626
J. Phys. Chem. 1994, 98, 626-634
Molecular Thermodynamic Model for the Solubility of Noble Gases in Water A. Braibanti,**fE. Fisicaro,? F. Dallavalle,; J. D. Lamb,* J. L. Oscarson,l and R. Sambasiva Raoll Institute of Applied Physical Chemistry and General and Inorganic Chemistry, University of Parma, I-43100 Parma, Italy, Departments of Chemistry and of Chemical Engineering, Brigham Young University, Provo, Utah 84602, and School of Chemistry, Andhra University, Visakhapatnam, 530 003 India Received: May 18, 1993”
The thermodynamic model based on the distributions of molecular populations among energy levels has been employed for the analysis of the solubility of noble gases in water at different temperatures. The solubility is expressed as the polynomial {ln X ~ T ={ln x 2 ) e (-AHapp/R)e(l/T- l/e) + (1/2)(AC*p,app/R)a(l/T- l/e)* + (1/6)(a(AC*,app/R)/a(l/T)Je(l/T- 1/e)3 + (1/24)~d2(AC*,a,/R)/a(1/T)’je(l/T- 1/el4. The apparent thermodynamic quantities of this expression are obtained from the coefficients of the polynomial fitting the experimental data. The whole system is considered as the convoluted ensemble (gc*c)e formed by a grand canonical ensemble, gce, and a canonical ensemble, ce, the latter corresponding to the solvent. The statistical distribution is described by a convoluted partition function, (GC*C)PF,which is the product of a grand canonical partition function, GCPF,and a canonical partition function, CPF. The apparent thermodynamic functions can be decomposed into the contributions of the separate partition functions. In particular, the apparent enthalpy (-AHapp]~ = -AHo - nwCp,,T is the sum of the enthalpy change due to the reaction between gas and water, -AHo, and the heat absorbed by the water molecules involved in the reaction AH, = nwCp,wT.The enthalpy term AHw, which varies linearly with the temperature, has been calculated by using the relationship of thermal equiualent dilution valid for the canonical ensemble, ce. By plotting the apparent enthalpy {-AHa& versus T, the value n, can be obtained from the slope of the line. Sets of data from different sources have been analyzed and yield congruent values of -AHo and n,. The values n, ranging from 1.5 for helium to 3.3 for xenon clearly depend on the size of the atoms of the noble gas and can be related to the formation of a cavity of water molecules in the solvent.
+
We have shown1.2 how theequilibria in solutioncan bedescribed by means of a grand canonical partition function, GCPF, applied to a system which is a grand canonical ensemble,gce, of statistical thermodynamics. The partition function is a moment generating function. The moments or derivatives of the partition function with respect to the variables, dilution, or concentration and reciprocal or logarithmictemperature correspond to experimental quantitiesdetermined by either potentiometry, spectrophotometry, calorimetry, or other methods. The relationships between concentration and temperature moments can be calculated as mixed derivatives of the partition function. The solvent can be represented as a canonical ensemble, ce, whose properties are described by a canonical partition function, CPF, which is independent of the concentration. For ce, only moments with respect to the variable temperature T o r to the variable reciprocal or logarithmic temperature can be measured. The properties of the ensembles gce and ce are related to models consisting of sets of quantized energy levels. The ensembles gce and ce differ from one another in the extent of the interlevel energy separation and hence in the different types of eigenvalues of the Hamiltonian assignable to each of them. The carboxylic acids in aqueous solution have been represented3 as a convoluted ensemble, (gc*c)e. The protonation process can be described on the basis of a convoluted partition function, (GC+C)PF = GCPFCPF, obtained as the product of a grand canonical partition function, GCPF, and a canonical partition function, CPF, the last one corresponding to the solvent. The various mathematical expressions used to represent the logarithms of equilibrium constants or kinetic constants as the Institute of Applied Physical Chemistry. General and Inorganic Chemistry, University of Parma. 8 Department of Chemistry, Brigham Young University. I Department of Chemical Engineering, Brigham Young University. I Andhra University. Abstract published in Advance ACS Abstracts, December 1, 1993. f
0022-3654 194 12098-0626SO4.50,IO ,
I
function of the temperature or reciprocal temperature are more or less correct representations of the partition of species over the different levels of the model, and as such they depend on the partition function appropriate to the system. The mathematical expressions have been thoroughly examined by Blandamer et ala4 The methods used to calculate the errors of the thermodynamic functions derived from the curves fitting the experimental data have been developed by Dec and Gills and Clarke and Glew.6 The shapes of the curves representing the logarithms of the solubility of gases as a function of temperature are similar to those of the logarithms of the protonation constants of carboxylic acids, and therefore the same validation proceduresu suggested for the equilibrium constants can be used for the solubility of gases also. On the other hand, because of this similarity of the plots, it seemed to usvery promising to try to extend to the solubility of gases the same analysis on the grounds of the molecular thermodynamic model applied previously to the effect of the temperature on the protonation of carboxylic acids.3 The solubility of gases in water has been widely studied because the gas-water interface is the region at which the hydrophobic bond is supposed to occur.’-g Particularly used for the interpretation of these solubilities is the scaled particle theory.10 The importanceof thesesystemsstemsfrom the fact that they represent models to study the more complex macromolecularand biological systemswhere the hydrophobic bond plays an important role.11-20 The solubility of gases has been reviewed by Wilhelm et a1.,21.22 and very accurate solubility data for gaseous hydrocarbons in water have been determined by Rettich et ~ 1and. for ~ noble ~ gases by Benson and Krause24.25 and at higher temperatures by Crovetto et The processing of the solubility data to obtain values of the Henry constant extrapolated to infinite dilution has been dealt with by Wilhelm et ~1.22and by Benson and Krause.2728 The application of GCPF, CPF, and convoluted ( G C C ) P F t o 0 1994 American Chemical Society
Solubility of Noble Gases in Water
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 621
Figure 2. The gas molecule in aqueous solution forms a cavity by expelling - n, water molecules rearrange, forming a cage. The cage is not a compound but rather a fluctuating variable structure which on the average depends on the size of the trapped particle (noninteger n,). n, water molecules. The x
I
I
Figure 1. Enthalpy and isobaric heat capacity of the levels along they coordinate of the thermodynamic space: First moment (mean, - ( M I ) ) and second moment (variance, -F ACPJ of the free energy distribution with respect to the variable 1/ T. The variance is represented as standard deviation. Note that ( H I J )= H I , ( H I J )= H I , and ( M I ) AH1 = HO -HI.
