Mobility and Ionic Association of Lithium and Quaternary Ammonium Salts in Propylene Carbonate and...

Mobility and Ionic Association of Lithium and Quaternary Ammonium Salts in Propylene Carbonate and /-Butyrolactone

Makoto Ue* Mitsubishi Petrochemical Company, Limited, Tsukuba Research Center, 8-3-1 Chuo, Ami, Inashiki, Ibaraki 300-03, Japan

ABSTRACT

The conductivities of lithium and quaternary ammonium salts have been measured in propylene carbonate and ~-butyrolactone at 25~ over the concentration range of I0 2 to i0 3 mol dm -3. The limiting molar conductivities and the ionic association constants were calculated by the expanded Fuoss-Hsia equation. The mobility of anions in both solvents decreased in the following order: BF4 > CIO~ > PF~ > AsF~ > CF3SO3 > (CF3SO~)2N- > C4F9SO~ > BPh~. The association constants increased in the following order: Li(CF3SQ)~N, LiAsF6 < LiPF6 < LiCIO~ < LiBF4 < LiC4FgSQ < LiCF3SQ, while those of the corresponding tetraethylammonium salts were leveled by electrostatic shielding effect of alkyl groups. These results were correlated with ionic radii obtained by MM2 calculation.

Propylene carbonate (PC) and ~-butyrolaetone (GBL) are important dipolar aprotic solvents used for nonaqueous electrolytes I-4 in high energy lithium batteries, 5 aluminum electrolytic capacitors, 6-8 and electrical double-layer capacitors. 9'~~ There have been numerous studies regarding the conductivity of nonaqueous electrolytes containing these solvents at high concentrations 11-19 and the eonducto- metric analysis in the dilute range was also performed in order to gain a better understanding about the solvent- solute interactions (20 to 35 for PC; 20, 28, 29, 31, 36 to 38 for GBL).

LiC104 has long been used as an electrolyte salt in lithium batteries, however, its undesirable safety characteristic requests a more stable and safer salt. 19'a9 Although lithium salts having a strong electron-withdrawing anion such as LiBF~, LiPF6, LiAsF~, LiCF3SO3, Li(CF3SQ)~N, and LiC~FgSO3 are commercially used in lithium batteries, the conductometric studies on these salts in dipolar aprotic solvents are very scarce compared with those of LiCIO4.

Since the conductometric parameters in pure solvents are important information for investigating the practical binary or ternary mixed solvent systems, a systematic study on the transport properties of these lithium salts in pure solvents is desirable. This kind of basic research has been carried out especially for propylene carbonate by several groups, however, a perusal of the literature shows re- markable deviations in the data.

The author has carried out a systematic conductometric analysis on these lithium salts in propylene carbonate and ~-butyrolactone in order to compare the mobility and ionic association tendency of each anion. The corresponding quaternary ammonium salts having the same anions, which are also important solutes for nonaqueous electrolytes in electrical double-layer capacitors, were also examined in both solvents for obtaining a more consistent set of the single-ion limiting molar conductivities and for comparing the difference between R4N + and Li + in the association constants with each anion.

In spite of the fact that the radii of ions and solvents are very important information for interpreting the obtained parameters such as single-ion limiting molar conductivity and association constant, different values have been adopted by the several authors. It seems that incorrect or inconsistent data have been used without precaution. The author has systematically calculated the radii of ions and solvents by using the molecular models constructed from both the crystallographic data and computer-aided molecular dynamics.

The mobility and ionic association of each anion were discussed from this consistent set of ionic radii.

Experimental Reagents.--LiCl04, Et4NCIO4 (Japan Carl i t Co.), LiBF4,

Et4NBF4, LiPF6, Et4NPF6, LiCF3SO3, Et~NCF3SO3 (Morita

* Electrochemical Society Active Member.

Chemical Industries), LiAsF6 (Lithco), Li(CF3SO2)~N (3M), LiC4FgSO 3 (Tochem Products Co.), Bu~NBPh4 (Aldrich Chemical Co.), and Bu4NCIO4 (Kodak Chemical Co.) were purified by recrystallization. Me4N(CF3SO2)2N and Et4N(CF3SO2),~N were prepared by the titration of bis(trifluoromethylsufonyl)imide 4e with tetramethyl- or tetraethylammonium hydroxide aqueous solution (SACHEM, Inc.) followed by recrystallization. All the salts were vacuum dried at 100~

Propylene carbonate and ~-butyrolactone (Mitsubishi Petrochemical Co., Battery-grade F-PC, and F-GBL) were used without further purification.

Electrolyte preparation.--An electrolyte salt was dis- solved in propylene carbona te and ~-butyro lac tone to make about 1 mol dm -3 solutions. The wa te r content in these mothe r solut ions (20 to ]00 ppm) was measured by a mois ture me te r (Mitsubishi Kasei Co., CA-06) and the con- centra t ions of the solutions were cal ibrated. The mothe r solutions were d i lu ted to about 10 -2 mo] dm -3 by the solvent to make the stock solutions, whose densit ies at 25~ were measured by a digi ta l densi ty mete r (PAAR, DMA 45).

Conductivity measurements.--The stock solutions were successively di lu ted and the mola r concent ra t ions C (tool dm 3) were ca lcula ted f rom the concent ra t ions m ' (tool kg-~-solution) and the densit ies d ( g c m -z) of each solut ion by C = m'd. The densit ies were ca lcula ted f rom d = do + Dm', where do is the densi ty of solvent and D is a charac ter i s t ic cons tant of the electrolyte.

