * k Hcalculated from solubility and corrected to 17.2 MPa by using In kH(P2.T)= In kH(P,,T)+ 13 ( V 2 ; " / R TdP, ) where is the partial molar volume at infinite dilution of the solute. kH calculated from heat capacity data in ref 3. d A k H = k , at 17.2 MPa derived from solubility data - kHat 17.2 MPa derived from heat capacity data. 'Solubility data from ref 19. fSolubility data from ref 27.
Registry No. H 2 0 , 7732-18-5; Ar, 7440-37-1; Xe. 7440-63-3; C2H4. 74-85-1. (33) Brelvi, S. W.; O'Connel, J. P. AIChE J. 1972, 18, 1239. (34) Pierotti, R.A. Chem. Reu. 1976, 76, 717. (35) Neff, R. 0.;McQuarrie, D. A. J. Phys. Chem. 1973, 77, 413. (36) Marshall, W . L.; Franck, E. U. J . Phys. Chem. ReJ Data 1981, IO, 295. (37) Frantz, J. D.; Marshall, W. L. Am. J. Sci. 1984, 284. 651 (38) Marshall, W. L. Rec. Chem. Prog. 1969, 30. 61.
in Figure 2. There is qualitative agreement between the results. The maximum is in the correct place, but the calculated values are too low a t temperatures below the maximum and too high above the maximum. Presumably, better agreement would be
Apparent Molar Heat Capacities of Aqueous Argon, Ethylene, and Xenon at Temperatures up to 720 K and Pressures to 33 MPa Daniel R. Biggerstafft and Robert H. Wood* Department of Chemistry, University of Delaware, Newark, Delaware 19716 (Received: June 23, 1987; In Final Form: October 1 4 , 1987)
The apparent molar heat capacities, Cp,m, of aqueous solutions of argon, ethylene, and xenon were measured at temperatures decreases from 300 to 720 K with a flow, heat capacity calorimeter developed in our laboratory. For all three gases CP,* as temperature is increased, goes through a shallow minimum at about 420 K, then rises to a maximum -5000 J mol-l K-' at about 665 K, and decreases to a minimum -5000 J mol-' K-' at 685 K for pressures near 33 MPa. This remarkable behavior at high temperatures is due to the proximity of the critical point of water. The behavior is qualitatively similar to ( d 2 p / d P ) , where p is the density of water.
-
Introduction A previous paper from this laboratory reported the first experimental measurements of the heat capacity of an aqueous solution of a slightly soluble gas. The measurements were on aqueous argon solutions from 300 to 600 K.' These results showed a remarkable behavior. The apparent molar heat capacity of aqueous argon increased to a very large and positive value at 600 K. Theoretical models predict that, at pressures above but near the critical pressure of pure water, the apparent molar heat capacity of slightly soluble gases will get very large and positive as the temperature increases, then suddenly decrease to a large and negative value, and finally, gradually increase again.*" Corresponding behavior for the enthalpies of gases in other solvents has been found?+ and similar behavior has been found at liquid-liquid critical points." This paper reports experimental measurements * A u t h o r to whom correspondence should be addressed ' Present addre>, Reishhold Chemicals, Dot er. DE I9901
0022-3654/88/2092-1994$01.50/0
of the heat capacity of dilute aqueous argon, ethylene, and xenon a t temperatures up to 720 K. The qualitative predictions of the ( 1 ) Biggerstaff, D. R.; White, D. E.; Wood. R. H. J . Phys. Chem. 1985, 89, 4378. (2) Wheeler, J. C. Ber. Bunsenges. Phys. Chem. 1972, 76, 308. (3) Chang, R. F.; Morrison, G.; Levelt, Sengers, J. M. H. J . Phys. Chem. 1984, 88, 3389. (4) Levelt Sengers, J. M. H.; Chang, R. F.; Morrison, G. Equation of State-Theories and Applications; Chao, K . C.; Robinson, R. L., Jr., Eds.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986. ( 5 ) Levelt Sengers, J. M. H.; Everhart, C. M.; Morrison, G.; Pitzer, K. S. Chem. Eng. Commun. 1986, 47, 315. ( 6 ) Chang, R. F.; Levelt Sengers, J. M. H. J. Phys. Chem. 1986,90,5921. (7) Morrison, G.; Levelt Sengers, J . M. H.; Chang, R. F.; Christiansen, J . J . Supercritical Fluid Technology; Penninger, J . M. L.; Radosz, M.; McHugh, M. A,; Krukonis, V. J., Eds.; Elsevier: Amsterdam, 1985; p 2 5 . (8) Christianson, J. J.; Walker, T. A. C.; Schofield, R.S.; Faux, P. W.: Harding, P. R.; Izatt, R. M. J. Chem. Thermodyn. 1984, 16, 445 and references cited therein.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1995
Apparent Molar Heat Capacities of Ar, CzH4, and Xe theoretical models are confirmed. The present results on the heat capacities can be combined with previous experimental measurements of the volumes as a function of temperature and pressure, together with the room temperature values of the Henry's law constant and the enthalpy of solution. This combination gives all of the thermodynamic properties of these solutions over the whole temperature and pressure range and allows the testing of theoretical models for these properties. We have investigated the utility of a perturbation theory to represent these results.