the analysis of the solubility data of gases opens, as will be shown below, new insights into the important problem of the hydrophobic interactions.
of each sublevel of the molecule. The distinction between enthalpy and entropy within one level cannot be recognized by experiment. The only change that can be measured in the most general cases of ce is the thermal energy change measured by a thermometer or any thermometric sensor. The joint probability of transition from level i to level 0 is given by the free energy probability exp(-AG,/RT) = exp(-AH,/RT) exp(hS,/R)
Molecular Thermodynamic Model The molecular model for the analysis of the solubility of gases in water is based on the assumption of the existence in the system of two discrete energy levels corresponding to different chemical states (Figure 1). Each of the two energy levels corresponds to a chemical state of the gas, namely state i = 0 for free gas in water and state i = 1 for gas trapped in a cage.g The probability of occupation of any level i as measured by the molecular populations of the levels follows a Boltzmann probability law, exp(-H/RT), whereH,is theenthalpyof thelevel. Theenthalpy, HI, of each level is the weighted mean of the contributions to the enthalpy of every vibrational, rotational, and translational state j of the chemical species i
where AH, = HO - Hi is the difference between two means calculated from eq 2. The heat capacity change of the reaction, AC,,, is the difference between the heat capacities of the two levels, and the variance of thedistribution is-PAC,,,.l We expect that in a system where the same reaction is maintained, the square root of the heat capacity, AC,,,, times the temperature is always much less than AH,so that reversal of the sign of AH,is not possible (cf. Figure 1). Reversal of the level order would imply a reversingor changing of the reaction. The free energy change for the formation reaction, AGG is related to the excess grand canonical partition function for a gas, Z G , by -AG, = R T In Z,
The weights wlare the populationsof thejstatesinmolar fractions. Each level i can be considered as a canonical ensemble or subsystem, ce, where no reaction is taking place. The dispersion or variance around the mean is related to the heat capacity of the state, Cp,,,of the level
(5)
(6)
with
z,
= 1 + 1/KS[l3l (7) where [g] is the concentration of free gas and K, is a specific dissociation constant between free gas and solvent. 1/K, can also be considered as a saturation partition function29 and the standard free energy for the dissociation reaction obtained from -AGO = R T In K,
The probability of occupation of the level depends also on the state multiplicity or entropy of the level, exp(Si/R), which in turn depends on the number of equivalent binding sites, mi, on the degrees,j , of rotational, vibrational, and translational motion, and on the dilution, di, of the species’
(3) I
The heat capacity is also related to the dispersion or variance of the entropy of the level, at d, = 0 and mi = 1, so to have a subsystem represented by a canonical ensemble, ce,
(8) On the other hand, the solvent water is considered in excess and therefore in a practically constant concentration, [W]. We can assume that the solvent is a ce1,2with CPF= [W]. Therefore theproduct [W](l/K,)can be treatedasaconuoluted(GCC)PF with a contribution to the free energy from the water molecules -AG, = R T ln[W] and a total apparent free energy -AG,,O
= -AGO
+ AGw= R T In K, - R T ln[W]
(9) (10)
Solubility as a Chemical Equilibrium The solubility of a gas such as He in water can be treated (Figure 2) as an equilibrium between He and W
This observable, CPi,yields two apparently different pieces of information, the variance of H,and the variance of SI. The equivalence for ce of entropy and enthalpy with respect to the temperaturecan be explained by considering a change of entropy as a change of dilution over more previously unoccupied sublevels and a change of enthalpy as a change of the fractional occupation
He
+ XW = Hew,-, + n,W
(11)
where XW is the portion of the bulk involved in the solubilization process and Hew,, is the gas molecule trapped in a cavity formed by (x - n,) water molecules. The n, water molecules are those expelled to form the cavity to host the gas molecule, The
Braibanti et al.
628 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
obtains at T = 0
constant of this equilibrium is
Ks = [HeWx-nJ [WI"./ [He1[WIx
(12)
This constant conforms to the proposal of W. L. MarshalPo who showed that (i) the power of the concentration of the solvent ought to be introduced into the calculation of equilibriumconstants and (ii) the exponent of theconcentration of the solvent represents the net change in solvation numbers between the product and reactant species of the process. At constant gas pressure, theconcentration [Hewx+] measures the solubility if we assume that [He] is constant. The solubility is usually identified with the molar fraction x2 a t unit pressure, where the index 2 indicates the solute. At unit pressure, the molar fraction x2 is the reciprocal of the Henry constant, KH. Sometimes the solubility is expressed as the Ostwald ratio L.22 which is the ratio of the actual volume of gas to the volume of solvent at the same temperature and pressure or Bunsen coefficient, 8, which is the volume of gas reduced to standard (STP) conditions dissolved in 1 L of the solution at 1 atm partial pressure.24 At high dilution, the solubility x2 can be converted into the Ostwald ratio.22 With these assumptions, eq 12 can be written as a solubility product,
P, = [W]"/K,
= x,[W]".
