XQCEOVSSOLUTIOSS OF SONPOLAR GASES
281
Aqueous Solutions of Nonpolar Gases'
by Robert A. Pierotti School of Chemistry. Georgia Institute of Technology, Atlanta, Georgia 50956
(Received August 3, 1964)
A theory of gas solubility in water has been developed using the scaled-particle theory to calculate the reversible work required to introduce a solute molecule into a fluid. I t yields very good agreement with experiment for the heats, entropies, arid niolar heat capacities of solution and for the partial molar volumes of the solutes. The thermodynaniic properties of aqueous solutions are discussed in light of the enthalpy and entropy of cavity formation. The theory shows promise as a method for investigating the thermodynamic properties of aqueous solutions of ions and the properties of gas solubility in niolten salts and molten metals. The theory has been used to deteriiiirie the Lennard-Jones (6-12) pair potential parameters for water and for benzene. The values for water are u = 2.76 A. and e / k = 85.3OK. The values for benzene are u = 5.24 A. and e l k = 498°K These parameters are discussed in relation to values reported elsewhere.
1. Introduction
The solubility of nonpolar gases in water has been a subject of considerable experimental and theoretical interest for many years. The most recent interest stems from the importance of hydrophobic bonding in proteins and from the process of the denaturation of proteins, both of which are related to the interaction of nonpolar groups in aqueous solutions.28tb Recently, a number of papers have appeared in which new and more precise data have been reported on the aqueous solutioris of various nonpolar gases. 3-g Except for the work of Sdmethy and Scheraga,'o few theoretical papers have appeared since the work of Eley11t12and Frank arid E v a ~ s . ' ~These papers emphasized that a fundamental difference exists between the solubility of gases in water arid in unassociated organic solvents. This difference is manifest in the large negative heats, negative entropies, and positive molar heat capacities of solution always observed for aqueous solutions. Frank and Evansl3 ascribed this difference to the formation of an ordered, hydrogen-bonded structure around the solute molecule. This model of water solvation is called the iceberg theory or the frozen-layer theory, but Frank and Evans were careful to point out that the ordered structure might not be ice-like. Semethy and Scheraga'O have approached the problem by developing a statistical thermodynamic theory of water based iipori the existence of five species of
water molecules differing in their number of hydrogen bonds and then considering the perturbation of the energy levels of these species by the presence of a solute molecule. They coriclude that the completely hydrogen-bonded specie is energetically favored in the first solvation layer and hence partial hydrogen-bonded cages develop around each solute molecule. It is interesting that nuclear magnetic resonance s t u d i c ~ ' ~ have been unable to substantiate the existence of an ice-like structure about nonionic solutes and do riot indicate any significant increase or decrease in the (1) This work was supported in part by the National Science Foundation. (2) (a) I. AI. Klots. Science, 128, 815 (1958); (b) W. Kauamann,
Adnan. Protein Chem., 14, 1 (1959).
T. J. Morrison and N.B. Johnstone, J . Chem. Soc., 3441 (1954). D. 11.Himmelblau, J . P h y s . Chem., 63, 1803 (1959). (5) C. E. Klots and B. B. Benson, J . .)farim Res.. 21, 48 (1963). (6) V. H. Konig, Z . lVaturforsck., 18, 363 (1963). (7) C. E. Klots and B. B. Benson, J . Phys. Chem., 67, 933 (1963). (8) E. Douglas, ibid., 68, 169 (1964). (9) D. N. Glew, ibid.. 66, 605 (1962). (10) (a) G. NQmethyand H. A. Scheraga, J . Chem. Phys., 36, 3382 (3)
(4)
(1962); ib) ibid., 36, 3401 (1962). (11) (a) D. D. Eley, Trans. Faraday Soc.. 35, 1981 (1939); (b) i h i d . , 35, 1242 (1939).
D. D. Eley, ibid.,40, 184 (1944). Frank and 3f. W. Evans, J . Chem. P h y s . , 13, 507 (1945). (14) (a) E. A. Balars, A. A. Bothner-By, and J . Gergely. J . .\fol. B i d , 1, 147 (1959); (b) F. A. Bovey, .Yatctre, 192, 324 (1961). (12)
(13) H. S.
