J. Phys. Chem. 1984, 88, 2405-2411
2405
Molecular Relaxation of Lithium Salts in 2-Methyltetrahydrofuran at 25 C M. Delsignore, H. E. Maaser, and S. Petrucci* Department of Chemistry, Polytechnic Institute of New York. Long Island Center, Farmingdale. New York 1 1 735 (Received: July 6, 1983; In Final Form: October 24, 1983)
Ultrasonic absorption data are reported in the frequency range 1-7 MHz and 10-500 MHz for the electrolytes LiAsF, and LiBF, and described by a single Debye relaxation for LiASF, and by two Debye relaxation processes for LiBF4. The observed processes are interpreted by a multistep Eigen type dimerization process leading to contact quadrupoles. In the case of LiAsF6 the products are solvent-separated quadrupoles and contact dimers, whereas for LiBF4contact species predominate. Literature data for LiClO, suggest that the dimers are the major species present. Hence, the extent of dimerization process for the three electrolytes seems to follow the order: LiAsF, N LiBF, < LiC104. Electrical conductance data at t = 25.00 OC for LiAsF6 and LiBF, in 2-methyltetrahydrofuran (2MTHF) are reported. The data in the concentration range 6 X 104-1.50 X lo-* M are interpreted by the Fuoss-Kraus theory of triple-ion formation yielding estimates of the ion-pair formation constant K p and triple ion formation constant KT. Microwave dielectric complex permittivities in the frequency range 0.6-90 GHz and concentration range 0.02-0.1 M for LiAsF6 and LiBF, in 2MTHF at 25 OC are reported. Two Debye relaxation processes, one related to the solute and one to the solvent, are able to describe the data. The solute relaxation is interpreted as due to the rotational relaxation of ion pairs. From the calculated apparent dipole moments it is surmised that on the average LiBF4 pairs exist more as contact species with respect to LiAsF, pairs.
Introduction In recent years research on lithium electrolytes in ether solvents has been focused on studies of solvation and ion association aimed at elucidating the structure and dynamics of the systems.’ The practical aspect of this current interest is the knowledge of the species in the same solution which are used in constructing secondary batteries with lithium electrodes.* The interpretation of the results is expected to give guidance toward choosing solutions which have stability toward the electrode and a minimum internal resistance (maximum conductivity). It is also our belief that only a diversified technology will be able to give an adequate description of these systems, no single method alone being sufficient to meet the above requirements. To this end a classical method such as electrical conductance has been used to study the status of association of the electrolytes. Also, two dynamic tools, ultrasonic relaxation and microwave dielectric relaxation, have been employed to study the rates of formation and dissociation Qf the species present and the rotational relaxation of the dipolar species in solution, respectively. A solvent particularly relevant for battery construction using lithium electrodes is 2methyltetrahydrofuran (2MTHF). This is due to its proven stability when in contact with lithium.* We have studied the two electrolytes LiAsF, and LiBF, and compared the results, when available, with LiC104, in an effort to provide a picture of the effect of the nature of the anion of the process of ion complexation. Experimental Part The equipment for the ultrasonic work consisted of a Matec (Warwick, RI) 6600 pulser-receiver unit, equipped with tuning heads covering the range 10-700 MHz. A Matec pulse comparator and Master synchronizer completed the unit. The twin-crystal interferometric ultrasonic cell was equipped with two lithium niobate coaxially gold-plated crystals cut +36O with respect to t h e y axis and furnished by Valpey Fisher (Hopkinton, MA). This new setup delivered ultrasonic decay echoes of much improved signal over noise ratio with respect to the previously described equipment using x-cut quartz crystal^.^ With water as an ab(1) (a) Wang, H.-C.; Hemmes, P., J . Am. Chem. SOC.1973,95, 5115; 1973,95,51 19. (b) Jagodzinski, P.; Petrucci, S., J . Phys. Chem. 1974,78, 917. (c) Onishi, S . ; Farber, H.; Petrucci, S . Ibid. 1980,84, 2922. (2) Goldman, J. L.; Manck, R. M.; Young, J. H.; Koch, V. R. J . Electrochem. SOC.1980, 127, 1461. (3) Darbari, G. S.; Richelson, M.; Petrucci, S . J . Chem. Phys. 1970,53, 859. Petrucci, S . J . Phys. Chem. 1967,71, 1174. Petrucci, S.; Battistini, M. Ibid. 1967, 71, 1181.