Experimental Section The flow, heat capacity calorimeter used has been described elsewhere'' as well as the sample preparation, sample loop loading, and concentration determination.'J2J3 The calorimeter measures the ratio of powers between the tube heaters in the sample and reference cells required to maintain a constant temperature rise in the fluid passing the heaters. The ratio of powers in the heaters, P,/Pw,is proportional to the ratio of the heat capacity flux through the two cells where the subscripts s and w refer to sample and water, respectively. The ratio is used to calculate a heat capacity , ~ ,usingI4 ratio, C ~ , , / C ~ by c p , , / c p , w = (1 + fAP/Pw)dw/d, (1) where AP/Pw = ( ( P , - P w ) / P w )d, is the density at the experimental pressure and temperature of the sample loop, and f is a heat loss correction factor which must be determined at each temperature.I5 The solution density, d,, was calculated by using d, = d , + [ A ( T - 298.15) + B ] m , where m is the molality, T i s the sample loop temperature in K, and A and B were determined from the experimental volumetric data.'J2 The values of A are 0.000, -0.20, and -0.029 kg m-3/(K.moEkg-') for argon, xenon, and ethylene, respectively, and the values of B are 9.32, 89.13, and -17.81 kg m-3/(mol-'.kg-') for argon, xenon, and ethylene, respectively. The heat loss correction factor,& was determined in one of two ways.I5 A known change in the heat capacity flux can be obtained by changing the flow rate in the sample cell by a known amount. The correction factor then was the ratio of the known ratio of heat capacity flux and the measured ratio of heat capacity flux. The other method is to use 3.0 mol kg-' NaCl as a chemical standard. This is applicable only over the range of temperatures and pressures that the power ratios are known for 3.0 mol kg-' NaCI. The apparent molar heat capacity Cp,+can be calculated from the heat capacity ratio, molality, m , solute molecular mass, M2, and heat capacity of water, cDIw, by using
CP,+ =
CP,W
[ "( G)+ ;( "1 CP.W
CPS
(2)
where Ac,/c,,, = cps/cpsw - 1. Note that if the units of c , , ~ ,M2, and m are in J g-' K'I, g mol-', and mol g-' of water, respectively, then Cp,+is in J mol-' K-I. The uncertainty in the heat loss correction factor is 1% up to 600 K but increases to 5% at 715 K. The uncertainty in the power ratio increases from f0.0002 to fO.OO1 as temperature increases and is estimated for each measurement from the signal to noise ratio during the experiment. The uncertainty in the molality is 1%. The uncertainty in the temperature varied from f O . l K at 300 K to f l . O K at 720 K. The uncertainty in the pressure was f 0 . 2 MPa. A typical measurement is made as follows." The isothermal shield, bottom lid heater, and preheater are all set to the desired (9) Worlmald, C . J. Ber. Bunsenges Phys. Chem. 1984, 88, 826 and references cited therein. (IO) Pleger, M.; Klein, H.; Woermann, D. J . Chem. Phys. 1981, 74, 2505. ( 1 1 ) White, D. E.; Wood, R. H.; Biggerstaff, D. R., submitted for publication in J . Chem. Thermodyn. ( f 2 ) Biggerstaff, D. R.; Wood, R. H. J . Phys. Chem., preceding paper in this issue. (13) Biggerstaff, D. R. Ph.D. Dissertation, University of Delaware, June ~~
1986. (14) Smith-Magowan, D.; Wood, R. H. J. Chem. Thermodyn. 1981, 13, 1047. (15) White, D. E.; Wood, R. H. J . Solution Chem. 1982, 11, 223.
temperature of the experiment, and the instrument is allowed to equilibrate with the flow off. The block is heated only by the surrounding. Once equilibrium is reached, the tube resistance thermometers are calibrated against the Burns resistance thermometer in the block by measuring the resistance of all thermometers at very low flow rates. With the resistance thermometers calibrated, the preheater temperature is changed as the flow is increased to keep the block temperature at approximately the same temperature. This ensures that if a temperature gradient is present, it will be small and that the incoming water is close to the block temperature. After the instrument reequilibrates the cell tube heaters are turned on. A steady state is established, and the resistance thermometers and tube heater voltage are measured. A base line is established, the sensitivity of the instrument measured, the heat loss correction factor determined, and then a sample injected. The calorimeter was shown to be functioning properly after the modifications by measuring the heat capacities of 0.015 or 3.065 mol kg-' NaCl at temperatures of 575, 668, and 718 K with various flow rates to give different heat loss correction factors. The same results were obtained at each temperature within experimental error."
Results The apparent molar heat capacities, Cp,+,for aqueous argon, ethylene, and xenon are given in Tables I-IV. These measurements, together with our earlier low-temperature measurements on argon,' are the first direct measurements of Cp,+,for these solutions; thus comparison to other data is very limited. The concentrations used in the present study are dilute enough such that Cp,+should be equal to the partial molar heat capacity, within experimental error at temperatures below 500 K. Values of (?p,2 have been calculated from both high precision solubility data below 373 K and the temperature dependence of enthalpies of dilution for comparison with the present values. The high precision solubility results of Benson and Krausei6 together with their fitting equation yield cp,2 = 240 and 166 J mol-' K-' at 319.05 and 392.28 K, in excellent agreement with the present results of Cp,+= 243 and 179 J mol-' K-I. The agreement at 392 K is remarkably good considering the measurements covered the range 274-318 K. Not surprisingly, at higher temperatures the differences become very large. The results of Benson and Krause for argon show the same excellent agreement with our earlier measurements on argon in the temperature range of both experiments.l Above 320 K the results diverge as before. The enthalpies of solution of Olofsson et al.I7 yield a value of 270 f 5 for at 298 K and 0.1 MPa, also in excellent agreement with the present results, Cp,+= 270 f 10 J mol-' K-' at 298 K and 3 1 MPa (extrapolation from measurements at 3 19 K). Battino'sI8 smoothed fit of solubility measurements is in reasonable agreement with the other results (Cp,+= 248 J mol-' K-l for from 273 to 348 K and 0.1 MPa). The pressure dependence of Cp,+at 298 K can be estimated from the volumetric data,I2 and it is less than the experimental error, so the above comparisons are justified. For ethylene, values of 190, 197, and 290 J mol-' K-I for cp2 were calculated from Wilhelm et al.,I9 Morrison and Johnstone,io and Glew and Moelwyn-Hughes,21respectively, at 298.15 K and 0.1 MPa. Dec and Gillz2 found 280 f 5 J mol-' K-' from the temperature dependence of AH,. A value of 265 f 20 J mol-' K-' for Cp,+was obtained by a short extrapolation (from 302 to 298 K) of our direct measurements of Cp,+at 18.2 MPa.
