498
S. K. SHOORAND K, E. GUBBINB
Solubility of Nonpolar Gases in Concentrated Electrolyte Solutions
by S. K. Shoor and K. E. Gubbins Department of Chemical Engineering, Untversity of Florida, Gainesville, Florida 3 Z 6 O l
(Received May 2 . 1988)
-4theory of the solubility of gases in electrolytic solutions is presented which is based on the scaled particle theory. The resulting equation for the solute activity coefficient is compared with experimental data for seven nonpolar gases dissolved in potassium hydroxide solutioiis over a wide range of electrolyte concentration and temperature, The salting-out effect is well predicted by the theory. Calculated heats of solution agree well with experiment for small solute molecules, but are larger than observed values for the larger molecules. This is attributed to a temperature dependence of the molecular hard core diameter, The proposed theory is superior to electrostatic theories in its description of both the concentration and temperature dependence of the solute activity coefficients for these systems. It is found that the salting-out effect is largely due to changes in the cavity work term, which can be calculated with reasonable confidence. Such changes arise primarily from nonpolar solute-ion interactions, and not from the ionic charges themselves,
Introduction Most of the theories proposed to explain the observed effect of salts on the solubility of nonelectrolytes are electrostatic in nature. The theory of Debye and McAulay’ provided an expression for the salting coefficient in dilute electrolyte solutions. X variety of attenipts to improve their theory have been made, and are discussed in A11 of these theoretical approaches are closely siniilar and treat the solvent as a coiitiiiuous dielectric medium containing ions and solute molecules, A quantitative test of these theories is difficult because the equations involved usually contain parameters which are not readily available. In addition, some qualitative aspects of salting effects are not satisfactorily explained by such theories even in dilute electrolyte solutions.2,3 Intel-nal pressure theories2N6adopt a different Jriewpoint to that of the electrostatic theories and succeed in explaining many of the observed experiineiital trends. Ilowever, predicted salting coefficients arp often a factor of 2 or more larger than experiniental values. At present nom of the theories of salting phenomena can be regarded as fully satisfactory, and it is of interest to examine alternntive approaches. Pierotti’,* has recently proposed n theory of gas solubility in iioiipolar solvents and in water which is based on the scaled-particle theory of Reiss, Frisch, Helfand, and L e b o w i t ~ . ~ *The ’ ~ theory predicts solubilities within a factor of 2 of experiment for a wide variety of solutes and solvents, and gives a good representation of the effects of temperature and pressure. In this approach it is not necessary to propose any special models for the structure of the solvent (e.g., hydrogen-bonding, “iceberg” models, etc.). However, the theory provides no information concerning the solution density, which must be available from experiment. As pointed out by Pierotti,” T h e Journal 01Physical Chemistry
it is the use of experimental densities that allows the theory to avoid such structural concepts. I n this paper an approach similar to that used by Pierotti is adopted for the solubility of gases in electrolyte solutions. The solvent medium may be regarded as the electrolyte solution itself, and thus consists of a mixture of several species. It is assumed that the system contains m components, one of which is the solute gas (component 1 ) , l 2 whereas the remaining (ni - 1) componeiits comprise the solvent species and mag include water molecules, ions of different types, undissociated electrolyte, etc. The scaled-particle theory has been extended to mixtures by Lebowitz, Relfand, and Prae~tgaard.’~
Theory The Equation f o r Solubility, Assuiiing that the total potential energy is the sum of pair potentials, the following equation can he obtained for the partial molecular Gibbs free energy of the solute gas in the (1) P, Debye and J. McAulay, P h y s i k . 2.. 26, 22 (1925). (2) F. A. Long and W. P. McDevit, Chem. Rev., 5 1 , 119 (1952). (3) H . 9. Harned and B . B . Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd e d , Reinhold Publishing Gorp., New York, N.Y., 1958, pp 80-88. (4) B . E. Conway. Ann. Rev. Phys. Chem., 17, 481 (196G). (5) Reference 3, p 534. (6) W. F. McDevit and F. A . Long, J . Amer. Chem. Soc., 74, 1773 (1952). (7) R. A . Pierotti, J . Phys. Chem., 67, 1840 (1963). (8) R . A . Pierotti, {bid., 69, 281 (1965). (9) H. Reiss, H . L. Frisch, and J. L. Lebowitz, J. Chem. Phys., 3 1 , 369 (1959). (10) H. Reiss, H. L. Frisch, E. Helfand, and J. L. Lebowitz, ( b i d . , 32, 119 (1960). (11) R. A. Pierotti, J. Phys. Chem., 71, 2366 (1967). (12) The usual convention of labeling the solute as component 2 is not convenient here because the solvent medium contains several species. (13) J. L. Lebowitz, E. Helfand, and E . Praestgaard, J. Chem. Phys., 43, 774 (1965).
