91 7
Electrical Conductance and Ultrasonic Relaxation for LiC104
Electrical Conductance and Ultrasonic Relaxation for Lithium Perchlorate in Tetrahydrofuran Paul Jagodzinski' and Sergio Petrucci* Department of Chemistry, Polytechnic lnstitute of New York. Brooklyn, New York 7 7201 (Received July 2, 7973; Revised Manuscript Received December 13, 1973)
Electrical conductance (10-5 to 1 M) a t 25, -15, and -30" and ultrasonic absorption (0.05-0.5 M ) in the frequency range 5-330 MHz a t 25, -15, and -30" have been measured for LiC104 in tetrahydrofuran (THF). The electrical conductance data reveal the presence of a minimum that can be interpreted and evaluat,ed by the Fuoss-Kraus theory of triple ion formation (up to -0.1 M ) . Numerical evaluation of ion pair and triple ion pair constants were made. The ultrasonic relaxation data are interpreted as due to the process of formation of triple ions, the barrier of activation energy for the process being comparable to the one for viscous flow.
Introduction Electrolytes solutions in media of low dielectric constants were investigated by audio-frequency electrical conductance methods in the 1930's by Fuoss and Kraus in a series of classic papers.a Later, Daviesa reexamined some of the same systems by dielectric relaxation techniques giving qualitative evidence of the molecular complexities of these systems through the appearance of distribution of relaxation processes. It was of interest to study the same system by classical and by modern kinetic tools, in order to determine the molecular dynamics involved. For this purpose, the system of LiC104 in THF has been investigated by both audio-frequency electrical conductance and by radio frequency ultrasonic absorption measurements. The results of each method will be discussed. Experimental Section Materials. Tetrahydrofuran (Matheson Coleman and Bell) was distilled under a nitrogen atmosphere over a 60% K-Na alloy in a 3-ft Vigreaux column. It was found that distillation over Na was ineffective, the melting point of Na being higher than the boiling point of the solvent. A 60% K-Na alloy is liquid a t the boiling temperature of the solvent and provides good surface contact between liquid and metal. The central third of the THF distillate was collected and stored under nitrogen. The density of the distilled T H F a t 25.0" was 0.8819 f 5 X g/cc; the literature ~ a l u e s range ~ 3 ~ from 0.880 to 0.883 g/cc. Anhydrous reagent grade LiC104 (Smith, Cleveland, Ohio) was dried to constant weight a t -190". It was stored in sealed bottles; contact with the atmosphere for weighing lasted only 30 sec or less. Redrying the material a t -190" for 1-2 days after being weighed gave no changes in weight. Further, LiC104.3HzO reagent grade (Smith, Cleveland, Ohio) was dehydrated a t 75" under 0.1 mm pressure. A 0.25 M solution of this material in THF gave conductance values within 0.2% of the values reported in Figure 1. Solutions for conductance work more concentrated than 50 X 10- M were prepared individually by weighing both salt and freshly distilled solvent. This was done to avoid possible changes of conductance due to aging or formation
of peroxides. Each value of the reported conductance a t c = 50 x lo4 M and higher was obtained from individual solutions. For the solutions of concentration lower than 5 X M, a "concentration" method was used. The stock solution was added by weight burets to the solvent in the conductance cell. A drybox filled with purified Nz was used in all the operations involving cell filling and solution transfer. Procedure and Instrumentation. The density of the solutions at 25" was required to convert molalities to molarities. At both -15 and -30" the difference between the density of the solutions and the pure solvent was within f0.2% of the values at 25.0'. A special pycnometer with a capillary stem and a bulb (acting as a reservoir to allow for the liquid expansion upon thermal reequilibration to room temperature) was used for the density measurements a t - 15 and -30". Each reported value of the density was obtained by averaging a double determination. The average error was g/cc using thermostats with f 2 X found to be f 5 X of precision and a Mettler H-6 balance. The capacity of the pycnometers was approximately 50 cc. The equipment for the conductance work has been described elsewhere.5 Conductance measurements were made a t three temperatures: 25.000 0.001, -15 f 0.01, and -30 f 0.01". At 25" the previously reported5 fluctuation of f0.005" were decreased by circulating water