where Pa includes the conversion factor deriving from the inhomogeneity of the concentration scales of gas and water. By taking the logarithms of eq 13, we can write In xz = In P,- n, ln[W]
(14)
In the affinity thermodynamic space,' the different concentration scales should produce a displacement of the origin of the x = U / R axis. The solubility can be expressed in the thermodynamic space (In x2 = -AGapPo/RT)as the function of the inverse temperature by a polynomial
+
n-N
{ln x21T= {In x218 C ( l / n ! ) ( d n ln x2/ n= 1
8(1/T)'l&1/T- l / W (15) where 0 is a reference temperature. The maximum degree N depends alsoon the precision of the experimental data. Blandamer e? ala4have shown how the maximum acceptable degree of the polynomial can be inferred from the statistical analysis of the data. Thevalues of the momentsor derivativesof the thermodynamic function can be identified with experimental quantities with maximum degree N = 4 {lnxzl,=
hx2l8 +
(-mapp/R)&l/T-
1/61 +
(1/2)(AC*p,app/R)Al/ T - l/o)' + 1/ T - 1/e)' ( 1/6){a(AC*p,app/R)/d( 1/
+ 1/el4
(1/24)(d2(AC*p,,pp/R)/d(l/T)2)e(l/ T (16) where AC*P,,pp = 42ACP,a,is thederivative of -AHHapp with respect to 1 / T at T = 0.192 If the polynomial fitting the experimental data In x2 =f(l / T ) is written as
+ +
+
+
y = a bx cx2 dx3 ex4 (17) then the quantities of eq 16 can be obtained at any temperature T from the coefficients of polynomial 17. By deriving eq 14 with respect to the inverse temperature, one
xJd(l/T)],q = PlnJ'S/d(l/T)len,(d ln[WI/d(l/T)b (18) which can be identified for a combination of first moments and therefore of enthalpies (19) - ( m a p p l , = -{Mob + Wwle The term (AHaPp)o can be obtained from eq 17 as the tangent at T = 0 of the function In x2 =f(l/T). The enthalpy {AHo]# can be obtained by assuming the hypothesis that at a first approximation the solubility product Pasatisfies the van't Hoff equation with (AHo]e= AH". The enthalpy (AHw)ecan be obtained from the dependence of ln[W] upon T or thermol equivalent dilution which can be expressed2 as
-n, d ln[w]/a In T = nwCp,/R
(20) where CP,,is the molar heat capacity of water. By transformation into the derivative with respect to 1/T, eq 20 yields n, T d ln[W]/d(l/T) = {-AZf,/R),=
nwCP,,T/R (21)
By combining eqs 19 and 21, one obtains, if the constant Pafollows thevan't Hoffequation with constant enthalpy, M ,andconstant heat capacity of water, C,,,, k m a p p / R l i j = -Mo / R - nWCp.$/R (22) which shows how the apparent enthalpy, AHapp, is a linear function of the variable T = 0 with slope -nWCp,,. According to eq 22, the contribution of the solvent molecules to the apparent enthalpy should depend linearly upon the temperature and the slope of the line should give the number of water molecules. This behavior is shown in general by the gases examined, as will be shown in the next paragraphs, and makes possible the determination of the number of water molecules, n, involved in the reaction. The plot of the apparent enthalpy against the absolute temperature T with extrapolation to T = 0 provides evidence for the properties of the CPF for the subsystem ce. In fact, the reference state of C P F with respect to temperature is the limit a t which the kinetic energy of the water molecules is null. Therefore, an asymmetric convention must be assumed for the referencestates ofthe two factors of (GC+C)PFwithextrapolation to 1 / T = 0 for GCPF and to T = 0 for CPF, respectively. The derivatives of In P, with respect to the reciprocal temperature are the sum of terms depending on the reaction enthalpy of the grand canonical ensemble, gce, and of terms depending on the heat absorbed by the solvent of the cesubsystem. Therefore, we expect that the convoluted ensemble, (gc*c)e, behaves in a manner different from that of the pure gce, and in particular, we can envisage a combination of terms of the moments of the pure partition functions, GCPF and CPF, respectively, leading to a reversal of sign in some terms of the convoluted (GC*C)PFand hence of (-AHapp/R]e in eq 15 at some temperatures. Therefore, both the first and second moments of the free energy change (ie. enthalpy and heat capacity, respectively) in the solubilization of gases in water are highly dependent on the heat absorbed by the water molecules taking part in the reaction. This explains why Wilhelm et state that "the values of partial molar entropy and enthalpy reported so far for gaseous solutes are substantially lower than for other solvents, and the partial molar heat capacity anomalously large".
Data Sources and Handling The experimental data for the solubility of the noble gases have been taken from different sources. They can be presented as molar fractions,2OOstwald ratios,21v22 Bunsen coeffi~ients,2~J~ or Henry ~0nstant~.24J5,2~J* The most accurate sets of data for noble gases are apparently those of Benson and K r a u ~ e . 2They ~.~~
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 629
Solubility of Noble Gases in Water -5.12
I a
-5.16 I -0.8
-0.6
-0.4 -0.2 ( 1 /T- 1 /T*)'lOOO
I
I
-0.0
0 #2
determined the Henry constant by manometric method. The data were reported in terms of the Bunsen coefficient, 8. The corrections for real gas were applied in the relationship between the Henry constant, KH,and the Bunsen coefficient, @, as
KH=(1/8) 124.4 142pZJeX 1O4
(23) where 20is the compressibility factor at 0 "C and 760 Torr, Z, is the compressibility factor under equilibrium condition of the experiment, and p is the density of water at the equilibrium temperature. Some years later, Benson and Krause2' applied to the same data a more sophisticated treatment in order to extrapolate the values at infinite dilution at a given pressure p and temperature T. Benson and Krause28 in their search for a smoothing function capable of fitting the solubility data at higher temperatures, adopted a reduced temperature, T+ = T/Tcl, where Tcl is the critical temperature of the solvent. This choice had been suggested31 by the finding that the "reference solubility" in an infinitely dilute solution at the critical temperature Tcl of the solvent is equal to the reciprocal pressurepcl. The value 647 K was used as the critical temperature of water. They compared three types of equations called Mark I, Mark 11, and Mark 111. The equation Mark I11
+ a,T*
(24)
is believed to be probably the most accurate for the range 0-60 OC. Another set of values for the solubility of noble gases in water has been reported by Wilhelm et ~ 1 They . ~searched ~ in the literature for data of reasonable accuracy. Unless the data used from one source were sufficiently precise to warrant independent treatment, all the experimental data for a particular gas were handled as a group. The reported data were converted where required to Ostwald coefficients, mole fractions at 1 atm partial pressure of gas, and Kelvin temperatures. Correctionsfor nonideality and chemicaleffectswere not made. All the data for each gas were submitted to an initial screening using scattergrams. Data which were grossly out of line were discarded after double checking the original paper for its level of precision and the reliability of the work. The final fitting and selection of data were done by Wilhem et al. using a least-squaresprogram. The temperature dependence of the solubilitywas then accounted for by fitting to an expression of the form
R In x2 = A
280
300
320
340
360
T/K
Figure 3. Solubility of noble gas (He) as a function of reciprocal temperature (data from ref 22).
