Dielectric Relaxation in Dimethyl Carbonate
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978 545
orbitals for the bis Ir(II1) complexes are likely single ring delocalized orbitals, hence a similar situation for the other (tris, bisesquis) Ir(II1) complexes would not be surprising.
T. Sali and S. Aoyagui, J . flectroanal. Chem., 58, 401 (1975). N. Tanaka and Y. Sato, flectrochim. Acta, 13, 335 (1968). N. Tokel-Takvotyan, R. E. Hemingway, and A. J. Bard, J. Am. Chem. Soc., 95, 6582 (1973). S. Roffia and M. Ciano, J. flectroanal. Chem., 77, 349 (1977). J. Van Houten and R. J. Watts, J. Am. Chem. Soc., 98,4853(1976). T. Saji and S. Aoyagul, J . flectroanal. Chem., 63, 31 (1975). F. Zuloaga and M. Kasha, Photochem. Photobiol., 7,549 (1966). N. Tanaka and Y. Sato, Bull. Chem. Soc., Jpn., 41, 2059 (1968). G. Kew, K. DeArmond, and K. Hanck, J. phys. Chem.,78,727(1974). K. DeArmond and J. Hillis, J . Chem. Phys., 54, 2247 (1971). K. Hanck, K. DeArmond, G. Kew, J. Kahl, and H. Caldararu in "Characterization of Solutes in Non-Aqueous Solvents", Plenum Press, New York, N.Y., 1977. R. Watts, J. Harrington, and J. Van Houten, J . Am.'Chem. Soc.,
Summary The redox sequence for the Ir(II1) complexes are similar to those sequences found for the analogous Fe(II), Ru(II), and Os(II1) complexes rather than those found for the tris and bis Rh(II1) complexes. Such a sequence is consistent with the delocalized redox orbital character postulated for these d6 starting materials. Ultimately, ESR s t ~ d y ~ofl v ~ ~ the stable reduced species may be useful in determining the relative amount of metal and ligand character for the reduced species and in describing the interaction between the ligands within a complex.
99,2179 (1977). C. Flynn and J. Demas, Jr., J. Am. Chem. Soc., 96, 1959 (1974). S.Roffia and M. Ciano, J. flectroanal. Chem., in press. &.Martin and G. Waind, J. Chem. Soc., 4284 (1958). B. Martin, W. McWinnie, and G. Waind, J. Inorg. Nucl. Chem., 23,
207 (1961).
Acknowledgment. This research was supported by the National Science Foundation (CF 40894 and CHE7605716).
B. Chriswell and S. Livingstone, J. Inorg. Nucl. Chem., 26, 47 (1964). R. Watts and G. Crosby, J . Am. Chem. Soc., 93, 3184 (1971). R. Bull and G. Bull. Anal. Chem., 43, 1342 (1971). R. Nicholson, Anal. Chem., 37, 1351 (1965). R. Wopschall and I. Shain, Anal. Chem., 39, 1514 (1967). A. Vlcek, Rev. Chim. Miner., 5, 299 (1968). A. Vlcek, Electrochlm. Acta, 13, 1063 (1968). A. Vlcek, Proc. Int. Coord. Chem., 14, 220 (1972). G. Kew, K. Hanck, and K. DeArmond, J. Phys. Chem., 79, 1828
References and Notes N. Tanaka and Y. Sato, Inorg. Nucl. Chem. Lett., 2, 359 (1966). N. Tanaka and Y. Sato, Bull. Chem. SOC. Jpn., 41, 2064 (1968). S.Musumeci, E. Rizzarelli,I. Fragah, S. Sammartano, and R. Bonomo, Inorg. Chim. Acta, 7,660 (1973). S.Musumeci, E. Riuarelli, S. Sammartano, and R. Bonomo, J. Inorg. Nucl. Chem., 36, 853 (1974). T. Saji and S. Aoyagui, Chem. Lett., 203 (1974). T. Saji and S Aoyagui, J. flectroanal. Chem., 60, 1 (1975). T. Saji, T. Yamada, and S.Aoyagui, J. flectroanal. Chem.,61, 147
(1975).
I. Hanazaki and S.Nagakva, Bull. Chem. Soc. Jpn., 44, 2312 (1971). R. Ballardlni, G.Varani, L. Moggi, V. Balzani, K. R. Olsen, F. Scandola, and M. Hoffman, J. Am. Chem. SOC.,97,728 (1975). R. J. Watts, M. J. Brown, B. S.Griffith, and J. S.Harrington, J . Am. Chem. SOC.,97,6029 (1975). R. J. Watts, B. B. Griffith, and J. S.Harrington, J. Am. Chem. Soc.,
(1975).
98,674 (1976).
J. Demas, Jr., and G. Crosby, J. Am. Chem. SOC.,93, 2841 (1971). K. DeArmond, Acc. Chem. Res., 7,309 (1974). G. Crosby, Acc. Chem. Res., 8,231 (1975). N. Tanaka, T. Ogato, and S.Niizuma, Bull. Chem. SOC.Jon., 46,
N. Tanaka, T. Ogata, and S. Nilzuma, Bull. Chem. SOC. Jpn., 46,
3299 (1973). H. Caldararu, M. K. DeArmond, K. W. Hanck, and V. C. Sahini, J. Am. Chem. Soc., 98,4455 (1976).