cp,2,
cp,2
cp,2
(16) Benson, B. B.; Krause, D., Jr. J . Chem. Phys. 1976, 64, 689. (17) Olofsson, G.; Oshodj, A. A,; Quarnstrom, E.; Wadso, I . J . Chem. Thermodyn. 1984, 16, 1041. (18) Battino, R. Solubility Data Series; Clever, H. L., Ed.; Paragon Press: New York, 1979; Vol. 2, p 135. (19) Wilhelm, E.; Battino, R.; Wilcock, R. J. Chem. Rev. 1977, 77, 210. (20) Morrison, T. J.; Johnstone, N. B. J . Chem. SOC.1954, 3441. (21) Glew, D. N.; Moelwyn-Hughes, E. A. Discuss. Faraday SOC.1943, 15, 150. (22) Dec, S. F.; Gill, S. J. J . Solution Chem. 1985, 14, 827
Biggerstaff and W o o d
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988
1996
TABLE I: Apparent Molar Heat Capacities of Aqueous Argon" TIK ATIK PIMPa mblmol kn-l C..'/J mol-' K-' 0.0787 520 (15) 12.26 32.09 606.77 730 (38) 10.44 32.19 0.0767 629.57 679 (38) 10.44 32.19 0.0767 629.57 2474 (46) 0.0862 65 1.47 9.15 32.06 2454 (46) 0.0862 9.15 32.06 65 1.47 3390 (127) 0.0906 3 1.99 659.02 7.52 0.0906 3464 (127) 3 1.99 7.52 659.02 0.0959 5964 (162) 32.29 668.69 6.26 0.0959 6220 (162) 32.29 668.69 6.26 0.08 17 5537 (161) 32.39 4.85 672.47 5731 (161) 0.08 17 32.39 672.47 4.85 0.0888 -1867 (207) 32.12 679.34 5.33 -1839 (207) 32.12 0.0888 679.34 5.33 -5215 (200) 32.36 0.0843 5.60 686.41 -4916 (200) 32.36 0.0843 686.41 5.60 0.0961 -4660 (89) 31.58 8.10 690.35 -4872 (89) 3 1.58 0.0961 8.10 690.35 32.29 0.0792 -3834 (96) 9.17 697.24 0.0792 -3984 (96) 9.17 697.24 32.29 -1254 (95) 708.72 32.06 0.0702 1 1.40 708.72 32.06 0.0702 -1400 (95) 11.40 32.06 0.0975 10.24 -798 (55) 720.37 0.0975 -743 (55) 32.06 10.24 720.37
f
P.lP,d 1.0045 (2) 1.0056 (4) 1.0050 (4) 1.0189 (4) 1.0188 (4) 1.0243 (10) 1.0248 (10) 1.0342 (10) 1.0358 (10) 1.0188 (6) 1.0195 (6) 0.9880 (10) 0.9881 (10) 0.9709 (10) 0.9724 (10) 0.9711 ( 5 ) 0.9700 ( 5 ) 0.9779 ( 5 ) 0.9771 ( 5 ) 0.9918 ( 5 ) 0.9910 ( 5 ) 0.9906 (5) 0.9911 ( 5 )
T//K 302.15 302.35 302.35 302.85 302.85 302.1 5 302.15 303.15 303.15 300.65 300.65 301.95 301.95 302.65 302.65 301.65 301.65 302.1 5 302.15 301.55 301.55 302.35 302.35
1.06 1.12 1.12 1.18 1.18 1.14 1.14 1.16 1.16 1.43 1.43 0.90 0.90 0.84 0.84 1.oo 1.oo 1.07 1.07 1.33 1.33 1.37 1.37
T$/K 308.15 308.35 308.35 308.35 308.35 308.35 308.35 306.45 306.45 308.15 308.15 306.45 306.45 306.45 306.45 305.05 305.05 306.25 306.25 306.45 306.45 306.45 306.45
PRh/MPa 7.57 7.37 7.37 8.44 8.44 8.95 8.95 9.31 9.31 7.90 7.90 8.50 8.50 8.00 8.00 9.13 9.13 7.41 7.41 6.48 6.48 9.50 9.50
'The density of the solution, ds, in kg m-3 at the experimental pressure, p , was calculated by using ds = d, + m[A(TL - 298.15) + B ] , where d , is the density of water at the experimental pressure and the temperature of the sample loop, TL, m is the molality, and A and B are -0.00 kg m-3/(Kmobkg-l) and 9.32 kg ~n-~/(mol.kg-'),respectively, as determined from experimental volumetric data.12J3 T is the average temperature of the experiment, and A T is the temperature rise during the experiment. bThe molality was calculated from the temperature of the sample cylinder bath, TB, and partial pressure of the 'The number in parentheses is the estimated uncertainty in the least significant digit of the apparent molar heat capacity due to the uncertainty in the power ratio only. The apparent molar heat capacity was calculated by using eq 1 and 2. dThe number in parentheses is the uncertainty in the least significant digit of the power ratio determined from the signal to noise ratio. eThe heat loss correction factors were determined by flow-rate experiments as described by White and and are the average of the infusion and withdrawal values. The values measured for the experimental temperatures, the infusion and withdrawal heat loss correction factors, and the flow rates were as follows: 606.77 K, 1.03, 1.09, 0.687 mL min-I; 629.57 K, 1.08, 1.14, 0.688 mL min-I; 651.47 K, 1.15, 1.20, 0.5402 mL min-I; 659.02 K, 1.11, 1.17, 0.588 mL min-I; 668.69 K, 1.13, 1.19, 0.505 mL min-I; 672.47 K, 1.44, 1.41, 0.486 mL min-'; 679.34 K, 0.87, 0.94, 0.416 mL m i d ; 686.41 K, 0.77, 0.90, 0.397 mL min-I; 690.35 K, 0.97, 1.03, 0.326 mL min-I; 697.24 K, 0.97, 1.17, 0.318 mL min-I; 708.72 K, 1.26, 1.39, 0.318 mL min-l; and 720.37 K, 1.32, 1.41, 0.4012 mL min-I, respectively. The flow rate was changed by rt0.0515 mL min-I. f T L is the sample loop temperature. gTs is the sample cylinder equilibrium temperature. hPBis the total pressure of the sample cylinder at equilibrium. I
I
1
I
ETHYLENE
-0.6
6
4000
-0.4
40001 4000
2000
-0.2
6000
O
8
O
v
l
-
0
.
6
-
2000
-
l-
r 1-
O
N
:
'Y m
l-
0
O
?
2 0
'E m
k . f
N
-
.8 0
NP
m
-2000
0.2
-4000
0.4
600
700
650
TIK
-2000
-
-4000
-
I 600
I
1
I
650
Y
I
I
700
TIK
Figure 1. Apparent molar heat capacity, CP,+,plotted versus temperature for ethylene: 0, experimental values near 32.2 MPa; -, ( a 2 p / a p ) ,at 32.2 MPa.
Figure 2. Apparent molar heat capacity, Cp,+,plotted versus temperature for argon: X, experimental values near 32.2 MPa; -, ( a 2 p / a p ) , at 32.2 MPa.