SOLUBILITY OF GASES IN ELECTROLYTE SOLUTIONS
499 Putting eq 4 and 8 in eq 7 and rearranging
liquid phaseI4
The mole fract’ion (solubility) of the solute gas in the solution is $1 =
and pj is number density of component j, ulj is the pair potential between a solute gas molecule and a solvent molecule of species j at a distance r, and gli is the radial distribution function for 1 - j pairs. The term .$ is a coupling parameter15which allows the solute molecule t o be coupled with the solvent. The lower and upper integration limits of 5 in eq 1 correspond to complete uncoupling and complete coupling of the solute molecule, respectively. The real molecules are assumed to possess hard cores of diameters a ] , a2 ... a,),,so that the pair potential is of the form ulj(r, 4 ) =
+
uljh(r,[h)
ulis(T,
k)
(2)
where ulih and ulJsare the hard-core and soft potential interactions, respectively, given by
In charging up the hard-core contribution the coupling parameter &h varies from 0 to alj, and in charging the soft potential contribution tsvaries from 0 to 1. When the two coupling parameters are zero the solute molecule is decoupled from the system. If the two coupling processes are imagined to take place separately eq 1 becomes plL = kT In p 1 h 3 glh glS (4) where
+ +
flii, &) 4nT2dr
(6)
For a vapor and liquid phase in equilibrium
(7) and the chemical potential of the solute in the gas phase is given by16 I.110
plG =
where
flG
IcT In
=
PlL
(g)+
kT In flG
is the gas-phase fugacity of the solute.
C Pj
(10)
j
so that eq 9 may be written
At low pressures the fugacity in eq 11 may be replaced by partial pressure, and the last term on the righthand side of this equation may be calculated if the density is known. It remains to evaluate the terms 0lh and gla, Evaluation of Oxh. This term represents the free energy of introducing a hard sphere of diameter al into the solvent (electrolyte solution), Alternatively, it may be thought of as the work required to make a Lebowitz, et aZ,,13 cavity of this size in the s ~ l v e n t . ~ have shown that this cavity work is given by
where m2
Cn =
tn
C
~ i ( ~ 3 ) n
i=1
and P is pressure. EiiaZuation qf @. The term due to the soft part of the potential may be thought of as the free energy needed to introduce the solute molecule into the cavity, and may be written g1’
x yij(1., &h
P1 -
=
+ Piils - T81’
where ela, fila, and :la are partial molecular interiial energy, volume, and entropy. Following Pierotti,’ it is here assumed that the terms Pols and TgLaare much smaller than the internal energy term and may be ignored. While the first of these terms is known to be small at low pressures, the approxiniation concerning slSmay lead to errors for some solutes. With (14) N. Davidson, “Statistical Mechanics.” McGraw-Hill Book C o . , Inc., New York, N . Y . , 1962. p 481. (15) T. L. Hill, “Statistical Mechanics,” McGraw-Hill Book C o . , New York. N.Y., 1956,p 180. (16) Reference 15, p 130. Volume 75, Number 5 March 1960
8, K, SHOOR AND K. E, GUBBINS
500 these assumptions
2 JI=, , m~
=
BIS
ulj(r)gl,(r) p,4nr2 dr
(13)
The radial distribution function is not readily evaluated. As an approximation we assume the solvent particles to be uniformly distributed about the solute molecule so that
(r > a d
gl,(r) = 1
Equation 13 becomes
assumed uniform, there is on the average a spherically symmetrical charge distribution about the solute molecule. For such a distribution the field, and hence also the ion-induced dipole interaction of eq 18, is zerot20 For the real solution it is clear that the solute molecule will experience some small fluctuating field Ef due to the surrounding ions, and since this term is squared in eq 18 there will be a finite ion-induced dipole interaction whose time average is not zero. This contribution is assumed small and is neglected here. Its inclusion would make the calculated gl8 value more negative. The only ion-solute interactions included in eq 14 are therefore nonpolar contributions. Substituting the above expressions for uld into eq 14 gives