3299 (1973).
Microwave Dielectric Relaxation of Some Lithium Salts in Dimethyl Carbonate D. Saar,+ J. Brauner, H. Farber, and S. Petruccl" Department of Nectrlcal Engineering and Chemistry, Polytechnic Institute of New York, Brooklyn and Farmingdale Campuses, Brooklyn, New York 11201 (Received August 30, 1977)
Complex permittivities (real and imaginary parts) of the solvent dimethyl carbonate (DMC) and of solutions of LiBr, LiSCN, and LiC104 in DMC at 25 "C, in the frequency range 0.45-67 GHz (wavelength 67-0.45 cm) are reported. The solvent shows a relaxation process which appears to be of the Debye type, within experimental error, with a relaxation frequency of 22 GHz. The electrolyte solutions show an additional relaxation process also of the Debye type, within experimental error, with relaxation frequencies in the range 1-1.5 GHz. Infrared spectra for LiSCN solutions show this salt to be completely associated to contact ion pairs or higher aggregates previously interpreted as ion-pair dimers. The correlation between the quantity (to - c m ) / p 2(using literature values for the dipole moments of the ion pairs) and the calculated concentration of the pairs, C,, is linear. It is concluded that, at the concentration investigated, the additional relaxation process observed in the electrolyte solutions is mainly due to rotational relaxation of contact ion-pair dipoles.
Introduction Lithium salts show remarkable solubility in relatively polar media of quite low permittivity. These systems are relevant to the development of energy sources such as batteries using nonaqueous solvents, and alkali metals as electrodes. Unfortunately, our knowledge of the structures This work is part of the thesis of D. Saar, in partial fulfillment for the requirements of Doctor in Philosophy (Chemistry),Polytechnic Institute of New York.
0022-3654/78/2082-0545$0 1 .OO/O
and dynamics of electrolytes in these media is quite limited, the most common misconception being that free ions are the preponderant species of these media, or that (at the other extreme) these systems may be treated as solvated fused ionic salts. Recently, Chabanel et a1.l have reported infrared spectra of LiSCN in the solvent dimethyl carbonate. Later2 dielectric constant measurements of the same system, together with other Li+ salts in DMC have been published. These data have been interpreted as indicating that the salts exist as contact ion pairs and 0 1978 American Chemical Society
546
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
ion-pair dimers (quadrupoles), free ions being almost nonexistent. It was decided in this laboratory to engage in microwave dielectric measurements of these same systems to ascertain whether the dynamics of the molecular species in the solution would confirm the information obtained from the static dielectric datal and the structural inferences of the vibrational spectraS2 (In fact, one of the spectra was repeated as shown below.) The coaxial instrumentation covering the range 0.3-9 GHz as reported previously3 has been supplemented by wave guide instrumentation reaching 66 GHz.
Experimental Section Equipment. The coaxial instrumentation consisting of a General radio bridge (0.3-1.5 GHz) and two reflectometers (2-4 and 8.5 GHz) has been described el~ewhere.~ Three new reflectometers have been built covering the KU band (with measurements taken at 16-17 GHz), the R band (measurements at 34 GHz), and the W band (54-66 GHz). The sources consisted of klystron oscillators. For the KU band a Varian X-12 klystron fed by a Polytech Res. Co. Type 809 klystron power supply was used. For the R band a klystron OK1 35V10 was used, whereas for the W band two klystrons OK1 55Vll and OK1 70V10 for 54 and 66 GHz, respectively, were used. For the R and W bands a Polytech Res. Co. Type 801-A universal klystron power supply was employed. The rest of the instrumentation consisted of commercial components such as, waveguides no. 91, 96, and 98, respectively. For each reflectometer instrument the sequence of components was power supply: klystron; ferrite isolator: rotary vane attenuator; frequency meter; two-directional couplers (to monitor both incident and reflected power) connected to crystal detectors and square-law meters; cell with reflector. The latter was a quarter wave transformer as described in the l i t e r a t ~ r e . ~ The cells were constructed from a piece of waveguide assembly surrounded by a jacket through which thermostatted water could be circulated. The reflector was driven up and down the guide by a micrometer transport mechanism with the reflector displacements shown by a height gauge which could be read to f 5 X in. Temperature was maintained within zt0.05 "C and checked (before each experiment) directly in the liquid contained in the wave guide by a thermistor probe connected to a calibrated dial-temperature indicator. The solution-air in. thick; interface consisted of a mica window, 3 X the window was held between waveguide flanges at the cell bottom. The density of dimethyl carbonate was determined at 25.0 " C by a pycnometer after calibration of its volume with distilled water. The resulting density was p = 1.063 g/cm3. The solvent viscosity was determined by a calibrated suspended level viscometer (Cannon no. 0). The viscosity was 7 = 0.00585 P at t = 25.0 "c. Materials. Dimethyl carbonate (Aldrich reagent grade) was distilled twice in an all-glass apparatus with a 3-ft column, connected to the outside atmosphere through bottles containing anhydrous Mg(C10& No grease was used in the ground-glass joints. LiBr (Fisher reagent grade) and LiC104 (Smith Cleveland) were dried in an oven a t 110 "C to constant weight. LiSCN.xH,O (Alfa Inorganics) was first dehydrated to the monohydrate LiSCN.H20 by keeping it under vacuum at room temperature. The temperature was then raised very slowly over a period of 1week still under vacuum until the temperature reached 110 "C and a constant weight of the sample was obtained. The sample, after transfer to a desiccator to cool, was
Petruccl et al.
