Dielectric Relaxation of LiCIO, and Bu,NBr Solutions
The Journal of Physical Chemistry, Vol, 83, No. 18, 7979 2419
Dielectric Relaxation of Lithium Perchlorate and Tetrabutylammonium Bromide Solutions. A Model of Ion Pairs Hubert Cachet, Alain Cyrot, Mohamed Feklr, and Jean-Claude Lestrade” Groupe de Recherche No. 4 du CNRS “Physique des Liquides et Electrochimie”, associ6 B I’Universit6 Pierre et Marie Curie, 4 place Jussieu, 75230 Paris, Cedex 05, France (Received February 5, 1979)
The dielectric relaxation spectrum of an electrolyte solution originates in the reorientation of the solvent molecules, on one hand, and in the motion of the ions, on the other. In this paper, the latter process is discussed under the assumption that it can be described as the rotation of polar ion pairs. Hunt and Powles’ extension of Glarum’s model applies to this case if we consider that the rotation is perturbed by random collisions with diffusing defects which are the ion pairs themselves; nevertheless, it is shown that the model has to be modified in order to take into account the finite size of the ion pairs. The model reproduces the experimental data with no more parameters than the usual empirical equations such as Cole-Davidson’s. Among these parameters, the diameter of the equivalent rotation sphere yields information on the closest distance of approach of the anion and the cation forming the ion pair. This distance can also be deduced from an analysis of the conductivity in a wide concentration range (typically 10-4-1 M) which we propose as an extension of Fuoss and Kraus’ model of triple ions and other high order aggregrate formation. Previously published relaxation data, on lithium perchlorate solutions, are reexamined in the light of the above model of rotating ion pairs, along with new data on tetrabutylammonium bromide solutions. Conductivity measurements on both types of solutions are also reported. The characteristic distances deduced from the two sets of data (dielectricrelaxation and conductivity) are shown to be consistent with one another. They are also consistent with the dipole moments of the ion pairs determined from the same experiments and/or given in the literature.
Introduction In previous papers, we have reported complex permittivity measurements on electrolyte solutions of lithium perchlorate in various polar solvents: ethyl acetate,l tetrahydrofuran (THF),2,3and T H F benzene mixture^.^$^ We proposed an interpretation of the data based on a translational diffusion m ~ d e l , ~ions , ~ moving ,~ according to a Brownian linear motion interrupted by random collisions. Soon after, Farber and Petrucci,? for the THF solutions, proposed another model based on the existence of rotating polar ion pairs, associated with Glarum’s theory.8 The rotation is perturbed by defects which diffuse in a one-dimensional system; in the present case, the defects would be the ion pairs themselves. We have discussed e l ~ e w h e r ethe ~ , ~theoretical basis of these models which are both “dielectrics”, in the sense that none of them is able to predict the low frequency conductivity which has to be deduced from another theory and both rely on a postulate about the state of the solute. Interest in the translational diffusion model is to demonstrate that the dielectric relaxation data can be quantitatively reproduced without assuming the existence of any species with a permanent dipole moment; as a consequence, the evidence for dielectric relaxation cannot be considered as a proof of the existence of such species. However, Farber and Petrucci’s model and ours have important qualitative features in common; they describe a collective motion of the ions, where the term “collective” does not simply refer to an ion plus a cation since ion pairs interact. They also both incorporate a diffusion process. Since experiment cannot decide which model, in fact which description of the state of the solute, is better, it seemed advisable to reexamine all the data about LiC104 solutions with the assumption of ion pairing, and more generally of aggregating into definite species of different orders. In order to discuss the results in a quantitative way, we have found it necessary to give the model a formulation for a three-
+
0022-3654/79/2O83-2419$01 .OO/O
dimensional system, starting from Hunt and Powles’ theorylO instead of Glarum’s. Conductivity measurements over a wide concentration range [10-4-1 MI have been performed on THF and ethyl acetate solutions and a discussion of the data is proposed in terms of ion pairs, triple ions, and quadrupole formation. Conductivity and dielectric relaxation data are also reported for tetrabutylammonium bromide (Bu,NBr) solutions in acetone (A), A carbon tetrachloride, THF + carbon tetrachloride, and T H F benzene mixtures.