Volume 69, Szimher 1
Janiiary 1 . W ;
282
ROBERTA. PIEROTTI
degree of hydrogen bonding in water as a result of the solution process. The solubility of gases in unassociated solvents was treated with very good success by an application of the scaled-particle theory of fluids. l5 I t seemed desirable to attempt to treat aqueous solutions in essentially the same manner and without explicitly introducing any assumptions about the structure of liquid water. The scaled-particle theory makes it possible to calculate the reversible work required to introduce a hard sphere into a dense fluid and has been shown to be applicable not only to nonpolar fluids, but also to molten salts, liquid metals, and polar fluids including water.I6 One would expect the major difference between aqueous and nonaqueous solutions to be in the process of creating a position in the solvent suitable to accommodate the solute molecule. I t is therefore the purpose of this paper to investigate whether or not the observed difference between the thermodynamic properties of gases dissolved in the two classes of solvents can be accounted for by means of a single theory and without recourse to models dependent upon the unique structures of solvents.
2. Theory Thermodynamic Equations. The solution process is considered for convenience to consist of two stepsI5: (1) the creation of a cavity in the solvent of suitable size to accommodate the solute molecule and (2) the introduction into the cavity of a solute molecule which interacts with the solvent. Associated with each step is a set of thermodynamic functions in terms of which the solution process can be described. It can be shown (see ref. 15) that for extremely dilute solutions
+ G,/RT + In ( R T / V )
spectively, and a p is the coefficient of thermal expansion of the solvent. The molar heat capacity change for the solution process is given by ACp
=
(T) dAH
=
C
+ Ci - R + 2apRT +
P
RT2
(z)
(4)
P
where ccand Ci are the partial molar heat capacities for cavity formation and interaction, respectively. The partial molar volume of the solute is given by
Vz
=
Vc + VI + PRT
(5)
where is the isothermal compressibility of the solvent. The Expression for Gc. The partial molar Gibbs free energy of creating a cavity in a fluid of hard spheres ~ ~ obtained the exwas derived by Reiss, et ~ 1 . l They pression
+ KIUIZ+ K Z ~ I Z+’ K3a1Z3
KO
(6) where the K’s are functions of the density, temperature, pressure, and hard sphere diameter of the fluid and a12is the radius of a sphere which excludes the centers of the solvent molecules. The K’s were evaluated to be G c
KO
=
=
R T { - ~(1 -
v) + ( 9 / z [ ~-/ (y)12) l (nPa13)/6
K~
=
- ( R T / u ~ ) ([ 6 ~ i ( l Y)I
+
1 8 [ ~ / ( 1- y)Iz]
K~
=
(RT/~~ [ ~w) (( i- Y)I
+ nPai2
+ 1 8 [ ~ / (-1 y ) i 2 ) -
(1)
27rPUl
where K is the Henry’s law constant, Gc and di are the partial molar Gibbs free energy for cavity formation and interaction, respectively, V is the molar volume of the solvent, T is the absolute temperature, and R is the gas constant. Henry’s law is defined here as
(7) K3 = ( 4 / 3 ) ~ P where y = ( ~ ~ 1 ~ p ) p/ 6is , the number density of fluid molecules, al is the hard sphere diameter of the fluid molecules, P is the pressure, R is Avogadro’s number times the Boltzmann constant, and T is the absolute temperature. The radius of the sphere (alz) which excludes the centers of solvent molecules is equal to (a1 a 2 ) / 2 ,where az is the diameter of the cavity to be created.
In K
=
G,/RT
pz
=
Kxz
(2)
where p 2 is the equilibrium pressure of the solute over the solution and x 2 is the mole fraction of the solute in the solutlion. The molar heat of solution is given by AH
=
b In K (---) b 1/RT
= p
Rc + Ri - RT
nc
+ crpRT2
(3)
where arid Ri are the partial molar enthalpies associated with cavity formation and interaction, reThe Journal of Physical Chemistry
+
(15) R. A. Pierotti, J . Phvs. Chem., 67, 1840 (1963). N o t e : a typographical error exists in eq. 22 of this paper. I t should read A H , = RT apRTz[(l 2 y ) l / ( l - Y ) ~ I . This equation is not applicable to water. (16) (a) H. Reiss, H. L. Frisch, and J. L. Lebowitz, J . Chem. Phys., 31, 369 (1959): (b) H. Reiss, H. L. Frisch, E. Helfand, and J. L. Lebowitz. ibid.. 32, 119 (1960); ( c ) H. Reiss and S. W. Mayer, ibid., 35, 1513 (1961); (d) S. W. Mayer, ibid.,38, 1803 (1963).
