J. PADOVA
796 considered significant; however, the possibility exists that other molecules may also show emission properties incompatible with the currently accepted value of 0.55 for the fluorescence yield of quinine bisulfate. The
Ion-Solvent Interaction. VI.
authors hope that additional results from other researchers working with different molecules will help to evaluate the validity of the observations reported in this work.
A Thermodynamic Approach
to Preferential Solvation in Mixed Solvents’“
by J. Padovalb Radiochemistry Department, Israel Atomic Energy Commission, Soreq Nuclear Research Center, Yavne, Israel (Received M a y $9,1967)
The thermodynamic treatment of a mixed fluid in an electrostatic field is applied to the calculation of preferential solvation in mixed solvents a t infinite dilution. The preferential solvation is interpreted in terms of partial molar free energies of solvation and is compared with literature data.
Introduction The preferential solvation shown by electrolytes in mixed solvents is a direct consequence of the specific interaction between an ion and one of the components of the mixed solvents. It may be defined as the relative change in the composition of the mixed solvent in the vicinity of the ion and may be expressed by
where n1/n2is the mole ratio of the components in the vicinity of the ion, nI0/nz0 is the initial mole ratio of the components, and a will be called the index of preferential solvation. It is obvious that for a > 0, the ion is selectively solvated by component 1; for a < 0, the ion is selectively solvated by component 2 ; and for a = 0 there is no preferential solvation. Various attempts have been made to derive theoretical expressions for eq l , 2 - 5 assuming either an ideal mixture for the mixed solvent and the electrostatic contribution for the free energy to be given by Born’s expression,2 or an empirical correction for the change in the dielectric constant of the ~ o l u t i o n ,or ~ other experimental corrections. 4t5 It was now found possible to apply the thermodynamic treatment of a mixed fluid in an electrostatic field* t o the problem of calculating the preferential solvation, assuming the mixed solvent’ to be sorted The Journal of Physical Chemistry
under the influence of the electric field caused by the presence of ions.
Theory From electromagnetic theory, it is possible to show that the isothermic, isobaric electrostatic work done on a fluid is given by the expression7 I
W
/-e
JJ Edbdv
= -
4n
where E and b are the electric field and electric displacement vectors, respectively, and dv is the elementary volume. From expression 2 , the elementary electrical work done on the fluid is obtained as
L ( @ )
dw = 4n
dv
(1) (a) Presented in part at the XXXIII Xeeting of the Israel Chemical Society Beler Sheva, Dec 1963; cf., J. Padova, Israel J . Chem., 1, 258 (1963). (b) Oak Ridge National Laboratory, Chemistry Division, Oak Ridge, Tenn. 37830. (2) P. Debye, 2. Physik. Chem., 130, 56 (1927). (3) G . Scatchard, J . Chem. Phys., 9, 34 (1941). (4) (a) J. E. Ricci and G. J. Ness, J . Am. Chem. Soc., 64, 2306 (1942); (b) H. L. Clever and F. H. Verhoek, J . Phys. Chem., 62, 1961 (1958). (5) I. F. Efremov, T. A. Probof’eva, and Yn. P. Symikov, Zh. Fiz. Khim., 38, 2258 (1964). (6) H. S. Frank, J . Chem. Phys., 23, 2023 (1955). (7) E.g., A. Prock and G. McConkey, “Topics in Chemical Physics,” Elsevier Publishing Co., Amsterdam, 1962, p 11.