0022-3654/84/2088-2405$0 1.50/0
sorbing medium, it was possible in preliminary tests to reach the 57th harmonic of the crystal (570 MHz) with an acceptable signal over noise ratio signal. Coupling the crystal to the delay line was accomplished with a film of nonaqueous grease. The remainder of the pulse equipment was basically the same as previously de~ c r i b e d .For ~ the measurements below 10 MHz, specifically for 1-4 MHz and for 6-7 MHz, an ultrasonic cylindrical resonator cell of new construction and a sweeper-receiver set were used. The main design of the cell is similar to the one of Eggers and Funck4 with some minor changes which basically consist in using a homemade Teflon bellow to space the two x-cut, 1-in., 5-MHz fundamental quartz crystals. The bellow allows for sufficient elasticity for micrometric orientation of the two crystal holders and also holds the liquid under study without leakage. Teflon washers pressed in between the crystals and the lips of the holders maintain a tight seal. The rest of the cell in contact with the liquid is made of stainless steel. We are grateful to Drs. T. Funck and F. Eggers of the Max-Planck Institute for having shown their cell during a summer visit of one of us to Gottingen, West Germany. Details of the cell construction will be reported e l ~ e w h e r e . ~ Thermostating of the resonator cell was provided by a circulating bath which was maintained at 25.0 OC by the aid of a Bailey proportional thermoregulator. The equipment and procedure for the electrical conductance6 and microwave dielectric complex permittivities’ have been described elsewhere. For the materials, 2MTHF was distilled under reduced pressure over sodium and benzophenone. Initial refluxing until the solution showed a permanent blue color was performed. The permanence of blue color was taken to indicate the absence of peroxides. Only the middle portion of the distilled liquid was collected. The distilled solvent was used within 1-2 days after distilling it. LiAsF6 was obtained from Agri-Chemical Co., Atlanta, GA. It was found that the product from this manufacturer did not show a yellow discoloration in 2MTHF even after 1 day at variance with a product from Aldrich which contained some impurities and was therefore discarded. LiASF6 was redried in vacuo at 60-70 OC for 1-2 days before use.* LiBF, (Aldrich) was subjected to a (4) Eggers, F. Acustica 1967168,19,323. (5) Chen, C.; Wallace, W.; Eyring, E.; Petrucci, S . , J . Phys. Chem., in
press. (6) Petrucci, S.; Hemmes, P.; Battistini, M. J . Am. Chem. Soc. 1967,89, 5582. (7) Saar, D.; Brauner, J.; Farber, H.; Petrucci, S . J . Phys. Chem. 1978, 82,545. Farber, H.; Petrucci, S . Ibid. 1975, 79, 1221. (8) Farber, H.; Irish, D. E.; Petrucci, S . J . Phys. Chem. 1983,87,3515. (9) Saar, D.; Brauner, J.; Farber, H.; Petrucci, S . J . Phys. Chem. 1980, 84, 341.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 11, I984
2406
Delsignore et al. TABLE 11: LiBF, in ZMTHF at 25 "C
140r LiAsFg 0.10M 2MTHRtz25'C
120
m 01
in 0.15, 0.25 0.34 0.41 0.47
r
I
5100-
r-
r
E
185 250 300 400 450
22 27 30 32 34
8 8 8 9.5 9
185 250 300 370 440
50 55 61 55 60
1.209 1.216 1.210 1.215 1.214
80-
J
t
where pFI = (1/2)A,~f,~ and pmII= (1/2)A1,u&11. The same data could have been expressed in a less illustrative way by
Y
60 -
405
20
10
I
I
I
J
50
100
200
500
f (MHd LBF, 0 . W in 2MeM
t=26"C
1
fl \
400 '0°[
(b)
In other words, the data can be best described by the sum of two Debye relaxation processes. The function for a single relaxation I is unable to describe the data adequately. In order to emphasize this point, the ultrasonic pulse data for the system 0.34 M LiBF4 in 2MTHF at 25 OC are presented in Figure 2 in the form of a vs. f,where aeXc = (a- BY). AcMikhailov plot as Cf2/aex,) cording to Mikhailov,Io for a single relaxation process, a straight line should represent the data according to the function
_ -WZ
aexc
1
5
20 f(MHz)--
100
500
Figure 1. (a) u/f vs. frequencyffor LiAsFXO.10 M) in 2MTHF; t = 25 OC. (b) g vs. frequencyffor LiBF, (0.34 M) in 2MTHF; t = 25 "C. TABLE I: LiAsF, in ZMTHF at 25 "C c,
M
0.52 0.44 0.33 0.32 0.20 0.10
f,, MHz
110 100 80 80 59 40
1 0 i 7 ~ ,1 0 ~ ~ ~ 3I O, - S U , cm" s' cm-' s' cm s-'
105pm
52 52 52 53 52 53
846 712 376 360 242 164
125 118 78 75 68 68
1.230 1.206 1.206 1.201 1.204 1.204