The a p p a r e n t molar heat capacities, CP,+,of t h e t h r e e gases exhibit a r e m a r k a b l e behavior. The C,,# initially decreases a n d a t approximately 420 K begins t o increase. A s h a r p m a x i m u m
(about 5000 J mol-' K-l) and minimum (about -5000 J mol-I K-I) occur a t 6 6 5 & 2 a n d 6 8 5 & 2 K, respectively, for pressures near
The Journal of Physical Chemistry, Vol. 92, No. 7, 1988 1997
Apparent Molar Heat Capacities of Ar, CzH4,and Xe
TABLE 11: Apparent Molar Heat Capacities of Aqueous Ethylene at 18 MPa" T IK
AT/K
302.10 302.10 346.52 346.52 348.99 378.99 348.99 379.39 379.39 446.60 446.60 496.54 496.54 545.51 545.51 570.60 570.60 596.09 596.09
2.25 2.25 3.25 3.25 8.84 8.84 8.84 8.25 8.25 7.40 7.40 6.39 6.39 4.99 4.99 4.69 4.69 3.62 3.62
P/MPa 18.10 18.10 18.27 18.27 18.20 18.20 18.20 18.30 18.30 18.30 18.30 18.40 18.40 18.24 18.24 18.30 18.30 18.40 18.40
mb/mol kg-I 0.1737 0.1737 0.1767 0.1767 0.1737 0.1737 0.1737 0.1727 0.1727 0.1737 0.1737 0.1737 0.1737 0.1727 0.1727 0.1717 0.1717 0.1727 0.1727
ps/pwd
f
TLf/K
TBgl K
PBh/MPa
1.0023 ( 10) 1.0035 (10) 1.0019 (2) 1.0021 (2) 1.0014 (3) 1.0018 (3) 1.0016 (3) 1.0005 (2) 1.0008 (2) 1.0006 (2) 1.0005 (2) 1.0011 (4) 1.0009 (4) 1.0019 (4) 1.0016 (4) 1.0048 (7) 1.0040 (7) 1.0150 (7) 1.0144 (7)
1.04 1.04 1.06 1.06 1.05 1.05 1.05 1.03 1.03 1.03 1.03 1.02 1.02 0.99 0.99 0.98 0.98 0.99 0.99
297.85 297.85 298.15 298.15 298.15 298.15 298.15 298.65 298.65 298.55 298.55 298.15 298.15 296.15 296.15 296.45 296.45 297.65 297.65
305.65 305.65 305.65 305.65 305.85 305.85 305.85 305.65 305.65 305.85 305.85 305.75 305.75 305.65 305.65 305.65 305.65 305.65 305.65
7.26 7.26 7.66 7.66 7.36 7.36 7.36 7.24 7.24 7.33 7.33 7.32 7.32 7.20 7.20 7.07 7.07 7.14 7.14
C,,,C/J mol-I K-l 246 277 237 244 225 235 230 206 212 212 210 236 230 281 271 389 364 803 782
(24) (24) (5) (5) (7) (7) (7) (5) (5) (5) (5)
(IO) (10) (11) (11) (21) (21) (24) (24)
.
+
+
"The density of the solution, ds, in kg m 6 at the experimental pressure, p , was calculated by using d, = d , m[A(TL - 298.15) B ] , where dw is the density of water at the experimental pressure and the temperature of the sample loop, TL, m is the molality, and A and B are -0.029 kg m-3/(Kmol.kg-1) and -17.81 kg m-3/(mol.kg-1), respectively, as determined from experimental volumetric data.I7 Tis the average temperature, and A T is the temperature rise during the experiment. bThe molality was calculated from the temperature of the sample cylinder bath, TB, and partial pressure of the gas.'2.44 'The number in parentheses is the estimated uncertainty in the least significant digit of the apparent molar heat capacity due to the uncertainty in the power ratio only. The apparent molar heat capacity was calculated by using eq 1 and 2. dThe number in parentheses is the uncertainty in the least significant digit of the power ratio determined from the signal to noise ratio. 'The values of the heat loss correction factor, f,were determined by using 3.0 mol kg-' NaCl as a chemical standard as recommended by White and W00d.l~ The values measured for PJP, corrected to 3.000 mol kg-I NaCl at the experimental pressures and temperatures of 302.10, 346.52, 348.99, 397.93, 446.60, 496.54, 445.5 1, 570.60, and 596.09 K were 0.9519, 0.9514, 0.9504, 0.9401, 0.9250, 0.9000, 0.8507, 0.8108, and 0.8360, respectively. fTL is the sample loop temperature. g T B is the sample cylinder equilibration temperature. hPBis the total pressure of the sample cylinder at equilibrium.