The conduct iv i ty measurements were carr ied out at 25 _+ 0.1~ by a br idge (Toa Electronics, CM-25E) and a LCR mete r (Ando Elect r ic Co., AG-4311) equipped wi th a s t andard- type conduct iv i ty cell wi th a cell cons tant 0.1017 cm -~ (Toa Electronics, CG-2001PL).

Molecular modeling.--The sizes and volumes of molecules were ca lcula ted f rom the molecu la r models con- s t ructed by a molecu la r model ing system CHEMLA B-I I (Molecular Design Co.) on a VAX8350 (Digital E q u i p m e n t Corp.). The MM2 program was used for s t ruc ture op t imiza- tion. All g raphica l d iagrams were genera ted by the CSC Chem3D Plus sof tware (Cambridge Scient i f ic Comput ing) on a Macin tosh Quadra 840AV (Apple Computer, Inc.).

Table I. Physical properties of the solvents at 25~

Solvent do (g cm -3) 'q (cP) e,. K (S cm -1) q (nm)

PC 1.1998 2.513 ~ 64.92 ~ 0.9 • 10 -7 0.432 GBL 1.1248 1.727 b 41.77 b 2.5 • 10 -7 0.671

a Ref. 23. b Ref. 29.

3336 J. Electrochem. Soc., Vol. 141, No. 12, December 1994 �9 The Electrochemical Society, Inc.

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J. Electrochem. Soc., Vol. 141, No. 12, December 1994 �9 The Electrochemical Society, Inc.

Table II. Molar conducfivities in PC at 25~

I03 C A (tool dm 8) (S crn 2 tool -I) 103 C A

LiCIQ Et4NCIQ

9.7362 24.220 9.0763 29.099 7.2990 24.594 6.8071 29.486 4.8663 25.052 4.5385 30.122 2.4338 25.766 2.2698 30.668 1.7056 26.009 1.5985 31.035 0.9767 26.324 0.9078 31.407

LiBF4 Et4NBF4

10.6130 24.461 10.6761 30.302 7.0786 25.430 8.0065 30.751 3.5394 26.406 5.3386 31.396 2.4835 26.963 2.6705 32.040 1.4210 27.348 1.8732 32.464 0.9472 27.617 1.1627 32.733

LiPF6 Et4NPF6

9.4052 23.382 9.7288 28.052 7.0557 23.741 7.2784 28.703 4.7040 24.152 4.8650 28.985 2.3518 24.730 2.4327 29.650 1.6519 25.098 1.7046 29.983 0.9443 25.321 0.9737 30.257

LiAsF 6 Et4NCF3SO3

9.9775 23.093 9.8549 27.185 7.4817 23.566 7.3907 27.604 4.9870 23.924 4.9296 27.996 2.4938 24.504 2.4655 28.721 1.7483 24.687 1.7282 28.996 0.9977 25.018 0.9877 29.372

LiCF3SO~ Me4N(CF~SOf)~N

10.3714 20.413 9.9101 25.833 7.7770 21.128 7.4314 26.228 5.1853 21.949 4.9541 26.627 2.5920 23.068 2.4785 27.278 1.8176 23.482 1.7339 27.660 1.0394 24.032 0.9928 27.812

Li(CF~SQ)~N Et4N(CF~SO~)~N

9.7428 20.067 9.5686 24.926 7.3039 20.374 7.1747 25.271 4.8716 20.755 4.7815 25.684 2.4449 21.301 2.3963 26.315 1.7063 21.573 1.6760 26.617 0.9776 21.902 0.9594 26.954

LiC~F~SO~ Bu4NC10~

9.5203 17.501 9.6754 24.889 7.1394 18.238 7.2550 25.212 4.7587 18.757 4.8379 25.695 2.3809 19.577 2.4191 26.377 1.6680 19.952 1.7084 26.680 0.9531 20.366 0.9705 27.007

Bu4NBPh~

10.1579 14.906 7.6219 15.234 5.0831 15.691 2.5433 16.164 1.7816 16.395 1.0181 16.708

Results and Discussion The re levan t p roper t ies of p ropy lene ca rbona t e and ~-

bu ty ro lac tone used in th is work are s u m m a r i z e d in Table I, whe re do, ~, er, K, and q are density, viscosity, re la t ive per- mitt ivity, e lectrolyt ic conductivi ty, and Bje r rum cri t ical d is tance , respectively.

M o l a r c o n d u c t i v i t i e s . - - T h e mola r conduct iv i t ies A were ca lcu la ted f rom the expe r imen ta l e lectrolyt ic conduc t iv i - t ies ~ a f te r cor rec t ion for the electrolyt ic conduc t iv i ty of the solvent K. The mola r conduct iv i t ies over the concen- t r a t ion range of 10 2 to 10 -~ tool dm 3 in p ropy lene ca rbon- ate and ~]-butyrolactone are given in Tables II and III, respectively.