previously thermostated at 24.00 f 0.05" through a 40-ft spiral of 1-in. i.d. copper tubing (immersed in the oil bath close to the thermostat walls). The room temperature was maintained as 23 f 1"by a York %ton air conditioning unit. The conductance cells used had constants of 0.1142 6 X, and 0.2794 + 12 X, cm-1 with X , the total specific conductance of the solutions. For the solutions of concentration below 50 x M it was necessary to insert a 30-kfl shunt (GR510 Series) in parallel with the cell. The resistances of the solutions were measured a t 1000, 2000, and 5000 Hz. For the diluted solutions linear extrapolation to zero frequency was performed since parallel losses may be the main sources of error. At higher concentrations, extrapolation of the resistance to infinite frequency by a R us. l/f1I2 plot was made, where f is the frequency
+
The Journal of Physical Chemistry, Voi. 78, N o . 9. 7974
Paul Jagodzinski and Sergio Petrucci
918
in Hz. Electrode polarization becomes the increasingly important source of error with increasing electrolyte concentration. In all cases the differences between either Ro, Rm,or any of the R's were not larger than k0.170. Solutions for ultrasonic measurements were prepared by first weighing the salt and then diluting in volumetric flasks. Only fresh solutions were used. Duplicate measurements were always made with newly prepared solutions. Whenever differences larger than the sensitivity of the method were noticed, the results were rejected. The equipment and experimental details have been previously d e ~ c r i b e d . For ~ , ~ measurements above 200 MHz at 25" a new interferometric cell with two 1-in. delay lines was used. The moving crystal connected to the delay line traveled over a high-precision horizontal slide (Automatic Gages, Rochester, N. Y.) ensuring maintenance of parallelism within few seconds of an arc for travel of 1 in. A precision micrometer (Boeckeler Instruments, Tucson, Ariz.) allowed readings of these displacements to within i l X l o L 4in. Temperature was maintained at 25" by circulating thermostated water through a jacket surrounding the cell. The temperature was measured by a thermocouple probing directly into the solution during the measurements. Plots of attentuation in decibels us. distance increments a t the frequency of 330 MHz were perfectly linear and repetitive runs gave results for the absorption coefficient of sound within 12%. At the lowest frequencies, 9 and 5 MHz, it is important to avoid diffraction errors. Accordingly, the distance between the sending and receiving crystal was kept within the Fresnel zone. A calibrated Kai attenuator (Model 1/432C) that could be varied by fractions of 1 dB (down to 0.1 dB) was used. The attenuator was put in series with the cell and changes in the attenuator settings were compensated by changes in the distance between the crystals of the cell. In this way, the maximum distance between the two crystals could be kept within the limit 0.5R2/X, with R the radius of the sending crystal and X the wavelength. Further, plots of the attenuation (in dB) us. distance were linear for all the data reported in this work. Results and Discussion 1. Electrical Conductance. After the molalities of the stock solutions used for conductance work were determined in a non-linear least-squares program was fitted to the following data at 25": m = 0, 0.11464, 0.28773, 0.57898, and 1.17988 mol/kg of solvent and p = 0.88196, 0.89115, 0.90376, 0.92480, and 0.96690 g/cc. The program fits the data according to the relation p(g/cc) = 0.88206 0.0758rn - 0.00353m2, from which the molalities of the solutions used for the conductance work were converted to molarities c at 25". For the lower temperatures it was found that the ratio CDtsolution / p 25801ut10n = CDtsoivent /p25so~vent) within f0.29'0. A similar functional form of p on m was therefore assumed for the solutions at the lower temperatures. In Figure l A , the plots of log A us. log c are shown a t 25, -15, and -30". Although the data a t low concentrations ( e I9 X M ) approximate the requirement7 of slope = -0.50 (required by an electrolyte heavily associated to a neutral ion pair), they diverge positiuely from this slope at increasing concentrations.* Eventually the log h us. log c plot goes through a minimum, a sign of the existence of conducting triple ions. Also above 0.5 M at 25", where data are available, the log h us. log c plot
+
The Journal of Physical Chemistry. Vol. 78. N o . 9 , 1974
Figure 1. Log A VS. log c (A) THF at 25, -15, -30'.
