1604
Charles F. Mattina and Raymond M. Fuoss
Conductance of the Alkali Halides. XIII. Cesium Bromide, Lithium-7 Chloride, and Lithium-7 Iodide in Dioxane-Water Mixtures at 25’ Charles F. Mattina’ and Raymond M. Fuoss* Department of Chemistry, Yale University, New Haven, Connecticut 06520 (Received January 20, 1975) Publication costs assisted by the Office of Water Research and Technology
Conductance data for cesium bromide, lithium-7 chloride, and lithium-7 iodide in dioxane-water mixtures covering the range 78.35 2 D 2 10 in dielectric constant are presented. Values of limiting conductance Ao, association constant KA, and pairing distance R are computed, using the three-parameter conductance equation A = A(c; Ao, KA, R ) . The model on which the equation is based defines ion pairs as those which are uniquely partnered with center to center distances r in the range a I r 5 R, where a is the contact distance. Ion pairs are formed by the diffusion of partner anions into spheres of radius R centered on the cations of the pairs (Debye-Eigen model) and dissociate by outward diffusion. For the systems of higher dielectric constant, R is somewhat larger than the sum of the lattice radii, corresponding to an average over a population consisting of contact pairs and solvent separated pairs; as D decreases, R increases, corresponding to the increasing range of electrostatic attraction. For the lower range of dielectric constant, the ion pairs are predominantly contact pairs, as shown by the relative insensitivity of the association constant to the numerical value of R.
In previous papers of this series2-6 the conductances of a number of the alkali halides in dioxane-water mixtures covering the approximate range 78.35 1 D 2 10 in dielectric constant have been reported. The data have been analyzed by several variations of a three-parameter conductance equation of the general form A = Y[AO- L ( c Y ) ]
(1)
where A is equivalent conductance, c is stoichiometric concentration, c y is the free ion concentration, and L ( c 7 ) represents the decrease in conductance with increasing concentration produced by the long-range interionic forces. The function L ( c y ) contains a distance parameter whose physical significance depends on the model chosen to represent the electrolytic solution; for the primitive model (rigid charged spheres in a continuum), this parameter is the center-to-center distance a of ions in contact. An equilibrium is assumed between free ions and nonconducting ion pairs, such that the fraction (1- y) of solute present as pairs is given by the mass action equation
where K A is the association constant. For the primitive model7
K A = (4aNa3/3000) exp(c2/aDkT)
(3)
if the pair population is defined to consist solely of pairs in contact. According to (3), a plot of log K A against reciprocal dielectric constant should be linear. For all of the systems which have been studied over a wide range of dielectric constants, the log KA-D-’ plots are not linear but concave down for the water-rich mixturess, and approach linearity only in the range of dielectric constants below about 25. The various equations of type (1)fit the data well within experimental error; assuming that no blunders have been made in the theoretical derivation of L ( c y ) , the discrepancy between theory and observation can only mean that the primitive model inadequately represents real physical systems, especially in the higher range of dielectric The Journal of Physical Chemistry, Vol. 79, No. 15, 1975
constants. It is also in this range where the Bjerrum model fails; although it leads to the observed shape of the log KAD-’ curve, it incorrectly predicts that the association constant drops to zero at a critical value of dielectric constant, above which association abruptly ceases. The virtue of the Bjerrum definition of ion pairs is the counting as pairs not only cation-anion pairs in contact but also those which are relatively near to each other; by definition, the interaction of unpaired ions then is weak enough to justify the classical time average treatment of Debye and Huckel. The false prediction just mentioned is merely the consequence of Bjerrum’s choice of the pairing distance: he defined as pairs those ions for which the center-to-center distance r lies in the range a 5 r I c2/2DkT = PI2
(4)
If this artificial restraint is removed by definings as pairs those ions for which a 5 r IR, where R is a distance parameter to be determined from the data, the problem of a “critical” dielectric constant vanishes. Furthermore, this definition avoids the unrealistic property of the primitive model for which r can vary continuously. In real solutions, ion pairs are pairs in contact and solvent-separated pairs; R is to be considered as the average radius of the spheres containing the partner ion for paired ions and the diameter of the spheres around free ions outside of which the solvent may be approximated by a continuum (Gurney’s cospheres). The purpose of this paper is to present conductance data for cesium bromide, lithium-7 chloride, and lithium-7 iodide in dioxane-water mixtures covering the range 78.35 I D 2 10 in dielectric constant and to derive the conductance parameters Ao, Ka, and R for the model described above. It will be shown that the model approaches two limiting forms: for solvents of higher dielectric constant, R appears as the radius of the Debye-Eigen diffusion sphere while for solvents of lower dielectric constant, most of the pairs are contact pairs, corresponding to the primitive model.