ION-SOLVENT INTERACTION
797
(")bP
whence the increment in internal energy, dU, is obtained through the application of the first law of thermodynamics to an infinitesimal process dU = TdS -. Pdv
+ 4n
dv
+ &dnt
(3)
We may now define a characteristic function, analog to the enthalpy
H*,
dP+ E,Nz,T
(g)
P,Ni,T
dE+
Since the changes in local composition produced by the field should be at constant chemical potentials p1 and pz of the binary solvent, inside and outside the field, we obtain for dp1 = dpz = 0
+ and Making use of the Gibbs-Duhem relations and the analog Gibbs free energy, G* dG* = -SdT
+ vdP - VE -dD + Cptdnl 47r
introducing thLe differential dielectric constant, dD/dE, we obtain the fundamental equation dG* = --SdT
+ vdP - -dE 4a
VedE
it may be shown that €d
+ &dnr
=
Rewriting eq 11 as (4)
from which the desired relationships will be obtained in the usual way. An earlier treatment6 assumed that the whole of the fluid sample under discussion is subject to the same (instantaneous) field strength. This has now been generalized, and eq 4 may be applied to a dielectric which is nonuniform in E , since eq 2 allows for infinitesimal variations of E and in the elementary volume, dv . We are interested in the change of local composition in a mixture of solvents produced by the field of an ion. Equation 4 for: a two component system is written as dG* = -SdT
and dividing by N1 gives
b In N1
EV bad = ~ G ) ~ , E , T , ~ L ~ + J ~ : E , T
(13)
with a similar expression for
( b y ) P1W
where the relative change of composition per unit volume under the charge of field strength dE is
+- vdP which gives for the change of composition in the vicinity of the ion (r = re) as compared to the composition outside the field (r = a )
hence by cross differentiation
1 b(vedE) =
(%).P,T,ninz
-~
[ ~ ] P , T , E , n z
(6)
or in terms of mole fractions N1 and N Z or
+
where v m = v/(nl nz). At constant temperature, pl, P , E, and N1 may be taken as a mutually dependent set of quantities, so that eq 8 follows. Volume 72,Number 3 March 1968
J. PADOVA
798 which is the general expression for the sorting of mixed solvents by ions where N,O is the value N t outside the field, re is the intrinsic radius of the ion, and E, is the field strength at the distance r. Various cases may be consideredin evaluating eq 16. First Case. The dielectric constant, E, and the volume, v, are assumed to be independent of the field strength (which is true for weaker fields only); the binary mixture is assumed to form an ideal solution. Under these conditions, it is possible to integrate the right-hand side of eq 16, since pi = pio
+ RT In N1 and
E
=
td
dD dE
= -
NI NiO -1n= N2 NZO
case, pi
= p10
Ni NZ
In - - In
+ RT In N 1 and eq 16 is transformed into
NiO -= N 2’
1 --
2RT
which may be interpreted as follows. The ionic free energy of solvation in a mixed solvent at infinite dilution, AG, may be formally described as where D,is a partial free energy of solvation of the ion in component i of the mixture. It has been showns that the integral
ln-
represents the electrostatic part of the ionic solvation free energy. The part 1/z~rrEdd(E2)may be con-
Substituting
sidered as the ionic-solvation free-energy density and the differential where vm
= v/(n1
+ nz) gives
which is identical with the expression obtained by Frank.6 Substituting E = (Ze)/(Er2) in eq 17 gives
as the partial ionic-solvation free-energy density in component 1 of the mixture. Hence the right side of eq 21 may be considered to represent the difference between the partial ionic-solvation free energies in components 1 and 2 of the binary solvent, and we may write
which may be compared with the similar equationZ
vi In
Ni NZO - - vz In NiO N$
=
Z2e2 v1 8nRTe2r4(
e - vz ”) (20) bm
where Z is the valence of the ion, e the electronic charge, and r the distance from the ion; ql and 712 are the number of molecules per milliliter of the two substances, and v1 and vz are molecular volumes defined by qlvl
+
77202
= 1
Franke has already shown that eq 20 is less general than eq 18 and that, qualitatively, it is possible to predict that in the immediate neighborhood of an ion, N1/Nz could change by several powers of ten, a rather complete sorting. This treatment, however, neglects electrostriction and dielectric saturation, even at infinitely dilute solutions, and cannot be used for finite salt concentrations. Second Case. We shall now solve eq 16 for the case of organic solvents containing a large percentage of water, where it has been shown that the activity of the water is very nearly given by the water mole fraction in the solvent. However, the dielectric constant will be taken to be a function of the field strength. In this The Journal of Physical Chemistry
In
Ni NiO 1 - - In - = - -(AG1 NZ N LO RT
-
nz) (24)
From eq 24, the ionic index of preferential solvation, obtained
a,is
(25)
We may now combine the ionic contributions of an electrolyte (va, anions, and v,, cations) and obtain thereby a mean selective index, il, given by E =
vasa
+
vcac
= -
V
+
aG1 - aGz 2.3vRT
zt
with v = va V , and is the partial molar free energy of solvation of the electrolyte in component i of the binary solvent. It follows that the preferential solvation depends only on the difference between the partial molar free energy of the electrolyte in each conponent of the solvent mixture, since the sign of E determines the solvent selectivity. (8) J. Padova, J . Chem. Phys., 39, 1652 (1963).