similar treatment before use. Solutions were kept in a desiccator and used as soon as possible after preparation. No solution was used after more than 1 day after preparation. Figure l a shows a representative plot of the quantity a/f vs. the frequencyffor LiAsF, in 2MTHF at 25 O C . The solid line is the calculated Debye function for a single relaxation process: a
A +B 7= 1 + (f/f,)2 where a is the absorption coefficient of sound expressed in N p cm-',f, the relaxation frequency, A a relaxation parameter, and B the value of a/f2 at f >> f,, not necessarily equal to the value of ao/f2 for the solvent. Table I reports the relaxation parameters A,f,, and B together with the sound velocity u (cm s-]) and the maximum excess sound absorption per wavelength p,,, (= (1/2)Auf,) for LiAsF6 in 2MTHF at 25 O C for the concentrations c (mol dm-3) investigated. Figure l b reports a plot of the quantity p = (a- Bf2)(u/f), the excess sound absorption per wavelength, plotted vs. the frequencyf, for LiBF, in 2MHF at 25 OC. The solid line corresponds to the sum of two Debye processes according to the function
2u3 r(u-2
- uo2)
+
2U3rW2 (u-2
- U2)
where w = 27rf, T = (27rf,)-l, uo and urnare the low- and highfrequency values of the sound velocity (with respect tof,). Attempts at interpreting the data by a single straight line result in the calculated straight line (by linear regression) showing a poor correlation with the data. Alternatively, Figure 2b shows an attempt to fit the p vs.fdata by a single relaxation function centered at f = 15 MHz with p = 500 X The relaxation spectrum appears to be too broad to be described by a single Debye function, whereas two Debye processes centered atfI = 30 M H z andfII = 8 MHz with pI = seem to be able to describe the 300 X and pII= 300 X data adequately. Figure 2c,d, show the same sequence for 0.47 M LiBF4 in 2MTHF at 25 OC. The same considerations and conclusions hold also for this case and the other concentrations studied. Figure 3 reports an additional run for 0.40 M LiBF4 at 25 O C . The inset shows the tail of the relaxation in the form of (a/f) V S . ~ : It is clear that the value of the parameter B cannot be raised above the used figure B = 52 X lo-'' cm-' s2. With this value and the sound velocity u = 1.219 X lo5 cm s-I, the pulse p vs. fdata have been calculated and reported in Figure 3 together with the resonator data. The spectrum can be interpreted by two Debye processes with parameters pI = 370 X 10-5,fI = 33 MHz, pII= 340 X andfiII = 10 MHz. Alternate attempts at fitting the spectrum with a single Debye function ( p = 630 X 10-s,f, = 18 MHz) do not give a satisfactory fit as the dashed line in Figure 3 indicates. This should ensure the reliability of the B parameters and the fact that the width of the relaxation spectrum is not an artifact due to the choice of the B parameter. Table I1 reports the parameters pmI,frI, pmII, frII,and B for LiBF, in 2MTHF for all the concentrations investigated at 25 OC. The sound velocities u (cm s-l) are also reported for the same systems. Figure 4 reports the equivalent conductance vs. the concentration c (mol dm-' = M) expressed as a log-log plot for LiAsF6 and LiBF, in 2MTHF at 25.00 O C . One may see that the data are separated by 1 order of magnitude, LiAsF, being apparently a better conductor than LiBF, in 2MTHF. Figure 5 reports the coefficient E' of the real part and the coefficient 6'' of the imaginary part of the complex permittivity E* = E' - Jd' plotted vs. the frequencyf(GHz) for a representative concentration of LiAsF6 and LiBF, in 2MTHF at 25 OC. The (10) Mikhailov, I. G. Dokl. Akud. Nuuk SSSR 1953, 89, 991.
Lithium Salts in 2-Methyltetrahydrofuran at 25
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 2401
O C
300 -
70 t=25 C LiEF4 Mikhailov 0.34M plotin 2MTHF
(Pulse
0
7
E
I'
Data)
,/
1.25 C
/'
200-
___---_-#r
0
regression
U Y c
f
50-
2
40-
(Pulse Data)
X 0
X 0
N'
Mikhailov plot
60 - LiBF, 0.47M in 2MTHF
(a)
100-
1
OO
300
200
100
400
500
1
600
250
f 2 x 1 0-14
500
LiBF40.47M in Z M T H F 400
p 300 X
i 200
LiBF40.34M in 2MTHF
100
,
I
10
5
20
,
1
100 200
50
5
500
20
10
50
100
250
f(MHz)
f(MHz)
Figure 2. (a) Mikhailov plot for LiBF4 (0.34M) in 2MTHF; r = 25 OC. (b) p vs.fand one or two relaxation fits for LiBF4 (0.34M) in 2MTHF. (c) Mikhailov plot for LiBF4 (0.47 M ) in ZMTHF; t = 25 OC. (d) p vs,ffor LiBF4 (0.47M ) in 2MTHF; t = 25 OC. 700
1
LiBF, 0.40M in 2MTHF 8oo t-25'C
YE
"'1
70
P
D
.