1
J
05
I0
-
20
l5
1 mrh
25
d
40
4
1 C
31
02
03
@a
1 inch
+
05
06
0 7
9
Figure 1. Power profile A (decibels) vs. distance /(Inch) of the reflector from the window for 0.1 M LiBr in dimethyl carbonate at t = 25 "C and frequencies f = 16.27 and 66.6 GHz.
dissolved in the dimethyl carbonate, the addition of the salt taking 1-2 s. LiSCN is extremely hygroscopic and tends to decompose if dehydrated too quickly with the appearance of the characteristic yellowish color of sulfur. All solutions were kept in glass stoppered volumetric flasks until use. Filling the dielectric cells took 5-10 s. For the waveguide instruments the filling was effected with syringes. At least two solutions of each salt were prepared to check the reproducibility of the dielectric data.
Results and Calculations Figure 1shows representative plots of the power profile for frequencies of 16.27 and 66.6 GHz. The abscissa, 1, represents the distance in inches of the reflector from the mica window which defines the liquid-air interface. The linearized data calculated from the maxima and minima positions and the corresponding attenuations are displayed in Figure 2. In the ordinate, is the voltage reflection coefficient at a maximum or minimum at some distance 1, where y1/2
= p exp(-O.l1512AA)
and p is the voltage reflector coefficient at infinite distance. p is calculated from the relation
with ho the free space wavelength and X is the average wavelength in the liquid as calculated from the averaged distance between minima and between maxima (X/2); aX is an estimated quantity to be refined by an iterative procedure; AA = A - A , where A is the attenuation at the distance 1, A , is the attenuation at such large distances that interference between the power coming from the reflector and the power reflected from the window is not present. In other words, A, measures only the power from the window. The slopes of the lines in Figure 2 give the quantity aX/2 which together with the determined X gave a, the attenuation constant expressed in neper cm-'. Then, the real and imaginary parts of the permittivity are, for the waveguide measurements El Ell
= ( X o / X ) 2 [ 1 - (.X/2?7)2] = (Xo/X)2aX/7r
+ (X/X,)2 (1)
where A, is the cutoff wavelength for the waveguide, with
Dielectric Relaxation in Dimethyl Carbonate LiBr O.IM in DMC, t = 25'C\ f
e q
,
,
,
,
0
2
4
6
8
,
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978 547
individual experiment of power reflection with the liquids under study. A rotary vane calibrated attenuator was used for the crystal calibrations. The following systems were investigated: the solvent DMC, and the following solutions in DMC: LiBr (0.1 M), LiC104 (0.1 M), and LiSCN (0.09 M), all at 25 "C. A study of the concentration dependence of the dielectric measurements on the electrolyte solutions was not practicable because of apparatus limitations and in order to avoid possible complications at higher concentrations, as shown in the Discussion section. Concentrations lower than 0.1 M would have resulted in poor precision in the determination of relaxation parameters due to the small size of the effects and the limited selnsitivity of the method. Concentrations much higher than 0.1 M may correspond to formation of larger aggregates than ion pairs and dimers. Further 0.1 M corresponds to the upper limit of the Chabanel et a1.2 static dielectric investigation. Figure 3 shows c1 and dl for the solvent DMC plotted vs. frequency and also the Cole-Cole plot of el1 vs. 4 at various frequencies. The solid line is a plot of values of 2 and obtained from the Debye functions for a single relaxation freq~ency:~
16 274 GHz
,
,
,
, ]
IO
12
14
16
n _. I
I
LiBr O.IM In DMC; 1 = 2 5 O C i f=66.66 GHr
Ot
L
a
o
,
0
2
4
6
8
n
IO
12
I 14
where eo and are the static and high (with respect to f R ) frequency permittivities, respectively, and f R is the relaxation frequeqcy. The dashed line corresponds to a plot of values of c1 and el1 obtained from the Cole-Davidson expressions which apply to an asymmetric distribution of relaxation frequencies:
16
Figure 2. Quantities -In [(I""- p ) / ( l - pI"'*)] vs. n odd (maxima) and -In [ ( p =F 1"'2)/(1 = ,OI'~'~)] i vs. R even (minima) for 0.1 M LiBr in DMC at 25 OC and frequencies f = 16.27 and 66.66 Wz,respectively.
A, = 2b and b is the length of the longer side of the rectangular cross section of the guide. The values of A and A , in decibels were corrected for deviations from square law behavior by the crystal detector. To do this, readings of the response to calibrated power variations were made. These were done after each
where p is a parameter related to the asymmetry in the
LiBr O.IMIn OMC t.25-C
t I
2,
20
Cde-Cole plot
05
-
I 5-
0
Flgure 3. Plots of e' and e'' vs. frequency f(GHz) and Cole-Cole plots of 6" vs. E' for the solvent DMC at 25 OC and for 0.1 M LiBr in DMC at 25 OC. Solid lines are calculated functions f9r one or two Debye relaxation functions as indicated. Dashed line corresponds to a Cole-Davidson distribution function.