Conductance of the Alkali Halides
Experimental Section
Materials. Cesium bromide was used as received from the Harshaw Chemical Co.; the material was random cuttings from single crystals grown from melts of purified salt. Lithium chloride was obtained from the Isotopes Division of the Oak Ridge National Laboratory; certification reported 99.9926% 7Li. The salt was used as received to make master solutions as described below. Lithium iodide (99.93 7Li, 0.007 6Li) was also from Oak Ridge; the sample was pinkish tan in color and contained a small amount of unidentified organic material. The certificate reported 0.0448% carbon and 0.0291% nitrogen in the lithium from which the iodide was prepared. The salt was purified by recrystallization as the trihydrate, LiI-3Hz0, from water which had been thoroughly deoxygenated by boiling and then sparging with nitrogen while cooling; 10 g of salt as received was dissolved in 30 ml of water, and after filtration under nitrogen to remove some insoluble material, the solution was evaporated under vacuum to about 12 ml and cooled. Long glass-clear crystals formed; these were filtered out under nitrogen and washed on the filter with several portions of ice-cold peroxide-free diethyl ether. The crystals were stored in an evacuated desiccator, where they gradually lost some water of crystallization; this change in composition was unimportant, because the material was used to prepare master solutions which were then standardized. Lithium iodide, either in solution or as trihydrate, oxidizes in air very readily; it is therefore essential to handle it under nitrogen. The conductance of solutions in water or in dioxane-water mixtures remains constant within 0.01% for days if air is carefully excluded; after even a few minutes exposure to air, however, the yellow color of the triiodide begins to develop. Dioxane was first refluxed over potassium hydroxide to condense aldol impurities; it was stored over sodium-lead alloy under nitrogen, and distilled as needed, with a gentle flow of nitrogen through the still, condenser, and receiver during distillation. It is, of course, essential to avoid formation of peroxide in the dioxane. Methods. All solutions were made up by weight and volume concentrations (c equivalents/liter) were calculated using the appropriate densities. Cesium bromide was dried overnight at 80'; samples weighed on the microbalance in platinum cups were dropped into solvent in the conductance cells (after determination of solvent conductance) to give the initial solution for a conductance run. Other points were obtained by successive dilutions, by adding solvent by nitrogen pressure from the solvent storage flask. In some runs, in order to cover a wide concentration range, solution was removed after four or five dilutions from the cell using an evacuated pipet, and the (weighed) solution remaining in the cell was then further diluted. Since the lithium salts are extremely hygroscopic, direct weighing was not practical. Instead, master solutions in water were prepared, and standardized by potentiometric titration against silver nitrate solutions which had in turn been standardized against pure potassium chloride for the lithium chloride work and against pure potassium iodide for the lithium iodide work. The titrations were made with chilled solutions, with minimum light in the room, using a silver wire as the measuring electrode and a Corning glass electrode as reference. Solutions were acidified by a few drops of halogen-free nitric acid. Approximately 0.1 N solutions from weight burets were used to reach about 98-99% of complete titer; the titration was then carried through the endpoint using three-
1605