9
400
TABLE 111: Relaxation Parameters E ~ , E . = . , ~ , E ~frl, ~ , frII, and Electrical Conductance X for LiAsF, and LiBF, a t the Concentrations Investigated in ZMTHF at 25 "C
v)
,
100
200
500
I(MHz1
LiAsF,
/ln
0.05 0.035
0.023
8.0 7.6 7.2
6.0 6.0 6.1
1.4 1.4 1.4
40 40 40
0.7 X 10"
2.4
1.1 1.1
2.6
1.1
45 40 45
1.9 X lo-, 0.81 X lo-, 0.76 X l o +
2.5 2.5 2.5
3.0 X IO-' 1.2 X l o - '
LiBI',
0.10 0.072 0.05
7.2 7.2 6.9
6.0 6.2 6.2
2.4
Discussion - L
5
10
20
50
100
200
500
f(MHz)
Figure 3. p vs.ffor LiBF4 (0.40M ) in 2MTHF; t = 25 OC.
Previous studies from this laboratory14 and previous data" seem to convey the following picture for the process of ionic complexation of ethereal lithium salt solutions:
A+
solid lines are the fitted functions, sum of two Debye relaxation processes
where tois the static permittivity of the solution, em, and em, are the permittivities for f >> fI and f >>AI, respectively, and AI are the two relaxation frequencies bound to the respective decay times of the polarization by the expressions = (27r-)-' and T~~ = (27rA1)-l. e/ = (1.8 X 101*)X/fis the conductance contribution to the loss; X is the specific conductance. Table I11 reports the values of the parameters eo, e,,, em2,fI,fiI, and of the conductance X for the solutions investigated.
+ Bk
+ k
4
k
A,B
2AB & k-2 (A,B),
k-I
AB
.=$(AB), k
where A,B and AB are the outer-sphere and inner-sphere ion pairs, respectively, whereas (A,B), and (AB), are the corresponding species for the dimers or quadrupole species. In a way, processes VI and VI1 are extensions of the Eigen multistage scheme of ionic association. It is clear that at a given concentration, in a given solvent, the predominant species (and hence the likelihood of observing a given process) will depend on the nature of B, namely on its structure, donor ability, and the (1 1) Maaser, H. E. Doctoral Thesis, Rutgers University, New Brunswick, NJ, 1982. (12) Von Srnoluchowsky, M. Z. Phys. Chem. (Leipzig) 1917, 92, 179. Petrucci, S. In 'Ionic Interactions"; Academic Press: New York, 1971; Vol.
11.
2408 The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 Electrical conductance of electrolytes in 2MTHF
Delsignore et al. AS,+
+ BS;
2
ZABS,
-3
-1
-2
7
S,_,ABS,
(ASBS,,),
+S
(ABSy&
+S
(VIIa)
Again, the "nature" of the solvent encompasses properties such as permittivity, donor number, steric effects on solvation, etc. Because of the relatively high concentration (necessary to observe an ultrasonic relaxation process) and of the magnitude of the equilibrium constants (few free ions in solution), step one of (VI) is generally unobservable in lithium ethereal solutions. The other steps have been at times reported in previous ~ o r k s . ' ~ ~ . ' ' In what follows we are going to interpret the reported data for LiAsF6 and LiBF4 according to (VII). A comparison of these findings with LiC10411and some generalization on the effect of the anion will be made. Scrutiny of Table I reveals that the relaxation frequency for LiAsF, depends on concentration. It is, in fact, linearly dependent on c within experimental error. Further, then In pmis proportional with In c. Linear regressions applied to these data give r2 = 0.91 and slope = 0.99. It has been indicatedIc that for a dimerization process of the type
'.