The nonpolar part of the pair interaction between a solute molecule and a solvent molecule of type j is assumed to be given by the Lennard-Jones (6-12) potential
where the mixture potential parameters el, and ul, are related to the pure component parameters by the approximate mixing rules1’ ~ i r= j
3(UI
f
;
~ j )
€13
= ( € 1 ~ )‘1’
(1G)
It is now assumed that the electrolyte is coinpletely dissociated, and the only species present iii solution are solute molecules (1), water molecules (2) , and positive and negative ions (3 and 4). In addition, the solute molecule is assunied nonpolar. The treatment for polar solutes, or of electrolyte solutions containing additional species (e.y., undissociated clcctrolytc, various water structures, ctc.) is an obvious extensioii of what follows. After avcragiiig the interaction between the permanent dipole of the water molecule and the solute induced dipole over all orientations,18 and ignoring terms due to higlier niultipole inoinents, the solute-water pair potential is
where z ~ is ~given ~ by ~ cq~ 1.7,” p2 is thc dipolc niomcnt of a water molecule, aiid a1 is the solute polarizability. Thc total ion-induced dipole interaction of a solutc molecule with all of the surrounding ions may bc written19 U(C,ind c)
=
-
[
vl(ind)
. dE’ = -ial&2
(18)
where p1(jnd)is the induced dipole for the solute and E is the electric field at the position of the solute molecule that is produced by all of the surrounding ions. The field E depends upon the distribution of ions about the neutral molecule. If, as above, the distribution is The Journal of Physical Chemistry
On performing the integrations and following Pierotti7m8 in taking al, = ulj
Comparison With Experiment Shoor, Walker, and GubbinsZ1have reported experimental solubilities for a wide variety of nonpolar gases in postassiiuiii hydroxide solutions, and over a wide rangc of KOH concentration and temperatures. These values are compared below with those predicted by the scaledparticle theory. The experimental data were reported as activity coefficients (n), the standard state being the hypothetical liquid state referred to the behavior of the solute at, infinite dilution in water, Under these conditions
where K1° is the Henry constant for the gas in pure water. For a partial pressurr of 1 atm and an ideal vapor phase, the solubility is XI = 1/ ( K I ~ ~ I ) . Molecular Parametem. Values of the IdennardJones u and e/lc for solutc gases were those obtained from second virial coefficient data and arc shown together with polarizabilities in Table I. These parameters were taken from Hirschfelder, Curtiss, and (17) J. 0.Hirschfelder, 0.F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids.” John Wiley & Sons, Inc., New York, N.Y.,1954, p 168. (18) Reference 17, p 29. (19)Reference 17, p 852. ‘(20)S. G . Starling and A. J. Woodall, “Physics,” Longmans, Greon and Co. Ltd.. London, 1950, p 940. (21) 5. K. Shoor, R. D. Walker, Jr., and K . E . Gubbins, J. Phlls.
Chem., 73, 312 (1969).
SOLUBILITY OF GASES IN ELECTROLYTE SOLUTIONS Table I: Molecular Parameters for Solutes
He Ha Ar 02
CH4 SFa C (CHa) 4 a
b
eilk, OK
2.63 2.87 3.40 3.46 3.82 5.51 7.44'1
6.03 29.2 122 118 137 200.9 232.5'1
ui,
Solute
a1 x 1034 cm*/molecule
0 204 0.802 1.63 1.57 2.70 6.21" 10.36'1 I
T.M. Reed, J. Phys. Chem., 59,428 (1955). J. H. Bae, Ph.D.