In ( K , / p ) = u-,(T*)-~+ a-,(T*)-'
-260
+ B T ' + C In( T / K ) + DT
(25) where the inclusion of the fourth term depended on the overall precision and the number of points. These authors state that the advantage of eq 25 over polynomial fits with an equal number of coefficients is that it correctly correlates solubility and temperature with a significantly smaller standard deviation. Occasionallythey encounteredsystems for which only very limited
x He
0
Ne
Figure 4. Enthalpy change of noble gases for the transformation from gas trapped in a cage to free gas as a function of temperature (data from
ref 22).
or somewhat less accurate data were available, and in those cases they used the simple relation
R In x2 = A
+BT'
(26) The relationship between molar fraction and Ostwald coefficient was used in the form
+
x2 = {RT/(V,0LLp2) 1) (27) on the assumption that the second virial coefficient B22 is much smaller than RT/p2. They report solubilitydata in terms of mole fractions, x2, and Ostwald coefficients, L,at 1 atm partial gas pressure at selected temperatures (with 5 K intervals) calculated by e q ~25-27. The logarithms of solubilities at some representative temperatures are reported in Table 1 together with values of solubilities reported by Benson and Krause.28 We note that the data of Wilhelm et al. are interpolated values obtained as averages from different experimental sources, whereas the data of Benson and Krause are very accurate experimental data. We have applied the treatment according to the model (gc*c)e here proposed to both sets of data and also to the smoothed data given by Benson and Krause from the Mark I11 equation (24). Another set of data considered by us is that of Crovetto et They have determined the solubility of inert gases in H2O and D2O between room temperature and 600 K. The calculation of Henry constants KH from the solubility data has been analyzed in detail; if due account is taken of the nonideality of the gas phase, they can be determined unambiguously up to 520 K. Above this temeprature, the ambiguity in KH increases sharply as contributionsof third and higher virial coefficients to the equation of state of the gaseous mixture become important. The data of Crovetto et al., however, do not comprehend enough values below the boiling point of water, and it is not certain that the isobaric heat capacity of water, Cp,w,remains constant above 100 OC. Therefore their data were not introduced in the calculations. Data Analysis The solubility of He in water at different temperatures from Wilhelm et a1.22 plotted in Figure 3 as log x2 versus 1 / T shows a minimum which occurs at T,,, = 305.4 K. The data are fitted by the polynomial logx, = -5.122
+ (0.176 X 103)x + (0.197 X 106)x2(0.058 X 109)x3- (0.014
X
1012)x4 (28)
(regression coefficient, R = 1.00) with x = 1 / T - l/O with O = 273.1 5 K. In order to calculate the tangent to the curve at each temperature T, we calculate the first derivative of polynomial 17
630 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
Braibanti et al.
TABLE 1: Solubility of Noble Casea at Selected Representative Temperatures’ gas He
273 278 288 298 308 318 328 333 ref logxz -5.122 -5.133 -5.148 -5.157 -5.156 -5.153 -5.144 -5.138 22 l0gp2 -5.114 -5.127 -5.143 -5.150 -5.15Ob -5.144 -5.134‘ -5.126 28 Ne logx2 -4.998 -5.023 -5.062 -5.107 -5.090 -5.115 -5.115 -5.1 12 22 logp2 -4.995 -5.020 -5.058 -5.083d -5.102 -5.111 -5.112‘ -5.1 11 28 -4.366 -4.422 -4.519 Ar lOgx2 -4.599 -4.664 -4.7 15 -4.754 -4.770 22 bgp2 -4.363 -4.420 -4.518 -4.597 -4.659 -4.708 -4.745 -4.760 28 -4.054 -4.123 -4.246 -4.347 Kr lOgx2 -4.431 -4.498 -4.551 -4.572 22 logpi -4.049 -4.133’ -4.242 -4.342 -4.423 -4.489 -4.539 -4.562 28 -3.853 Xe logx2 -3.774 -3.992 4 1 10 -4.292 -4.209 -4.359 -4.388 22 lOgp2 -3.741 -3.826 -4.019 -4.111‘ -4.1921 -4.261‘ -4.349 -4.377 28 Rn logx2 -3.373 -3.469 -3.637 -3.776 -3.889 -3.980 -4.050 -4.078 22 log P2 x2 = molar fraction; p2 = partial vapor pressure; log x2, values calculated by Wilhelm, Battino, and Wilcock22 as averages of various sets of experimental values; log p2, experimental values of Benson and Krause.28 A t T = 309. c At T = 327. d A t T = 297. e A t T = 329. / A t T = 279.r At T = 327. h A t T = 291. ‘ A t T = 299.jAt T = 307. At T = 315.
TABLE 2
Interpolation Polynomials for the Solubility of Noble Cases in Water Tm (K)
gas He
coefficients“ lode
1v3 b
a
10-9 d
10-12
e
ref
+0.176 +0.198 305.4 -5.122 -0.058 -0.014 22 (log x2) -5.1 14 +0.207 +0.281 -0.057 -0.074 28 (log P2) 302.5 -2.486 +5.234 X lO-’ +9.986 X 10-6 -2.068 X lt9 28 (log f32’lb 302.5 -4.998 +0.394 +0.300 -0.037 +0.026 22 Ne (log x2) 323.1 +0.401 +0.366 +0.07 1 +0.048 28 326.0 -4.995 (log P2) +1.704 X lv3 +8.331 X 10-6 -1.992 X lt9 326.0 +2.226 28 (log P2’Y -4.366 +0.873 +0.359 -0.031 +0.036 22 Ar (log x2) 369.2 -4.363 +0.897 +0.413 -0.115 -0,098 28 (log P2) 379.1 +2.372 +0.958 X lO-’ +9.175 X 10-6 -2.578 X lv9 28 (log P2’)b 379.1 -4.054 +1.083 +0.390 -0.069 +0.035 22 Kr (log x2) 374.1 +1.105 +0.489 +0.006 +0.030 28 383.1 -4.049 (log P2) -0.989 X 10-3 +8.570 X 10-6 -2.680 X lv9 383.1 +4.478 28 (log P2’Y +1.222 +0.404 -0,030 +0.043 22 396.1 -3.114 Xe (log x2) -3.740 +1.338 +0.479 -0.245 -0.176 28 (log P2) 396.4 +9.856 -5.632 X lv3 -6.741 X 10-6 -2.646 X 10-9 28 (log P2’Y 396.4 +1.503 +OS96 4,054 +0.056 22 371.5 -3.373 Rn (log x2) (log P2) (log P i ) aCoefficientsofeq 17,y=a+ bx+cx2+dx3+ex4,withx=(1/T- 1/273.15). (logx2): Valuescalculatedbyeq 17fromtheresultsofWilhelm, Battino, and Wilkwk.22 (log p2): Values calculated by eq 17 from the experimental data of Benson and Krause.28 b Coefficients obtained by Benson and Krause28 calculated by y = a+ + b + cx + dx2 with x = T,/T (cf. eq Mark 111).