+
+
AQUEOUS SOLUTIONS OF NONPOLAR GASES
283
The Expression for ci. The partial molar Gibbs free energy for the interaction term is taken to be equal to the molar interaction energy. This approximation amounts to neglecting PYi and TSi in the interaction process and should be valid for the systems considered here (see ref. 15). The interaction energy of a nonpolar solute molecule with a polar solvent can be described in terms of dispersion, inductive, and rcplusive interactions. The dispersion and repulsive interactions can be adequately approximated by a Lennard-Jones (6-12) pairwise additive potential while the inductive interaction energy is given by an inverse sixth power law. The total interaction energy per solute molecule is then given by
where r p is the distance from the center of the solute molecule to the center of the Pth solvent molecule, Cdis is the dispersion energy constant, Cind is the inductive energy constant, and u12 is the distance a t which the dispersion and repulsive energies are equal in magnitude. In order to calculate ~ i it, is assumed that the solute molecule is immersed in the solvent. The solvent is assumed to be infinite in extent and uniformly distributed according to its number density p around the solute molecule. l7 The number of molecules contained in a spherical shell a distance r from the center of the solute molecule is. then equal to 47rpr2 dr where dr is the thickness of the shell. Combining this with eq. 8, dividing by kT, and replacing the summation by an integration gives
ZilkT
=
Ei/RT = t.i/kT
=
-5.3&*dis/kT - 8sOO€*ind/kT (12) The value of Cdia may be estimated by means of the Kirkwood-Muller formulala
}
CKM = 6mc2
where m is the mass of an electron, c is the velocity of light, al and a2 are the molecular polarizabilities of the solvent and solute, respectively, and x1 and x2 are the molecular susceptibilities of the solvent and solute. Alternately, Cdis may be evaluated in terms of the empirically determined Lennard-Jones (6-12) energy parameters usingig
+
CLJ = 4eizUiz6 = 4(€iE~)~’*[(Ql uz)/2]’
=
-(470/kT)
JRm
{(cdis
Cind = l.rl2ff2 (15) where p1 is the dipole moment of the solvent and az is the polarizability of the solute.
3. Results The Determination of 12 into eq. 1 yields
Q,O.
Cind )r-* dr
The substitution of eq.
(?‘,/RT (9)
where R is the distance from the center of the solute molecule to the center of the nearest solvent molecule. The integration yields
ei(R’)/kT =
(~*dis
-I- ~*lnd)(2/R’)~/kT(8/3)(E*di,/kT)(1/R’)g (10)
=
~pC,/GkTui2~
(11)
and R’ is equal to R/ui2. The minimum in e,(R’)/kT is determined by differentiation of eq. 10. A t the minimum R’ is equal to the sixth root Of €*dls/(€*dls €*lnd). SiIlCe t * , n d is always a small fraction of €*dls, R’ is very nearly unity and heiice
+
(In K )
lim uz -* 2.58
e,*/kT
+
+ In (RTIV)
(16)
Since both €*disand €*ind are related to the polarizability of the solute through eq. 1.7 and 1.5, a plot of In K 2’s. az for spherical solutes (the rare gases) should be a smooth curve. It was shown in ref. 15 that the extrapolation of such a curve to zero a2 was equivalent to obtaining the solubility of a hard sphere of diameter 2.58 A. This was expressed as uz-0
where
(14)
where el and el are the energy parameters for the solvent and solute, respectively, and ul and u2 are the distance parameters of the solvent and solute. Since the potential energy is rising very rapidly with decreasing distance a t u, the values of u1 and u2 are also effectively the values of al and a2. The value of Cind is given byl9
In K = -S.t%e*dls/kT - 8.OO€*ind/kT
ci(R)/kT
(13)
=
In KO
=
d,/RT
+ In (RTIV)
(17)
A.
(17) X-Ray diffraction studies indicate that this is a good approximation for water; see M.D . Danford and H. A. Leby, J. Am. Chem. Soc.. 84, 3965 (1962). It is interesting to note that the number of nearest neighbors calculated in this manner for Ar is 20 as in the argon clathrate, whereas for He and Ne it is between 7 and 8. No clathrates of He or Ne have been found. (18) A. Muller, Proc. Roy. SOC.(London), A154, 664 (1936). (19) J. 0.Hirshfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids.” John Wiley and Sons, Inc., New York, N . Y.,1945.
Volume 69. Number I
January 1966
ROBERTA. PIEROTTI
284
where KO is the Henry's law constant for hard spheres of diameter 2.58 1. Figure 1 shows such a plot for the solubility of the rare gases in water at various temperatures. Since the hard sphere diameter of water is not known with certainty from other sources, it seemed theoretically consistent in the present context to determine it from the value of KO obtained from Figure 1. Table I contains the necessary physical properties of water, while Table I1 contains the physical properties and parameters for a number of solute gases. The dipole moment of water was taken to be 1.84 D. 0
2
I
a Temperature, OC.--------. 40
V, cc./mole"
18.02 0 ... 2.75 85.3
X 105, deg.-l" P x 106, atm.-Ib ap
IT,
A.c
elk,
OK."