Thesis, University
of Florida, Gainesville, Fla., 1966.
Birdz2except where otherwise indicated, and are the same values as used by Pierotti7v8except in the cases of sulfur hexafluoride and neopentane. Values of u, €/IC, and dipole moment p for water were those used by Pierotti8 (Table 11) .28J4
501
ionic mobility for the hydroxyl ion is due to the special mobility ("proton-jump") mechanism of these Although experimental values of elk are not available for ions, several theories provide expressions for the potential interaction due to dispersion forces.26~~~ The Navroyannis-Stephen theory2* gives for the dispersion interaction
where a0 = 0.5292 A is the Bohr radius, e is electronic charge, 2 is the total number of electrons in the particle, and & and rq are the polarizabilities for the two species in the mixture. Comparing with the dispersion term of the Lennard-Jones ( G , 12) potential
For like-pair interactions, after substituting values for a0 and e, this equation gives
Table 11: Molecular Parameters for Solvent Species Solvent species
u, d
a,
b.
HnO 2.75" ... K+ 2.60b 2.66 OH- 3.30'1 3.52
a,
Ad
... 2.50 0.92
a x lo:( e / k , OK cma/molecule
85.3" 239& 137.2'1
I . ,
p,
D
1.84
0.835* 1.83b
Values from ref 8. Calculated in this work. c Diameters from crystal radii, ref 23 and 24. dDiameters from ionic mobility, ref 24. e Reference 31.
l'alues of u and e/k do not seem to have been reported for ions. Although crystal radii should provide approximate values for u, such radii are difficult to determine accurately, and considerable disagreement is shown between values reported by various workers. In view of thc seiisitivity of the calculated solubilities to the u values, a procedure similar to that used by Pierottia in determining u for water was used. Experimental values of In (yX10) were plotted against polarizability of the solute molecules at 25' for 10 and 20% by weight KOH solutions. The value of In (YIK?) extrapolated to zero polarizability is the experimental hard-sphere value. This may bo COMpared with the theoretical hard-sphere value from eq 11 and 21
where Blh is given by eq 12. As the other molecular parameters are known, the resulting two equations may be solved for the u values for the two ions. The values arc shown in Table 11, and are secn to be in good agreeinent with crystal diameters. The abnormally low value of the diameter calculated from
where a and u are in cgs units. The MavroyannisStephen theory gives E values which are in considerably better agreement with values obtained from experimental data than those calculated from most previous theories.2g Equation 25 was used to calculate the e l k value for the K+ ion shown in Table 11. For OH- no polarizability value could be found in the literature, and the value given in Table I1 mas calculated from the relation between polarizability and mole refraction R3O (26)
where T' niid N are volume a i i d number of molecules and I I is the index of refraction. Nole refraction data were obtained from the Landolt-Hornstein tables.31 Test of Theorg. From eq 11 aiid 21
(22) Reference 17, p 1110. (23) B. E. Conway, "Electrocheniical Data," Elsevier Publishing Co., New York, N. Y., 1952. (24) E. R. Nightingale, J.Phys. Chem., 63, 138 (1959). (25) J. D.Bernal and R. H . Fowler, J. Chem. Phys., 1, 515 (1933). (26)H.Margenau, Philosophy of Science, 8, 603 (1941). (27) D.D.Fitts, Ann Rev. P h y s . Chem.. 17, 59 (1966). (28)C. Mavroyannis and M . J. Stephen, Mol. Phys., 5 , 629 (1962). (29) T . M. Reed, 111, University of Florida, personal communication, 1987. (30) L. Pauling and E. B . Wilson, "Introduction t o Quantum Mechanics," McGraw-IIill Book Co., Inc., New York, N.Y., 1935, p 227. (31) LandoltiBornstein, "Zahlenwerte uiid Funkion aus PhysikChemie-Astronomie-Geophysik-Technik,"Vol. I , Part 1, 1950.