TABLE 3 gas He Ne
Apparent Enthalpies (AH,, (kJ mol-’)) at Selected Representative Temperatures’
AHapp(x2) AH.pp@2) AHlpp@2’) A&@2) AH.pp(P2)
AH*pp@2’) Ar
AHHlpp(x2) AHapp@~)
Kr
AH&2’) AHapp(x2) AH&2)
Xe
AHHlpp(x2)
AH8pp(p2‘)
AHapp(p2) AHnpp@2’)
Rn
A&,&)
273
278
288
298
308
318
328
333
ref
-3.369 -3.963 -3.955 -7.543 -7.677 -7.5 8 7 -16.71 -17.16 -17.25 -20.73 -2 1.1 5 -21.25 -23.39 -25.60 -25.75 -28.77
-2.856 -3.242 -3.233 -6.777 -6.771 -6.734 -15.80 -16.11 -16.11 -19.73 -19.92 -19.99 -22.31 -24.35 -24.33 -27.26
-1.811 -1.833 -1.818 -5.263 -5.130 -5.128 -14.01 -13.97 -13.94 -17.73 -17.58 -17.59 -20.36 -21.70 -21.63 -24.28
-0.759 -0.5 15 4.537 -3.759 -3.653 -3.642 -12.25 -11.91 -1 1.92 -15.70 -15.37 -15.34 -18.39 -19.05 -19.06 -21.35
+0.282 +0.670 +0.622 -2.256 -2.291 -2.266 -10.49 -9.994 -10.04 -13.65 -13.26 -13.24 -16.43 -16.53 -16.61 -18.44
+1.301 +1.700 +1.669 4.749 -1.003 -0.988 -8.735 -8.255 -8.288 -1 1.57 -1 1.23 -1 1.25 -14.47 -14.22 -14.28 -15.53
+2.289 +2.562 +2.613 +0.765 +0.236 +0.198 -6.971 -6.717 -6.649 -9.459 -9.265 -9.383 -12.49 -12.18 -12.05 -12.61
+2.770 +2.930 +3.048 +1.525 +0.844 +0.760 -6.086 -6.026 -5.869 -8.397 -8.301 -8.488 -1 1.so -11.27 -10.97 -11.15
22 28 28 22 28 28 22 28 28 22 28 28 22 28 28 22
M8pP@2) AH.pp@2’) a Akfapp(X2): values calculated by us from results of the smoothing functions obtained by Wilhelm, Battino, and Wilcock.22 AHlPp@2): values calculated by us from experimental values of Benson and Krause.28 AHHlpp@2‘): values calculated by Benson and Krause28 by Mark 111. (-Mapp/R)T
= b + 2cx
+ 3dX2 + 4ex3
(29)
This isequivalenttocalculating thecoefficientbof the polynomial by changing each time the reference temperature 8. According to equation 22, the tangent to the curve at each
temperature 8, i.e. (-AiYapp/R)e, should change linearly with the temmrature 8 = T. The values for He and Ne dotted against T (Figure 4) confirm the hypothesis. The slop;: of the {ne is nwC,,, from which one obtains nw = 1.35 and 2.01 for He and Ne, respectively, by considering Cp,w= 75.378 J. There are a
The Journal of Physical Chemistry, Vol. 98, No.2, 1994 631
Solubility of Noble Gases in Water 4
"r 2
/
p'
/
II
x
x
loo
0
b.p.
BBK I281
x
200 K
x,W.B.w.[22]
Figure 5. Relationship between n, obtained from the statistical thermodynamic model and the boiling point in noble gases. Note how the set of data credited from the maximum precision (Benson and Krause, ref 28) fit the regression line.
-40
'
g -60
'
\
3 b\ -80 2
a
-100
-120
'
1
2
3
"**
4
Figure 6. Relationshipbetween the extrapolatedenthalpy,AHO,and the number of water molecules, n,, in noble gases. The equilibriumconstant is for dissociation. Full squares arc from data of Benson and Krause?* and the empty square is from data reported by Wilhelm et 0 1 . 2 ~
TABLE 4
Enthalw Functions'
~
_
_
_
_
____
gas He +0.102 +o.iia +o.i17 1.353 1.565 1.552
Ne
Ar
Kr
Slope (kJ K-l mol-')* +o.i5i +o.i77 +0.207 +o.i4i +o.ia7 + o m + o m +o.i89 +0.212 2.003 1.871 1.84
nw
2.348 2.481 2.507
2.746 2.839 2.812 AHO (kJ mol-') (log x2) -31.295 -48.790 44.945 -77.476 (loa DZ) -35.979 -45.798 6 7 . 8 6 4 -79.259 p?) -35.689 -45.246 4 8 . 6 4 2 -78.922
(1s
Xe +o.i98
+om
Rn
+om
+0.246 2.627 3.237 3.264
3.887
-77.341 -108.817 -91.970 -92.623
Our analysis according to the model (gc*c)e has been applied also tocorrected data28of Benson and Krause. Values of the first and second derivatives, with respect to temperature, of Mark 111 which correspond to (-m,,/R)eand (ACPm/R)e, respectively, as quoted by the authors, have also been analyzed. The coefficients of the polynomial interpolation functions for all the solubility data examined are given in Table 2. The values of T m have been calculated from the minimum of the solubility curve for each gas. For this purpose, third-degree polynomial expressions have been used as smoothing functions. Care must be taken that the curve is well behaving around T, because it is very sensitive to numerical approximations. The values of T,, obtained from the molar fraction, x2, and from the Ostwald coefficients,L,respectively, differ because of the different scales for the equilibrium constants, although not exactly by the same amount, as one would have expected because of the same extrapolation procedure followed for all the gases. The values of the apparent enthalpy, AHappobtained by means of the interpolation polynomials at some representative temperatures are reported inTable 3. Thevaluesoftheapparententhalpy result in linear functionsof the absolutetemperature T, as expected according to eq 22. For every set of data, the apparent enthalpies as a function of Thave been calculated. The results of the extrapolationprocedure for all the noble gases are reported in Table 4, where the values of the enthalpy, AHo, the difference in the number of water molecules, n,, obtained from the slopes, and the temperature of the minimum of the curve, Tm, are given. At Tm, the apparent enthalpy, AHapp is null because of the compensation between the two terms on the right side of eq 22. At this temperature, the free energy change is equal to the entropy change. Moreover, the temperature T, can be calculated from the enthalpy function of eq 22 by setting AHapp= 0 and by dividing AiF by the slope of the line. The values of T, reported in Table 4 can be compared with the corresponding values of Table 2, obtained from the minimum of the solubilitycurve. The correspondenceis in general good, except in a few cases were the differences are relevant. The values of A P and n, obtained from the molar fraction, x2, and from the Ostwald coefficients, L, respectively, are in good agreement. The contribution from the water molecules affects also the apparent isobaric heat capacity, AC'@p,awThe dependence on reciprocal temperature is more conveniently transformed into a polynomial dependence on the direct temperature T. To do so, values of A C , , , are calculated at all the temperatures as second derivatives of polynomial 17 for each gas and then transformed into AC,. Values of AC,, at some representative temperatures for all the sets of data analyzed are reported in Table 5. The values of AC, are then expressed as a function of ( T - Tm)
Tm (K)
(log x2) 306.814 323.1 13 366.921 374.280 390.61 1 371.389 (log p2) 304.907 324.809 362.909 370.369 376.926 (iogpz') 305.034 325.51i 363.185 372.274 376.516 a (log x2): equationcoefficientscalculated by us by eq 22 from results of the smoothing functionsobtained by Wilhclm, Battino, and WilCoCk22 (logpz): cquationcoefficientscalculatedby us byeq22 fromexperimental values of Benson and Krause.28 (logp2'): equationcoeficicntscalculated by us by eq 22 from the values calculated by Benson and Kraus@ by Mark 111. Slope = n.,CR.,.