-4
2 % k
S,-lASBSy
0
logloc
3r
2AB
kr
(AB),
(VIII)
one would expect the relaxation time T to correlate with the forward and reverse rate constants by the linear relation ( T - I = 2rfr): T-' = 4kf[AB] + kr (IX) whereas pm should correlate with [AB] and [(AB),] according to
LiBF4 n
n V I
r-.
v
d
0 0
50
100
I
J
150
200
4
(I-& c x 1 0 4
Figure 4. (a) Log A vs. log c for LiAsF, and LiBF, in 2MTHF t = 25.00 OC. (b) Ag(c) c1I2vs. c(1 - Ao/Ao) for LiAsF6 and LiBF4in 2MTHF t = 25.00 "C. 81
which, by the introduction of the quadrupole equilibrium constant K , = [(AB),]/[AB]* reads ?r
13
pm =
2
-
_ x W I W
I
1
0.6
1
2
5
10
20
50
0 100
f(GHz)
LIBFd 0 . 0 7 2 M in 2MTHF
7 6
(AV,),
RT
Kq[AB12 1 + 4Kq[AB]
(XI)
where = (pu)-' is the adiabatic compressibility, p the density, and AVs the adiabatic volume change due to process VIII. In a medium of low permittivity such as 2MTHF (to = 6.2) if ( K , is not large), one can set the total concentration equal to the concentration of the dominant species, namely the ion pairs AB. Then, from eq XI one would expect the slope of In pmvs. In c to be between 1 and 2 (depending on whether 1 > or < 4K,[AB]), as observed in the present case. If one accepts the hypothesis that (VIII) describes the observed process for LiAsF6 in 2MTHF, in a first approximation, setting [AB] N c gives from linear regressions of 7-l vs. c 3 = 0.996, slope = 1.07 X lo9, and intercept = 1.5 X lo8, from which one calculates kf = 2.7 X 10' M-I s-l, k, = 1.5 X lo8 s-', and K, = 1.8 M-l. Similarly, by transforming eq XI, one gets
2_x
,
-u 'Iu
5
W :
4
1
3 2
0.6
1
2
5
10
20
50
0
100
f(GHz)
Figure 5. (a) e' vs.fand e" vs. ffor LiAsF, (0.05 M) in 2MTHF; t = 25 OC. (b) e' vs.fand e'' vs.ffor LiBF4 (0.072 M) in 2MTHF; t = 25 "C.
like. Further, for a given electrolyte, the same equilibrium chain will be shifted to the left or right depending on the nature of the solvent, as we ought to write the above reactions as
Setting again [AB] = c, linear regressions of the left part of eq XI1 vs. c gives 9 = 0.84, intercept = 3.2 X lo", and slope = 4.82 X lov2,from which AV, = 9.1 cm3/mol and Kq = 3.8 X lo3 cm3 mol-3 or K , = 3.8 M-I, of the same order of magnitude of the figure above, Kq = 1.8 M-l. Parts a and b of Figure 6 show the plots of T-' vs. c and of the quantity (m2/2&,,RZ9, respectively. Figure 6c shows the calculated and experimental pmvs. the concentration c. It is interesting to compare the above results with data" of LiC104 in 2MTHF. In this system, an ultrasonic relaxation process, independent of concentration (and assigned to the equilibrium outer sphere ==inner sphere quadrupoles), was observed. Hence, for LiC104, the last step of (VII) is the only observable process around c = 0.1 M. The ion-pair concentrations
Lithium Salts in 2-Methyltetrahydrofuran at 25
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 2409
O C
S=
LiAsF6in ZMTHF
~ 1 - lN
4k2[AB]
+ k-2
and
zl------
'
TI;l
/
4 t
7;
01 0
I
,
1
0.2
0.1
--
0.3
0.5
C,M
Farber plot
2, 1
4k2[~~1 + k3 4k2[AB] + k-2
+
t=HC
+
,
0
s
In the situation where k-3 >> k3, qcl = k-3 independent of c. It is clear that because of the closeness of the T ' S , we cannot apply the condition ~ 1 - l>> q f l . However, by using as first approximation [AB] N c, we can plot S vs. c and P vs. c. Linear regressions applied to the quantity S = ( ~ 1 -+ l q - l ) vs. c gives r2 = 0.99, slope = 4kz = 2.63 X lo8, and intercepts = k-2 k3 + k-3 = 1.50 X lo8;hence, k2 = 6.6 X lo7 M-I s-l. Further linear ) r2 = 0.95, regressions of the quantity P = ( T < ~ T ~vs.~ -c ~gives slope' = 4kz(k3 + k-3) = 1.79 X 10l6,and intercept' = k-2k-3 = 4.0 X 1015. From the above -slope' - - k3 k-3 = 6.8 X lo7 slope
I
1
0.4
P = k-3
0.2
0.1
I
I
J
0.3
0.4
0.5
and 'Oo0 800
1
slope' k-2 = intercept - -= 1.50 X lo8 - 0.68 X lo8 = slope 8.2 x 107 s-1
LiAsFgirl 2MTHF t:25"C
0
Also
0
0.1
0.2
0.3
and
I
0.4
0.5
k3 = ( k 3
cxl03(mo1e/cm3)
Figure 6. 7-l vs. c for LiASF6 in 2MTHF; t = 25 'C. (b) (7r$/2p&RT) vs. c for LiASF6 in 2MTHF; t = 25 O C . (c) p,,, vs. c for LiAsF6 in 2MTHF; t = 25 O C . The solid line is the calculated function.