+
+
Experimental Section Conductivity Measurements. For conductivities higher than 20 X lo4 ohm-l cm-l, the cell is made of two parallel plates (bright platinum) about 1 cm2 and 1 cm apart (Tacussel, Model CM03). The conductance of the cell is measured as a function of the angular frequency o,between lo3 and 2 X lo4 Hz, and extrapolated to infinite frequency as a linear function of u-’/~. This conductance is measured within 1% with a 1608-A General Radio impedance bridge. The cell is calibrated with KCl standard aqueous solutions. The overall accuracy, including calibration and extrapolation, is estimated to be about 2-4%. For conductivities lower than 20 X lo4 o h d cm-l, a stainless steel coaxial cell is used, identical with that used for complex permittivity measurements, with a slotted line between 0.15 and 3 GHz.” This cell is Calibrated as a condenser, using benzene as a standard. Its conductance is measured with a transformer ratio bridge (General Radio, 1615-A) between lo3 and lo5 Hz. The accuracy is about twice as bad as on the other device, but the error increases with decreasing concentration and it can reach 15% a t very low concentration where the conductivity of the solution is not higher than ten times the residual conductivity of the solvent. Complex Permittivity Measurements. The procedures have been described elsewhere.’l They are based on slotted 1979 American
Chemical Society
2420
The Journal of Physical Chemistry, Vol. 83, No. 18, I979
coaxial line measurements between 0.15 and 3 GHz, and use of waveguide interferometers at 9.4 and 35 GHz. Additional measurements at 8,26, and 106 MHz have been performed for very low conductivity solutions by using a twin-T bridge built in this laboratory after the method of Woods" with the same coaxial cell as in slotted line measurements.
Cachet et al. h 10-~
10'~
1
lo-'
CM
1
I
Conductivity The equivalent conductivity, A, of LiC104 solutions in T H F (at 30 O C ) and in ethyl acetate (at 25 "C) is plotted vs. salt concentration c in Figure la. In Figure l b the same quantity is plotted vs. l / t , for LiC104 solutions in T H F + benzene mixtures at 25 " C and constant salt concentration (0.6 M); E, is the static permittivity (dielectric constant) of the solution, i.e., the extrapolation to zero frequency of the real part t' of the overall measured complex permittivity E' - it". The data of Figure l b are taken from ref 5. The solid lines in Figure 1refer to values of A calculated with a model derived from the Fuoss-Kraus theory of triple ion formation13in a way analogous, though not identical, with that of Jagodzinski and Petrucci.14 The model runs as follows in the present case. The species assumed to be present in the solution are single ions A+, B-, ion pairs AB, triple ions ABA+, BAB-, and quadrupole ABBA. The concentrations, alc, a2c,age, and a4c, of each species, a t a given salt concentration c, are deduced from the following equilibria with constants K2, K3, and K 4 corresponding to the formation of the heavier species in each equilibrium:
A+ 4- BAB
~i AB
+ A+ + ABA+
+ B- is BABAB + AB ~1 ABBA AB
K2 K3 K3
K4 If we assume the activity coefficient to be unity for every species, a', a2,cy3, and cy4 are deduced from the following set of equations: a1 + a2 + 3a3 t 2a4 = 1 a Z / a l 2= K2c (1) 0(3/0(10!2 =K ~ c =K ~ c with the obvious conditions 0 I aiI 1. The crude assumption about the activity coefficients, especially for the conducting species, is partly compensated by the assumption that, whereas K 4 is considered as a constant, K z and K3 decrease with increasing concentration through a dependence upon the static permittivity of the solution
Flgure 1. Equivalent conductivity A (ohm-' cm2 M-')of LiC104 in ethyl acetate (A),and THF benzene mixtures tetrahydrofuran (THF, O), (O)? The solid lines results from the model of association given in the text. I n (a), is plotted vs. the salt concentration c ; in (b), c = 0.6 M and A is plotted vs. l/t, where t, is the static permittivity of the solution.