A n a l y s i s o f c o n d u c t i v i t y d a t a . - - T h e data were pre l imi- na ry analyzed in terms of the Shedlovsky equation. ~ How- ever, very small or negative association constants were of- ten obtained, presumably due to the neglect of short range

3337

interactions in Onsager limiting law, 3 Therefore, the data were analyzed by means of the Fernandez-Print expan- sion ~2 of the Fuoss-Hsia equation 43

A = A0 - S(C~) I/2 + E(Cw) in (Cw) + JI(Cw)

- J2(C'y) ~/2 - KA(C'y)f~A

S = ~Ao + B E = E1Ao - E2

J1 = okAo + ~2 '/2 = %Ao + ~r4 [1]

where A0, KA, % and f . are the l imi t ing mola r conductivi ty, the associa t ion cons tant , the degree of d i ssoc ia t ion and the m e a n mola r act ivi ty coefficient, respectively. All t he coeffi- c ients ~, ~, E~, Ez, a~, af, a3, and a~ d e p e n d on the solvent p roper t ies (er, ~) and the charge on the ions. a,, af, a3, and a4 also d e p e n d on the d i s tance R of closest a p p r o a c h of ions.

The m e a n mola r ac t iv i ty coeff ic ient was ca lcu la ted by the Debye-Hi icke l equa t ion 3

Table Ifl. Molar conducfivities in GBL at 25~

103 C A (mol dm -3) (S cmfmol -~) 1O ~ C A

LiCIO4 Et~NCIO4

9.5897 35.992 10.3057 40.381 7.1911 36.705 7.7294 41.239 4.7939 37.600 5.1540 42.345 2.3979 39.033 2.5756 43.854 1.6799 39.436 1.8048 44.466 0.9614 40.253 1.0337 45.274

LiBF 4 Et4NBF4

10.1799 36.263 10.0073 42.594 7.6317 37.312 7.5056 43.468 5.0014 38.579 5.0029 44.624 2.5482 40.322 2.5036 46.194 1.7834 41.112 1.7456 46.819 1.0189 42.055 1.0060 47.662

LiPF6 Et4NPF6

10.0067 34.812 10.0110 39.232 7.4990 35.452 7.5069 39.943 4.9974 36.269 5.0020 40.954 2.5022 37.477 2.5035 42.321 1.7517 37.963 1.7563 42.875 1.0036 38.560 1.0018 43.672

LiAsF6 Et4NCF3SQ

9.5542 34.147 10.0793 37.349 7.1634 34.781 7.5590 38.120 4.7754 35.568 5.0414 39.146 2.3898 36.740 2.5188 40.595 1.6740 37.245 1.7668 41.090 0.9582 37.779 0.9810 42.050

LiCF~SO3 Me4N(CF3SQ)fN

10.2819 30.028 8.4817 36.579 7.7048 31.078 6.3538 37.183 5.1396 32.503 4.2368 37.942 2.5696 34.363 2.1212 39.035 1.8050 35.319 1.4369 39.479 1.0285 36.092 0.8481 40.090

Li(CF3SQ)fN Et4N(CF3SOf)~N

9.5764 29.369 10.6250 33.931 7.1799 29.896 7.9614 34.649 4.7914 30.607 5.3106 35.448 2.3930 31.592 2.6555 36.641 1.6787 31.929 1.8652 37.171 0.9591 32.531 1.0648 37.753

LiC4FgSO3 Bu4NCIO4

10.0028 25.678 8.9950 36.148 7.4976 26.509 6.7464 36.827 4.9991 27.575 4.4976 37.742 2.4997 29.064 2.2495 39.098 1.7546 29.636 1.5769 39.603 1.0015 30.383 0.9251 40.319

Bu4NBPh4

10.4833 20.752 7.8644 21.279 5.2452 21.896 2.6410 22.927 1.8393 23.345 1.0500 23.856



3338 ,1. Electrochem. Soc., Vol. 141, No. 12, December 1994 �9 The Electrochemical Society, Inc.

Table IV. Derived parameters for PC and GBL electrolytes at 25~

Salt Ao

PC GBL A0 (GBL) log/CA (GBL)

/CA era Ao /CA ~ Ao (PC) log/CA (PC)

LiC1Q 27.35 2.8 0.063 42.44 LiBF~ 28.83 8.4 0.047 44.64 LiPF8 26.31 2.1 0.019 40.73 LiAsF 8 26.00 1.1 0.038 39.91 LiCFsSO3 25.33 16.1 0.088 38.86 Li(CFsSQ)~N 22.80 1.5 0.036 34.43 LiC~F~SQ 21.45 12.3 0.017 32.65 Et~NC104 32.43 3.1 0.013 47.77 Et4NBF4 33.96 3.0 0.014 50.19 Et4NPF~ 31.33 2.0 0.029 45.99 Et~NCFsSQ 30.39 2.3 0.062 44.30 Me~N(CFsSQ)eN 28.90 1.8 0.007 42.07 EqN(CF~SQ)~N 27.92 2.0 0.031 39.98 Bu4NCIQ 28.02 2.8 0.037 42.48 Bu4NBPh4 17.61 2.4 0.031 25.70

8.4 0.026 1.55 2.0 16.7 0.086 1.55 1.3 5.7 0.049 1.55 2.3 5.7 0.055 1.53 - -

25.0 0.018 1.53 1.2 3.7 0.034 1.51 3.1

18.1 0.084 1.52 1.2 9.5 0.062 1.47 2.0 9.9 0.053 1.48 2.1 7.8 0.100 1.47 3.0 9.5 0.045 1.46 2.7 4.8 0.069 1.46 2.7 5.6 0.039 1.43 2.4 8.8 0.042 1.52 2.1 6.3 0.032 1.46 2.1

A0 in S em z mol -~,/CA in dm 8 tool -~.