Volume 75. Number 3 March 1060
S, K, SHOORAND K, E, G U B B I N ~
602 Table 111: Predicted and Experimental In
VaIues for KOH Solutions
(-&lo)
---Solute
He
Hi? Ar 0 2
CHI 8F8 C(CHs)a
He Hz Ar
02 CH4
SFs C (CHs) 4 0
0%
10%
--_Exptl --
Theor
11.86 11.17 10.62 10.66 10.68 12.38 11.51
11.62 11.10 10.27 10.47 10.68 10.75 11.80
12.25 11.85 11.38 11.50 11.46 13.88 12.99
11.98 11.85 11.04 11.28 11.38 12.38 13.55
13.10 12.41 12.30 12.46 12.49 15.58 14.81
11.74 11.24 11.10 11.13 11.14 13.17 12.37
11.42 10.67
12.39 11.90 11,75 11.81 11.80 14.30 13.61
12.11 11.89 11.60 11.72 12.05 13.99 16.43
13.18 12.55 12,55 12.68 12.70 15.72 15,15
Exptl
Theor
--
% KOHa20%
Exptl
30%
Theor
Exptl
Theor
16.43
14.05 13.17 13.40 13.54 13.65 17.52 16.85
13.62 13.42 13.13 13.43 13.66 16.25 19.87
12.83 12.67 12.50 12.66 13.10 15.79 19,40
14.10 13.28 13.50 13.63 13.69 17.36 16.91
13.71 13.62 13.54 13.83 14.35 18.10 23.10
7
40%
10.81 11.04 11.18 12.49 14.42
14.10
--
----
25'
12.85 12.63 11.98 12.25 12.48
50% Exptl Theor
Theor
Exptl
...... 14.13 14.63 14.73 15.10
. . . . . .
14.63 14.47 14.82 15.17
15.2
16.07 16.59
.....
......
*..
I
.
,
. . . . . .
. . . . . .
. . . . . . 14.25 14.60 14.76 14.80
16,15
......
......
14.73 14.92 15.13 15.79
15.29 16.03
......
16.07 16.82
. . . . . . . . . . . .
......
......
. . I
8
,
.
Weight per cent KOH.
Predicted and experiment alZ1valued of lii ( ylKlO) are comparrd in Tahlc 111 for 7 nonpolar solutes in KOH solutions at two temperatures. The solute molecule9 considered exhibit a wide range of (T values, as seen from Table I. Hard and soft contributions to the chemical potential in eq 27 were calculated from eq 12 and 20 using the inolccular parameters of Tables I and 11 together with experimental densities from the literat~re.3~ For the solutes Hc, Hz, Xr, 02, and CHA at 2j0 the agreeinent between theory and experiment is very good for all KOH concentrations. Values of (y1KI0)agree mithin a factor of about 2 or better even at the highest concentrations, and in most cases the discrepancies are substantially snialler than this. The agreenient for these solutes at 80" is slightly poorer; however, it should be recalled that Q values for the ions were determined at 23". Discrepancies between theory and experiment are larger in the cases of sulfur hexafluoride and neopentane. Because of the large hard-core diameters for these molecules, the calculated (y&lO) are very sensitive to the value taken for Q of solute gas. Considerable uncertainty is involved in evaluating the Lennard-Jones u parameter, and it is possibIe that the values used were in error. Similar considerations apply to the B / I Z parameters for these molecules, although these have less effect on calculated (y1Kl0) values. Because intermolecular interactions for sulfur hexafluoride and neopentane are both large and acentric, it also seems likely that the assumptions of s l s = 0 and of a uiiiform molecular distribution around the solute are poor approximations in these cases. The experimental and predicted temperature deare) compared in Figure l for pendences of In ( Y ~ K ~ O The Journal of Physical Chemistry
argon in 20% KOH solution. Thc partial molal heat of solution AA,, provides a quantitative measure of the temperature dependence of solubility and is given by AR1 d In (TI&~) = ---
(28)
RT2
dT
Observed and predicted AR1 values are shown in Tables IV and V. From Table I V the predicted AI?I values are seen t o be too large, particularly for temper-
13
h
"2-
;s'
-c
12
I
2-a
293
I 33
I
I
333
353
T (OK)
Figure 1. Effect of temperature on the activity coefficient of argon in 20% XOH solution: solid line, observed values; - _ ,-scaled-particle theory with constant u ; - * - *, scaled-particle theory with temperature-dependent u. ( 8 2 ) M. K. Tham, K. E. Gubbins and R. D. Walker, Y. Chem. Eng. Data, 12, 525 (1067); (b) Solvay Technical Bulletin No. 15, Allied Chemical Corp., New York, N. Y., 1960.