+
Strictly speaking, these values depend slightly on the way E' is extrapolated to zero frequency, so that they depend on the model chosen for the ionic relaxation. In fact, anticipating the results given below, the values found in this work are very close to those already calculated with other relaxation models and given in ref 1-6. A dependence of K 2 upon the dielectric constant of the solute had already been proposed by Cavell15who showed that this assumption could explain the increase of A with c without assuming the existence of triple ions; the underlying idea is that, as long as c is low enough, the salt associates; at higher concentration, however, the ion pairs become close enough for their stability to decrease, so that they dissociate. In order to calculate A from al and a3,we assume it to be inversely proportional to the viscosity, q , of the solution. If the conductivity of triple ions is arbitrarily chosen as 113 that of single ions, h is written as
-i = (Aoqo/r)(ai
E,:
K 2 = (4~N~a:/3000) exp(e2/a2E,hT - y2)
(2a)
K3 = ( ~ N ~ a , 3 / 1 0 0e0x)p ( e 2 / 2 u , ~ , k T- y2)
(2b)
where NA is Avogradro's number, k Boltzmann's constant, T the absolute temperature, e the charge of the electron, a2 the closest distance of approach of A+ and B-, and u3 a distance characteristic of the A+/AB and B-/AB interactions. These expressions are those of Jagodzinski and Petrucci in ref 14, apart from the meaning of which they take as the dielectric constant of the solvent, as usual. It does not make a great difference at concentrations lower than about 0.01 M (e, = 7.24 for T H F at 30 "C, e, = 6.0 for ethyl acetate at 25 "C). However, at the highest concentrations, t, reaches 17.95 for the T H F solution at 0.8 M, and 16.3 for the ethyl acetate solution at 1 M.
+ a3/3)
(3)
where hois the limiting equivalent conductivity of the salt in a solvent of viscosity qo. Choosing such a dependence of A upon 7 takes into account, a t least qualitatively, electrophoretic effects, but eq 3 ignores the influence of electrostatic interactions between ions upon their mobility. However, the concentration of the conducting species is far too high for any theory to apply. LiC10, Solutions. The A(c) data of T H F and ethyl acetate solutions have been fitted with a common set of parameters (a2,u3,K4)neglecting the differences between solvated ions in different solvents. For A. and qo, we took the values already chosen in ref 14 from Justice and Treiner's data,16 namely, ;lo = 156.1 ohm-' cm2 mol-' in T H F at 25 " C , qo = 0.470 cP. The fitting procedure consists of minimizing the sum of the squares of the
The Journal of Physical Chemistry, Vol. 83, No. 18, 1979 2421
Dielectric Relaxation of LiCIO, and Bu,NBr Solutions
differences between calculated and measured values of A, properly weighted by the estimated accuracy of measurement; the minimization is performed by the strategy of the simplex given by Nelder and Mead,17 which does not necessitate the calculation of partial derivatives. The values found are u2 = 3.86 X
cm u3 = 5.00 X K4 = 16.4 M-'
cm
I
I
(4)
If the solutions in T H F and in ethyl acetate are considered separately the best fit values of u2 and u3 differ from those just given by 2% for u2 and 15% for u3. The values of K4 are respectively 35 and 3.4 M-l, indicating that this parameter is not well defined. However, calculations with K4 = 0 yield a much poorer fit of the data; the higher concentration parts of the curves are well reproduced at the expense of the lower concentration parts which are not. For the best fit values (eq 4), as shown in Figure la, the discrepancy is outside error limits but the shape of the curves, the position and value of their minima, and the order of magnitude of the A's are correct for both T H F and ethyl acetate solutions. The same set of parameters has also been used to try and predict the conductivity of LiC104 solutions in T H F benzene mixtures at constant salt concentration. The corresponding A( l / t J curve is drawn as a solid line in Figure Ib. A undergoes a 100-fold decrease when changing the solvent from pure T H F to the less polar mixture. The predicted change is this