50'
25'
18.08 25.5 45.5 2.75 85.3
18.25 45.9
Table 11: Properties and Parameters for the Solutes u1,
A.
2.58" 2.63b 2.78' 3 . 40b 3 . 60' 4 . lob 4.36" 2.87' 3 . 70' 3 . 46b 3 . 76' 3.17' 3 . 82' 4.70b 5.88' 4.22' 4.23' 4.42' 5.27'
Hard sphere He Ne
Ar Kr Xe Rn Hz Nz 0 2
CO NO CH, CF4 CC14 CzH2 CzHa CJL C6H6
'
ai X
m/k, OK.
0 6.03 34.9 122 171 221 290 29.2 95 118 100 131 137 153 327 185 205 230 440
cc
1024,
./molecule 0
0.204" 0.393" 1.63" 2.46" 4.00" 5.86" 0.802" 1.74d 1.57d 1.93d 1.70d 2 . 70d 2.53" 10.10d 3. lgd 3 . 70d 4.33d 9 . 8gd
" See ref. 15. From second virial coefficients, ref. 19. From G. A. Miller, J . Phys. Chem., 64, 163 (1960). E. A. SfoelwynHughes, "Physical Chemistry," Pergamon Press, New York, N. Y., 1957, p. 373. * S. Glasstone, "Textbook of Physical Chemistry," 2nd Ed., D. Van Nostrand Co., New.York, N. Y., From viscosity data, ref. 19. 1951.
The Journal of Physical Chemistry
x
~
4
3
d
~
~
~
.
~
Figure 1. Experimental In K values plotted us. the polarizability of the rare gas solutes dissolved in water at various temperatures.
... 2.75 85.3
a N. A. Lange, "Handbook of Chemistry," 9th Ed., Handbook Publishers, Inc., Sandusky, Ohio, 1956. E. A. MoelwynHughes, "Physical Chemistry,'] Pergamon Press, New York, Determined as described in the text. N. Y., 1957.
Solute
I
I
Table I : Physical Properties and Parameters of Water
The values of aH20determined in thisomanner were 2.75 8. a t 4, 25, and 50" and 2.74 A. a t 70°.20,21 These values are in good agreement with values of aH,o obtained by means of an application of the scaled particle 'theory to compressibility data of water.21 In that case, the values found were 2.71 8. a t lo", 2.72 A. a t 30°, 2.71 8. a t 50", and 2.69 8. a t 80". Values of 5H1o from gas phase studies are 2.71 and 2.65 A.23 The oxygen-oxygen distance in ice is 2.76 A.24 Other values ranging from 2.50 up to 2.93 A.21have been used or reported, but the bulk of the values center around 2.7 A. In all subsequent calculations in this paper the value OH,O = aH,o = 2.75 8. was used. The Determination of E H , O / k . As in the case of UH,O, no single best value for eH2O,'k can be found in the literature. Values ranging from 167°K.25*26 up to 775"K.22 have been reported, but none of these seems consistent with the present treatment. The values obtained from gas phase studies using a Stockmayer potential22are invariably very high because they (20) A small temperature dependence for a should be expected and the value of a should decrease with increasing temperature: H. Reiss, to be published. (21) S. W. .Mayer, J . Phys. Chem., 67, 2160 (1963). (22) From transport properties: L. Monchick and E. A. Mason, J . Chem. Phys., 35, 1676 (1961). (23) From the second virial coefficient: J. S. Rowlinson, Trans. Faraday Sac., 45, 974 (1949).
(24) L. Pauling, "The Nature of the Chemical Bond," 3rd. Ed., Cornell University Press, Ithaca, N. Y.,1960. (25) J. H. van der Waals and J. C. Platteuw, Advan. Chem. Phya., 2, 1 (1959). (26) V. McKoy and 0. Sinano& J. Chem. Phys., 38, 2946 (1963).
~
~
~
AQUEOUSSOLU't'IONS
OF SONPOLAR
GASES
contain contributions from more than simply dispersion forces. I n addition, gas phase pair potentials are expected to be too high for use in calculations dealing with condensed phasesn The value 167°K. was obtained by van der Waals and PlatteuwZ6by fitting the equilibrium properties of the argon clathrate to a statistical thermodynamic theory of clathrates using u H 2 0 = 2.50 8. I n order to compensate for the very small u H , ~ , a high value of eH,o/k is required. Thus we should expect a value of e H , O / k smaller than 167°K. An estimate of the gas phase value of the potential parameter for water can be obtained from the polarizability and ionization energy of water using the London e q ~ a t i o n . ' ~This turns out to yield eH,o/k = 130°K.