SOLUBILITY OF GASES IN ELECTROLYTE SOLUTIONS
Table IV: Partial Molal Heats of Solution for Ar in 20y0KOH Solution
--
Temp, OC 25 40
60 80 a
AB,, cal/g mol--
Exptl
Theorya
-2100 -1210
-2380 -2100 -1780 - 1430
-400
- 150
Assuming u independent of temperature.
Table V: Partial Molal Heah of Solution for Ar a t 25' W t 70 KOH
0 10 20 30 40 a
--
AZ,, cal/g mol
Exptl
Theorya
-2800 -2700 -2100 - li00 - 1240
-2750 - 2640 -2380 -2110 - 1770
-
Assuming u independent of temperature.
atures other than 23". .An examination of Table 111 indicates that for other solutes the predicted and observed temperature dependences are in agreement) for helium and hydrogen, but discrepancies of the type shown in Figure 1 are found for the remaining solutes and become larger as the hard-sphere diameter increases. These errors appear to arise from the fundamental assumption that a pair potential of the form given by eq 2 may be used. In practice it is necessary to choose appropriate constant values for the hard-core diameter of each species. However, the mal particles do not possess hard cores, and the effective core diameter may be expected to decrease with rising teniperature because of both the increase in average particle kinetic energy and averaging over molecular orientations. This temperature dependence of the diameters has been discussed recently in connection with applications of scaled-particle theory to gas ~ o l u b i l i t i e sand ~ ~ ~surface ~~ tensi0ii.5~ The predicted temperature dependence for both of these properties is improved if u is allowed t o decrease with rising teniperature. I n determining u for water, Pierotti* found that the best value was 2.74A at 70" as opposed to 2.75 a t 23'. To illustrate the effect of a small decrease with temperature of the u values for the various species, calculations were performed for argon in 20% KOH using the following diameters (A) a t 25 and 80"
503 tained by linear interpolation. The resulting predicted temperature dependence of In (ylK1O) is shown in Figure 1 and is seen to be much improved. It is instructive to conipare the relative magnitudes of the various terms in eq 27, and Table VI shows the contributions for argon at 25". The free energy of cavity formation is seen to be the dominant term in eq 27 and is affected by the addition of ions to a much larger extent than is the term. The success of the theory is probably due in large part to this fact, for the calculation to obtain glh may be performed with considerably greater confidence than that needed to obtain Bls. The last term on the right-hand side of eq 27, which corresponds to the free energy for the fixed solute molecule to wander within the solvent, is seen to vary little with KOH concentration, Of the various contributions to o15 the electrostatic interaction between solute and water niolecules is a relatively m a l l contribution, whereas that of the nonpolar interactions is large. Nonpolar interactions between solute and ions become appreciable at high concrntrations. Similar trends arc observed for the other solutes and temperatures. For sulfur hexafluoride and neopentanc the relative magnitudes of the various terms are similar to thoscl in Table VI, but because of the larger magnitudes of g?' and glS together with their opposed signs the possibility of errors in the predicted values is increased.