drop portions of 0.01 N solutions from an ordinary buret. Concentrations of master solutions were frequently checked; they held constant within f0.01%. For the conductance runs, secondary master solutions were made up by weight by adding aqueous master solution to a mixture of dioxane and water in a weight buret, amounts of solution, dioxane, and water chosen to give the desired dioxane-water ratio, and a salt concentration such that the concentration for the initial point would be apD 3 ( D = dielectric constant of solproximately 2 X vent). This upper limit is set by the range of validity of the conductance-concentration equation used to analyze the data. Then, knowing the composition of the solution in the weight buret, water was added to dioxane in the solvent storage flask (or vice versa) until the mixture in the latter had within f0.02% the same dioxane-water ratio as the solution in the weight buret. The various compositions are listed in Table I (see paragraph a t end of text regarding miniprint material), where w is weight percent dioxane in the mixture, D is the dielectric constant (measured a t 25' and 1 MHz2), and 1007 is viscosity (centipoises; measured in an Ubbelohde viscometer with 470.8 sec water-flow time). Conductance parameters are summarized in Tables IV and V. Electrical equipment and general measuring technique have already been described? Several conductance cells were used: three with cell constants 8.3234 f 0.00008 cm-l, 5.1331 i 0.0005, and 1.76025 f 0.00013 were calibrated by comparison with the 1.76 cell, using dilute aqueous solutions of sodium bicarbonate. Calibrations were, of course, checked at intervals during the work. All conductance measurements were made at 25.000 f 0.003'. Resistances were measured at 0.5, 1, 2, and 5 kHz and extrapolated to infinite frequency. The conductance data are summarized in Tables I1 and I11 (see paragraph a t end of text regarding miniprint material), where c is concentration (equivalents/ liter) and A is equivalent conductance (cm2 equiv-' ohm-l). Discussion As preface, a critical examination of conductance theory appears appropriate. The original statement of the problem about a century ago was deceptively simple: why does equivalent conductance h decrease with increasing concentration c? Arrhenius provided a partial answer by his hypothesis that free current-carrying ions were in mass action equilibrium with nonconducting entities; the relative concentration of the latter increased with increasing concentration and therefore A decreased with increasing c. The other part of the answer was given by Debye and Huckel in 1923 when they showed that the long-range effects of electrostatic forces between ions produced a relaxation field AX and an electrophoretic countercurrent Ah,, both of which led to a decrease in A, initially proportional to the square root of ionic strength. The Arrhenius theory gave a good account of the conductance of weak electrolytes where ionic concentrations were low and where, therefore, the long-range effects could be practically neglected; the Debye-Huckel theory was equally successful in predicting the conductimetric behavior of strong electrolytes at low concentrations in solvents of high dielectric constant. A general theory, rather than two special theories, was finally made possible by Bjerrum's proposal in 1926 that shortrange ion pairs would play the same role in solutions of strong electrolytes as did the neutral molecules of the Arrhenius theory in solutions of weak electrolytes. The Journal of Physical Chemistry, Vol. 79, No. 15, 1975
I606
Charles F. Mattina and Raymond M. Fuoss
TABLE V: Derived Constants for Lithium-7 Chloride and Iodide
TABLE IV: Derived Constants for Cesium Bromide Solutions
D
*O
KA
R
78.35 74.75 71.90 '70.28 62.42 5b.91 55.58 49.58 46.37 45.91 43.70 41.74 38.59 30.48 24.17 20.11 16.73 16.54 14.57 14.08 18.83 13.43 12.50 12.41 11.42 10.76 10.19 9.36
155.37 142.84 133.46 128.48 108.64 102.05 93.25 82.54 77.30 76.69 73.47 70.75 66.98 58.07 52.27 48.80 45.82 45.90 43.82 43.29 43.29 42.54 41.08 41.16 40.42 39.41 39.25 33.80
0.695 0.748 0.823 0.870 1.55 1.59 2.14 2.56 4.36 4.27 5.73 6.62 9.35 26.7 74.0 154 381 431 884 1070 1212 1340 2110 2155 3850 6070 9200 13200
6.69 6.71 6.75 7.12 8.40 8.01 8.58 7.68 9.09 8.74 9.89
103~
6 4 2 10
16 2 7 9 16 12 10
10.10
8
10.91 13.73 14.05 13.59 12.88 13.55 17.48 16.94 16.54 15.75 20.85 18.43 21.33 23.67 24.44 26.02
10
9 19 6 59 63 22 31 26 62 48 36 185 329 214 32
~,,q
1.383 1.394 1.395 1.392 1.378 1.381 1.366 1.342 1.325 1.321 1.301 1.287 1.261 1.160 1.051 0.952 0.870 0.868 0.806 0.788 0.784 0.765 0.726 0.725 0.699 0.668 0.655 0.550
D
*O