285
I
I 25 -
20
-A
Solvent
-
I
I
-
I
I
-
Water
-
25'C
-
,5-
I 500
1500
io00
-
pair potential parameter for water in liquid water should be between 66 arid 75% of the gas phase value. Therefore, a crude approximation yields that ea,o/k should be between 86 and 98°K. It is possible, using the present theory of dilute solutions, to extract the value of eH,o/k from solubility data and a t the same time test further the validity of eq. 7 for water. Equation 16 can be rewritten and rearranged using eq. 11, 12, and 14 to give
The data presented in ref. Id were used for this purpose. The value of u+H calculated from the intercept of the In K us. uz curve was found to be 5.24 A., while the value of e,H/k calculated from the slope of the was 498°K. Table I11 comcurve - A us. (~Z/k)'/~u12~ pared the values of the above parameters with those determined in various other ways. The agreement is excellent with those methods which utilize the properties of the liquid state.
e+HIIC.
~~~~
Table I11 : Comparison of Pair Potentials for Benzene Obtained from Various Sources
where eZ is the Lenriard-Jones energy parameter for a given solute arid p is the number density of the solvent. The left-hand side of eq. 18 which we will designate as A can be calculated from experimental solubilities a t a given temperature along with the known physical properties of the solutes and the solvent including u H 2 0 equal to 2.75 A. A plot of - A us. (tZ/k)1'2u.123 should be a straight line with a slope given by ~ 1 1 . 1 7 ~ ~ / ~ ) ( e ~ , 0 Figure / k ) ~ ' ~2. shows such a graph for a large number of solutes dissolved in water a t 2.5" and indeed a good straight line is obtained. The only gas that does not fall on or very near the line is acetylene and perhaps this is not too surprising. The value of eH,cIk obtained from the slope of the line is 85.3"K. This is lower than the estimate made earlier, but considering the crudity of the theory and of the estimate, the agreement must be considered good. In order to substantiate the present method of using gas solubilit>ymeasurements for estimating the LennardJones parameters for a solvent, the solubility of a number of gases in bcwzerie was used to determine U+H and
u,
A.
elk, OK.
5.24" 498"
5.22b
501'
5 22" 504"
5.27d 440d
5.26= 494"
a Obtained from gas solubilities in the manner described in the text. *Obtained from the liquid state using the equation of state and the vapor pressure. Obtained from the liquid state using the equation of state and the entropy. Obtained from gas viscosity measurements. e Obtained from the cell theory of liquids of Salsburg and Kirkwood. This table except for the first column was taken from Y. Kobatake and R. J. Aider, J. Phys. Chena., 66,645 (1962).
The Thermodynamics of the Solution Process. Calculated arid experimental values of a number of thermodynamic properties are compared in Tables IV, V, and VI. Table IV compares values for the Henry's law constants a t 25" obtained from eq. 1. I n all cases ex(27) N. K.Kestner and 0. Sinanoglu, J . Chem. Phys., 38, 1730 (1963). (28) The continuum approximation (eq. 18) was used in these calcu-
lations.
Volume 69,Sumbtr 1
January 1.96.5
ROBERTA. P I E R O ~ I
286
cept CC14, C2H2,and CzHsthe predicted value is somewhat lower than the observed value. Only in the case of C2H4 and CzHa does the predicted value of K differ from the experimental value by more than a factor of 2. In all fairness t o the theory, it should be pointed out that it does much better in predicting the Gibbs free energy change associated with the solution process. The heat of solution is given by eq. 3 in which . P i is given by eq. 12 and GC(see ref. 15) by
Table IV : Henry Law Constants and Gibbs Free Energies of Solution at 298°K.
Solute
He Ne Ar Kr Xe Rn HZ Nz 0 2
co
NO C& CF4 CClr CzHz C2H4 CzHa CsHs
Theoret. K x io', stm.
10.40 6.18 3.10 1.23 1.oo 0.453 6.58 8.35 4.10 9.05 1.60 3.94 16.8 0.276 3.10 1.75 1.38 0.020
Exptl.
K
x
Theoret.
atm.
14.71" 12.36" 4.10" 2.35" 1.30" 0. 605b 7.10" 8.63d 4 . 3gd 5 . 82c 2.89" 3 . 928 27.6" 0 . 157b 0.134" 1,14" 2,90" 0.036
Exptl.