ION-SOLVENT INTERACTION Table I : Values of
Salt
HCl HBr HI HClOi NaCl NaBr NaI NaOH NaNOs LiCl LiBr LiI KCl KBr
KI RbCl RbBr RbI CSCl CaBr CsI
AgCl AgBr
Ad CaClz ZnClz CdClz
-
799
ti for Various Solvents as Compared to
Water 50%
_ _ I -
Solvent6CzHsOH CiHoOH
NHa
NzHi
CHaOH
-5.58 -6.73 -7.58
-2.67 -3.78 -2.67
2.02 2.02 1.47
2.56 2.56 2.56
1.28 1.28 0.73
1.83 1.83
1.47 0.32 -0.55
1.09 0 1.10
0.92 -0.23 -1.08 4.76 3.61 2.76 2.93 1.80 0.92 4.76 3.57 2.73 -5.50 -6.65 -7.50 6.98 -5.99 -4.03
1.65 0.55 1.65 1.10 0 1.10 1.28 0.18 1.28 0.92 -0.18 0.92 -6.23 -7.33 -6.23 2.44 -3.66 -4.37
1.83
1.28 1.28 0.73 1.28 1.28 0.73 1.83 1.83 1.28 1.83 1.28 2.20 2.20 1.65
2.19 2.19 2.19 2.38 2.38 2.38 2.93 2.93 2.93 2.67 2.67 2.67 2.38 2.38 2.38
4.40 3.98
5.86 4.51
1.83
m2
The difference g1 could be rigorously obtained by applying the Backhuis-Roozeboom procedure to the free energy of solvation of the electrolyte per mole of mixed solvent. However, because of the paucity of data for mixed solvents, the molal solvation free energy of the electrolyte in the pure component i of the mixed solvent has been used, since this has been showngt o introduce only small errors in the calculation of the difference Dl The calculated values of B are listed in Table I. All the values of the solvation free energy were taken from Izmailov,’O except those for the mixture of 50% dioxane-water, taken from Grunwald et al., l1 who give a correct value for the difference - E2. From the table, it may be seen that, generally, salts will solvate water preferentially, except for mixtures of NH3 H20, N2H4 H20, and HCOOH H20. The negative values of B in these cases indicate that chemical interaction is taking place (e.g., ammine formation), which probably leads to complex formation; this tallies with the fact that lower free-energy states are more stable. This assump tion of selective solvation in interpreting data on conductance, 12-15 s o l ~ b i l i t ythermodynamic ,~~~ association constant, l6 viscosity,l7 density,ls transference numbers,lg molar refraction,2O ultraviolet and visible spectra,21 vapor-pressure data, l1 solvolysis
mz.