and k'/' vs. c
(B)
for LiC104 in
shows a downward concavity possibly indicating the appearance of quadrupoles. The conductance data were treated as follows. At low concentrations (c I9 X M ) the Fuoss-Kraus method% for association to ion pairs has only been tentatively applied. Plots of F / A us. c Afi2/F are shown in Figure 2. These plots are approximately linear. Least-squares analysis should give 1 / A o as intercept and &/A$ as slope. It is apparent that the intercept gives the wrong value of ho. In fact at 25" it gives A 0 = 17.88 0 - l cm2 equiv-l whereas from the data of Szwarc, et al.,"' Xo(Li+) = 36.6 Q - lcm2 equiv- According to Justice and Treiner X0(c104-)is 119.5 giving AO(LiC104) of 156.1 Q - l cm2 equiv-l. (The value h0(C104-) = 119.5 was obtained as follows. Justice and TreinerlO determined AO(Bu4NC104) = 162.6 Q - l cm2 equiv-l in THF a t 25". Michelettill reports h0(Bu4NBPh4) = 82.97 and AO(iAm3BuNBPh4) = 79.66 3-l cm2 equiv-l a t 25" in THF. Taking XO(iAm3BuN) = XO(BPh4) one calculates h0(Bu4N) = 82.97 - 39.83 = 43.14 and hO(C104-) = 162.6 - 43.14 = 119.5 Q - l cm2 equiv- I.) Szwarc, et al,% worked in the concentration region from 10- 6 to 10- M therefore avoiding the complication of triple ions. They reported that when data a t higher concentrations were plotted according to the Fuoss-Kraus method a downward trend appeared. The failure t o describe the ionic equilibria in terms of ion pairs and free ions alone was also predicted theoretically by Fuoss7J2 for concentrations higher than co = 3.2 X 10-7D3. In our case, D = 7.39 at 25" which gives co = 1.3 X M , almost at the lowest limit of our concentrations. Tentative runs at concentrations below 0.4 X M were abandoned because the resistance was too high to be measured with accuracy. Correction of the solvent conductance would have implied a several per cent correction to the
919
Electrical Conductance and Ultrasonic Relaxation for LiCiOL TABLE I: Triple Ion Analysis of LE104 in THF t, OC
25 - 15 - 30 a
AO, cm2 equiv -1
K A X 10-7, M -1
KT,
156.1" (97.3) (78.6)
4 . 8 4 i0 . 1 9 1 . 5 0 =!= 0 . 1 2 0 . 9 3 i.0 . 1 0
153 rt 3 139 i 5 133 i.9
$2-1
Value calculated from literature data. See text.
'"1
M
-1
UT (eq 3) X 108, cm
aT (eq 4) X 108,cm
aT (es 5) X 108,cm
5 . 2 7 i0 . 0 2 5.14 i0.03 5 . 1 0 i0 . 0 7
3 . 6 3 =k 0 . 0 2 3.53 i0.02 3.50 f 0.04
4 . 6 2 =!= 0 . 0 2 4 . 4 7 i.0 . 0 3 4.44 f0.06
Values calculated from the Walden rule retaining A' = 156.1 at 25'.
FUOSS-KRAUS PLOT
I20
i
concentration of 0.1 M after which a strong positive deviation from linearity appears.l3 Linear least-squares analysis (solid line) of the linear portion of the plots give the slopes and intercepts from which Ka and KT are calculated as reported in Table I. Correlation of the KA's in the form of log KA US. 1/DT according to the Fuosslh function
70 -
t 0 60-
2 50'
-: 40 30 I
(CAf~/F)xlO'-
Figure 2.