78.35 62.25 49.99 40.47 33.84 27.18 21.97 18.88 18.13 18.07 16.23 14.45 13.48 11.69
114.99 81.47 64.68 53.92 49.13 44.49 41.11 39.29 38.91 38.98 37.87 36.24 36.61 34.45
78.35 44.21 29.30 19.25 16.15
116.03 55.26 45.89 41.67 40.09
KA
1 0 3 ~
R
Lithium -7 Chloride 7.95 9 0.81 9.06 9 1.63 9.39 13 2.98 15.56 16 7.50 12.89 17 13.6 13.64 15 34.4 15.02 18 55.8 22.00 14 174 18.51 19 203 16.76 5 209 20.19 6 356 111 642 27.70 18.07 253 1090 20.84 157 2310 Lithium-7 0.62 3.17 16.1 62.8 170
Iodide 8.76 12.39 14.43 11.64 12.48
1 - y = K,\cy2f2
h = y[ho(l
- A X / X )- Ahe]
(5)
where the fraction (1 - y) of solute existing as ions pairs is given by We Jollrhai of Physical Chemistry, Vol. 79, No. 1s: 1975
11
59 94
1.024 1.051 1.040 0.996 0.959 0.896 0.818 0.764 0.751 0.752 0.715 0.665 0.660 0.597 1.033 0.974 0.917 0.807 0.750
(6)
Here, Ka is the conductimetrically determined association constant and f is the activity coefficient. (Ka = KA, the ion pair association constant, if the ions of a pair retain their identity as ions in the pair. If the ion pair rearranges to a neutral molecule, e.g., acetate-hydronium pair rearranging to acetic acid-water, then K,i = K a ( l 1IKR) where K R is the rearrangement constant of the neutral molecule. For present purposes, we shall assume that K,, = K A . ) Equation 1 answers in principle the original question, but it is an implicit equation and therefore useless in the analysis of conductance data to find the parameters A0 and K A . Theory can develop an explicit equation only on the
+
Symbolically, conductance is now described by the equation
12 14
A,V
1607
Conductance of the Alkali Halides
basis of a model. The situation thus poses two new problems: first, what shall the model be? Obviously, it must be simple enough to be amenable to theoretical treatment; ideally, it should correspond as closely as possible to a real physical system. However, the latter desideratum would involve a niultiplicity of system-specific parameters such as ionic size, shape, polarizability, and charge distribution, and analogous parameters characteristic of the solvent. This catalog leads to t,he second problem, a practical one: how many parameters can be obtained from conductance data? The answer is three, and no more than three. I t has been shown8 that the following semiempirical equation will reproduce conductance data for 1-1 salts in water up to about 0.1N
,I = d o - Sc112+ Ec In c
+ Ac + Bc3I2
(7)
where S and E are theoretically predictable coefficients. In principle, the empirical coefficients A and B are determined by the second and third derivatives of A(c112). T o determine a fourth parameter would require data of sufficient precision to guarantee a reliable value for the fourth ~ an inaccessible goal. Therederivative of the A - C ~ ’ curve, fore the explicit function must be a three-parameter equation; two, A0 and K A , are already present. The net result is that we must choose a model which can be described by one more parameter; dimensional arguments show that this parameter must be a distance R. The conductance function therefore will be A = A(c; Ao, KA, R ) and all system-specific parameters must be contained in the conductimetric parameters &, K A , and R. Based on the above arguments, the following model has been proposed:8 surround each ion by a sphere of diameter R, outside of which the solvent is describable by its macroscopic dielectric constant and viscosity. If the sphere for a given cation overlaps an anionic sphere, the two ions are counted as a nonconducting pair, and their electrostatic interaction with unpaired ions is approximated by zero. The spheres for the unpaired ions are by hypothesis immersed in a charged continuum, such that the integral of the space charge is equal and opposite to that of the central ion
where pi i s the Debye-Huckel charge density. The relaxation field and electrophoresis have been calculatedlo for this model and given as explicit functions of the variables t and 7,where
t = KR T
= P K /= ~ (t2/RkT)(~/2)
(9)
(10)
These equations are summarized in the Appendix. The Debye-Huckel parameter is calculated using the concentration of unpaired ions K~
= (4~/DkT)Z,n,yt,~
(11)
The activity coefficient is given by -In f = 7/(1