+
’
+
+
HCOOH
CsHiiOH
(CHs)zCO
2.89 2.02 2.57
3.12 3.19 2.20
3.12 1.84 2.75
2.94 4.22 4.58
3.07 2.2 2.75
4.71 4.78 3.79
6.78 5.50 6.42
-1.10 0.18 0.55
1.97 1.10 1.65 3.98 3.12 3.66 5.63 4.77 5.31 3.26 2.39 2.93
2.02 2.09 1.10 4.22 4.29 3.30 4.22 4.29 3.30 3.67 3.74 2.75
2.38 1.10 2.02 4.95 3.67 4.58 5.50 4.22 5.13 3.85 2.57 3.49
-0.73 0.54 0.92 -0.55 0.73 1.10 0.55 1.78 2.20 -1.78 -0.52 -0.13 -2.20 0.92 -0.55 0.49 1.84 0.98
Dioxane water
2.98
0.21 5.06
6.45 3.59 4.21
5.06 4.14 3.11 4.81
4.58
reactions,22and emf1° is confirmed by results listed in Table I. In the cases where it was possible to prove (9) J. Padova, to be published. (10) A. N. Ismailov, “Electrochemistry of Solutions,” Kharkov University, 1959. (11) E. Gmnwald, G. Beughman, and G. Kohnstam, J . Am. Chem. SOC.,82,5802 (1960). (12) (a) G. Atkinson, ibid., 82, 818 (1960); (b) C. J. Mallada and G. Atkinson, ibid., 83, 3759 (1961); (c) G. Atkinson and C. J. Mallada, ibid., 84, 721 (1962); (d) G. Atkinson and S. Petrucci, ibid., 86, 7 (1964). (13) (a) N.Goldenberg, Ph.D. Thesis, University of Kansas, Kansas City, Kan., 1961; (b) A. Fratiello, Ph.D. Thesis, Brown University, Providence, R. I., 1962. (14) A. Campbell and G. Debus, Can. J . Chem., 34, 1232 (1956). (15) J. B. Hyne, J. A m . Chem. Soc., 85, 304 (1963). (16) (a) H. K. Bodenseh and J. B. Ramsey, J. Phys. Chem., 67, 140 (1963); (b) Y.H.Imami, H. K. Bodenseh, and J. Ramsey, J. Am. Chem. Soc., 83, 4745 (1961); (c) A. d’Aprano and R. M. Fuoss, J . Phys. Chem., 67, 1704, 1871 (1963). (17) J. Padova, J. Chem. Phys., 38, 2635 (1963). (18) J. Padova, ibid., 39, 2599 (1963). (19) (a) J. 0.Wear and E. 9. Amis, J . Inorg. Nuel. Chem., 24, 903 (1962); (b) R.K. Kay and A. Fratiello, J. Phys. Chem., in press; (c) H. Schneider and M. Strehlow, Z . Elektrochem., 66, 309 (1962); (d) H.Strehlow and H. M. Koepp, ibid., 69,674 (1965);(e) H. Schneider and H. Strehlow, ibid., 69, 674 (1965); (f) H.Schneider and H. Strehlow, 2. Ph2/sik. Chem., 49, 44 (1966). (20) (a) S. Minc and W. Libus, Roczn. Chem., 29, 1073 (1965); (b) J. Padova, Can. J. Chem., 43, 458 (1965). (21) (a) 8. Minc and W. Libus, Roczn. Chem., 30, 945 (1956); (b) I. S. Pominov, Zh. Fiz. Khim., 31, 1926 (1957); (c) R.F.Pasternack and R. A. Plane, I n o r g . Chem., 4, 1171 (1965). Volume 7% Number S March 1968
Y. BOTTINGA
800 that selective solvation is taking place, as in the nmr study of electrolytes in 50% dioxane-water all salts examined showed preferential solvation for water, contradicting the results obtained by Grunwald," but in complete agreement with the results for iic in the table. Additional confirmation was obtained from the use of epr for l\;lnS0414diii water-dioxane mixtures, the study of rate of exchange of solvent molecules for Cr(III) , 2 4 high-resolution nmr of Cu2+, &'In2+,and in ethanol-water and in methanol~ ~ a t eand r ,in~pyridine-~ater;~~b ~ ~ ~ ~ all the nmr data23 confirm the results listed in the table for the solvents usually examined, Le., methanol, acetone, ethanol, and dioxane. It has been possible to predict preferential solvation in aqueous organic mixtures by using the thermody-