The coefficients of these polynomial expressions are reported in Table 6. From the first term {ACpapp)~m = a of the equations, the contribution a' = n, C,, of the water molecules can be subtracted togeta"= a - a ' = AC,,,, which is the real isobaric heat capacity of the reaction at the temperature T,. The values AC,,r, come out to be much smaller than (ACpapp)~,, as expected.
Discussion different number of water molecules involved in the solubilization process depending upon the size of the atoms. The calculations of the apparent enthalpy as the function of T have been extended to all the data concerning the noble gases reported by Wilhelm et ~ 1 The . calculations ~ ~ have been applied to both series of data, molar fraction solubilities,x2,and Ostwald coefficients, L. The results, however, are reported for molar fractions only.
Thevalueof A V is the trueenthalpyof reaction 11. It indicates the heat exchanged in the reaction ideally extrapolated to T = 0 where the heat absorbed by the water molecules involved in the reaction is null. The value of AiF is negative, and its absolute value increases with the size of the gas element. The number of water molecules, n,,increases with the atomic number of the element. In order to prove that n, is a true molecular parameter and not a mathematical artifact, we have
632 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 TABLE 5
Apparent Isobaric Heat Capacities ( AC,,
gas He
ACp,pp(X2)
ACP,,(PZ) ACp,pp(P2’)
Ne
ACp,pp(~z) ACP,W@2) ACp+pp(Pz’) ACp,&) ACP,W(P2) ACpcPp(P2’) ACp,pp(XZ) ACPSpp(P2) ACp,pp(Pz’) ACp,app(xd ACPSW(P2) ACapp(PZI) ACpcpp(x2) ACP,.pp@l)
Ar Kr Xe Rn
273 -101.6 -144.2 -156.2 -154.0 -187.7 -173.9 -184.2 -2 12.0 -238.8 -200.1 -250.9 -257.0 -207.3 -246.1 -286.4 -305.9
(J
278 -103.5 -143.7 -148.6 -1 52.4 -174.8 -167.1 -181.2 -214.4 -225.1 -200.2 -241.8 -248.2 -203.4 -258.7 -278.4 -301 .O
K-1
Braibanti et al.
mol-’)) at Selected Representative Temperatures. T (K)
288 -105.2 -137.2 -134.6 -1 50.7 -154.9 -1 54.4 -177.3 -210.9 -209.0 -201.6 -226.9 -232.0 -198.5 -267.8 -263.6 -294.7
298 -104.9 -125.6 -121.8 -150.2 -141.2 -143.0 -175.7 -199.6 -194.5 -203.9 -215.6 -217.5 -196.4 -260.6 -250.4 -291.8
308 -103.2 -111.0 -110.1 -150.5 -131.9 -132.6 -175.4 -183.4 -181.4 -206.6 -206.7 -204.3 -196.0 -242.8 -238.6 -290.9
318 -100.5 -94.8 -99.4 -151.1 -125.9 -123.1 -175.9 -164.2 -169.4 -209.4 -199.7 -192.4 -196.6 -218.1 -228.1 -291.3
328 -97.1 -77.8 -89.5 -151.8 -122.2 -1 14.4 -176.9 -143.4 -158.6 -212.0 -194.0 -181.6 -197.8 -189.3 -218.6 -292.3
333 -95.3 -69.3 -84.9 -152.1 -121.0 -1 10.4 -177.4 -132.8 -153.5 -213.1 -191.6 -176.7 -198.5 -174.1 -214.2 -292.9
ACP,PP(PZI)
ACp, (xz): values calculated by us by the second derivative of eq 17 from results of the smoothing functions obtained by Wilhelm, Battino, and ACp,,(P2): values calculated by us by the second derivative of eq 17 from the experimental values of Benson and Krause.28 ACpJpp(pZI): WilcoCk.2qP values calculated by Benson and KrauseZ8by Mark 111. TABLE 6 gasb He
Intermlation Polynomials for Apparent Isobaric Heat Capacities of Noble Gases in Water (AC,,) coefficientsa Tm‘ n, U a‘ a” 102 b 104 c (log x2)
(logP 2 ) Ne
(logpi)
(log x2)
(log P2)
Ar Kr Xe Rn
(log p2’) (log x2) (log P2) (logpz’) (log x2) (log PZ) (logp2’) (log x2) (log P2) (logp2’)
(log x2)
305.4 302.5 302.5 323.1 326.0 326.0 369.2 379.1 379.1 374.1 383.1 383.1 396.1 396.4 396.4 371.5
1.353 1.539 1.552 2.003 1.871 1.844 2.348 2.481 2.507 2.746 2.826 2.8 12 2.627 3.210 3.264 3.887
-103.817 -1 19.764 -1 16.557 -151.436 -123.001 -116.131 -176.919 -96.816 -68.679 -214.556 -152.582 -135.872 -180.694 -225.189 -168.793 -296.034
-101.99 -1 16.01 -1 16.99 -150.98 -141.03 -139.00 -176.99 -187.01 -188.97 -206.99 -213.02 -211.96 -198.02 -241.96 -246.03 -292.99
-1.83 -3.74 0.43 -0.46 18.03 22.87 0.07 90.19 120.29 -7.57 60.44 76.09 17.33 16.77 77.24 -3.04
20.008 145.251 117.499 -8.31 1 32.661 84.437 25.253 -1 16.768 286.301 24.844 135.009 72.124 1 1 1.004 -7 17.425 69.302 9.778
59.124 137.149 -52.787 -0,351 -51.812 -35.207 93.254 -558.651 301.931 88.812 174.788 -6.378 19 1.650 -1440.903 14.308 79.322
lo6d -72.237 -244.259 22.145 51.892 220.785 21.158 77.714 -354.575 172.059 49.410 121.234 25.691 96.606 -685.505 28.794 79.866
(log P2)
(logP2’) Interpolation polynomial (cf. eq 30): ACpcPP= a + bx + cxz + dx3 with x = T - T,. a’ = -nwCP,,. a” = a - a’, a” = ACp at Tm.C,,, = 75.378 J K-’ mol-’. (log x ~ ) :coefficients in eq 30 calculated by us from results of the smoothing functions obtained by Wilhelm, Battino, and WilCoCk.22 (logpz): coefficients in eq 30 calculated by us from experimentalvalues of Benson and Krause.28 (logpz’): coefficients in eq 30 calculated by us from the values calculated by Benson and Krause2*by Mark 111. Tm = temperature in kelvin at which log x2 (or log p2) is a minimum. free gas to gas trapped in water is composed of two contributions, searched for any relationship of n, with some parameter which one endothermicand one exothermic. The exothermic component depends on the size of the element. To this end, we plot n, against the boiling points of the elements (Figure 5 ) . A relationship is linearly related to n,. The constant endothermic contribution between the two parameters clearly exists. It allows also a could be attributed to energy spent to rearrange the environment. discrimination between the sets of data with different precision The specific enthalpy for cavity formation, Ah-, can beconsidered and accuracy. The points obtained from the most precise set of as an experimental evaluation of the energies involved in the two data of Benson and Krause show an exact linear dependence steps, gas-water attraction and cavity formation, postulated by upon boiling point whereas the data obtained as averages of the