Inf._ = - A (C~/)4n/[1 + BR (C]) ~ ] [2]

whe re A a n d B are t h e coeff ic ients d e t e r m i n e d by t he sol- ven t p rope r t i e s (e~) a n d the cha rge on the ions.

The a s soc ia t ion c o n s t a n t was e v a l u a t e d acco rd ing to t he mass ac t ion e q u a t i o n ~

~:~ = (1 - ~) /C~f~ [3]

w h e r e t he ac t iv i ty coeff ic ient for the n e u t r a l ion p a i r is a s s u m e d to equa l unity.

A l t h o u g h the d i s t ance p a r a m e t e r s R~ a n d R~, w h i c h are de r ived f rom J~ a n d J2, m u s t be c o m p a t i b l e w i t h R~ in Eq. 2, t he re exis ts a n i ncons i s t ency b e t w e e n d i s t ance p a r a m e - t e r s Y Jus t i c e r e c o m m e n d e d t h a t Rz~ a n d R~ be set equa l to the B j e r r u m cr i t i ca l d i s t ance q. in Eq. 4. ~ F e r n a n d e z - P r i n t sugges ted f u r t h e r t h a t the be s t r e su l t s for s l ight ly associ - a t ed e lec t ro ly tes are o b t a i n e d by f ix ing these p a r a m e t e r s , a n d t hey are set equa l to q, w h e n a s soc ia t ion is a s s u m e d to be of a n e l ec t ros ta t i c n a t u r e ~

q = z~'e~/2e~kT [4]

w h e r e z, e, k, a n d T are cha rge number , e lec t ron ic charge , B o l t z m a n n cons t an t , a n d abso lu t e t e m p e r a t u r e , r espec- tively.

A d o p t i n g th i s a s sum p t i on , the t w o - p a r a m e t e r Eq. 1 was so lved b y a n i t e r a t ive l eas t s q u a r e m e t h o d s imi la r to t h a t desc r ibed b y KayJ ~ A BASIC p r o g r a m was u sed to solve the above procedures , w h e r e a n in i t i a l va lue of A0 was ca lcu- l a t ed f rom the S h e d l o v s k y e q u a t i o n a n d i t e r a t i ons were c o n t i n u e d u n t i l t he e r rors in % A0, a n d KA b e c a m e less t h a n I0 ~, ]0 -~, and 10 -~, respectively.

Derived parameters are summarized in Table IV with the standard error of estimate ~, which is defined by the following equation

~ = [ Y ~ ( A o ~ - A o ~ ) ~ / ( n - 2 ) ] ~/~ [ 5 ]

where n is the n u m b e r of d a t a po in ts . The s t a n d a r d er rors of e s t ima te were less t h a n 0.1 a n d s t a n d a r d dev ia t ions in A0 and/CA were less t h a n 0.1 a n d 1, respect ively.

Limi t ing molar conduct iv i t ies . - -The r a t ios of the l imi t - ing m o l a r conduc t iv i t i e s in G BL to those in PC [A0 (GBL)/ A0 (PC)] are also g iven in Table IV. They r a n g e d f rom 1.51 to 1.55 for l i t h i u m sa l t s a n d f rom 1.43 to 1.48 for t e t r a e t h y l - a m m o n i u m salts , respectively. S ince t he r a t i o of the viscos- i ty of GBL to t h a t of PC is 1.46, h i g h e r va lues obse rved for l i t h i u m sa l t s sugges t the ex i s tence of a s l igh t ly s t ronge r Li§ i n t e r a c t i o n in PC t h a n in GBL.

Single-ion l imiting molar conduct iv i t ies . - -Conduct iv i ty d a t a of Bu~NC104 a n d Bu~NBPh~ were also a n a l y z e d in o rde r to spl i t a l im i t i ng m o l a r conduc t i v i t y in to two s ing le - ion l i m i t i n g m o l a r conduc t iv i t i e s , X~ a n d kS. Xo (BPh$) in PC a n d GBL were ca l cu l a t ed f rom empi r i ca l ly ad ju s t ed e q u a t i o n 4~

)to = 0.8204/~q[5.35 - (0.0103 e~. + 0.85)] [6]

The o b t a i n e d k : (PBh~) gave X~ (Bu4N +) f rom A0 (Bu~NBPh4), w h i c h t h e n g e n e r a t e d ko (C10~) f rom A 0 (Bu4NC1Q). The r e su l t i ng X$ (C10~) = 18.93 S cm ~ mol -~ in PC was c o m p a r a b l e w i t h r e p o r t e d va lues (18.78, 22 18.44, 23 18.40, 3~ 18.89 35], a n d the ca l cu l a t ed t r a n s p o r t n u m b e r t~ (Et4NCIQ) = 0.416 was a l m o s t the s ame as t he e x p e r i m e n - t a l ly obse rved va lue (0.417 47). A l t h o u g h no re l i ab le conduc t i v i t y d a t a in p u r e GBL exist , t h e r e su l t i ng ko (CIO~) = 28.45 S cm 2 tool 4 was s imi l a r to the r e p o r t e d va lues [26.23, 2~ 27.20, 3~ 27.77 37] a n d the ca l cu l a t ed t~ (LiC1Q) = 0.330 was close to t he e x p e r i m e n t a l l y obse rved va lue (0.36 ~). F u r t h e r m o r e , c a l c u l a t e d t~ (Bu4NCIQ) = 0.330 was very close to t h a t in PC (0.324) in th i s w o r k a n d those in o the r d ipo la r ap ro t i c so lvents (DMF, 0.326 2; DMSO, 0.329 z).