+ KR)
(12) The boundary conditions used in the theoretical development are nonspecific and valid for electrolytes in general: electroneutrality, continuity of all functions a t r = R and nonsingularity of functions for Kr = 0. Given a set of N conductance data points (cl, AI; j = 1, . . . N) and eq 5, 6, and 12, values for the three parameters
h o , K A , and R must be found by the method of successive approximations, because eq 5 is nonlinear in two of the parameters. The conductance can be expanded as a truncated Taylor’s series in AAo, AK, and AR around preliminary estimated values .io‘, KA‘, and R’ to give A, = A,(cl; Ao’, KA‘,R’)
+ (aB,/aAAo)AAo 4- (aA,/aAK) -4(aA,/aAR)AR
(13)
which is linear in the increments Ado, AK, and AR. This set of equations is then solved to find the values which minimize E2 = Zl[ii,(calcd)
- A,(ob~d)]~
(14)
In effect, the procedure is to locate the minimum in the four-space ( 2 ;Ao, K A , R ) , as illustrated in Figure 1 which shows two-dimensional cross sections of the surface for the data for a typical example (cesium bromide in the dioxanewater mixture with R = 74.75). The ordinate is the standard deviation (15) expressed as percentage of limiting conductance. Curve 1 shows the values of u corresponding to a sequence of values 0.8 5 P/R I1.3 and curve 2 the corresponding values of ‘20; minimum cr appears a t P/R = 1.12. Curve 3 shows values of u for the perpendicular cross section for the sequence 0.0 5 K A 5 1.0 and curve 4 the corresponding values of Ao; here, minimum cr comes a t K A = 0.75. Projecting up from the minima to curves 2 and 4 gives the value A0 = 142.84 corresponding to the minimizing values of K A and PIR; minimum u is 0.004%. The dashed line in the lower part of the figure is drawn a t u = 0.010%; it shows that values of /3/R between 1.05 and 1.20 and values of K A between 0.70 and 0.80 would still give a pretty good fit to the data, but the fit with the actual minimizing values is, of course, much better. The significant feature of curves 1 and 3 is the sharpness of the minima. This pattern is common to all the systems with solvents in the higher range of dielectric constant. A computer program14 designed to determine the three parameters simultaneously usually converges within three or four cycles (Le., reaches a value of h o which is changed by less than 0.01% by one more cycle of approximations) for such systems. However as the dielectric constant decreases, the minima lose their sharpness, and the computer futilely hunts through the shallow trough in the four-space without finding the lowest point. For these systems, a two-parameter program is used: the values of A0 and K A are determined which will minimize u for a sequence o f values of /3/R Examples of this type of calculation are shown in Figure 2 for eight of the cesium bromide-dioxane-water systems which are identified in the figure by rounded values of the dielectric constants. The vertical scale (0.0 Iu 5 0.10) is the same for the eight curves; the horizontal scale is compressed to one half for the mixtures at lower dielectric constants. Down to a dielectric constant of about 20, the rninimizing value of PIR can be located without ambiguity, but it will be noted that the minima become steadily more shallow as D decreases from 75 to 20, in the sense that a wider range oE values of PIR would fit the data within an acceptable precision. In other words, for the lower range of dielectric constants, the goodness of fit becomes quite insensitive to the value of R. The explanation lies in the form of the mass action eq 6 and in the fact that, at lower dielectric constants, the change of A with c due to ion pairing dorniThe Journal of Physical Chemistry, Vol. 79, NO. 15, 1975
1608
Charles F. Mattina and Raymond M. Fuoss (Here a is the contact distance.) The model has two advantages over previous models. The first is a practical one: removing the paired ions from the total ionic population guarantees that the mutual Coulomb energy t,tJIDrLJof all pairs of unpaired ions will be small compared to the thermal energy kT, with the consequence that the exponential Boltzmann factor in the charge density pI = Z n r ~exp(-tL$llkT) 2
.