scaled particle theory. The same energies should be involved different sets are dispersed around the line, probably due to in the formation of the hydrophobic bonds. experimental uncertainties. Alternative plots can be obtained of The number of water molecules, n,, is also correlated to the n, uersus the atomic polarizability or the distance of minimum value of the solubility, log p2.r. at the minimum. At this point, approach. All the plots confirm that the number n, can be the free energy equals the entropy multiplied by Tm. By plotting considered an atomic parameter which is peculiar for each element the values of log p2,rmagainst n,, one obtains the line and increases regularly with the atomic size. The value of the enthalpy, AHo, depends on n,. We have log pz,Tm= -5.800 + 0.397 n, (32) searched for a relationship between the enthalpy and the number The coefficients of this equation can be transformed into values of water molecules involved in the reaction. By plotting AHo of thermodynamic functions. In fact, at the minimum the uersus n, (Figure 6 ) , the linear relationship is obtained following equality holds p H 0 = A H 0 + “w Ahca” (31) logpz,Tm= -AGm0/2.3RT = ASmo/2.3R (33) with AH0 = +16.4 kJ/mol and Ah,,, = -33.6 kJ/mol/n,. From eqs 32 and 3 3 , one obtains the change of entropy Mm0= Equation 31 states that the enthalpy change of the process from
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 633
Solubility of Noble Gases in Water
--c
IVI
water molecules expelled, n,, is obtained as
I
wheres is the index of the type of cavity and tsis the total number of types. The variance of n, can be obtained by the equation
-50
2
1
4
3 RW
Figure 7. Isobaric heat capacity of noble gases in water as function of the number of water molecules, ?.,I Both sets of points are from the data of Benson and Krause.2s Full points are calculated by Bensonand Krause and empty points by us.
The variance of the model and the isobaric capacity for noble gases are compared in Figure 8b. The agreement is pretty good and confirms that at least the type of distribution of the sizes of cavities is probably correct. This does not imply that thesupposed distribution is, strictly speaking, the only one compatible with the observed heat capacities. Further terms of the smoothing function of Table 6 are difficult to be assigned a chemical interpretation because they highly depend on the residual experimental error and numerical rounding off.
Conclusions
4-
-
-t
4 -50
0
50
100
150
ACpl JlmollK
3
b)
Figure 8. (a) Models of cages in aqueous solutions of noble gases. The average size of the cage increases with the size of the included particle. n, (s, 34)
is the average number of water molecules per particle expelled
to form the cavity and depends on the size of the entering atom. (b)
Dispersion or fluctuation between different cage sizes of the model as evaluated by the variance of n, (s, eq 35), in accordancewith ACp found in the present work.
19.145(-5.800+0.397nW) =-111.04+7.60n,Jmol-I K-I,which also depends on the number of water molecules expelled from the cavity. From the slope of eq 32, the entropy change As,, = 7.60 J mol-' K-1 nw-*per water molecule expelled can be obtained. We have also searched for an explanation of the behavior of AC, as a function of n,. The values obtained refer to the data of Benson and KrauseZ8treated either by eq 17 or by eq 24 (or Mark 111). The values of AC, from Table 6 are plotted in Figure 7 as a function of n,. The results obtained by the two different mathematical interpolation functions givevery similar results, as it was expected because the Mark I11 polynomial (24) and polynomial 17 are both functions of the reciprocal temperature. Bothcan beconsidered togivea descriptionof populationsbetween levels of a quantized model of the system. The values of AC, according to this quantized model must be related to the dispersion of the molecules among the different sublevels.l.2 In this case, this could mean that the radius of the cavity can assume a range of values of n, and correspondingly expel1 a range of a number of water molecules which do not correspond to integers. The larger the size of the atom, however, the larger is the average radius of the cavity. A possible model distribution of species in accordance with the observed values of n, and AC, for the different noble gases is shown in the diagram of Figure 8a. The average number of
The dependence of the solubility of noble gases in water upon the temperature can be successfully interpreted on the basis of the statistical thermodynamic model where the whole solution is represented by a convoluted ensemble (gc*c)e. The analysis of thecorresponding convoluted partition function, (CG*C)PF, and its temperature derivatives has shown how the apparent enthalpy, AHappr is a balance between the actual enthalpy of the reaction, AHo,and the heat absorbed by n, water molecules expelled from the bulk to form a cavity. The enthalpy of the solubilization reaction, AHo,depends in turn on the number of water molecules, n,. It is found to be composed of a constant endothermic contribution and of an exothermic contribution proportional to the number of water molecules and the size of the dissolved gas. The values of n, range from 1.5 for helium to 3.3 for xenon (for radon -4). For larger molecules and macromolecules, thevalues might be as high as 90 or 150 water molecules per mole of macromolecule. These findings are reasonable for the energetics involved in the thermodynamics of hydrophobic bonding. On the whole, the capacity for explaining the solubility of noble gases in water can be considered as a further proof of the validity of the statistical model itself as a potent tool for the interpretation of the reactions in solution.