The r e su l t i ng cons i s t en t set of the s ing le - ion l im i t i ng mo- l a r conduc t iv i t i e s in PC a n d GBL are g iven in Table V. D e v i a t i o n s in the s ingle ion l im i t i ng m o l a r conduc t iv i t i e s ca l cu la t ed f rom the l i m i t i n g m o l a r conduc t iv i t i e s of Et4N + sal ts a n d Li § sa l t s were at mos t 0.03 a n d 0.11 S cm 2 tool -~ in PC a n d GBL, respectively. The ra t ios of t he s ing le - ion l im- i t ing m o l a r conduc t iv i t i e s in GBL to those in PC [)t o (GBL) / X0 (PC)] r a n g e d f rom 1.43 to 1.51 excep t for Li § Bu4N § a n d BPh$.

The obtained single-ion limiting molar eonductivities in PC agreed with the literature values within 0.5 S cm 2 tool -z in error except for BF~ and AsF~ as shown in Table V. As for GBL, there is no reliable data, and the values presented in Table V can be regarded as the first reliable set of the single-ion limiting molar conduetivities in GBL.

Table V. Single ion limiting molar conductivities in PC and GBL at 25~

PC GBL

Ion ko Literature ko Literature

LF 8.43 8.32, 48 7.86 38 13.99 12.95. 28 15.3031 Me4N + 14.50 14.17 48 21.52 25.07 28 Me3EtN + 14.16 20.77 Me2Et~N + 13.90 20.18 MeEt8 N+ 13.68 19.70 Et4N * 13.50 13.19 48 19.32 20.43 28 Bu4N + 9.09 9.00 48 14.03 11.75 28 C10$ 18.93 18.43, ~8 18.89 38 28.45 26.23, 28 27.20 81 BF~ 20.43 19.59 3. 30.77 PF~ 17.86 17.46 48 26.70 AsF~ 17.58 14.67, ~ 19.23 33 25.92 20.16, 28 31.06 81 CF~SO7 16.89 16.90 s0 24.93 (CF3SQ)~N- 14.40 14.90 38 20.55 C4F980~ 13.03 18.66 BPh~ 8.52 8.10 ~8 11.67 11.52 28

ko in S cm 2 mol -~.



J. Electrochem. Soc., Vol. 141, No. 12, December 1994 �9 The Electrochemical Society, Inc. 3339

Fig. 1. Space-filling models of ions and solvents constructed by MM2 calculation.




Table Vl. Geometric parameters obtained from crystallographic data and MM2 calculations.

Ion V

From crystallographic data From MM2 calculation

r a b c c /a V r a b c c /a

L i §

Me4N + 0.086 Me3EtN + Me2EtoN + MeEt3N + Et4N + 0.159 Pr~N § BurN + 0.290 C10~ 0.055 BF$ 0.049 PF~ 0.069 AsF~ 0.073 CF3SO ~ 0.080 (CF3SO2)~N- 0.146 C~FgSO~ BPh~ 0.310 PC GBL

0.076 . . . . . . . . . . 0.274 0.519 0.519 0.519 1.00 0.095 0.283 0.663 0.646 0.628 0.95

. . . . . 0.111 0.298 0.800 0.673 0.566 0.71

. . . . . 0.129 0.313 0.921 0.673 0.556 0.60

. . . . . 0.146 0.327 0.921 0.801 0.577 0.63 0.336 0.746 0.746 0.736 0.99 0.170 0.343 0.799 0.787 0.595 0.74

. . . . . 0.231 0.381 1.080 1.003 0.488 0.45 0.411 1.138 1.097 0.637 0.56 0.299 0.415 1.344 1.171 0.770 0.57 0.236 0.547 0.528 0.524 0.96 0.056 0.237 0.543 0.538 0.508 0.93 0.227 0.521 0.492 0.461 0.89 0.051 0.229 0.521 0.491 0.479 0.92 0.254 0.612 0.516 0.516 0.84 0.069 0.254 0.610 0.610 0.610 1.00 0.259 0.632 0.539 0.539 0.85 0.073 0.260 0.634 0.634 0.634 1.00 0.267 0.565 0.560 0.547 0.97 0.082 0.270 0.625 0.551 0.518 0.83 0.326 0.980 0.629 0.543 0.55 0.144 0.325 0.994 0.581 0.558 0.56

. . . . . 0.163 0.339 1.082 0.610 0.573 0.53 0.420 1.091 1.035 0.959 0.88 0.309 0.419 1.068 1.017 0.992 0.93

. . . . . 0.088 0.276 0.781 0.641 0.430 0.55

. . . . . 0.081 0.268 0.680 0.552 0.422 0.62

V in nm3; r, a, b, and c in nm.