Fuoss-Kraus plot: F / A vs. c A f , * / F
total conductance. In fact Szwarcs' association constant for LiBPh4 is ~ 1 . X2 IO4 ~ M - l , more than three orders of magnitude lower than the one for LiC104 as reported below. Therefore in THF, LiC104 is a weaker electrolyte than LiBPh4 and its electrical conductance is far lower a t c = 10- 4 to 10- 5 M . It should be noticed that using A 0 = 156.1 f2-I cm2 equiv-I a t 25", h o = 97.3 at -E", and A = 78.6 at -30" (from the Walden rule), the slopes of the Fuoss-Kraus plots yield a K,z of 4.57 X lo7 at 25", 1.53 X lo7 a t -E", and 0.98 X lo7 M - l a t -30", respectively. These figures are in surprisingly good agreement with the results given below and based on triple ion formation analysis. Therefore, it seems that the presence of triple ions rather than the slope of a Fuoss-Kraus plot affects the determination of90 (because of its functional form in the intercept). Analysis by the Fuoss-Kraus theory% of triple ion formation has been performed according to the relation C
where A,ro is the conductance of the two triple ions (iLTo = Xo(Li(C104)2-) + Xo(Li2(C104)+)),and KT is the formation constant of the triple ions
In the above a symmetrical approximation is assumed for the coordination of the triple ions.% Also ATO is assumed to be ATO = (1/3)A0. Plots of hc1I2 us. c are shown in Figure 1B. These plots are approximately linear up to the
gives a = l.& x 10-8 cm where a is the minimum approach distance between ions in the ion pair. Based on the Fuoss-Kraus ion pair analysis, one gets a = 1.95 x 10-8 cm. These figures are in reasonable accord with the sum of the crystallographic radius of Li+ rLi = 0.68 x 10-8 cm and the C1-0 distance, rclo = 1.65 X cm, namely, Z r = 2.33 X cm. Thus from the exponential part of eq 2 (see footnote 16a), one would conclude that some contact between ion pairs for LiC104 probably exists. The same conclusion was reached for LiBPh4 by Szwarc, et aL,a in comparing data for LiBPh4 with data for NaBPh4. However judging from the Stokes radius r+ = 4.9 x cm it would seem that Li+ is solvated in cm is in fair accord with THF, whereas r- = 1.5 X the rclo distance. Next the correlation of the triple ion association constants KT should be tried. There are two theories of association for triple ion formation. Both are based on the assumption of a continuum for the solvent and on ion-dipole interaction energy, but use different models. The first theory by Kraus and Fuoss of 1933% retains the Bjerrum model and conceives of a triple ion as an entity held together by electrostatic interactions. In particular the ion pair and the free ion are considered to be associated into a triple ion when the parameter bT = e2/ WDkT > 813. q is the distance of minimum approach between the ion and the center of the ion pair dipole of internuclear separation a. According to the Debye-Huckel model one would expect CQ = 1 . 5 ~ (Assuming . ion pairs a t distance a is consistent with the Bjerrum model a t low dielectric constants. It has been pointed out1* that the average distance for ion pairs ( r ) = a[l (l/b)] N a for large b, or, for small D.) According to the triple ion theory," it is not necessary that an ion and a dipole be in actual contact to be considered as a triple ion. In fact, triple ions were considered to exist% up to the minimum of their probability distribution function, namely, up to the distance R > e.The triple ion association constant according to the original 1933 theory% is given by the expression
+
where I ( ~ T is) a double integral which has been tabulated% for a wide range of values of b ~ . In 1959 Fuoss proposed7 a new theory for triple ion forThe Journal of Physical Chemistry, Vol. 78, No. 9 , 1974
920
Paul Jagodzinski and Sergio Petrucci 8.0,
L,CPO4
Triple ,on Formallon
i n THF
3(lsoAml.q
Log Ke,
YS
104/OT
e Fuoss - K r o u s
I
N NO,
in
H20-Dioxone, t:25'C A
4
/
43
--. 45
44 1041 DT
-
Figure 3. Log K A vs. 1 / D T for LiC104 in THF. The dashed line refers to the Fuoss-Kraus analysis shown in Figure 2. The solid line refers to the results of the triple ion analysis. mation based on the definition that only ions in direct contact with a dipole ion pair should be considered as triple ions. When an ion is in contact with an ion pair, the energy U becomes7 U = e2/2aD (for a 1:l electrolyte). The triple ion formation constant is given by the expression