Acknowledgment. This work has been supported by grants from the NATO-Scientific Affairs Division and the Italian National Research Council (C.N.R. Progetto Chimica Fine). List of Symbols canonical ensemble grand canonical ensemble convoluted ensemble canonical partition function grand canonical partition function convoluted partition function absolute temperature weight of sublevelj enthalpy of level i average enthalpy for sublevelsj of level i free energy of level i entropy of level i average entropy for sublevelsj of level i isobaric heat capacity of the level i multiplicityfactor depending on the number of equivalent binding sites dilution of the species i free energy change from ground level to level i
Braibanti et al.
634 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 enthalpy change from ground level to level i M i entropy change from ground level to level i Mi exp(-AG,/RT) joint probability of transition from level i to ground level exp(-AH,/RT) probability of transition from level i to ground level probability of transition from level i to ground level free energy change for formation of hydrated gas excess grand canonical partition function for hydrated gas contribution to free energy from the water molecules contribution to enthalpy from the water molecules concentration of solvent (water) concentration of free gas specificdissociationconstantbetween freegasandsolvent apparent free energy change for hydration of gas apparent enthalpy change at temperature T = 0 standard free energy change standard enthalpy change at temperature T = 0 environment enthalpy change (extrapolated at nw = 0) standard free energy change at Tm (at a minimum) standard entropy change at Tm number of water molecules involved in the formation of cage average number of water molecules per noble gas atom number of water molecules in species s total number of hydrated species molar fraction of solute partial pressure of solute at unit total pressure partial pressure of solute at unit total pressure at Tm (at a minimum) Henry constant cavity formation enthalpy enthalpy contribution for cavity formation per water molecule entropy contribution for cavity formation per water molecule molar isobaric heat capacity of water isobaric heat capacity change at Tm temperature at which log x2 (or any solubility) is at a minimum second virial coefficient reduced temperature critical pressure of water compressibility factor at 0 O C and 760 Torr compressibilityfactor under equilibrium condition of the experiment density of water
e AC,,,
L B p,
reference temperature derivative of -AH,pp with respect to 1/T Ostwald coefficient, ratio of the actual volume of gas to the volume of solvent at the same temperature and pressure Bunsen coefficient solubility product
References and Notes (1) Braibanti, A.; Fisicaro, E.; Dallavalle, F.; Lamb, J. D.; Oscarson, J. L. J. Phys. Chem. 1993,97,8054. (2) Braibanti, A.; Fisicaro, E.; Dallavalle, F.; Lamb, J. D.; Oscarson, J. L.J. Phys. Chem. 1993,97,8062. (3) Braibanti, A,; Fisicaro, E.; Dallavalle, F.; Lamb, J. D.; O ” n , J. L.; Ughi, F. X . J . Phys. Chem. 1993, 97, 8071. (4) Blandamer, M. J.; Burgess, J.; Robinson, R. E.; Scott, J. M. W.
Chem. Rev. 1976,82, 259. ( 5 ) Dec, S. F.; Gill, S . J. J . Chem. Educ. 1985,62, 879. (6) Clarke, E. C. W.; Glew, D. N. Trans. Faraday SOC.1966,62,539. (7) Hermann, R. B. J. Phys. Chem. 1971, 75, 363. (8) Hermann, R. B. J. Phys. Chem. 1971, 76, 2754. (9) McAucliff, C. J. Phys. Chem. 1966, 70, 1267. (10) Pierotti, R. A. Chem. Rev. 1976, 76, 717 and citations therein. (11) Kauzmann, W. Ado. Protein Chem. 1959, 14, 1. (12) Tanford, C. The Hydrophobic Effect: Formation of Micells and Biological Membranes; Wiley: New York, 1980. (13) Ben Naim, A. Water and Aqueous Solutions; Plenum Press: New
York, 1974. (14) Ben Naim, A. Thermodynamics of Solvation; Plenum Press: New York, 1980. ( ! 5 ) Brandts, J. F. In Conformational Transitions of Proteins in Water and in Aqueous Systems; Timasheff, S.N., Fasman, G. F.,Eds.;Structure and Stability of Biological Macromolecules; Dekker: New York, 1969. (16) Hopfinger, A. J. Intermolecular Interactions and Biomolecular Organization; Wiley: New York, 1977; pp 324-339. (17) Lumry, R. DynamicalAspectsof Small-MoleculeProtein Interaction; Braibanti, A., Ed.; Bioenergetics and Thermodynamics. Model Systems; Reidel: Dordrecht, The Netherlands, 1980. (18) Sturtevant, J. M. Biochemical Applicationsof Differential Scanning Calorimetry, Annu. Rev. Phys. Chem. 1987, 38, 463. (19) Privalov, P. L.; Gill, S. J. Stability of Protein Structure and Hydrophobic Interaction, Ado. Protein Chem. 1988, 39, 191. (20) Privalov, P. Thermodynamic Problems in Protein Structure, Annu. Rev. Biophys. Biophys. Chem. 1989, 18,47. (21) Wilhelm, E.; Battino, R. Chem. Rev. 1973, 73, 1. (22) Wilhelm, E.; Battino, R.; Wilcock, R. J. Chem. Rev. 1977, 77,219. (23) Rettich, T. R.; Handa, Y. P.; Battino, R.; Wilhelm, E.J. Phys. Chem. 1981,85,3230. (24) Benson, 9. B.; Krause, D., Jr. J . Chem. Phys. 1976,64,689. (25) Benson, B. 9.;Krause, D., Jr.; Peterson, M. A. J. Solution Chem. 1979, 8, 655. (26) Crovetto, R.; Fernandez-Prini, R.;Papas, M. L. J . Chem.Phys. 1982, 76, 837. (27) Benson, B. B.; Krause, D., Jr. J. Solution Chem. 1989, 18, 803. (28) Benson, B. B.; Krause, D., Jr. J. Solution Chem. 1989,18, 823. (29) Braibanti, A.; Dallavalle, F.; Fisicaro, E. Ann. Chim. (Rome) 1988, 78, 679. (30) Marshall, W. L. J . Phys. Chem. 1972, 76, 720. (31) Beutier, D.; Renon, H.AIChE J. 1978, 24, 1122.