TABLE III: Apparent Molar Heat Capacities of Aaueous Ethvlene at 32 MPaY _. T/K
AT/K
PIMPa
mb/mol kg-'
606.05 629.47 629.47 65 1.49 65 1.49 659.02 659.02 668.43 668.43 672.38 672.38 679.43 679.43 686.38 686.38 697.21 697.21 705.41 705.41 708.67 708.67 722.21 722.21
11.77 10.44 10.44 9.15 9.15 7.52 7.52 6.26 6.26 4.68 4.68 5.33 5.33 5.60 5.60 9.17 9.17 4.90 4.90 1 1.40 11.40 11.45 11.45
32.29 32.46 32.46 32.19 32.19 32.26 32.26 32.19 32.19 32.23 32.23 32.23 32.23 32.29 32.29 32.19 32.19 32.39 32.39 3 1.99 3 1.99 32.06 32.06
0.1697 0.1697 0.1697 0.1697 0.1697 0.1697 0.1697 0.1727 0.1727 0.1677 0.1677 0.1717 0.1717 0.1717 0.1717 0.1707 0.1707 0.1687 0.1687 0.1697 0.1697 0.1697 0.1697
C,,,'/J
mol-' K-l
461 (7) 656 (12) 633 (12) 2131 (23) 2140 (23) 3072 (66) 2971 (66) 4461 (90) 4610 (90) 3020 (78) 3134 (78) -1968 (106) -1750 (106) -4526 (99) -4546 (99) -2711 (44) -2689 (44) -1562(106) -1551 (106) -1021 (39) -1012 (39) -493 (34) -470 (34)
p,/pwd 1.0056 (21 1.0083 i 3 j 1.0078 (3) 1.0294 (4) 1.0295 (4) 1.0391 (IO) 1.0376 (IO) 1.0424 (10) 1.0440 (10) 1.0177 (6) 1.0186 (6) 0.9729 ( I O ) 0.975b ( I O ) 0.9451 (IO) 0.9449 (IO) 0.9625 (5) 0.9627 (5) 0.9786 (1 1) 0.9788 (1 1) 0.9813 (5) 0.9814 (5) 0.9877 (5) 0.9880 (5)
f
TLf/ K
TBSlK
PBhIMPa
1.09 1.12 1.12 1.18 1.18 1.14 1.14 1.16 1.16 1.38 1.38 0.90 0.90 0.84 0.84 1.07 1.07 1.45 1.45 1.33 1.33 1.51 1.51
302.15 302.35 302.35 302.65 302.65 302.15 302.15 303.15 303.15 300.65 300.65 301.95 301.95 302.65 302.65 302.15 302.15 301.65 301.65 301.55 301.55 302.35 302.35
308.35 308.35 308.35 308.35 308.35 308.35 308.35 306.45 306.45 308.15 308.15 306.45 306.45 306.45 306.45 306.25 306.25 308.15 308.15 306.45 306.45 306.15 306.15
7.84 7.78 7.78 7.80 7.80 7.81 7.81 7.62 7.62 7.24 7.24 7.44 7.44 7.44 7.44 7.35 7.35 7.53 7.53 7.24 7.24 7.10 7.10
+
+
"The density of the solution, ds, in kg m-3 at the experimental pressure, p , was calculated by using d, = d, m[A(TL - 298.15) B ] , where d , is the density of water at the experimental pressure and the temperature of the sample loop, TL, m is the molality, and A and B are -0.029 kg m-3/(Kmol.kg-1) and -17.81 kg m-3/(mol.kg-'), respectively, as determined from experimental volumetric data.12 Tis the average temperature, and AT is the temperature rise during the experiment. bThe molality was calculated from the temperature of the sample cylinder bath, TB, and partial pressure of the gas.12*45'The number in parentheses is the estimated uncertainty in the least significant digit of the apparent molar heat capacity due to the uncertainty in the power ratio only. The apparent molar heat capacity was calculated by using eq 1 and 2. dThe number in parentheses is the uncertainty in the least significant digit of the power ratio determined from the signal to noise ratio. 'The heat loss correction factors, f,were determined by flow-rate experiments as recommended by White and Wood'* and are the average of the infusion and withdrawal values. The values of the experimental temperatures, the infusion and withdrawal heat loss correction factors, and the flow rates at the experimental pressures were as follows: 606.05 K, 1.07, 1.10, 0.698 mL m i d ; 629.47 K, 1.08, 1.14, 0.688 mL min-I; 651.49 K, 1.15, 1.20, 0.540 mL m i d ; 659.02 K, 1.11, 1.17, 0.558 mL min-I; 668.43 K, 1.13, 1.19, 0.505 mL m i d ; 672.38 K, 1.40, 1.35, 0.495 mL min-'; 679.43 K, 0.87, 0.94, 0.416 mL m i d ; 686.38 K, 0.77, 0.90, 0.397 mL m i d ; 697.21 K, 0.97, 1.17, 0.318 mL m i d ; 705.41 K, 1.46, 1.43, 0.502 mL min-I; 708.67 K, 1.26, 1.39, 0.318 mL min-I; and 722.21 K, 1.46, 1.57, 0.354 mL m i d , respectively. The flow rates were changed by f0.0515 mL min-'. I T L is the sample loop temperature. gTB is the sample cylinder equilibrium temperature. *PBis the total pressure of the sample cylinder at equilibrium.
33 MPa. Figures 1-3 show the high-temperature behavior. Discussion
Low-Temperature Results (below 430 K ) . The apparent molar
heat capacities, Cp,@, of the three solutions have the same general behavior over the entire temperature range studied. The decrease in Cp,@ with increasing temperature at 298 K proposed first by Frank and Evans23was found. The Cp,#of all the solutions de-
1998 The Journal of Physical Chemistry, Vol. 92, No. 7. 1988
Biggerstaff and Wood
TABLE I V Apparent Molar Heat Capacities of Aqueous Xenon at 31 MPa" T/K ATIK PIMPa tnblmol kg-' CnhC/Jmol-' K-' 244 (1 3) 14.18 30.97 0.0676 319.05 242 ( I 3) 14.18 30.97 0.0676 3 19.05 14.47 30.94 0.0661 179 (13) 392.28 0.0661 179 (13) 14.47 30.94 392.28 184 (14) 16.40 31.18 0.0672 436.84 31.18 0.0672 184 (14) 16.40 436.84 190 (13) 15.14 3 I .24 486.52 0.0685 35.14 31.24 0.0685 190 (13) 486.52 11.53 0.068 1 332 (28) 569.48 31.55 0.068 1 332 (28) 11.53 569.48 31.55 932 (28) 11.96 0.0596 31.28 632.50 0.0596 987 (28) 11.96 632.50 31.28 2353 (74) 0.0606 10.64 648.25 30.60 2285 (74) 10.64 648.25 0.0606 30.60 9.91 2469 (72) 0.0600 31.55 65 1.01 9.91 2469 (72) 0.0600 651.01 31.55 5372 (171) 0.0587 6.29 663.74 31.14 5552 (171) 6.29 0.0587 31.14 663.74 6.60 -4352(154) 0.0590 30.43 685.51 -4151 (154) 6.60 30.46 685.51 0.0590 -246 (77) 10.80 30.46 0.0598 720.10 -385 (77) 0.0598 10.80 720.10 30.46