548
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
Petrucci et al.
45
;tu 30-
IO QI
'
02
1
05
IO
50
20
IO
20
50
0
100
f(GHz1-
4
LiSCN 0 0 9 M in DMC
LiSCN 009M in DMC
t=25'c
t 4 iu
3 3
2 2 I5Ol
02
05
IO
20
50
IO
20
50
100
f(GHz1-
Figure 4. Plots of E' and E" vs. frequency f(GHz) and Cole-Cole plots of E" vs. E' for 0.1 M LiC104 and 0.09 M LiSCN in DMC at 25 OC. Solid lines are calculated functions for two Debye relaxation functions as indicated in the text.
TABLE I: Dimethyl Carbonate Relaxation Parameters According t o a Debye Single Relaxation Function and a Cole-Davidson Distribution and LiBr, LiSCN, and LiClO, in Dimethyl Carbonate Relaxation Parameters According to Two Debve Relaxation Functions
DMC
3.12 3.12
2.35 2.35
22 20
0.35 0.29
1 0.9
0.22 0.27
Electrolyte in DMC
z: IecI z: i e c l l
Electro1Yte
€0
LiBr LiC10, LiSCN
3.45 4.25 3.95
fr,
fW1
3.20 3.30 3.30
2.40 2.40 2.45
f,
1.5 22 1.2 18 1.0 18
- € I /
0.24 0.64 0.47
- ell1
0.19 0.28 0.29
distribution of relaxation frequencies. In Table I, the calculated quantities eo, E,, f R , and /3 are reported for the Debye and Cole-Davidson equations together with the summations of the absolute deviations C~E:- ell and Clc:l - PI. The values of eo, E,, fR, and /3 (in the Cole-Davidson equations) finally accepted were those which minimized the above summations. It may be seen that the values of the summations are comparable for the two sets of equations, On the frequency scale (Figure 3) it seems that the very high frequency 2 points are fitted better by the Cole-Davidson equation whereas the intermediate frequency loss data are fitted better by the Debye equation. In total, since the addition of an adjustable parameter, P, to the Debye equations does not seem to improve the fit in a pronounced way, it may be satisfactory to conclude that the Debye equations are able to fit the data for the solvent within experimental error. Note, also, that the calculated value eo = 3.12 is within 1% of the measured static value of the permittivity reported by Chabanel et a1.,2 eo = 3.09. In Figures 3 and 4 the values of c1 and el1 for LiBr, LiC104, and LiSCN solutions in DMC are plotted vs. frequency and as Cole-Cole plots of d1vs. c1 at various frequencies. The solid lines are obtained by plotting values
where eo is the static permittivity, E,~,is the permittivity at f >> fR1, is the high frequency permittivity ( f >> f R 2 ) , and fR1 and f R 2 are the relaxation frequencies. The values of cm2, and f R 2 resemble the corresponding parameters for the pure solvent not necessarily being identical with them. The values of to and fR1 refer to an additional relaxation process due to the presence of electrolyte in the solution. In what follows it will be our concern to try to characterize the molecular source of this additional relaxation process.
Discussion Simple electrostatic theories predict that 1:l electrolytes (-0.1 M) in media of permittivities between three and four should be completely associated into ion pairs with the probable presence of larger aggregates. Although it has been shown3by statistical calculations that the ion pairs in media of low permittivity are largely of the contact type, it is always preferable to have experimental evidence for the structure of the system. Structure of the Ionic Aggregates. The infrared spectrum of a 0.2 M solution of LiSCN in DMC is shown in Figure 5. The spectrum was taken differentially with respect to the pure solvent in closely matched demountable cells with CaF, windows and a path length of about 0.055 mm. The two frequencies of 2044 and 2070 cm-I are qualitative indications of the positions of the two apparent maxima. The shoulder at higher wavenumbers may be due to incomplete matching of the solvent cell (the solvent has
Dielectric Relaxation in Dimethyl Carbonate
The Journal of Physical Chemistry, Vol, 82, No. 5, 1978 549
V ( cm-I 1
center frequency of the band, a is the variance, bound to the width of the function a t half intensity I:/2 by the 6 . the present case the expression ui = ( A ~ , ) ~ / ~ / 1 . 4For parameters used were vl = 2067 cm-l, v2 = 2052 cm-l, (AvJlj2 = (AvJlj2 = 13 cm-l. This last value has been found to be characteristic of the free -SCN ion at -2060 cm-1.6 The above fit ensures that another band is not hidden at -2060 cm-l; hence it confirms Chabanel et al.l findings of the absence of spectroscopically free SCN, for the present system. The above IR spectrum is in rather close agreement with the one obtained by Chabanel et a1.l on the same system (absorption maxima at 2068.2 and 2040.4 cm-l). These authors by comparison with many other solvent systems for the electrolyte LiSCN assigned the 2068-2075-~m-~ band to the CN stretching of the contact ion-pair LiNCS. They also assigned the 2040.4-~m-~ band to an aggregate which later, by analysis of the dielectric data,2 was identified as an ion-pair dimer. As proof of these assignments it was shown' that the similar band at 2041 cm-' in ethyl acetate ( e t 5 = 6.02) disappeared when the solvent N,N-dimethylacetamide ( e t 5 = 