0.9 /.O ,8/R 1.2 l.3 Figure 1. Determination of conductance parameters for mide-dioxane-water at D = 74.75.
08
r
cesium bro-
3*
r-
L LL r a / k 2
3
Figure 2. Variation of CT with PIR ratio for cesium bromide-dioxanewater at D = 74.75, 59.91, 49.58, 41.74, 30.48, 20.11, 13.83, and 9.36.
nates, the long-range AX and ALL,terms becoming relatively insignificant. The product f r appears in (6); if R is increased, more solute is counted as pairs, and y decreases. However more ions paired means fewer free ions and therefore f increases, and in such a way that the product f y remains nearly constant. Of course the values of R and K A are coupled in the sense that a change in R a t (nearly) constant CT around the minimum produces changes in KA. The coupling, however, is not tight a t lower dielectric constants: for example, a t D = 13.83, and P/R varies by a factor of 2 from 1.5 to 3.0, K A decreases by only about 20%,from 1340 to 1090; a t D = 9.36, K A decreases from 13,500 to 12,400 for the same twofold change in PIR. The ion pairs of the present model include both contact pairs7 and solvent separated pairs, because they are defined as those two-ion configurations in which the centerto-center distance r lies in the range a < r I R, subject t o the condition8J1 that the pairing is statistically unique. The Journal of Physical Chemistry, Voi. 79, No. 15, 1975
(16)
may be approximated by the first three terms of its series expansion, thereby enormously simplifying the mathematical treatment of the long-range interactions. The second is the implicit inclusion in the model of the particulate structure of real solutions as contrasted to the long-recognized artificiality of the primitive model which represents the ions as rigid charged spheres moving in a continuum. For the primitive model, all interionic distances rv > a are permissable. In real solutions, ions are either in contact, or separated by one or more solvent molecules; that is, interionic distances clustering around ( a md,) are much more probable than intermediate distances. (Here, m = 0 , 1, 2, . . . and d , is the diameter of a sphere which contains a solvent molecule plus the corresponding free volume.) From this point of view, R may be considered as the average radius of spheres surrounding the cations of the pairs and containing both the partner anions and the solvent in the immediate vicinity. In Figure 3, the values of R for the three salts are plotted against the Bjerrum distance p/2, which is inversely proportional to dielectric constant (0= 560.4 X 10-8/D2b cm). The parameter R appears to be a characteristic of the solvent rather than of the individual salts because the differences between points for different salts at the same solvent composition is no greater than the scatter of the points for a given salt. The averaging curve extrapolates to about 4 A a t PI2 = 0; this would correspond to the pairing radius in a hypothetical solvent of infinite dielectric constant where, due to chance collisions, the probability of finding pairs is not zero. As the dielectric constant decreases, the probability is enhanced by electrostatic attraction, corresponding to the increasing range of Coulomb forces; as expected, R is seen to increase. I t becomes equal to the Bjerrum radius around a dielectric constant of 20, and then increases a t a lower rate. I t should be mentioned that, with this model, association is possible a t all dielectric constants, is contrast to the original Bjerrum model which led to the unrealistic requirement that association abruptly cease when the parameter b = p/a is less than two. Unlike the R parameter, which (for these examples a t least) depends primarily on the solvent, the association constants are salt dependent, as shown in Figure 4. At a given dielectric constant, lithium iodide has a lower association constant than the chloride, which in turn i s somewhat less associated than cesium bromide. This is a reasonable sequence, if we assume that the lithium ion is hydrated and that the association constant contains the factor exp(e2/ a D h T ) or its asymptotic equivalent. The shape of the log K A - D - ~ curve suggests that the model may be approximated by one of two limiting descriptions, one valid for the range of higher dielectric constants and the other for lower. Debye12 and Eigen13 calculated theoretical association constants for a model closely similar to the present one: cations are surrounded by spheres of volume Au and pair formation and dissociation is treated as the diffusion of anions into and out of the volumes Au. If we identify Au with 4rR31 3, Eigen's equation becomes
+
1609
Conductance of the Alkali Halldes
80 40
20
J5
R 3 ) plots become nonlinear. In this range, the dominant
D IO
term in the probability function which describes ion pairing is the exponential exp(@/r).If other terms are neglected, the association constant is
30
KA = (4nN/1000)
-
0
0 l o p/2 2 0 30 Dependence of pairing radius R on solvent composition: upper abcissa scale, dielectric constant; lower abcissa scale, @/2, proportional to reciprocal dielectric constant; dashed line, R = Bjerrum radius; 0, CsBr; 9,LiCI; 6 ,Lil. Flgure 3.