must be low enough for the first step of (VII) not to be visible. It is difficult, from the isothermal data at hand, for LiAsFs in 2MTHF to decide whether the final product is solvated, contact quadrupole, or a combination of both. What is observed is that, at variance with LiC104, a second-order process is observable. The rate constant k f is lower than what expected on the basis of a diffusion-controlled rate constant between uncharged molecules, according to Smoluchowksy.I2 kD = 8 ~ R T / 3 0 0 0 7 (XIII) with 7 = 0.0047 P, the viscosity at 25 O C for 2MTHF. That LiAsFs is probably more shifted toward the left in (VI) and (VII) (in terms of the equilibrium concentrations) with respect to LiC104 can also be deduced from conductance results? showing the former electrolyte being much more conducting than the latter one in all the concentration ranges studied. The case of LiBF, in 2MTHF is more complicated. Here, the ultrasonic spectrum can be described as due to the sum of two close Debye relaxation processes. Inspection of Table I1 reveals that the relaxation a t high frequency is concentration dependent, whereas the one a t low frequency, within experimental error, is not concentration dependent. We will consider the hypothesis that we are observing both steps of (VII). Accordingly, as already shown: one would expect two relaxation times bound by the relation 71,J-l
= Y*(Sf ( S - 4 P ) q
(XIV)
where S =
7I-l
+
7f1
= 4k2[AB]
+ k-2 + k3 + k-3
and P = TL'TII-~= 4kz[AB](k3
+ k-3) + k4k-3
Notice in particular that if the condition then
(XV)
>> qr1were true,
T ~ - ~
+ k-J
- k-3 = 6.8
X lo7 - 4.9 X lo7 =
1.9 x 107 S-1
In summary, the rate constants are k2 = 6.6 X lo7 M-'
s-l
k-2 = 8.2 X lo7 s-l hence K2 = k 2 / k - 2 = 0.8 M-I k3 = 1.9 X lo7 s-l k-3 = 4.9 X lo7 s-I hence K3 = k3/k-3 = 0.4 The overall quadrupole formation constant K, is related to (VII) 2AB + (A,B) + (AB),
cz
c3
c4
by the relation
where K2 = C3/C22
and K3 =
c4/c3
Therefore, K = 1 . 1 M-l. Parts a and b of Figure 7 report the plots of S vs. c and of P vs. c, respectively. The conductance data will now be discussed. Figure 4b reports the quantity hg(c)c1l2vs. c(1 - A/&) according to the FuossKraus expression:
2410
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 +
3r
LiBF, in 2MTHF; t=2S0C
Delsignore et al. equiv-' l b and for T H F 7 = 0.0046.' Therefore, in 2MTHF XoLi = 35.7 Q-' cm2 equiv-'. This gives A'LiBF, = 121 0 - I cm2 equiv-' at 25 "C in 2MTHF. Hence, for LiBF4
K~ = 9.5
M-1
1
I
1
I
0.1
0.2
0.3
0.4
0.5
I
neglecting interionic terms (g(c) = 1) and A/Ao with respect to one. For LiAsF6 one obtains by linear regressions of A C ' / ~vs. c for the first run slope = 0.924, intercept = 0.025, and 1.2 and 0.98, from which
P= 4k2C (k,
+ k-,) + k-,
k-,
b
I
k-
i i o
K~ = 39
1
CM-
t
109 M-1
From the above it results that the reason LiAsF6 is a better electrical conductor in 2MTHF than LiBF4 is because of its smaller formation constant Kp rather than to differences in the extent of triple-ion formation. It is of theoretical interest to examine the g(c) term at this point. It was noticed16 that the use of the g(c) term seemed important at permittivity 6 = 7.0 and irrelevant a t = 2.7. The reason for this behavior is due to the fewer ions in the latter medium. Another check of this statement can be found by analyzing our present data in the same medium 2MTHF of E = 6.2 with the simple mass law expression
a 01 0
x
I
I
I
1
K~ = 2.70 x 107 M-1
KT = 55.2 M-I
For the second run of LiAsF6 in 2MTHF, slope = 0.922, intercept = 0.0268, and r2 = 0.99, from which
K~ = 2.35 x 107 M-1 exP(
giving as an average
p'
= (2.5 f 0.2) x 107 M-1 (XVII)
\