P
P,IP,d 1.0010 (2) 1.0010 (2) 1 .oooo (2) 1 .oooo (2) 1.0000 (2) 1.oooo (2) 1 .oooo (2) 1.oooo (2) 1.0016 (4) 1.0016 (4) 1.0045 (2) 1.0049 (2) 1.01 15 (4) 1.0115 (4) 1.0107 (4) 1.0103 (4) 1.0168 (6) 1.0175 (6) 0.9798 (6) 0.9806 (6) 0.9971 (4) 0.9963 (4)
1.06 1.06 1.03 1.03 1.12 1.12 1.03 1.03 0.96 0.96 1.22 1.22 1.26 I .26 1.31 1.31 1.32 1.32 0.80 0.80 1.61 1.61
TJIK 295.15 295.15 297.15 297.15 295.35 295.35 295.45 295.45 293.15 293.15 292.65 292.65 293.65 293.65 292.35 292.35 291.85 291.85 291.25 291.25 290.95 290.95
PRh / M Pa 2.10 2.10 2.03 2.03 2.09 2.09 2.14 2.14 2.10 2.10 1.91 I .97 1.98 1.98 1.90 I .90 1.83 1 .83 1.80 1.80 1 .I9 1.79
TR'IK 306.35 306.35 306.15 306.15 306.35 306.35 306.35 306.35 306.1 5 306.15 308.15 308.15 308.15 308.15 306.85 306.85 306.85 306.85 306.35 306.35 305.85 305.85
"The density of the solution, d,, in kg m-3 at the experimental pressure, p , was calculated by using d, = d, + m [ A ( T ,- 298.15) + B ] , where d, is the density of water a t the experimental pressure and the temperature of the sample loop, TL, tn is the molality, and A and B are -0.20 kg m-3((Kmol.kg") and 89.13 kg ~n-~/(mol.kg-'),respectively, as determined from experimental volumetric data.I2 T i s the average temperature, and AT IS the temperature rise during the experiment. b T h e molality was calculated from the temperature of the sample cylinder bath, T,, and partial pressure of the gas.'*," CThenumber in parentheses is the estimated uncertainty in the least significant digit of the apparent molar heat capacity due to the uncertainty in the power ratio only. The apparent molar heat capacity was calculated by using eq 1 and 2. dThe number in parentheses is the uncertainty in the least significant digit of the power ratio determined from the signal to noise ratio. 'The heat loss correction factors, were determined by flow-rate experiments as recommended by White and Wood15 and are the average of the infusion and withdrawal heat loss correction factors, and the flow rates were as follows: 319.05 K, 1.07, 1.06, 0.988 mL min-'; 392.28 K, 1.02, 1.04, 0.997 mL min-'; 436.84 K, 1.18, 1.07. 0.807 mL min-'; 486.52 K, 1.02, 1.05, 0.791 mL min-I; 569.48 K, 0.93, 0.99, 0.780 mL min-'; 632.50 K, 1.21, 1.24, 0.509 mL min-I; and 648.25 K, 1.31, 1.21, 0.411 mL min-'. The flow rates were changed by f0.0515 mL min-'. ITL is the sample loop temperature. gTB is the sample cylinder equilibration temperature. * P Bis the total pressure of the sample cylinder at equilibrium.
creased up to approximately 420 K. Frank and Evans proposed that water formed a more hydrogen-bonded structure around a hydrophobic solute leading to an increased Cp,+. This hydroXENON gen-bonded structure breaks up with increasing temperature, leading to a decrease in Cp,$with increasing t e m p e r a t ~ r e . ~ ~ S h i n ~ d aestimated ~~ the contribution of the hydrogen-bonded structure near the solute and concluded it is negligible . . above o o o 4000 6 -0.4 approximately 430 K. Studies of enthalpies of dilution confirm this c o n c l u ~ i o n . ~ ~ Gill, Dec, Olofsson, and Wadso26proposed a model for the heat capacity of nonpolar solutes which assumes the waters in the first solvation shell exist in states that are not cooperatively coupled to each other. If the model parameters are chosen to fit the data for methane, the heat capacity as a function of temperature for other nonpolar gases is predicted with reasonable accuracy I E (It10%).22s26.27Gill, Dec, Olafsson, and Wadso's model sucm 0 24 cessfully predicts the present experimental results (It- 10%) for argon to 350 K and for xenon and ethylene to 400 K. At higher temperatures the Cp,#of the three solutes we investigated begins u I / to increase, indicating that a new factor, the proximity to the '0.2 -2000 critical point, is becoming important. Although the apparent molar heat capacities, Cp,+of all three solutions have the same general behavior, the rate of decrease in Cp,+with temperature at 298.15 K and the depth of the minimum 0.4 -4000 are larger for aqueous xenon. The different behavior of aqueous xenon could be related to the fact that it forms the most stable I I I 1 I I I I clathrate. 700 600 High-Temperature Results. Above approximately 430 K the 650 TIK at 33 MPa increase to a sharp apparent molar heat capacities, Cp,@, Figure 3. Apparent molar heat capacity, C,@, plotted versus temperature maximum at approximately 665 K and then decrease rapidly to for xenon; A, experimental values near 31 MPa; -, (a2p/r3p), at 32.2
r2
I
.
I
4
MPa. (23) Frank, H. S.;Evans, W. J . Chem. Phys. 1945, 13, 507. (24) Shinoda, K. J . Phys. Chem. 1977, 81, 1300. ( 2 5 ) Mayrath, J. A.; Wood, R. H. J . Chem. Thermodyn. 1983, 15, 265. (26) Gill. S. J.; Dec, S. F.; Olofsson, G. 0.;Wadso, I. J . Phys. Chem. 1985, 89, 3758.
(27) Naghibi, H.; Dec, S. F.; Gill, S. J. J . Phys. Chem. 1986, 90, 4621: J . Phys. Chem. 1987, 91, 245.