1"
r 2 exp(@/r)d r
(19)
Using the values of K A and R derived from the data for cesium bromide, values of the lower limit in the above integral were calculated. The results are shown by the curve a t DI 20, the the lower right in Figure 4. Over the range 10 I values of a lie in the range 4.52 5 a 5 4.75, which is reasonably constant. For higher dielectric constants, unrealistically large values of a are found; not unexpected behavior, because the integrand in (19) is no longer a good approximation to the probability function appropriate for ion pairing in such solvents. For any dielectric constant, the total pair population consists of statistically paired ions (solvent separated pairs) and contact pairs; at higher dielectric constants, the former species is a significant fraction of the total, while a t lower dielectric constants, the latter predominates. Acknowledgment. This work was supported by Grant No. 14-01-0001-1308 from the Office Water Research and Technology.
Appendix The conductance equationlo used to obtain the parameters given in Tables IV and V is A(c) = y[Ao(l
+ RX) + HY]
(All
where the relaxation field is given by
RX = -@[exp(l
- q)t - 1]/3R(1 + t ) + ( ~ ~ / 3 Inf i t~ +) h2(FB2) + ( ~ @ T ~ / R ) ( F(A2) B~)
and the hydrodynamic terms by
+ + (7/2fi2)In t + H1 + H2]
HY = - @oc1/2y1/2[(1/(1 t )
(-43)
4
10 Figure 4. Dependence of parameters on dielectric constant: upper central curves, log KA; lower curve, A for cesicurves, log (&Id); um bromide; 0, CsBr; 9,LiCI; 6 ,Lil. KA = (4nNR3/3000) exp(@/a)
/I = t2/DkT
(A41
q = 2lI2 = 0.70711
(A51
(17)
At the upper left in Figure 4 are shown plots of log ( K A / R ~ ) against D-l. If the data conformed to (17), the plots would be linear, with a common intercept of log (4nN/3000) at D-l = 0, at the point indicated by the arrow on the vertical scale of Figure 4. The plots are linear, and give a values of 6-7 A; they have, however, different intercepts. This discrepancy can be covered by an ad hoc hypothesis, of course. The factor exp(P/a) in the Eigen equation describes rigid charged spheres in a continuum, for which the potential energy a t contact equals @/a.If the Eigen equation is rewritten
K A = (4rNR3/30O0) exp[p/a - u(a)/kT]
The symbols are defined as follows:
(18)
where u ( a ) is the potential of all the short-range forces, real and virtual, other than coulombic, which are involved in ion pair formation, the desired multiplicative factor can be included. This approach can account for the behavior a t high dielectric constants; in the lower range, the log (KA/
1/11 = e t / ( l
+t)
(A8)
/Io = 82.5/?(DT)lJ2
(A91
T h e terms given explicitly in (A2) and (A3) are the leading square root and c log c terms; the higher terms are summarized by the following interpolation formulas: FB2 = 0.0755 - 0.1089t
+ 0.0705t2 - 0.0569t3 0