It b e c a m e ev iden t t h a t mobi l i t i e s of t he a n i o n s in b o t h so lvents dec rease in the fo l lowing o rde r

BF~ > CIO~ > PF$ > AsF~ > CF3SO~

> (CF3SOs)2N- > C~F6SO~ > BPh~

In a p rev ious paper , 9 t he a u t h o r ha s s h o w n t h a t t he s in - g le - ion l im i t i ng m o l a r conduc t iv i t i e s of q u a t e r n a r y a m m o - n i u m ions can b e e s t i m a t e d f rom t h e i r f o r m u l a weight , because ionic mob i l i t y co r re l a t e s l i nea r ly w i t h t he r ec ip roca l of s q u a r e roo t of f o r m u l a w e i g h t J 6 The rev i sed va lues for u n s y m m e t r i c a l q u a t e r n a r y a m m o n i u m ions b e t w e e n te - t r a m e t h y l - a n d t e t r a e t h y l a m m o n i u m ion a re p r e s e n t e d in Table V. However , a s imi l a r r e l a t i o n s h i p was no t obse rved for t he a n i o n s due to t h e i r l a rge d i f ference in s t ruc tu re .

A s s o c i a t i o n c o n s t a n t s . - - D e r i v e d as soc i a t i on c o n s t a n t s g iven in Table IV s h o u l d be used on ly for the c o m p a r i s o n of each sa l t a n d so lven t in t h i s work , becaus e it is wel l k n o w n t h a t d i f fe ren t expres s ions in E, J1 a n d J2 in Eq. i give different/CA, a n d th i s effect is l a rge r w i t h sma l l e r a s soc ia t ion c o n s t a n t s (hA < 20 d m 3 mot- l ) . 42 E v e n w h e n t he same ex- p re s s ion was used, d i f fe ren t ion size p a r a m t e r R g e n e r a t e d fa i r ly different /CA p a r t i c u l a r l y for G BL solut ions .

Assoc ia t ion c o n s t a n t s in G BL were l a r g e r t h a n those in PC due to the di f ference of r e l a t ive pe rmi t t iv i t i e s . The ra - t ios of t he l o g a r i t h m of a s soc ia t ion c o n s t a n t s in G BL to those in PC [log KA (GBL)/ log/CA (PC)] a re also g iven in Table IV. They were close to t he r a t i o of r ec ip roca l of r e l a - t ive p e r m i t t i v i t y of G BL to t h a t of PC, [e[ ~ (GBL)/e[ ~ (PC)] = 1.6. I t b e c a m e ev iden t t h a t the a s soc ia t ion t e n d e n c y of the a n i o n s w i t h Li + in b o t h so lven ts inc reases in the following o rde r

(CF3SOs)2N-, AsF~ < PF$ < C]O~ < BF~ < C4F6SO~ < CF3SO~

However , a s soc ia t ion w i t h Et4N + was leveled by the elec- t ro s t a t i c sh i e ld ing effect of a lkyl g roups on n i t rogen . The a s soc i a t i on c o n s t a n t s for (CFsSO~),N a n d PF$ were s t i l l sma l l e r t h a n those for o t h e r anions .

C a l c u l a t i o n o f i o n s i z e a n d s o l v e n t s i z e . - - T h e r ad i i of ions a n d so lven ts give very i m p o r t a n t i n f o r m a t i o n for in - t e r p r e t i n g the de r ived p a r a m e t e r s f rom c o n d u c t i v i t y a n a l - ysis. However , i nco r r ec t or i n c o n s i s t e n t d a t a h a v e b e e n u sed f r e q u e n t l y w i t h o u t p recau t ion . The a u t h o r ha s sys- t e m a t i c a l l y d e t e r m i n e d ion size a n d so lven t size by com- p u t e r - a i d e d m o l e c u l a r model ing .

There is d i f f icu l ty in se lec t ing a r e p r e s e n t a t i v e va lue for m o l e c u l a r size, w h e n t he molecu le ha s a s h a p e fa r f rom spher ica l . Therefore , t he a u t h o r ha s a d o p t e d t he r a d i u s de- f ined by t he fo l lowing e q u a t i o n

r = (3V/4~r) ~/~ [7]

w h e r e V is the v a n de r Waals volume.

Fo r t he ca l cu l a t i on of the van de r Waals volume, t he v a n de r Waals r ad i i of a toms p r e s e n t e d b y B o n d i 51 were used. These are as fol lows H; 0.120, B; 0.190, C; 0.170, N; 0.155, O; 0.152, F; 0.147, C1; 0.175, P; 0.180, S; 0.180, As; 0.185 rim.

S ince the v a n de r Waals r ad ius of B was no t g iven by Bondi , i t was e s t i m a t e d f r o m Table I in Ref. 51.

The coo rd ina t e s of a toms in t he molecu le were o b t a i n e d by two me thods , c ry s t a l l og raph i c d a t a (x - ray or n e u t r o n d i f f rac t ion) a n d MM2 ca lcu la t ion .

The Car t e s i an coo rd ina t e s of each a t o m were g e n e r a t e d f rom the s t r u c t u r e s of Me4NI ~" for Me4N +, Et4NL 55 for Et4N +, Bu4NL54 for Bu4N +, Me~NC104 ~ for C10~, Me4NBF4 5~ for BF~, Me4NPF6 ~7 for PF~, (CICH2)2SAsF6 58 for AsF$, HaOCF3SO3 58 for CF3SO$, a n d Me4NBPh4 so for BPh$. The coo rd ina t e s of (CFsSQ)2N- were m a d e b y re- p l ac ing H a toms in H50(CH3SQ)2N 61 b y F a t o m s w i t h keep ing ave rage C-F d i s t ance 0.1305 n m obse rved in H3OCFsSO3.