Discussion I t is interesting to coiiiparc the scaled-particle theory with electrostatic theories of salt effects. As thew theories are all quite similar,2 only the theory of Debre and 31cAulay1 and the inore recent approach of Conway, Desnoyers, and Smith3j will be considered. 111 thr low (electrolyte) concentration limit the Tkhye-McAulay equation for the activity coefficient of solute is log y1 = k8C6
whcre C, is the molar concentration of salt and k , is the salting coefficient given by
0A-O
k -
' - 2.303 X 1000kTDo
CAI
25 80
3.40 3.39
Q10
2.75 2.74
UK
UOH
2.60 2.59
3.30 3.29
Diameters a t intermediate temperatures were ob-
,
a3
(30)
and AVOis Avogadro's number, DO is the dielectric constant of water, v j is the number of ionr of type j per mole of electrolyte, and e, and a3 are ionic charge and diameter. The term fl is related to the dielectric constant D of the nonelectrolyte solution by
D Temp, OC
(29)
Do(1 - fin)
Ben-Naim and H . L. Friedman, J. Phys. Chem., 71, 448 (1967). (34)9. W.Mayer, J. Chem. Phys., 38, 1803 (1963). (36) B. E. Conway, J. E . Desnoyers, and A. 0.Smith, Phil. Trans. Roy. Soc. London, A256, 389 (1984). (33) A.
Volume 79, Number 3 March 1069
504
S. K. SHOORAND K. E. GUBBINS
Table VI: Contributions to In (ylK10) for Argon in KOH Solutions a t 25'
-Soft contributions -------Nonpolar interactions-Wt % K O H
0 10 20 30 40 a
p / kT
ArHzO
Ar-K+
7.41 8.46 9.71 11.18 12 85
-3.71 -3.64 -3.52 -3.34 -3.11
...
...
-0.20 -0.44 -0.72 -1.03
-0.21 -0.47 -0.76 -1.09
I
AI-OH-
-
Electrostatic terma
0is/kT
In (kTZm)
-0.64 -0.63 -0.61 -0.58 -0.54
-4.35 -4.68 -5.04 -5.40 -5.77
7.21 7.26 7.31 7.35 7.39
Arising from the water dipole-solute induced dipole interaction term of eq 17.
where 41 is the nuiiiher of iiiolecules of nonelectrolyte solute per cubic centimeter of solution. Comparison of eq 29 with experiment is made difficult by the lack of experimental values for p. However, following the method used by C o n ~ a yit, ~ is ~possible to estimate this term from Kirkwood's theory of dielectrics. After some simplifying assuniptions are made it can bc shown that for a nonpolar solute and a highly polar solvent system36
where is partial molal volume of solute. Values of IC, calculated using eq 30 and 31 are compared with experiment in Table VII. Ionic diameters were calculated from crystal radii (Table 11); no values of 71 could be found for sulfur hexafluoride and neopentane, so that a comparison was not possible for these solutes, The more recent theory put forward by Conway, et is an improvenient of Debye's theory,a7 and takes into account dielectric saturation effects. Each ion of type i is supposed to have a hydration shell of radius r h ( i ) containing n, lyater molecules, in which the dielectric constant i s assumed very small. Outside of this shell the dielectric constant is assumed to be the value DO for pure water. Their equation for a non-
Solute
He Hz Ar 0 2
CHI
-k8-
P
where 81,SIoare the solubilities in electrolyte a i d pure water respectively, n3 and n4 are hydration numbers for the two ions, ~ h ( and ~ ) Th(4) are the corresponding radii of the primary hydration shells, d is density of solution, M , is the molecular weight of electrolyte, e is electronic charge, Pi is total molar polarization of solute, and R is a radius correqponding to the volume available per ion in the solution. I n the infinite dilution limit
This equation was used t o calculate salting coefficients for the systems in Table VI1 using the values in Table VI11 for the ionic parameters n and Th. The hydrativn number for the OH- ion was not available, and was
Table VII: Salting-Out Coefficients a t Infinite Dilution, k., for KOH Solutions a t 25"
VI,
polar gas in a 1-1 electrolyte is
7
cms/mole
Debye-McAulay eq 30
Conway eq34
7.75 19.4b 24.2n 32.05 36.0d
0.0157 0.039 0.049 0.065 0.073
0.081 0.094 0.099 0.108
0.112
Exptl
Table VI11
0.015 0.129 0.179 0.180 0.197
Reference 35. b R. Kobyashi and D. L. Xatz, Znd. Eng. Chem., 45,440 (1953). T. Enns, P. F. Scholander, and E. D. Bradstreet, J. Phys. Chem., 69, 389 (1965). W. L. Masterton and D. A. Robins, J . Chem. Phys., 22, 1830 (1954).