order of magnitude, and the predicted A's, though too low, are never less than four times the experimental values. The three parameters of the model can thus be given a unique set of values to reproduce the data in a wide range of concentration ( 10-4-1 M) and dielectric constants; that of the solvent varies from 2.4 to 7.4,5 that of the solution, more relevant in this model, ranges from about 5 to 18. In order to compare these results with those of Jagodzniski and Petrucci, we can calculate K2 and K 3 with es = 7.39, the dielectric constant of THF at 25 "C. One finds K2 = 4.82 X lo7 M-' and K 3 = 147 M-' while the values given in ref 14 are K , = (4.84 f 0.19) X lo7 M-l and K 3 = 153 f 3 M-l. It may be considered as a fair agreement since the values of ref 14 were deduced from A(c) curves at low concentration (c < 0.1 M), while in the present work, the h(c) curves are fitted up to 1 M. The value of u2, in ref 14, is not deduced from K2 at 25 "C (it would yield a value close to 3.86 X lo-@cm) but from the slope of the cm, log K , vs. 1/T curve; the value thus found, 1.8 X is surprisingly low, as noticed by the authors, when compared with the Stokes radius of Li+, which they give cm. The former is too low for Li' to be as 4.9 X considered as solvated, while the present value, in eq 4, is rather in favor of Li+ remaining solvated, even when involved in an ion pair. In the discussion of the dielectric data, we shall add further comments to this point. In order to realize what is the state of the solute, we show as an example, in Figure 2, the variations of aL,..., a4with c for the T H F solutions. Ion pairs and quadrupoles dominate a t c = M and above, and the salt can be considered as associated into neutral species to at least 95%. Association of ion pairs into quadrupoles is an increasing function of concentration. In all cases, single ions predominate among the charged species and, though a3 continuously increases, the large increase of A in the range 0.1-1 M (that of interest for dielectric relaxation measurements) is mainly due to the increase in q. Bu4NBr Solutions. In Table I, conductivity measurements are reported for Bu4NBr solutions in acetone and various polar mixtures whose composition is also given in
+
I
=, IO-^
10-2
10-'
lo-'
1
Figure 2. State of the solute vs. salt concentration for THF solutions of LiCIO,; a,, a*,as, and cy4 are the fractions of solute in the form of single ions, ion pairs, triple ions, and quadrupoles.
the table. The salt concentration is the same (0.3 M) for all the solutions. The dielectric constants of the solvents, E,, are given in column 2, along with the viscosity, 1, of the solutions in column 4. These quantities are directly measured with a capacitance bridge operated at lo3 and lo4 Hz for ,t (1615-A General Radio bridge) and with a calibrated capillary viscosimeter for 1. The dielectric constant, es, of the solution, given in column 3, is obtained by extrapolation to zero frequency of the complex permittivity, as discussed in the next section. We anticipate these results in order to give in the table all the elements necessary to operate the model of ionic association just discussed for LiC10, solutions. Conductivity measurements have also been performed in two mixtures of acetone and carbon tetrachloride as a function of concentration. The A values so obtained are plotted in Figure 3, along with the measurements in pure acetone by Cavellls who gives, in the same paper, complex permittivity data, with values for the dielectric constant of the solutions. These values are lacking for the other data of Figure 3. We admitted a linear variation of 17 as a function of the concentration in the range 0-0.3 M, and a quadratic variation of cs: Es
= ,t
+ b1c + b*C2
The limiting slope, bl, can be deduced from the Frohlich-Kirkwood formulalg written in the present case as
NAC
9000tokT
where to is the dielectric constant of vacuum and E, the high-frequency limit of the complex permittivity of the solution. The latter has been taken as the same as that of the solvent and calculated under the assumption of
2422
The Journal of Physical Chemistry, Vol. 83, No. 18, 1979
Cachet et al.