MM2 calculation was carried out by optimizing an initial two-dimensional structure. All unknown bending and tor- sional parameters were supplemented by the method desc r ibed by Yoshida. 62 Comple t e t e t r a h e d r a l or o c t a h e d r a l mode l s were used w i t h o u t c a l cu l a t i on for C1OL BF~, PF~, a n d A s F L The space- f i l l ing mode ls are dep ic t ed in Fig. 1 a n d t he v a n de r Waals vo lumes V a n d the de r ived v a n de r Waals r ad i i r are g iven in Table VI.

All the molecu les were o r i en t ed s u c h t h a t t h e i r p r i n c i p a l axes were co inc iden t w i t h e x t e r n a l X, Y, Z c o o r d i n a t e axes, a n d the longes t axis a n d the sho r t e s t axis were f i t to ex ter - na l X a n d Z coo rd ina t e axes, respectively. The p r i n c i p a l axes were d e t e r m i n e d by the c o m p u t a t i o n of t he m o m e n t s of ine r t i a I g iven b y the fo l lowing e q u a t i o n

Z = Era, l} [8]

w h e r e m a n d l are t he mass of a t o m a n d the d i s t ance f rom a n axis, respect ively. N o t a t i o n s a, b, a n d c in Table V[ are t he X, Y, Z l eng ths of a molecu le w h e n i t is f i t in to the smal l e s t b o x a n d c / a becomes an easy i n d e x of d e f o r m a t i o n f rom sphe r i ca l shape.

The ionic r ad i i o b t a i n e d by MM2 ca l cu l a t i on were very close to those o b t a i n e d by c r y s t a l l o g r a p h i c d a t a w i t h - in 0.003 n m in e r ro r excep t for r ad i i of t e t r a a l k y l a m m o - n i u m ions, because C-H d i s t ance can be gene ra l ly u n d e r e s - t i m a t e d by x - r a y c rys ta l lography. The a u t h o r h a s dec ided to use va lues o b t a i n e d by MM2 ca l cu l a t i on s u m m a r i z e d in Table VII, because some d i s to r t ion is obse rved for the mode ls f rom c ry s t a l l og raph i c da ta , a n d t he mode l s f rom MM2 ca l cu l a t i on are t h o u g h t to be c loser to t he r ea l s t a t e in solut ion.

The o b t a i n e d ionic rad i i for t e t r a a l k y l a m m o n i u m ion ag reed w i t h the va lues o b t a i n e d by a s imi la r m e t h o d 63a a n d by the m e a s u r e m e n t of p a r t i a l m o l a r vo lume, 6~ w h i c h are



J. Electrochem. Soc., Vol. 141, No. 12, December 1994 �9 The Electrochemical Society, Inc. 3341

Table VII. Estimated ionic radii. Table VIII. Contact distances and calculated association constants.

Ion This work Literature PC GBL

Salt ~ K~ KB/K. Ks KB/KA Li + 0.076 0.060, 68 0.078 4 MeaN + 0.283 0.283, 63 0.285, 64 0.347 Me3EtN ~ 0.298 0.305 ~ LiC1Q 0.313 1.2 0.4 11.5 1.4 Me~Et2N + 0.313 0.319 ~ LiBF 4 0.305 1.3 0.2 11.9 0.7 MeEt~N ~ 0.327 0.332 ~ LiPF~ 0.330 1.0 0.5 10.7 1.9 Et4N + 0.343 0.339, 63 0.348, 64 0.400 ~ LiAsF6 0.336 1.0 0.9 10.4 1.8 Pr4 N§ 0.381 0.381, 6~ 0.398, 6~ 0.452 ~ LiCF~SQ 0.346 0.9 0.1 9.9 0.4 Burn + 0.415 0.415, 6~ 0.437, ~4 0.494 : Li(CF3SO~)2N 0.401 0.3 0.2 7.5 2.0 C]O~ 0.237 0.240, 6~ 0.283 31 LiC~FgSQ 0.415 0.2 0 7.0 0.4 BF~ 0.229 0.232, ~ 0.278 3~ Et4NC1Q 0.580 0 0 2.2 0.2 PF~ 0.254 0.295 66 E t 4 N B F 4 0.572 0 0 2.3 0.2 AsF~ 0.260 0.384 ~ 0.326 ~ Et4NPF~ 0.597 0 0 1.8 0.2 CF~SO~ 0.270 0.602 ~ Et~NCF~SQ 0.613 0 0 1.4 0.2 (CF~SO~)~N- 0.325 0.450 ~ Me~N(CF~SO~)~N 0.608 0 0 1.5 0.3 C4F9SO~ 0.339 Et4N(CF~SO:):N 0.668 0 0 0.4 0.1 BPh~ 0.419 0.480, ~ 0.535 ~ Bu~NC104 0.652 0 0 0.7 0.1

Bu4NBPh4 0.834 0 0 0 0

r in nm. in nm, KB in dm 3 tool ~.

listed in Table VII. The traditional values given by Robinson and Stokes ~ are systematically larger than our values, because they considered the distance from the center of N atom, along the line of the C-C bond, to the surface of H atom.

The obtained ionic radii of CIO$ and BF~ agreed well with thermoehemical radii. 6~ It is evident that ionic radii of PF~ 66 and AsF~ ~7 are overestimated and those of CF3SO~ 31 and (CF3SO2)2N- 3~ are wrong.