namic approach to the continuum theory of electrolytes in mixed solvents. Aclcnowledgment. I wish to thank Professors H. S. Frank and A. Katchalsky for their reading of the manuscript and offering valuable comments. (22) For review, see ref 14; R. Hudson, Ber. Bunsenges Physik. Chem., 68, 215 (1964). (23) (a) A. Fratiello and D. C . Douglas, J . Chem. Phys., 39, 2017 (1963); (b) A. Fratiello and E. G. Christie, Trans. Faraday Soc., 61, 306 (1966); ( 0 ) A. Fratiello and D. P. Miller, Mol. Phys., 11, 37 (1966). (24) J. C. Jayne and E. L. King, J . Am. Chem. SOC.,86,3986 (1964). (25) In. Ua. Shamonin and S . A. Yan, Dokl. Akad. N a u k SSSR, 152, 677 (1963). (26) J. H. Swinehart, T . E. Rogers, and M. Taube, J . C h e w Phys., 38, 393 (1963). (27) T. E. Rogers, J. H. Swinehart, and M. Taube, unpublished results.
Calculation of Fractionation Factors for Carbon and Oxygen Isotopic Exchange in the System Calcite-Carbon Dioxide-Water by Y. Bottinga Scripps Institution of Oceanography, University of California, S a n Diego, L a Jolla, California 92037 (Received June 14, 1967)
Partition-function ratios for Hz018/Hz01G,C0182/C0162, CaCO1%/CaCO1%, C1302/C1202, and CaC130a/CaC1203 have been calculated. Water vapor and carbon dioxide are treated as ideal gases, but vibrational anharmonicitg effects are taken into account. The calcite partition function ratios were calculated by adop'ing the Giulotto-Loinger model for the acoustical, optical, and librational lattice modes; the orbital-valence force-field method was used for the carbonate ion vibrational frequencies. For oxygen isotopic exchange (1.0611 X between COz and calcite, the calculated fractionations can be expressed as 1000 In (YO = -3.2798 lO4/T) - (1.8034 x 106/T2),for the temperature interval 0-550'; for carbon exchange, the corresponding equation is 1000 In oC = -2.4612 (7.6663 X 10*/T) - (2.9880 X 10G/T2). The calcu'ated fractionation factors for isotopic exchange in the systems carbon dioxide-calcite and carbon dioxide-water agree with the observed ones. The agreement is less satisfactory for the oxygen isotopic fractionations between calcite and water above 100°.
+
+
I. Introduction A knowledge of isotopic fractionation factors is a prerequisite for understanding the stable isotope abundance patterns observed in nature. Fractionation factors for oxygen have been measured in several systems as a function of temperature.l Unfortunately, it is often difficdt to measure fractionation factors at temperatures of geochemical interest in systems containing solids. Therefore, it is frequently necessary to extrapolate measured data over a considerable temperature interval, A theoretical treatment of the problem should produce guide lines for extrapolation The Journal of Physical Chemistry
and may alleviate the need for experimental results in favorable cases. After Urey2 had published his important paper on isotopic exchange, several attempts were made to calculate the fractionation factors for isotopic exchange between condensed phases. McCrea3 published a brief account of calculations of the isotopic fractionation factors for carbon and oxygen exchange between calcite (1) H. P Taylor, Trans. Amer. Geophys. Union, 47, 287 (1966). (2) H. C . Urey, J . Chem. SOC.,562 (1947). (3) J. M . MoCrea, J . Chem. Phys., 18, 846 (1950).