37.8) was added. Eventually, the band at 2060 cm-l due to spectroscopically free SCN- appeared again upon further addition of the N,N-dimethylacetamide. It was, in fact, contended' that the absence of the 2060-cm-' band in DMC indicated the absence of solvent separated ion pairs as well as free SCN- since the infrared spectra of these two species would be indistinguishable. On the basis of the above information (supported qualitatively by our data), it was concluded' that LiSCN in DMC exists as "contact ion-pairs and contact aggregates with no appreciable amounts of free ions or solvent separated pairs present". Spectra taken at lower concentrations were said' to be qualitatively similar with the exception of the relative intensity of the two bands. Calculations of Distribution Functions. By regarding the above interpretation as correct, the first consequence in terms of analyzing the present dielectric relaxation data is that any theoretical approach should refrain from using theories which employ the concept of relaxation by transport of charges at long range. There are no free ions present and the IR spectra suggest the species present are in contact. Objections might be voiced, however, about the identification of individual ion pairs or even quadrupoles in a relatively concentrated ionic solution. In other words, it may be contested that ions do not belong to any single partner during the period of the applied field and that systems like the present ones should be treated as solvated fused ionic salts. This proposal would try to interpret the observed dielectric relaxation phenomen as due to the relaxation of the transport of charges at short ranges without the concept of ion pairs, that is, without the hypothesis of rotating dipoles. The above proposal would probably be acceptable a t high concentrations (above 0.5-1.0 M) in a electrolyte composed of spherical ions with nondirectional bonding which would be predominantly ionic. However, many systems are subjected to other forces and covalent interactions become the relevant ones at short range. In order to give a more quantitative discussion of the correctness of assigning an individual partner to any particular ion, that is, defining a stoichiometric mass law for ion-pair formation, the following treatment is presented. Fuoss7 in discussing ionic distribution functions predicted the existence of a minimum in the distribution
I
Figure 5. Infrared spectrum of 0.2 M LiSCN in DMC. Differential' spectrum with respect to the solvent DMC: cell depth, 0.055 mm; windows, CaF,; scan time, =30 min. Raman S p e c t r u m of L i S C N 024M i n Dimethyl carbon ate
-A 2100
2000
2060
2040
2020
Figure 6. Raman spectrum of 0.237 M LiSCN in DMC: slit, 3 Cm-'; scan speed, 0.1 cm-'ls; source, 19433-cm-' line of an argon ion laser; power, 1.4 W.
an absorption band around 2060 cm-l which hinders quantitative interpretation of the spectrum) or an additional small band due to the solute. Note also that the asymmetric stretching band of SCN due to the CN stretching vibration, commonly present a t 2060 cm-' is absent. As a further experimental check of the above, the Raman spectrum of 0.23, M LiSCN in dimethyl carbonate is reported in Figure 6. The spectrum was recorded by a Jarrell-Ash 25-100 1.0 m double Czerny-Turner monochromator, a RCA 31034 selected photomultiplier, a SSR Model 1105/1120 photon-counting system, and a W + W 1100 digital recorder. The source was the 19433-cm-l line of a Spectra-Physics 165 argon-ion laser. The recording conditions were as follows: slit, 3 cm-'; scan velocity, 0.1 cm-'/s; photon counter time constant, 10 s. (The spectrum was recorded a t the University of Waterloo, Ontario through the courtesy of Professor D. E. Irish). From Figure 6 two bands, one-centered at -2067 cm-' and the other at -2052 cm-l are visible, the band of free SCN at 2060 cm-' due to the antisymmetric stretching of the CN bond being absent. The advantage of having obtained a Raman spectrum is the possibility of a quantitative band analysis. Indeed in this frequency range the solvent DMC does not show any absorption band overlapping with the ones of the solute. Digitized Raman intensities from Figure 6 could be quantitatively reproduced, within experimental error, by the summation of only two Gaussian-Lorentzian product functions of the type5
where 3 is the wavenumber (frequency) in cm-l, vi is the
Petrucci et
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
550
al.
TABLE 11: Interparticle Potential Energies (U), Calculated Values of the Minimum in the Distribution Functions Corresponding Values of the Volume (V), and Maximum Concentrations (c,)"
(r-),
Ua Ion-ion -e /re Ion-dipole -ep /r2e Dipole-dipole
M
rrnimb cm e2/2ekT= 70.0 X lo-*
V,c cm3 1.437 X l o - '
1.16 x 10-3
(ep/ekT)'/2= 15.28 X lo-'
1.49 X l o - "
0.112
( 3 ~ ~ / 2 e k T )="8.36 ~ X lo-*
2.447
lo-''
0.679
(12E/kT)1/6u= 2.42 X l o - '
5.937 x 10-23
X
Cmax
jd
-p2/r3e
London
28.0
- 4E(u /r)6 a
U = -C/rn.
P =8X
lo-''
rmin = (nC/2kT)'/". V = (4/3)nrmin3. cm esu, E = 10 kcal/mol, u = 1 x cm.