with p' the Debye-Huckel activity coefficient term and S the Onsager limiting coefficient of the conductance equation. The solid lines in Figure 4b correspond to the calculated function XVI by linear regression. For LiAsF6 one obtains slope = 0.369, intercept = 0.0220, and r2 = 0.96 (first run) and slope = 0.318, intercept = 0.0237, and r2 = 0.99 (second run). In order to evaluate Kp and KT, one needs estimates of A. and noT.For the former quantity we have used A. = 178.2 52-l cm2 equiv-' in acetonitrile at 25 "C ( 7 = 0.003499 P).13 Hence, from the Walden rule Ao7 = 0.615. We have measured the viscosity of 2MTHF to be 7 = 0.0047 P at 25 "C by the use of a suspended level Cannon viscometer with certificate of calibration. Hence, bo= 130 0-I cm2 equiv-'. For AoTthe arbitrary position AoT= '/3Ao, consistent with previous work, has been retained. Hence, for the first run of LiAsF6 in 2MTHF K~ = 3.5 x 107 M-1 K~ = 25.1 M-1 For the second run Kp = 3.0 X lo7 M-' giving as averages K p = (3.25 f 0.25) X lo7 M-'
KT = 5 1.4 M-'
KT = 20.0 M-'
Ir, = 22.6
f 2.6 M-I
For LiBF4 linear regression gave slope = 0.0323, intercept = 1.24 X and r2 = 0.90. In order to evaluate A. of LiBF4 in 2MTHF, the following calculation has been performed. In nitrobenzene it has been reported14 that X"BF4- = 22.1 and, since for nitrobenzene 7 = 0.01823 P," it results A",,- 7 = 0.403. Hence, in 2MTHF XoBF4= 85.7 Q-' cm2 equiv-I. For Li+ in T H F XoLi = 36.6 Q-l cm2 (13) Hopkins, H. P. Jr.; Jahagirdar, D. V.; Norman, A. B. J . Solution Chem. 1979, 8, 147.
(14) Witschonke,C. R.; Kraus, C. A. J . Am. Chem. SOC.1947,69,2472. (15) Powell, A. L.; Martell, A. E. J . Am. Chem. SOC.1957, 79, 2118. (16) Farber, H.; Petrucci, S. J . Phys. Chem. 1981, 85, 1396.
K~ = 53.3 f 2 M-1
Similarly, for LiBF4 the same calculation gives slope = 0.0338, intercept = 0.001 25, and r2 = 0.91, from which one obtains
Kp = 9.31
X
IO9 M-'
KT = 40.4 M-'
One may see, by comparing the analysis by eq XVI and XVIII, that whereas there are significant changes for LiAsF6,the changes are insignificant for LiBF4. Since the two solutions are practically isodielectric, it is the ionic strength, hence the departure of g(c) from unity, that makes eq XVIII inaccurate with respect to eq XVI. The general conclusion from the above portion of the work is that the preponderant part of the electrolytes exist as ion pairs with only a minor fraction of triple ions. Some fraction of the electrolytes are, however, dimerized to quadrupoles as revealed above by ultrasonic relaxation techniques. We shall now discuss the dielectric data. From Table I11 one should point out that the difference eo E, as well as the process centered aroundfi, is due to the presence of the solute, the value of e,', cmZ, and &I being similar to the relaxation values of the solvent. We propose, as done in previous work, that the relaxation process centered at the lower frequency& is due to the diffusion rotation relaxation of ion-pair dipoles LiASF6 and LiBF,, respectively, in 2MTHF solutions. In order to extract more information out of the dielectric data, in Figure 8 there are plotted the quantities (eo - em1)((2e0 + 1)/3c0 vs. c X according to the Bottcher e q ~ a t i o n : ' ~
where the a polarizability-reaction field factorf, term (1 -,',a generally of the order of 0.9, has been assumed to be unity. Linear regression, forcing the intercept through the origin (50% statistical weight to the origin), gives for LiAsF6 slope = 2.85 X lo4, intercept = 0.012, and 1.2 = 0.998. This in turn gives for the (17) Bottcher, C. F. J. "Theory of Electric Polarization";Elsevier: Amsterdam, 1973.
2411
J. Phys. Chem. 1984, 88, 2411-2414 Bottcher plot for lithium electrolytes in PMTHF tz25’C
with e = 4.80 X 1O-Io esu, the charge of the electron. Equating Kp for LiAsFs to eq XX, one calculates a = 4.8 X IOw8 cm. Similarly for LiBF4, one obtains a = 3.45 X cm. Although this calculation is very approximate, still there is a clear indication that both conductance and dielectric data point to the same information, namely that AsF6- is a poorer competitor than BF4for the first-coordination sites around Li+. It is also noteworthy that whereas p measures the charge to charge separation in the ion pair, T gives a measure of the rotating dipole length. If one retains as a first approximation the Debye expression18
LiAsF6
/
1
0
0.0 1
0.03
0.05
0.07
0.09
71
1.1
~x103(mo1e/crn3)
Figure 8. Bottcher equation, (eo - c,)(2eo + 1/3eo) vs. c X LiAsF6and LiBF, in 2MTHF; t = 25 ‘C.