A(h98,
AGm,
cal./mole
cal./mole
6820 6530 6110 5700 5450 5000 6550 6590 6270 6750 5730 6250 7110 4700 6110 5810 5650 3130
7040 6930 6270 5930 5600 5130 6600 6610 6310 6480 6060 6250 7400 4350 4260 5520 6050 3490
104,
Re
?/)I( [6/(1 - ?/)][2(aiz/ad2W?//(l - ~)*II(aiz/a1)~-
apRT2[?//(1-
=
(adai)I
+
(a1z/aS
'
"International Critical Tables," Vol. 111, Mca See ref. 3. Graw-Hill Book Co., Inc., New York, N. Y., 1928. ' N . A. Lange, "Handbook of Chemistry," 9th Ed., Handbook PubSee ref. 8. ' W. F. Clauslishers, Inc., Sandusky, Ohio, 1956. sen and M. F. Polglase, J . Am. Chem. SOC.,74, 4817 (1952).
+
l/*I
+ 1]
(19)
Table V compares heat of solutions calculated from the present theory with observed heats a t 25" wherever such heats were available. Fairly good agreement is obtained for all solutes. It should be pointed out that many of the heats are not known to better than several hundred calories per mole. Included in Table V are calculated and experimental entropies of solution a t 25". Again good agreement is achieved for most of the solutes. The heat capacity of solution is given by eq. 4. As an approximation we take Ci to be zero. This is certainly not quite true, but the contribution to ACP will
Table V : Heats and Entropies of Solutions for Aqueous Solutions a t 298°K. Theoret. Theoret.
- AHze8.
Solute
He Ne Ar Kr Xe Rn Hz
Nz 0 2
co NO CH4 CF, CClr C2Hz CZH4 C2Ha CeH,
cal. /mole
Ha
500 1020 2445 3125 4395 5563 1008 2351 2423 2410 2432 3095 4046 9465 3985 4310 4873 9355
390 (400)b 1080 31OOb 3440' 4710 (4360)'
... 905 (962)b 2500 (2640)b 2850 (2950)' ...
... 3170 (3210)b
... ... ... ... ...
...
Exptl. - AHlP8, cal./mole F and EC G and Md
840 1880 2730 3550 4490 5500 1280 2140 2990 3910 2680 3180 ..
200 1910 2820 3710 4390 5120 1000 2650 2960 2690 2900 3350
..
6665 3590 3920 4830
3360 3790 4430
..
Others
530" 850* 2840'
... ... ... I
.
.
2520' 2890' ... ... 3052" , . .
-900oh ... ... 3983O ...
.
-ASzw
cal./moledeg.
24.6 26.3 28.7 29.6 33.0 35.4 25.3 30.0 28.5 30.7 27.4 31.4 37.4 47.5 33.9 34.0 35.3
-Exptl. EC
- A S m cal./mole-deg.7 G and Md
Others
26.5 28.8 30.2 32.3 33.6 34.3 26.0 29.8 31.3 29.8 29.4 31.8
23.8 29.2 31.1 32.9 35.8 36.8 26.0 32.1 31.8 31 . O 30.2 33.3
25.0" 26.9" 30.5' 31.5" 34.7"
..
..
...
.. 25.6 31.3 35.4
41.8 26.8 32.8 38.6
...
41.8
... 25.3" 31.0' 30.9' .
.
I
...
31.9"
... ... 33.6' ...
Frank and Evans, ref. 13. D. N. Glew and E. A. MoelHimmelblau, ref. 4. b Calcd. by Himmelblau using data from ref. 3. wyn-Hughes, Discussions Faraday Soc., 15, 150 (1953). * J. A. V. Butler, Trans. Faraday SOC.,33, 229 (1937). Klots and Benson, W. F. Claussen and M. F. Polglase, J . Am. Chem. SOC.,74, 4817 (1952). * P. N. Gross and J. H. Saylor, ibid., 53, 1744 ref. 7. (1931).
'
The Journal of Physical Chemistry
AQI;EOUSSOLUTIOXS OF SONPOLAR GASES
287
Table VI : Heat Capacities and Partial Molar Volumes for Aqueous Solutions at 298°K.
Solute
He Ne Ar Kr Xe Rn Hn ?u'?