-bT
(ar=91; 8/3
,
IO
20
30 I 102/Dl
40
-4
Figure 4. Log KT vs. l / D for i A m 4 N . N 0 3 in H2O-dioxane at 25". Solid sigmoid line is the Bjerrum-Fuoss-Kraus eq 3. Solid straight line is the Fuoss eq 4 . Dashed straight line is t h e calcu-
lated least-squares function. Arrow indicates the intercept calculated according to eq 5. where uz = r a 3 is the volume of the ion pair assumed to be a cylinder of radius a / 2 and height 4a. The factor e- 3 stands for exp( -3R/2) which correspond^,^ according to the authors to the entropy changes due to the loss of three degrees of translational freedom upon formation of a triple ion. Vibrational degrees of freedom were excluded by the model, because it was argued that if vibration occurs, the contact between the ions breaks.7 Apart from obvious misprints,7 in the assignment of the number of rotational degrees of freedom for both ion pairs and triple ions, we believe the factor should read r 3 I z in eq 4. If one accepts the statement7 that the net entropy change per mole is -3R/2 (3RT/2 for the entropy was clearly a misprint7) the factor to be introduced in the KT expression namely exp(AS/R) gives c 3 I z not as reported.7 We also believe that the same result for KT can be achieved unambiguously by modification of the model to include the vibrational degrees of freedom. Spectroscopic evidence15 for vibrational motion of ion pairs in THF has been documented in the 1 i t e r a t ~ r e . l ~ A linear diatomic molecule (ion pair) possesses three degrees of freedom for translation and 3n - 3 = 3 internal degrees of freedom. Of these, two are for rotation, and one for vibration. A linear triple ion will have 3n - 3 = 6 internal degrees of freedom. Of these, two are for rotation and four for vibration, specifically the symmetrical and asymmetrical stretching modes and the two degenerate bending modes. In terms of the corresponding average internal energy, this can be expressed by the relations for linear molecules. 3 13 RT (3n - 5)Rl' = -RT triple ion U = zRT 2 3 7 U = 2RT RT (3n - 5)RT = 5 R T ion pair
+ +
free ion
+
+
3
U = 5RT
The Journal of Physical Chemistry, Voi. 78. No. 9 , 7974
The change in internal energy upon formation of a triple ion from an ion pair and a free ion will be AU = 3RT/2, namely, a gain of 3RT/2 per mole of triple ion formed. The triple ion formation constant expression should then be multiplied by a factor exp(-AUIRT) = c 3 / 2 , since AUtot,, = AUeiectr + AUfree,,, = (e2/2aD) + 3RT/2 and KT a exp(-AUtotal/Ra3. In the above, the ions, ion pairs and triple ions are considered immersed in a continuum of permittivity 0 . 7 The solvent has been excluded from participating in the association process. Volume changes have also been neglected. This last approximation is equivalent to equalize the Helmholtz and Gibbs free energy changes AA = A G and to equate internal energy changes to enthalpy changes A U = AH. The expression for KT should then read
It is of interest to test eq 3-5 against experimental data. The only system where extensive data are available is tetraisoamylammonium nitrate in HzO-dioxane at 25".16 Figure 4 shows these datal6 displayed as log KT us. 104/D. The solid sigmoid line is the Fuoss-Kraus eq 3 (marked K F K ) showing the asymptotic limit to log KT --c --m for the value of D corresponding to bT = 8/3. The value of UT = 9 X cm has been retained from the original paper.% The dashed straight line is the function calculated by least-squares analysis assuming linearity between log KT and 1/D (eq 4). The intercept however is equal to 0.63 which is larger than the value predicted by eq 4, namely, 0.1155. This last figure is obtained by using a = 10.7 X 10-8 cm. This in turn is the calculated value from the computed slope (dashed line) of log KT us. 1/D (Figure 4). In Figure 4, the solid straight line represents cm. the 1959 eq 4 retaining the value of a = 10.7 X It is noteworthy that by using eq 5 with a = 10.7 X cm one obtains KTO = 0.52 in better accord with the extrapolated value 0.63 from the linear least-squares analy-