TABLE I : Conductivity Data for 0.3 M Bu,NBr Solutions in Polar Solvents
solvent
em
acetone acetone ( 9 M! t CC1, acetone (6 M ) + CC1, acetone (4.5 M ) + CCl, acetone ( 3 M ) t CCI, acetone (1.5 M ) t CC1, THF (6.15 M ) + CC1, THF ( 3 M ) + CCI, THF (6.15 M ) + benzene THF ( 3 M ) t benzene
(solvent)
eS
20.7 13.5 9.18 7.30 5.38 3.71 4.70 3.53 4.65 3.50
(soln)
Il,
22.6 17.9 13.3 11.0 8.57 5.42 7.00 4.71 7.11 4.96
cp
0.42 0.57 0.77 0.92 1.18 1.85 1.49 2.63 1.00 1.24
a2 = 5.13 X
A (exp?,
A (calcd ),
ohmcm2 M-’
ohm-’ cm2 M-’
24.7 11.3 4.50 2.50 1.13 0.29 0.38
27.1 11.2 3.22 1.37 0.51 0.29 0.32 0.23 0.49 0.46
0.57 0.27
cm
u3 = 4.69 X
cm (7)
K4 = 7.52 M-’
In pure acetone, this value of u3 is higher than the value u3,,, which minimizes K3 as a function of u3 in eq 2. In order to keep a physical meaning for this equation, we have limited its validity to u3 C u3,, and, for higher values, taken K3 as zero. In order not to introduce discontinuities in the K3(u3)curve, we have chosen for K3the following arbitrary function in the range ~ ~ ~ - 1 . 5 ~ ~ ~ :
K3 = K3, exp[-25(u
-
with
Ksm= ( ~ N ~ u ~ , ~ / l Oexp(3/2) 00) and 0 01
0.1
u3m = e2/3e,kT
1
Flgure 3. Equivalent conductivity A of Bu,NBr in acetone (e,ref 18) and acetone (0,6 M, and A, 4.5 M) carbon tetrachloride mixtures. Solid lines are from the model of association given in the text.
+
additive polarizabilities so that, for the mixture of two components with concentrations n, and n2, em can be deduced from (E,
- l ) / k + 2) =
[nl(e-l -
+ 2) + n2(cm2+ 2)l/(ni + nz)
1)/(tm1
l)/(cmz
(6)
The value of cml for carbon tetrachloride is its dielectric constant (2.228 at 20 OCZo).For acetone a value of 2.065 results from the assumption that the atomic polarization is 20% of the electronic polarization, the latter calculated from the optical refractive index. In any case, the limiting slope, bl, is not very sensitive to the choice of E,. It principally depends on gp2 where g is the Kirkwood factor which characterizes the correlation between the orientation of neighboring species bearing a dipole moment p , In the present case the first term on the right-hand side of eq 5 refers to the solvent and the second term to the solute so that p is the dipole moment of noninteracting Bu4NBr ion pairs and g = 1 in the limit of vanishing concentration. Taking p = 12.2 D as given by Bauge and Smith21 from the data of Geddes and Kraus,22we are led to bl = 31 M-l for the solutions in the more polar mixture of Figure 3, bl = 21.3 M-l for the solutions in the less polar mixture. The second coefficient, b2, follows from the value of E, a t c = 0.3 M, given in Table I. We are now in a position to apply the model of ionic association just described for LiC104 solutions and calculate A for every system of Table I and Figure 3. We have fitted the same set of parameters u2,u3,and K4 to all these data. The calculated values of A are given in the last column of Table I and are plotted as h ( c ) curves (full line) in Figure 3. The values of the parameters are