02
co
XO
CH, CF, CCl, C2Hn C2Ha C2Hs CBHG
Exptl. A C P , ~cal./moleTheoret. ACp, --deg.-Tbeoret. cal./mole-deg. G and Vz, cc./ 277OK. 298OK. Mb Others mole
21.9 24.7 33.4 34 9 46.0 51.0 25.1 38.4 34.3 42.8 29.7 40.7 58.5 84.5 48.4 48.5 52.4 71.6
20.5 23.3 31.4 32.8 43.3 48 0 23 6 36.2 32.2 40.3 28 0 38.2 54.9 81.6 45.5 45 7 49,l 67.0
24.2 24.0 37.8 54.2 65.6 79.3 25.7 40.9 40.3 41.3 41.0 49.2
40.7' 40.7" 33d (40.7)" 40.7' 40.7c , , .
25.2" 33d 33d ... ...
55
..
...
..
...
35.5 ... 6 4 . 5 42.2" 6 5 . 5 59.2' ..
...
17.9 20.4 30.5 33.5 46.2 54.1 21.1 36.5 31.5 39.6 26.2 39.3 63.5 110.8 49.3 49.7 55.0 78.8
Exptl.
F2,cc./moie
qualitatively for the systems considered here that it should be a small negative number for weakly interacting solutes and should become a larger negative number a s ~ t h einteraction energy increases.- The- partial molar volume of cavity formation can be evaluated from eq. 6 and is given by l5
15.5" ... ... ... ... ...
1 8 . 9 (26)' 32.5"(40)' 25.Se(31)' 28.5e(36)'
... 37. 3h (36)' . . . .
.. . . .. ..
51.2h 83O
Usually assumed independent of temperature. * D. N. Glew and E . A. Moelwyn-Hughes, Discussims Faraday SOC.,15, 150 Morrison and Johnstone, ref. 3. Klots and Benson, (1953). J. H. Hildebrand and R. L. Scott, ref. 7. " Eley, ref. 12. "Solutions of Non-Electrolytes," 3rd Ed., Reinhold Publishing W. L. MasCorp., New York, N. Y., 1950. E Glew, ref. 9. terson, J . Chem. Phys., 22, 1830 (1954).
'
be small. The effect of this approximation is to cause the calculated ACp values to be too small and for the error to become greater as e2/k becomes greater. The heat capacity associated with cavity formation is given by
where only the second term is nonzero for water a t 4". Values of ACP calculated a t 4 and 25" are shown in Table VI along with experimental values from a number of sources. The difficulties in obtaining heat capacities from the temperature dependence of gas solubilities account for the discrepancies in the experimental quantities. The agreement between the calculated arid observed values must be considered good, and, indeed, the calculated values may well be better than many of the reported values. The partial molar volume of the solute, P,, is given by eq. 5 . Although Pi cannot be evaluated, it is clear
Vc = 82.O5(fi/ap)(XJc/RT)
+ ( T / ~ ) @ ~ N(21)
where RCis the partial molar heat of cavity formation from eq. 19, CYPand p are the coefficient of thermal expansion and the isothermal compressibility. respectively, and N is Avogadro's number. The factor 82.05 is the molar gas constant in units of cc.-atm./deg. The results of calculations of V2are given in Table VI and they are seen to be in good agreement with experiment in the limited number of cases where data are available. 4.
Discussion
The solubility of gases in water is adequately described by the same theory which successfully described the solubility of gases in organic solvents. Table VI1 compares theoretical and experimental data for the solubility of argon and nitrogen in benzene and water a t 25". The agreement between theory and experiment for both gases in both solvents is very good even though the magnitudes of the thermodynamic properties vary widely from benzene to water. It is clear that the scaled-particle theory gives a good approximation for the reversible work of cavity formation in both aqueous and nonaqueous solvents. A few comments deserve to be made concerning the present theory and that of E1ey.l' Eley, by applying thermodynamic relations for the bulk properties of a homogeneous system to the process of cavity formation, concludes that the enthalpy and entropy of cavity formation for water a t 4" are zero. Thus in Eley's treatment the free energy of cavity formation is zero, which is clearly i m p o ~ s i b l e . ~The ~ present theory uses the scaled-particle theory to determine the free energy of cavity formation and then splits this up into its appropriate enthalpy and entropy terms. A t 4" for water it is found that the enthalpy of cavity formation is indeed zero as determined by Eley, but that the entropy of cavity formation is a large negative value. Some typical thermodynamic values for cavity formation associated with argon in water a t 25" are G, = 4430 cal./mole, Hc = 690 cal./mole, and S, = -12.5 e.u. The corresponding values for argon in benzene a t 25" are B, = 3610 cal./mole, H = 3520 cal. /mole, and S = -0.30 e.u. It is the entropy of cavity formation that (29) H. H. Uhlig, J. Phys. Chem., 41, 1215 (1937). rhlig attempted to calculate the work required to create a cavity by using bulk surface tensions and the surface area created by the cavity.