for
apparent dipole moment p = 22.0 X esu cm. If one takes a rigid-sphere model, ~1 = ae and a = 4.6 X lod8cm. Similarly, for LiBF4 in ZMTHF, linear regression of (eo E,~)((+ ~ E1)/3eo) ~ vs. c X giving 50% statistical weight to the origin, gives r2 = 0.99, slope = 9014, and intercept = 0.011, from which one calculates p = 12.1 X esu cm and a = 2.5 x loM8cm. Judging from the above, one would conclude that in 2MTHF LiBF, pairs, on the average, exist more as contact species with respect to LiAsF6 pairs. This notion is reinforced by comparing the formation constants of the two electrolytes LiAsF6 and LiBF4 in 2MTHF with the Fuoss-Jagodzinski ion-pair formation constant:
4Ta; =kT
‘
neglecting differences between the microscopic relaxation time T’ and the decay of the polarization T and retaining 7 = 0.0047 P the viscosity of the solvent, it results for LiAsFs a, = 4.3 X and for LiBF, a, = 4.7 X cm two comparable figures at variance with the a’s calculated from the p’s. This may be because a, reflects the rotating solvated entity whereas a, reflects the charge separation in the ion pair. Acknowledgment. We are grateful to the U S . Army Research Office, Research Triangle Park, NC, for their generous support under Grant No. DAAG-29-82-KO048, Registry No. LiASF6, 29935-35-1; LiBF4,14283-07-9;ZMTHF, 9647-9. (18) Hill, N. In “Dielectric Properties and Molecular Behavior”; Van Nostrand Reinhold: London, 1969.
Effect of Temperature on the Enthalpy of Dilution of Strong Polyelectrolyte Solutions G. Vesnaver, M. RudeZ, C. Pohar, and J. Skerjanc* Department of Chemistry, Edvard Kardelj University, Ljubljana, Yugoslavia (Received: August 16, 1983; In Final Form: November 14, 1983)
The enthalpies of dilution of aqueous solutions of poly(styrenesu1fonic acid) (HPSS)and its alkali-metal salts were measured at 0, 15,25, and 40 O C in the concentration range from about 0.2 to 0.002 monomol/L. The enthalpy of dilution decreases approximately linearly with the decreasing temperature. The experimental data are compared with predictions from the electrostatic theories based on the cell and infinite line charge models. A satisfactory agreement is obtained for the acid and its lithium salt, whereas for the sodium, potassium, and cesium salts agreement is worse.
Introduction Various thermodynamic and transport properties of polyelectrolyte solutions have been accumulated in the period from the early 195Os, when this field of polymer chemistry became attractive for many scientists, to the present. A dependence of these properties on the nature of the polyion, its charge, concentration, and molecular weight, on the ionic radius of counterions and their valency, the dielectric constant of the solvent, the concentration of added simple salt, etc., has been investigated.l Accumulated data have frequently been used for evaluation of the existing polyelectrolyte theories. Most of the data have been obtained at a given temperature, usually at 25 O C , and there has been little interest to study the effect of temperature on any of these properties. A very few articles in which temperature dependence has (1) Armstrong, R.W.; Straws, U. P. In ‘Encyclopedia of Polymer Science and Technology“; Interscience: New York, 1968; Vol. 10, p 78 1.
0022-3654/84/2088-241 l$Ol.SO/O
been studied showed that the osmotic coefficientZand the apparent molal volume3 of highly charged polyelectrolytes depend only slightly on temperature, whereas recent measurements with two polysaccharides4 disclosed a definite effect of temperature on the enthalpy of dilution, AHD.The present work was undertaken with the purpose to examine the temperature dependence of AH, of a typical strong polyelectrolyte. The polyelectrolyte chosen was poly(styrenesu1fonic acid), and measurements of AH, were performed with the acid and its lithium, sodium, potassium, and cesium salts a t 0, 15, 25, and 40 OC.
Experimental Section Poly(styrenesu1fonic acid) (HPSS) and its alkali-metal salts were derived from a single sample of sodium poly(styrenesu1fonate) (2) Ise, N.; Okubo, T. MacromolecuZes 1969, 2, 401. (3) Ise, N.; Okubo, T. J . Am. Chem. SOC.1968, 90, 4527. (4) Hales, P. W.; Pass, G . Eur. Polymn. J . 1981, 17, 657.
0 1984 American Chemical Society