The parameters (eq 7) have been determined with this additional assumption about K3. As it can be seen from Table I and Figure 3, the experimental data are reproduced in the same sense as for LiClO., solutions, i.e., the order of magnitude of h and the shape of the h ( c ) curves are correct. The very value of u3, higher than for the LiClO, ion pair, will also be discussed after we have additional information from dielectric relaxation data. Dielectric Relaxation The data to be discussed have been partly published el~ewhere.l-~In the present work, complex permittivity measurements have been performed on the Bu4NBr solutions defined in Table I. As an example of the accuracy of these measurements, the errors on the real and imaginary parts o f t = 4 - i t f f are given in Table I1 for one of those systems, along with the values of ef and E” at each frequency. These errors increase with increasing E~ and e ” / e f . Because of the conductivity this ratio is too high, in the three more polar solvents, for t to be measured at frequencies below 0.14 GHz. The complex permittivity data for the other solutions defined in Table I are given in Table 111. Complex permittivity measurements have also been performed on some mixed solvents. They are reported in ref 5 for the THF-benzene mixtures. In Figure 4 ef and df are plotted vs. frequency for two acetone-CC1, mixtures. In the frequency range of the present measurements, only the beginning of the relaxation spectrum can be observed, as in pure acetone.23 For that reason, it is possible to interpolate, without much error, the values of t’ and 8’ to have these values at the frequencies of Tables I1 and I11 for any mixture. As in previous works: we have analyzed the dielectric spectra of the solutions under the assumption that 4 w ) ,
The Journal of Physical Chemistry, Vol. 83,
Dielectric Relaxation of LiCIO, and Bu,NBr Solutions
No. 18, 1979 2423
TABLE 11: Complex Permittivity and Errors for a Solution of Bu,NBr (0.3 M) in a Mixture Made of Acetone (4.5 M ) and Carbon Tetrachloride f , GHz Et A€' E" A f" 11.4 10.9 10.5 10.1 9.62 9.60 9.04 8.44 8.01 7.57 6.66 5.07
0.0266 0.1062 0.140 0.265 0.375 0.385 0.625 1.0606 1.6667 2.825 9.335 34.720
0.6 0.25 0.11 0.10 0.10 0.10 0.09 0.08 0.08 0.08 0.07 0.05
0.30 0.08 0.07 0.04 0.05 0.03 0.03 0.02 0.02 0.02 0.03 0.06
49.5 13.2 10.7 6.39 5.06 5.00 3.62 2.75 2.22 1.82 1.67 2.15
the overall measured complex pera,,tivity, can be wi .,ten as €(a)= ern(w) AE(u) + ~ / i w t o (8)
01
05
02
2
1
5
10
20
Figure 4. Real (E') and imaginary (E") parts of the complex permittivity 6 M, and A, 3 M) ivs. frequency for two mixtures of acetone (0, carbon tetrachloride.