(roughly equivalent to the minimum in the Bjerrum distribution q = IZ+Z-le2/2DkT)and of a maximum also roughly equivalent to the Debye 1/X distance between the central ion and its atmosphere (at the maximum of its charge density). It was also predicted7 that the maximum and the minimum coverge to an inflection point by increasing the concentration thereby having the distinction between free ions and individual pairs lose significance. One should remember, however, that in all these calculations only electrostatic forces, and hence, electrostatic potential energies were retained. Mathematically, if one assumes the simple Bjerrum distribution function A P = ( 4 n r 2 A r ) ( C L / 1 0 0 0 )e x p ( - U ( r ) / k T )
(VI
to get the value of r which minimizes the expression in the usual way (assuming Ar is fixed to a small value, e.g. 0.1 A) one obtains dAP
o=--
dr
4nLC
d A r- [r2 exp(- U(r )/ k 2') ] 1000 d r
--
One sees that the minimum shifts to smaller values of r , the greater the inverse power of r in the expression for U(r). Specifically, by setting U = C/r" one obtains rmin = (nC/2kT)lIn. Since the Coulombic potential energy depends on l/r, all other interactions such as ion-dipole, dipole-dipole, and London forces depending on higher inverse powers of r the minimum shifts to small values of r in the latter cases. The importance of rmhlies in the fact that at distances beyond rminfrom the central particle by definition all association ceases. To illustrate the above let us begin with the familiar Coulombic potential energy U = -ez/rt. In Table I1 the value of rmin = qBjerrum = e2/2ekT = 70 x cm is reported. This has been calculated using a typical (for our systems) value of E = 4 at T = 298.15 K for a 1:l electrolyte. corresponding to rminis also The volume V = 4/3~rmin3 reported in Table I1 together with cmax,the maximum concentration of the electrolyte such that the ions on the average will approach no closer than rmin,In the above (1/V) = cmaXxLx where L is Avogadro's number. In this calculation mutual interactions have been neglected and a random distribution of particles has been assumed which is at variance with more precise calculations. Table I1 reports the corresponding calculations for the ion-dipole potential energy function U = -ep/r2t (the first entity being a triplet for a pair to lose its identity). Again e = 4 and the chosen 1.1 = 8 X esu cm have been retained. It results that cmax= 0.112 M. By extending the calculation to point dipole-point dipole 0.679 M interactions with U N - p 2 / r 3 e one has c,, (Table 11). This means that above a concentration of
cmax = l/(lO-'VL).
e
Parameters used e = 4, T = 298.15 K,
electrolyte of 0.68 M single ion pairs no longer act as independent species and hence the simplest structures must be regarded as quadrupoles. Note that this last example is relevant to our case. I n LiSCN there are no free ions which can form pairs or triplets but in accord with the IR spectra only pairs and quadrupoles.li2 T h e n the relevant mutual potential energy in the system should be for dipole-dipole interactions and our concentrations, 0.1 M , are well below the critical cmax. Finally, it may be of interest to use a Lennard-Jones potential energy retaining only the London attractive component which varies as the inverse sixth power of r. Then U = - 4 E ( ~ / r )where ~ , t is taken as one, in the spirit of the short range of the potential energy under consideration. E is the depth of the potential energy well and cr is the finite distance where U ( r ) = 0 (for the complete Lennard-Jones potential energy, i.e., by retaining the attractive component of U(r)as well). In this case we find rmin= (12E/kT)1/6a.Using the reasonable values of E = erg/molecule and u = 1 X 10 kcal/mol or 6.94 X cm one obtains rmin= 2.42 X cm. This means an enormous concentration is required before the character of a bond loses individuality because of packing effects. The above calculations add support to the IR spectra evidence for the existence of discrete species in the system 0.1 M LiSCN in DMC. Elaboration of Relaxation Parameters. Assuming that ion pairs and quadrupoles are the major species present in the LiBr, LiSCN, and LiC104 solutions in accord with Chabanel et one notes that the difference, to - tml, which depends on dipole concentration and dipole strength, increases (Table I) for the three electrolytes LiBr, LiSCN, and LiC104 being 0.25, 0.65, and 0.95, respectively. Bottcher8 has developed an equation which relates EO eml to the dipole concentration and dipole moment. The relation i n a form expressed by Knightg reads )=
( E o --E
w1
4n (10-3)LC,U, 3~0 (1 - a f ) 2 3 h T ( 2 ~ to 1)
where C, is the concentration of ion pairs or dipoles, f = (2to - 2 ) / ( 2 c 0 + l ) a 3is the reaction field factor, a is the cavity radius, and 01 is the polarizability of the dipole having a dipole moment p located in the center of the cavity. Chabanel et a1.2 have reported several formation constants for quadrupole or ion-pair dimer formation related to the equilibrium 2P + P, K,= ~ i / [ 2 ( 1 - ~ i ) ~ C ] C(1
- a)
(al2)C
For LiBr K,(DMC) = 90 M-l and for LiSCN K (DMC) = 20 M-l. Thus at C = 0.1 M, 01 = 0.79 for LiBr an% C,(LiBr) = (1- 01)c= 0.021 M.