a sharp minimum at approximately 685 K. The absolute magnitude of the maximum and minimum are approximately the same for each solute (about 5000 J mol-' K-'). The magnitude of these results is illustrated by the fact that the enthalpy of solution increases by about 5 kJ for a 1 K change in temperature at 33
Apparent Molar Heat Capacities of Ar, C2H4,and Xe MPa and 665 K and decreases by about 5 kJ for a 1 K change in temperature at 33 MPa and 685 K. It is difficult to draw any conclusions about the differences in behavior of the three solutions near the maximum and minimum because of the large experimental errors in C,# in this region (-20%). The Occurrence of the maximum and minimum can be predicted by Wheeler's theory2 and by use of both classical and nonclassical critical point ~ c a l i n g . ~ - Wheeler ~ ~ ~ * showed using a lattice gas model that, for a solute where the solute-solvent interactions are repulsive or only slightly attractive compared to the solventsolvent interactions, the partial molar enthalpy at infinite dilution will approach positive infinity at the critical point of the solvent proportional to the isothermal compressibility of the pure solvent, K~ Since I?,~" = ( a H 2 0 / d T ) pwe conclude that cp,20 diverges as ( d ~ ~ / a O and , the partial molar heat capacity, C P , * O , will approach positive infinity as the critical point of the solvent is approached from a lower temperature and negative infinity as the critical point is approached from a higher temperature. For a pressure greater than the critical pressure, the maximum in K~ is finite and is shifted to a temperature higher than critical temperature. Under these conditions, has a maximum and minimum and is continuous. We expected and found similar at finite concentrations although is expected behavior for to diverge as x - ~at the critical p ~ i n t . ~ - ~ The present results for C,# have the same quantitative features as (a~,/aT),, but the temperature derivative of the compressibility goes through zero about 9 K higher in temperature. We have empirically found that our results behave more like (a2p/dP),, where p is the density of the solvent (see Figures 1-3). At the ) , exactly critical point of the solvent ( d * p / a P ) , and ( a ~ ~ / a Thave the same scaling law, so the theoretical predictions cannot tell which will be better away from the critical point. It should be remembered that the theoretical predictions are for CP,@"and solution nonideality is probably very large in this region (V# is as high as 3000 cm3).I2 The same behavior but with the opposite sign is expected for solutions where the solute-solvent interactions are much more attractive than the solvent-solvent interactions, e.g., aqueous solutions of electrolytes. This has been experimentally confirmed in our laboratory with 0.0150 mol kg-' NaCI." These results also behaved more like (a2p/@), than ( a ~ ~ / a T ) ~ . Pratt and Chandler29have developed a theory for the solubilities of gases in aqueous solutions that uses the experimental radial distribution function for water, together with a mean spherical approximation type of closure to the Omstein-Zernicke equation. We have not investigated the ability of this theory to reproduce the present data because the experimental radial distribution function for water is not available at high temperatures. The present data allowed us to explore the successes and limitations of scaled particle theory and perturbation theory in calculating the solubility of gases in liquids over a wide temperature range. The scaled particle theory as adopted by Pierotti30 for the calculation of the solubility of gases in liquids is a relatively simple and remarkably successful theory. This theory breaks the dissolution process down into a cavity formation term which is the free energy of forming a cavity in the liquid and adds to that an interaction term, giving the free energy of interaction of the solute with the solvent when it's in the cavity. Although several problems with the theory as applied to aqueous solutions have been pointed out,3'332the theory is successful in reproducing some of the remarkable properties of aqueous solutions near room temperature. Neff and M ~ Q u a r i ehave ~ ~ applied the perturbation theory of Leonard, Henderson, and Barker34to the calculation of the sol-
cp,20
cp,2
cp,2
Rosen, A. M. R u n . J . Phys. Chem. (Engl. Transl.) 1976, 50, 837. Pratt, L. R.; Chandler, D. J . Chem. Phys. 1977, 67, 3683. Pierotti, R.A. Chem. Reu. 1976, 76, 717. Ben-Naim, A.; Friedman, H. L. J . Phys. Chem. 1967, 71, 448. Stillinger, F. H. J . Solurion Chem. 1973, 2, 141. Neff, R. 0.; McQuarrie, D. A. J . Phys. Chem. 1973, 77, 413. Leonard, P. J.; Henderson, D.; Barker, J. A. Trans. Faraday SOC. 56. 2439.
The Journal of Physical Chemistry, Vol. 92, No. 7 , 1988
1999
ubility of gases and liquids. With certain simplifying assumptions, this theory becomes identical with the scaled particle theory as used by P i e r ~ t t iand , ~ ~the perturbation treatment gives additional terms that can be used for more rigorous calculations. Fernandez-Prini and c o - w o r k e r ~have ~ ~ successfully used the first few terms of this theory to reproduce the heat capacity of aqueous argon up to 575 K. We have investigated how well these theories can fit all of the thermodynamic data (Henry's law constants, enthalpies of solution, volumes, and heat capacities) over the whole temperature range from room temperature to above the critical point of the s01vent.l~ We wanted to see if we could understand the reason for some of the successes of these theories and, in addition, explore the applicability of these theories near the critical point of the solvent. In the theory of Neff and M c Q ~ a r i the e ~ ~Henry's law constant is given by In kH =
kHid + In kHHs
+ hl kHCor
(3)
where kHidis the ideal contribution to the Henry's law constant given by (4) and V , is the molar volume of the solvent. The hard-sphere reference term, kHHS,is given by In kHHs = p 2 H S / k T
(5)
where pZHSis the chemical potential of the hard-sphere reference fluid at density p. The third term gives the first-order perturbation theory correction to the hard-sphere reference value of the Henry's law constant. The equation for the enthalpy of solution AHs and partial molar volume V2 are similarly given by sums of ideal, reference, and correction parts. The important point for the purposes of this discussion is that the equation for AHs contains terms multiplied by the coefficient of thermal expansion of the refrence fluid a and that these terms occur in the ideal, reference, and correction parts of the equation. These terms give rise to terms in ( d a / d T ) , in the equation for Cp,#". In a similar way the equation for V2 contains terms multiplied by the compressibility of the reference fluid fl and these terms occur in the ideal, reference, and correction parts of the equation. The equations for the scaled particle theory have the same form as the above equations, except that some of the correction terms are missing or approximated in the scaled particle theory and all of the terms in V2c0rare missing.30 As expected, the main problem with these theories at high temperatures occurs in their handling of the infinities in the derivatives of the Henry's law constant as the critical point of the solvent is approached. The terms containing a and fl lead to major problems in applying the scaled particle theory at high temperatures. It is essential to use the experimental density of the solvent and, thus, the experimental values of a and p, rather than solvent densities calculated for the hard-sphere reference fluid. The reason for this is that the hard-sphere reference fluid does not have a critical point, and so its a and p's do not go to infinity at the critical point of water. By using the experimental value of the density, expansibility, and compressibility of the solvent, we are building into our hard-sphere reference fluid some of the properties of a solvent with a critical point, and this is essential in order to get the qualitatively correct behavior. For instance, there would be no sharp maxima at high temperatures in the predictions of V2 an no sharp maxima then minima in the predictions of CP,@" if the hard-sphere reference values of p , a , and p were used. If we do use these experimental values for p , a , and 0,then the correct maxima and minima in V2 and CS,+"are built into the equation because a and p both go to infinity at the critical point of the solvent. This creates a problem because the infinities in the enthalpy of solution and the volume occur in all three terms-the ideal term, as well as the reference and correction terms. Perturbation theories are only accurate when the correction terms (35) Fernandez-Prini, R ; Japas, M. L. J . Phys Chem. 1986, 90, 1395