Although the most frequently used ionic radius of LF is 0.060 nm, 68 it was estimated to be 0.076 nm by averaging the effective ionic radii for different coordination numbers presented by ShannonP 9

Applicability of Stokes Iaw.--Stokes showed t h a t the force f ac t ing on a spher ica l par t ic le of rad ius r moving wi th un i fo rm veloci ty of v in a con t inuous f luid of viscosi ty ~ is given by the equa t ion ~

f = 4~r~rv + 2 ~ r v = x ~ r v [9]

where the f i rs t t e rm represen t s a force due to pressure bui l t up in f ron t of it and the second t e r m is a f r ic t ional force para l le l to its surface. If the b o u n d a r y b e t w e e n the par t ic le and the f lu id is slippy, the second t e rm d isappears .

However, th is S tokes equa t ion mus t be modi f i ed by us ing a numerical factor x less than 6 in Eq. 9, when applied to very small molecules (r < 0.5 nm). 6~'7~

The author has attempted to determine an empirical correction factor, because it has practical utility. Walden prod-

0.6

perfect slip / / PC 0 cation 0.5 _ / �9 anion

GBL [] cation �9 anion

0. 0.4

0 E

0.3 effect stick

O9 ... Bu4N + [] ~" Li+ O

0.2 Ph4B

0.1

0 I I I I I I

0 2 4 6 8 10 12 14

r-1 / nm-1

Fig. 2. Walden products as a function of the reciprocal of ionic radii.

ucts k0~ of each ion are plotted against the reciprocal of ionic radii r in Fig. 2.

The theoretical behavior calculated from Stokes law in Eq. 9 is given by the bottom line for perfect stick (x = 6) and by the top line for perfect slip (x = 4). All ions except Li § MeaN § Bu4N § and BPh$ approached the behavior for perfect slip and average factor was 4.3. This result suggests that ionic mobilities of the most ions discussed here are determined only by ion size. A slightly higher factor for Mean § (x = 4.7) indicates the existence of weak solvationJ 9 Higher factors for Bu4N § (x = 5.0) and BPh~ (x = 5.7) are well understood, since larger ions approach the behavior for perfect stick. I

Assuming the correction factor for solvated Li § is 5.7, the solvation number N~ was estimated by the following equation I

N~ = 4~(r~ - r~)t3V [10]

where rs, ro, and V are S tokes r ad ius of so lva ted Li*, crys ta l - lographic rad ius of LF and the molecu la r vo lume of solvent, respectively. The solvat ion n u m b e r s were 3.2 for PC and 2.3 for GBL.

Applicability of Bjerrum theory.--Within the f r a m e w o r k of Jus t ice ' s t r e a t m e n t (R = q), the assoc ia t ion c o n s t an t is s t r ic t ly def ined for e lec t ros ta t ic in te rac t ions by the Bier- rum equa t ion 3

KB = 4~rNA f : exp (e2/erkTR)R2dR [11]

where NA is Avogadro number , d r epresen t s the con tac t d is- t ance for two spher ica l ions, and q is some cr i t ical d i s t ance at wh ich ion pa i r ing no longer is r ega rded to exist . For a con tac t ion pa i r & is s imply the sum of the ionic rad i i (r§ + r_) and for a s o l v e n t - s e p a r a t e d ion pa i r d is the sum of the ionic radi i and the d i a me t e r of the solvent (r. + r + 2r~).

Theoret ica l assoc ia t ion cons tan t s KB were ca lcu la ted f rom Eq. 11 us ing the contac t d i s t ance ~. The resul ts are given in Table VIII, w h e r e the ra t ios of the theore t ica l assoc ia t ion cons tan t to the observed one are given. All observed assoc ia t ion cons tan t s in PC were la rger t h a n calcu- la ted ones, wh ich indica tes con tac t ion pa i r s are fo rmed in PC. On the other hand, LiCIO4, IAPF6, LiAsF6, and Li(CF3SQ)2N in GBL showed smaller association constants. Salomon et al. have obtained similar results, and they concluded that alkali metal salts form solvent-separated ion pairs in GBL, even though they used a too small diameter (0.2 64 nm) for GBL. 28 If the solvent-separated ion pairs are formed, all lithium salts should be completely dissociated by Eq. Ii.

The author considers that this kind of argument is diffi- cult for slightly associated electrolytes, because the present conductance theories do not have such completeness as to afford an exact association constant (ambiguity of the coef-




ficients in Eq. i, inconsistency of ion size parameter, and validity of Bjerrum theory itself).

Conclusion 1. The order of mobil i ty in PC and GBL is BF~ > C104 >

PF~ > AsF~ > CF3SO3 > (CFaSQ)~N > CaFgSO~ > BPhg. 2. The order of dissociation abi l i ty in PC and GBL is

Li(CF~SQ)2N, LiAsF6 > LiPFG > L iCIQ > LiBF4 > LiC4FgSQ > LiCF3SO3.

3. Ionic radi i obtained by MM2 calculation are sui tably applicable to Stokes law.

4. Association constants can be reasonably compared, even though ambigui ty exists in the absolute values of association constants.

Acknowledgments The author would like to thank Ms. S. Sekigawa for

preparat ion of imide salts and Mr. S. Yoshida for his help in calculat ing unknown MM2 parameters. Also he thanks Professor M. Ueno and Professor K. Ibuki at Doshisya Uni- versity for providing a Fuoss-Onsager program.

Manuscript submit ted Apri l 15, 1994; revised manuscript received July 13, 1994.

Mitsubishi Petrochemical Company, Limited, assisted in meeting the publication costs of this article.

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