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978 551
Dielectric Relaxation in Dimethyl Carbonate
TABLE I11 : Macroscopic and Microscopic Relaxation Times, Radius of the Spherical Dipole a from Eq VIII, and Internuclear Distances r Calculated from Dipole Momentsa for LiBr, LiClO, , and LiSCN in Dimethyl Carbonate at 25 "C
Dielectric Correlation Piot LI Salts in Dimethyl Carbonate T= 2 5 O C
-
!i u
t
Electrolvte LiBr LiCIO, LiSCN
LiBr
a
cp(*)
-
Figure 7. Plot of the quantity [(eo - emr)/pU2] vs. 3Cpe0/(2c0-t 1) for LiBr, LiSCN, and LiC104 in DMC: t = 25 OC.
Similarly, for LiSCN a t C = 0.09 M, a = 0.544 and C,(LiSCN) = 0.037 M. For LiC104no equilibrium constant for quadrupole formation in DMC has been reportedS2 However, the constant has been determined in DEC (diethyl carbonate) to be Kq(LiC104)= 90 M-l. It has been pointed out2 that Kp is a measure of the associating power of the solvent, Le., the order of increasing Kq is independent of the salt; the associating power seems to be an intrinsic property of the solvent. Since it is known2 that in DEC K,(LiSCN) = 140 M-l and Kq(LiBr) = 650 M-' one may form the ratios: K,(LiSCN) in DMC = o.143 K , (LiSCN) in DEC and K,(LiBr) in DMC = o,138 K, (LiBr) in DEC Taking 0.14 as an average ratio one may form a similar ratio for LiC104: K,(LiC104) in DMC --= 0.14 K , (LiC104) in DEC 90 M-' from which one finds K (LiC104)in DMC = 12.6 M-l. By retaining tentatively t i i s value one can solve for the concentration of pairs of LiC104 in DMC. With C = 0.1 M, a = 0.54 and C, = C(l - a ) = 0.046 M. From eq VI, neglecting individual differences in the values of (1- an2for LiBr, LiSCN, and LiC104,one would expect the quantity (eo - eml)/p2to be roughly proportional to 3C,e0/ (260 + 1)if the observed solute relaxation process is a t all related to the rotational relaxation of ion pairs. The values of the dipole moments have been reported by Chabanel et aL2to be p(LiBr) = 8.1 D, p(LiSCN) = 10.5 D, and p(LiC10,) = 10.6 D (1D = 1 X 10-ls esu cm). The plot of the quantity (eo - e,,)/p2 vs. 3C,to/(2eo + 1)is shown in Figure 7. The solid line has been calculated by linear regression. The intercept = -0.0095 X slope = 1.314 X and the correlation coefficient r2 = 0.954. The line goes through the origin within the experimental error as expected. The quantity
4n(10-3)L 1 3kT (1 -
- 6.129 x
-
1034
(1-
The average of the quantity (1- an2for the three salts calculated from polarizabilities2 is 0.887 assuming a cavity
Reference 2.
10"~.s 10'O~'.s
1.06 1.33
1.03 1.23
1.59
1.51
108a,
lO*r(p),
cm
cm
3.8, 4.1, 4.3,
2.49 2.88
Equation VIII.
of radius a = 4 X cm, so that [ 4 ~ ( l O - ~ ) L / 3 k T ] [ l / ( l - an2]= 6.9 X which is within a factor of 2 of the slope in Figure 6. In view of the approximate nature of the existing theories correlation among the relaxation times of the three electrolytes is more uncertain than the one shown above for the quantity eo - eml/p2. The microscopic relaxation time (or autocorrelation time) T~ may be calculated (among several options)1° by the Powles-Glarum relation
where T = (2rfR)-l. T' in turn can be recalculated from the expressionll = (/2kT, with 5 the frictional coefficient. By treating the molecule as a spherical dipole of radius a and by retaining the bulk viscosity of the solvent as the one surrounding the dipole, one writedl (VIII) This relation is known to be approximate and to give results for a of the proper order of magnitude, at best. By introducing Table I11 reports the values for T,,;. these values in eq VIII, using the measured solvent viscosity 7 = 0.00585 P at 25 "C, one obtained the values of a reported in Table 111. It may be seen that these values are somewhat larger than the ones quoted by Chabane12 from calculated dipole moments. The differences, however, in view of the limited value of eq VIII, do not warrant speculations about solvation. Similarly introduction of model corrrections (prolate or oblate ellipsoid)12for the ion pairs (in order to rationalize the differences) do not seem to be justified in view of the approximations involved in eq VIII. Conclusions The conclusion emerging from the above analysis is that (in the frequency range investigated) the major source of the observed dielectric relaxation associated with the solute is due to rotational relaxation of ion-pair dipoles, the ion-pair dimers only making a minor contribution. The minor contribution of the ion-pair dimers for the systems investigated may be due to their lower concentration2 (LiC104),to their low2dipole moment (LiBr), or because of a combination of the two factors.
Acknowledgment. The authors are indebted to Professor B. Senitzky, Director of the Long Island Graduate Center of Farmingdale of PINY, for use of the waveguide equipment necessary to perform this work. References and Notes (1) M. Chabanel, C. Menard, and M. Guiheneuf, C. R. Acad. Sci. Park, 272, 253 (1971). (2) D. Menard and M. Chabanel, J . Phys. Chem., 79, 1081 (1975).