921
Electrical Conductance and Ultrasonic Relaxation for LiC104 L i C 9 0 4 in THF
1.25.C
i 6 . Cr =O.O50 M 150 -
140 -
130-
120-
140
u-
100
90
90
140
I20 110-
-
"\,
90 80-
\\
80-
f(MHz)-
Figure 5. a / f zvs. the frequency f(MHz) for LiC104 in thf at 25, -15, -30". sis. In Figure 4 the calculated intercept KTO = 0.52 is indicated by an arrow. It is noteworthy that eq 5 eliminates the need to introduce7 further parameters line the ion-solvent energy Es. This quantity was introduced7 in eq 4 in the form of exp(-E,/kT) to account for the experimental intercept 0.63. It is significant that Es resulted as Es = 1.7kT, namely, el.7, This value is close to the factor e 1 5 which distinguishes eq 5 from eq 4. Fitting eq 3 to the results of KT for LiC104 in THF (Table I) gives the values for a~ reported in Table I which average to Q = (5.2 f 0.7) X cm. Similarly, the fit of eq 5 to the data of LiC104 in T H F gives as an average a~ = (4.5 f 0.1) x cm (Table I). Equation 4 gives Q = (3.55 f 0.07) X 10- cm. All these figures are larger than expected on the basis of the axiom% = 1 . 5 ~ 3.0 X 10-9 cm.1& 11. Ultrasonic Relaxation. The results of ultrasonic absorption measurements, a(cm- neper), for LiC104 in T H F are reported in Figures 5 and 6. These data are expressed both as a/f2 us. the frequency f(MHz) and as aexcX US. the frequency f. a e x c is the excess absorption
-
TABLE 11: Ultrasonic Properties of THFa (orom x 1017, t, o
c
- 50 - 30 - 15 25 35 45
cm-1
secz
62 f 1 . 6 79 f 1 90.6 & 2 132 f 3 140 f 3 160 f 4
u, m/sec
1531 1457 1269 1232 1178
f 19 f3 f9 f 15 f 12
Frequency range, MHz
15-145 10-155 15-145 15-290 10-230 15-125
a The values of aa/fz are averaged over the frequency range reported. No systematic trend was noticed in this range. ao is the sound absorption coefficient (neper cm-1); u the sound velocity (meter/sec).
coefficient, aexc = a - am where am is the high-frequency value of a . A is the wavelength X = u/f with u the sound velocity in solution approximated to the one in the pure solvent a t the same temperature. The reason for disregarding the solvent a o / f 2 values (Table 11) is because LiC104 causes large depressions of sound absorption a t high frequency with respect to the absorption of the solvent, an effect increasing with concentration (Table 111).I7 The relaxation parameters were calculated as follows. The Journal of Physical Chemistry, Vol. 78, No. 9, 1974
922
Paul Jagodzinski and Sergio Petrucci
TABLE 111:. Ultrasonic Results for LiC104 i n THF
25
0.509 0.258 0.150 0.050 0,264 0.158 0.111 0.278 0.161 0.113
- 15 - 30
144 83 46 28.5 42 26 17 42 25 15
106 118.5 118 120 83
198 124 102 86 119 95 70.5 103 78.4 75
88
89 71 73 76
(1.60 f 0.10)
(0.81 f 0.28)
(1.02 f O . 0 8 ) (0.92 i O . 1 3 )
a The values reported for the B's are reliable to within ztl X 10-11 cm-1 st?c.2 The values reported for the fr's and pmax are reliable within zt5-10% tending toward the latter figure a t the lowest concentration. f~ is the relaxation frequency, B the background sound absorption, and the maximum excesa sound absorption coefficient times the wavelength a t f = fR.
500r
LiCP04
zoor
in
01 A f' - 1 + ( f / f R ) '
THF
+ B = 12cLrnaxllu.f +
01,
R
(f/fR)'
+
7
(7)
Another check on the estimated parameters was performed by recalculating B as the difference (a/f2)expt (A/[1 + ( f / f # ] ] . The fit was accepted only if the calculated B was independent of frequency and if the averaged B agreed with the estimated starting value of B to within 1X cm2 sec-I. The estimated uncertainty in A and f~ is *5-10%. The observed relaxation process was interpreted as follows: the conductance data indicate the presence of two major complex species in the concentration range of this ultrasonic study, namely, ion pairs and triplets. One could advance the hypothesis that the observed relaxation is due to ion pairs in equilibrium with free ions. This hypothesis, however, cannot be sustained for the following reasons. A very high association constant KA, as in the present case, would require approximate linearity18 between r - and c1I2 (7 is the relaxation time, 7 - l = 27rfR). This is shown below schematically.
2OOr
-
A + B