Ion
K+ OHa
Reference 35.
rhi
A
n
2.72. 3.00b Reference 24.
5
4.1a 5.35
Calculated.
a
The Journal of Phgsical Chemistrg
(36) J. T. Edsall and J. Wyman, "Biophysical Chemistry," Vol. I, Academic Press Inc., New York, N. Y . , pp 364, 370. (37) P. Debye, Z.Phus. Chem. (Lelpzig), 130, 56 (1927).
SOLUBILITY OF GASES IN ELECTROLYTE SOLUTIONS
505
The scaled-particle theory has thc advantage that calculated from the experimental partial molal entropy of hydrationz3 using the method proposed by U l i ~ h , ~ the ~ expression for solute chemical potential is derived from the equations of statistical mechanics by a series It is apparent from Table VI1 that neither of the of well-defined approximations. Salting-out effects electrostatic theories predicts the observed salting-out are accounted for within the framework of a more coefficients satisfactorily. Figure 2 compares the general theory that describes the solubility of gases in organic solvents and water7,*and provides a simple 17 I I : I I I model of the solution process. The theory explains the effect of solute species and electrolyte concentra16 tion on salting-out in KOH solutions, where salt effects are large. In contrast to the electrostatic theories, it is possible to calculate the solubility of the 15 gas in pure water. Furthermore, the molecular parameters needed are more readily obtained than those 14 involved in electrostatic theories. From eq 27 it is seen that the scaled-particle theory predicts salting-in G for systems in which the magnitude of #lB/lkTexceeds .! /' Scaled Particle E 13 : /-Theory the sum of the other two terms. Such behavior will occur if the "soft" interaction is very strong, and under 12 such conditions the assumptions that ( P D P- 2's~)FZ 0 and that the molecular distribution is uniform may be poor approximations. 11 Of the approximations involved in the scaled-particle theory the basic assumption that molecules possess hard cores (eq 2) seems the most serious, and may lead 10 10 m 30 40 50 to predicted heats of solution that are too high. In 'Wt. "/a KOH fairness to the scaled-particle theory, however, it Figure 2. Theoretical and experimental activity should be pointed out that electrostatic theories cannot. Coefficients of argon at 25'. usually attempt such a calculation because the necessary parameters are not available. The validity of the observed coiiceiitratioii depeiideiice of the activity approximations that sls = 0 and that the molecular coefficient for oxygen at 27' with the predictioiis of the distribution is uniform is difficult to evaluate 1)ecausc scaled-particle theory and the two electrost atic theorieq. thc change of Ola with electrolyte conrentration is The Debye-Mcbulay theory gives results much below sniall compared to that for olh. It SeeniY unlikely that experiment. While the theory of Conway, et al., these assuniptioiis lead t o large errors for sniall nongives better results atj low electrolyte concentrations, polar molecules, but this may not he the caw for larger it predicts negative solubilities for higher KOH connonpolar or polar solutc molecules. centrations and is thus invalid in this region. The Unlike niany of the electrostatic theories, the scaledfailure of the electrostatic theories to predict correctly particle theory iiiakes no appcnl to assumptions coneither the effect of solute species or electrolyte concerning solvent structure, ionic hydration, etc. In centration on tho activity coefficient can br attributed addition, the ionic chargc has little direct influence on to the assumptions made in these theories. It s e e m salting-out . The iiiost important effect of such chargrs probable that it is an oversimplification t o treat thc appears to 1)e in determining the density of the elecsolvent as a continuous dielectric medium, and that trolyte solution. observed salt effects call be adequately accounted for only if the solute-solvent inolecular interactions are explicitly introduced into the theory. (38)HaUlich, 2.Elektrochem,, 36, 497 (1930). 0 -
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Volume YS,Number S March 1989