552
A. K. Thakur, A. Rescigno, and C. DeLisi
The Journal of Physical Chemistry, Vol. 82,No. 5, 1978
(3) H. Farber and S. Petrucci, J . Phys. Chem., 79, 1221 (1975). (4) W. E. Vaughan In "Dielectric Properties and Molecular Behaviour", Van Nostrand, Reinhold, New York, N.Y., 1969, p 146. (5) A. R. Davis, D. E. Irish, R. B. Roden, and A. J. Weerheim, Appl. Spectrosc., 28, 384 (1972). (6) D. E. Irish, S. Y. Tang, H. Talts, and S. Petrucci, to be published. (7) R. M. Fuoss, Chem. Rev., 17, 27 (1935); J. Am. Chem. Soc., 57, 2604 (1935). ( 8 ) C. J. F. Bottcher, "Theory of Electrical Polarization", Elsevier,
Amsterdam, 1952. (9) E. A. S. Cavell, P. C. Knight, and M. A. Sheik, Trans. Faraday Soc., 67, 2225 (1971). (10) M. Davies in "Dielectric Properties and Molecular Behavlour", Van Nostrand, Reinhold, 1969, p 298. (11) P. Debye, "Polar Molecules", Chemical Catalog Co., New York, N.Y., 1929. (12) F. Perrin, J. Phys. Radium Paris, 5 , 497 (1934); E. Fischer, Physica, 7, 60, 645 (1939).
Stochastic Theory of Second-Order Chemical Reactionst Ajlt K. Thakur," Aldo Resclgno, and Charles DeLisl Laboratory of Theoretical Biology, National Cancer Institute, National Institutes of Health, Bethesda, Maryland 200 14 (Received April 25, 1977; Revised Manuscript Received September 21, 1977) Publication costs assisted by the Natlonai Cancer Institute
A stochastic theory of the reversible bimolecular reaction 2A C is formulated and its thermodynamic and kinetic properties investigated in detail. The distribution function was obtained exactly by an analytical method with the system in equilibrium, and by numerical solution of the Kolmogorov equation for the more general time-dependent situation. The solutions provide a standard against which various approximation procedures of general interest are evaluated; in particular, calculation of the moments and inference of the distribution by an expansion over an appropriate orthonormal basis set. The determinants of an efficient approximation procedure in terms of constraints on the moments are delineated in a general way and then illustrated by application to other commonly studied bimolecular systems.
I. Introduction The stochastic theory of compartmental analysis in general and first-order chemical reactions in particular is well developed and has made possible a clear understanding of a number of simple chemical reactions both near and far from eq~ilibrium.l-~ The stochastic theory of second- and higher-order chemical reactions, on the other hand, is considerably less developed and consequently such processes are not as fully understood. Exact expressions for the stochastic mean and the generating function at equilibrium have been found for a number of reversible reaction ~ c h e m e s ~and , ~ expressions *~ for time evolution of the stochastic mean and variance as well as the entire distribution have been obtained for some irreversible r e a ~ t i o n s . ~ J -Darvey l~ and Ninham'l have investigated some aspects of the reversible kinetic problem A + B F? C D based on different constraints and firstand second-order perturbation methods. Besides that, little seems to have been done in the way of a complete analysis of both the equilibrium and kinetic stochastic properties of even a relatively simple reversible secondorder chemical reaction. The purpose of this paper is twofold. First, we present a detailed investigation of the reaction
+
2A* C
as a function of time, system size, and kinetic parameters. We obtain the exact distribution function at equilibrium by a combinatorial method which for this particular problem appears to be simpler and more direct than a +Partsof this work were presented at the 21st Annual Meeting of the Biophysical Society held at New Orleans, La., 1977 and appeared as an abstract. * Present Address: Endocrinology and Reproduction Research Branch, National Institute of Child Health and Human Development, NIH, Bethesda, Md. 20014. This article not subject to US.Copyright.
generating function approach. Far from equilibrium we have solved the master equation numerically. The other aspect of the problem is the use of these exact solutions as a standard against which the utility of approximate computational procedures of general interest are evaluated in particular, the calculation of moments and the inference of the distribution from an expansion over a Gaussian basis set.
11. Formulation of the Equations Let N 2C + A be the total number of units in the system. We take N to be time independent so that it is necessary to keep track of only one type which we choose as A. Let P,(t) be the probability that the system is in state i at time t ; i.e., Pi(t) is the probability of finding i units of type A in the system a t time t. Then if aAt and @At are, respectively, the transition probabilities for the formation and dissociation of C in the small time interval At
P i ( t + A t ) = a('
+
') Pi+ ,(t)A t (/3/2)(N- i 2)Pi.z(t)At + [l - a ( $ ) A t - ( / 3 / 2 ) ( N i ) A t ] P j ( t )+ O ( A t )
...
+ (1)
The first term on the right is the probability that a system in state i + 2 will undergo a transition to a system in state i as the result of a pairwise interaction between two A's to form a C; the second the probability of a transition from state i - 2 to i as the result of dissociation of a C; and the third the probability that a system in state i remains in that state during At. Other terms, which are not written explicitly, represent multiple associations and/or dissociations and are proportional to terms in At higher than the first order. Rearranging eq 1 and letting At go to zero gives the forward Kolmogorov differential-difference equation for the system: Published 1978 by the American Chemical Society