+
where €,(a) is the contribution of the solvent molecules, and Ae(w) the contribution of the ions apart from the losses arising from a frequency independent conductivity u. We insist that, for the moment, splitting the ionic contribution into two parts does not imply any assumption upon the microscopic processes, and can always be done since, whatever its definition, the conductivity of an electrolyte solution always has a limit u when the frequency goes to zero, so that the last term of eq 8 can always be subtracted from the total ionic contribution. For a model of ionic relaxation of the "dielectric" typeg At(w) can be written as (9) A~(w= ) (€5 - ern)&[-d(t)I
&[-4(t)] = -Lm4(t) exp(-iwt) dt
"dielectric" model. For the LiC104 solutions already studied, it had been shown6 that no information could be drawn from the data about E,(w), apart from a change in E, which decreased with increasing salt concentration. In particular, no change in E- or the relaxation time of the solvent molecules can be observed because this time is too short when compared with the highest frequency used, and because the molecular relaxation is obscured by the ionic relaxation. However, since the second one is slower than the former, the decrease of em can be quantitatively determined under the assumption that no other change in the relaxation of the solvent molecules does happen. The same is true for the solutions of Bu4NBr studied in the present work. This assumption about E,(@) can be quantitatively used if we write
is the low-frequency limit of E,(w), E, the low-frequency , M(t) limit of E(@), and $(t)= (M(0) M ( t ) ) / ( W ( O ) )where is the electrical moment of a sphere embedded in its own medium. The time correlation function $(t)goes to zero when t goes to infinity by definition of what we call a
where emO(w) is the complex permittivity of pure solvent with a zero frequency limit E,., The ionic relaxation is the point of interest in the present work, and we shall discuss it under the assumption
with
,E
TABLE 111: Complex Permittivity of Bu,NBr (0.3M) in Acetone and Mixed Solvents acetone acetone acetone acetone (9 M ) + CC1, ( 6 M ) + CCI, f, GHz 0.140 0.265 0.375 0.385 0.625
a
E'
E'
El'
22.9 22.6 22.8 22.6
51.8 38.8 36.9 23.4
18.1 17.7 17.7 17.1
E'
E"
45.7 24.7 17.9 17.6 11.9
13.1 12.9 12.4 12.4 11.6
acetone
acetone
( 3 M)t CCI, -
(1.5 M ) + CC1,
f, GHz
E'
0.0082 0.0266 0.1062 0.140 0.265 0.375 0.385 0.625 1.0607 1.6667 2.825 9.335 34.72
8.76 8.49 7.98 7.64 7.26 6.92 6.97 6.56 6.18 5.93 5.66 5.05 4.10
Reference 35.
E"
69.69 22.0 6.28 5.13 3.24 2.51 2.53 1.91 1.52 1.26 1.07 1.03 1.31
E'
5.57 5.45 5.07 4.92 4.72 4.54 4.58 4.40 4.25 4.12 3.97 3.66 3.24
E"
18.5 5.92 1.84 1.58 1.07 0.85 0.88 0.70 0.60 0.54 0.49 0.49 0.62
f, GHz
E"
18.6 10.7 8.09 8.10 5.76
1.0606 1.6667 2.825 9.336 34.72
THF (6.15 M ) E'
7.19 7.01 6.46 6.28 5.98 5.86 5.81 5.64 5.42 5.28 5.12 4.79 4.19
+
CCl, E"
24.9 8.03 2.53 2.18 1.45 1.20 1.17 0.94 0.77 0.67 0.59 0.65 1.12
E"
E'
22.1 21.15 19.8 17.3
15.3 11.2 8.30 5.47
THF ( 3 M ) + CCl, f'
4.82 4.69 4.39 4.31 4.19 4.14 4.12 4.03 3.92 3.84 3.75 3.55 3.24
E"
6.12 2.11 0.78 0.68 0.49 0.44 0.43 0.38 0.35 0.32 0.31 0.35 0.56
aceton e + CCl,
acetone
( 9 M ) t CC1,
( 6 M)
--
E'
E)'
f'
16.1 15.1 14.2 12.4 9.02
8.41 6.40 4.91 3.70 5.20
10.9 10.2 9.61 8.42 6.15
E"
4.28 3.42 2.68 2.31 3.08
THF
THF
(6.15 M ) t C6H,
( 3 M ) t C6H6
E'
7.33 7.14 6.54 6.30 6.02 5.87 5.83 5.65 5.43 5.28 5.10 4.76 4.20
E"
37.8 11.9 3.57 2.99 1.90 1.49 1.49 1.14 0.90 0.76 0.66 0.66 1.10
E'
5.09 4.97 4.61 4.49 4.31 4.27 4.22 4.14 4.01 3.92 3.82 3.58 3.27
E"
17.7 5.61 1.75 1.48 0.96 0.79 0.76 0.61 0.49 0.43 0.38 0.39 0.55
2424
The Journal of Physical Chemistry, Vol. 83, No. 18, 1979
that & ( w ) originates in the reorientation of ion pairs. In most ~ a s e s , l -the ~ data can be reproduced by the ColeDavidsonZ4expression