Vol?ims 69, Number 1
January 1966
ROBERTA. PIEROTTI
288
Table VII: A C o m p a r i s o n of t h e Solubilities of Ar a n d Nz in Benzene a n d W a t e r a t 298°K. A S , cal./deg.
Ar in CsHa Ar in H 2 0
Theoret . Exptl. Theoret.
Exptl.
N2 in C6H6
Nzin H2O
Theoret. Exptl. Theoret. Exptl.
K , atm.
AG, cal./mole
A H , cal./mole
940 1140 31000 41000 2200 2260 83500 86300
4060 4170 6110 62'70 4560 4570 6590 6610
+278 420 - 2445 - 2820 $812 980 -2351 - 2520
gives rise to the large negative entropies associated with aqueous solutions. I n aqueous solutions in the vicinity of room temperature or below, the free energy of cavity formation is almost entirely manifest in the entropy term, wh(>reasin the common organic solvents it is manifest ttlmost entirely in the enthalpy term. This implies that the cavity formation process for water is dominated by a structural change in the solvent accompanied by only small changes in the internal energy of the solvent, while for organic solvents very minor structural changes take place and the process is dominated by changes in the internal energy of the solvents. Since the present theory is thermodynamic in origin, no direct information can be obtained concerning the structural changes involved. While one could make many speculations as to the nature of the changes taking place, let it suffice to say that with some minor alterations the pictorial models of Frank and Evans13and of N6methy and Scheraga'O seem adequate. The most important point to be made along these lines is that if this degree of hydrogen bonding in the solvation layer is increased, then the degree of hydrogen bonding outside of the solvation layer must be decreased since there is too small an enthalpy change to account for any significant change in the extent of hydrogen bonding.30 Since the scaled-particle theory is concerned with the packing of' molecular hard cores16'20 in fluids, its success. in the prcbsent context raises some interesting points. (1) Liquid water behaves as though it were made up of molecular cores of diameter 2.75 confined to a volume determined by its density. Although the intermolecular interactions including hydrogen bonding determine the volume available to those cores, their packing is in accord with the scaled-particle theory at least insofar as the introduction of an additional hard sphere is concerned. I 2) The thermodynamic properties of gas solubility in watrr and organic solvents are explainable
The Jownal of Phymkal Chemistry
+
+
mole
-12 -16 -28 -30 -12 -12 -30 -30
A C p , cal./deg.mole
8 6 7 5 6
31.4 33 18.8
1 0 7
36 33
16 9 ..
. .
-
Vz, cc.imole
52 43 30.5 ..
61 53 36 32
in terms of one theory and this theory involves no assumptions concerning the structure of the solvent. (3) Since the scaled-particle theory has been shown to be applicable to molten salts and liquid metals, l 6 the solubility of gases in these fluids should lend itself to the same treatment as developed here except for the necessity of modifying the interaction energy term. (4) The thermodynamic properties of infinitely dilute solutions of ions in water or other solvents should be amenable to study by including terms involving ion-dipole interactions and the reversible work of charging a particle in a dielectric medium.31 Along these lines, it should be possible to predict the free energy change associated with the transfer of a solute molecule from one solvent to mother solvent, a term related to the medium effect of Bates, et al.,32in their discussion of a generalized pH scale for aqueous and nonaqueous solutions. 5. Appendix The thermodynamic properties associated with cavity formation in water a t 25" are given by the simple formulas
+
Gc = 1 0 0 0 ~ 2~ 0~0 8~ ~ 1 ~ 1141 (cal. mole)
Rc = 16.422 (cal. mole) S, = (Re- G C ) / T
+
where 2 = 1 0 . 6 8 ~- ~2~3 .~9 1 ~ 1 ~ 14.30 and ulZ = (2.75 uz) /'2 The value of ccat 25" is complex, but at 4"it is given by 0.7912 cal. deg.-mole.
+
a.
(30) F r a n k and Evans13 took account of this possibility in their discussion of the solvation of ions in water. They consider a number of concentric solvation spheres in which ordering and disordering can take place. (31) R. H. Fowler and E. A. Guggenheim. "Statistical Thermodynamics," Cambridge University Press, London, 1939, Chapter IX. (32) (a) R. G. Bates, &I. Paabo, and R. A. Robinson, J . P h y s . Chem., 67, 1883 (1963): (b) R. G. Bates, "Electrometric pH Determinations," John Wiley and Sons, Inc., New Tiork, N. Y., 1954.