2000
J . Phys. Chem. 1988, 92, 2000-2007
are relatively small compared with the ideal reference terms, so these theories cannot be accurate very near the critical point of water. In applying these theories to solutes that have strong attractive forces for the solvent, the problem becomes worse because the ideal term has the opposite sign to the correction term and both go to infinity at the critical point of the solvent. In spite of these clear problems with these theories, they do give large maxima and minima in the correct places, so they may be useful in fitting experimental data, provided that the range of data fit does not come too close to the critical point of the solvent. In a preliminary attempt to fit the present data to what is essentially the scaled particle theory appr~ximation,'~ we found that the maxima and minima in heat capacity and volumes were not predicted at exactly the right temperatures, so we could not get good fits to the data. The predicted temperatures of the maxima were about 2 K too high for Cp,#and 5 K too low for V$. The magnitude of the maxima and minima could be fit qualitatively. At this point we do not know whether the failure to get the right temperatures for the maxima is due to the fact that our results are for finite concentrations and the theory is for infinite dilution, or whether there are errors in the theory or even temperature errors in the measurements. As expected, the theory could give quite reasonable fits if the data were limited to results below the maximum and only a reasonable number of parameters were necessary to give these fits. It seems probable that the theory would also be reasonably good at predicting thermodynamic properties above the maxima and minima. It has been surprising that scaled particle theory and perturbation theory work so well for water where many anomalous effects occur. There has been some controversy over the reason for this success. Fernandez-Prini et al.36conclude that "The dissolution of small nonpolar solutes in water does not involve significant reorientation of the neighboring water molecules. If this were the case the hard-sphere perturbation theory would not be capable of describing the thermodynamics of the process over a wide temperature range." Pierotti30 points out that the use of the (36) Fernandez-Prini, R.; Crovetto, R.; Japas, M. L.; Laria, D. Arc. Chem. Res. 1985, 18, 207.
experimental value of the thermal expansion coefficient introduces implicit information about the liquid structure of the solvent into the theory. The ability of these theories to predict the correct qualitative behavior near the critical point is another example of the implicit introduction of information about the liquid structure. Just as the use of experimental densities at high temperatures builds in some of the properties of a reference fluid with a critical point, the use of experimental densities at low temperatures builds in some of the anomalous properties of water at low temperatures. In particular it builds in a temperature of maximum density ( a = 0 at 4 "C) which is due to the open three-dimensional structure of water with its tendency to coordinate tetrahedrally. Eley3' showed that the anomalous a of water allows prediction of the high heat capacities of nonpolar solutes in water. This does not mean that this process is not accompanied by significant reorientation of the water molecules. We note that simulations using model potentials for the solute and solvent show significant reorientations about isolated nonpolar solutes and pairs of nonpolar sol~tes.~*-~~
Acknowledgment. This research was supported by the National Science Foundation under G r a n t s CHE8009672 and CHE8412592. We also thank Dorothy E. White for helpful discussions about experimental technique, Patricia S. Bunville for preparing the figures, and William E. Davis for help with some of the calculations. Registry No. Ar, 7440-37-1; Xe, 7440-63-3; ethylene, 74-85-1. (37) Eley, D. D. Trans. Faraday SOC.1938, 35, 1281. (38) Geiger, A.; Rahman, A.; Stillinger, F. H. J . Chem. Phys. 1979, 70,
263.
(39) Pangoli, C.; Rao, M.; Berne, B. J. J . Chem. Phys. 1929, 71, 2975, 2982. (40) Rapaport, D. C.; Scheraga, H. A. J . Phys. Chem. 1982, 86, 873. (41) Zichi, D. A.; Rossky, P. J. J . Chem. Phys. 1985, 83, 797. (42) Awicki, J. C.; Scheraga, H. A. J . Am. Chem. SOC.1977, 99, 7413. (43) Swaminatham, S.; Harrison, S . W.; Beveridge, D. L. J . Am. Chem. SOC.1978, 100, 3255. (44) Battino, R. Solubility Data Series; Clever, H. L., Ed.; Paragon Press: New York, 1980; Val. 4, p 2. (45) Bradbury, E. J.; McNulty, R.; Savage, R. L.; McSweeney, E. E. Ind. Lng. Chem. 1952, 4 4 , 2 11.
A Poisson-Boltzmann Approximation for Strongky Interacting Macroionic Solutions Clifford E. Woodward* and Bo Jonsson Department of Physical Chemistry 2, Chemical Centre, LTH, S-221 00 Lund, Sweden (Received: July 7 , 1987; In Final Form: October 14, 1987)
Disregarding many (23)-body interactions, the effective pair potential between macroions is derived as the difference in free energy between adding two macroions at fixed separation and at infinite separation to the dispersion, assuming other macroions do not respond to the added particles. The free energy difference is obtained by using the overlap approximation, within the framework of a free energy functional, obtained by simplifying a full Poisson-Boltzmann treatment of the dispersion. The resulting pair potential has a screened Coulombic form, but with an apparently renormalized macroion charge and Debye screening length. The model thus gives the appearance of ion binding, but no binding is assumed. The theory gives excellent agreement with experimental structure factors for dilute dispersions of micelles and colloids, where the usual DLVO potential fails, provided the surface charge density and volume fraction are not too large.
1. Introduction Solutions containing highly charged aggregates are frequently encountered in chemistry. Micellar solutions, colloidal dispersions, and biomolecules are a few examples of such systems, which in many cases are of great technical importance. Theoretical work in this area has been stimulated by recent progress in liquid-state physics and, nowadays, it is usual for a report on structural data determined by, for example, light or neutron scattering, to include a theoretical analysis of the structure factors based on some
integral equation approximation.'s2 In these analyses it is usual that a McMillan-Mayer3 approach is adopted, where the macroions only are treated as a fluid interacting via an effective potential, which is just the free energy of the mobile species in (1) Medina-Noyola, M.; McQuarrie, D. A. J . Chem. Phys. 1980,73, 6279. Senatore, G.; Blum, L. J . Phys. Chem. 1985, 89, 2676. Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109. (2) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982, 46, 651. (3) McMillan, W. G.; Mayer, J. E. J . Chem. Phys. 1945, 13, 276.
0022-3654/88/2092-2000$0 1.5010 0 1988 American Chemical Society