A€(@) =
(E,
- Cm)/(l
+
(11)
iW70)’
L
P(1) = 4rp12 exp --pl
3J
where p is the number of defects per unit volume. P(I) dl is the probability that the nearest defect is at a distance between 1 and 1 dl. It can be shown that this applies in the case where the probability of finding a defect at any distance from a dipole is uniform. In other words, the radial distribution function, g(r), defining this probability has to be independent of r , and equal to unity. Another consequence of Hunt and Powles’ assumption about P(1) is that a defect can be situated at a distance of a dipole less than R,, so that, at time zero, a fraction of the dipoles reorient instantly. In the present case we assume, following Farber and P e t r ~ c c ithat , ~ the defects are the ion pairs themselves. It seems unrealistic to assume these entities are dimensionless, so that their closest distance of approach is finite, their radical distribution function g(r) is not uniform, and P(1)has to be recalculated. We propose the following estimation of P(I). Consider a volume V with N = pV entities and a volume u = (4n/3)13 1)and the 8 term decreases ( a > 0). In Figure 9 we have plotted log [l - P(8)]vs. B1/* calculated with eq A7 and also with eq A2. The two formulas lead to the same limiting slope; this shows that the short time behavior of +(t),in the model, is actually controlled by the nearest neighbor. The numerical values of [, in the examples of Figure 9, correspond' to systems such as a 0.6 M LiC104 solution in THF ([ = 0.02) and a 0.3 M Bu4NBr solution in the mixture acetone (3 M) + carbon tetrachloride ([ = 0.1). These values of [ also correspond to the examples of Figure 5. The long time behavior of +(t)is governed by the rotational term -6(R/Ro)%' for both treatments, but a -6[d term has to be added, coming from log [ l - P(8)l in the case of Bordewijk's solution. Here is the contribution of the nonnearest neighbors which, of course, does not depend on the details of g ( l ) . Neglecting this extra term can be considered as an error in the estimation of the parameter Ro, which should be corrected by a factor of about (Ro/ R)2[/2. This correction amounts to about 3% for the less dense system ([ = 0.02, R,/Ro = 0.6), 8% for the other one (6 = 0.1, R,/Ro = 0.8). These deviations are well inside error limits in the evaluation of Ro from experiment.
\le
0.3
83,No. 18, 1979 2429
e8 -
References and Notes Flgure 9. Time correlation functions in Hunt and Powles' (HP) treatment and Bordewijk's (8)treatment. Top (a and b): in the absence of rotation, short time behavior. Bottom (c and d): overall correlation function, long time behavior. Left (a and c): [ = 0.02, R,/Ro = 0.6 (for instance 0.6 M LiCIO, in THF). R i h t (b and d): = 0.1, RJRo = 0.8 (for instance 0.3 M Bu4NBr in acetone (3 M) CCI,).
+
inside V, and multiply the N possibilities 1 - P,(t);with the assumption that they are independent from one another, and in the limit of N and V going to infinity a t constant p = N/V, one finds 1 - P ( t ) = exp[-NPl(t)]
(-44)
In terms of 8 and of the packing fraction 5, it can be written as log [l - P(8)] = -24EJmx g ( x ) erfc(
5) dx
(A5)
The integral is easily evaluated if we admit for g ( x ) the approximation x g(x) = x + Ag exp[-cu(x - l ) ] (A61 with Ag = g(R) - 1 and a chosen in such a way that the first derivative of g ( x ) is the Percus-Yevik solution of hard sphere fluid at contact (x = 1). Then log [ l - P(8)l = -24[[(
!)"' + 7r
+
"[ a
1-
exp(
$) erfc (a.)1 d 1
(A7)
The first two terms are Bordewijk's solution (for Ag = 0). The third term, which vanishes when 8 goes to infinity, characterizes the nonuniform spatial distribution of defects. At short times, the expansion of log [l - P ( @ ]in terms of e1I2 reads
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