Indian Journal of ChemistryVol. 25A, February 1986, pp. 116-122

Ion-pairs in Aqueous Solutions & Transition Model of Electrolytic Behaviour

P P SINGH·, H P DAHIYA & Y K SHARMA,Department of Chemistry, Maharshi Dayanand University, Rohtak 124001

Received 10 December 1984; revised and accepted 5 September 1985

Equivalent conductance (A)data at 25°C of the aqueous solutions of RbCI04• CsCIO•• RbCIO). CsCIO). KCIO•• KCIO),KNO). AgNO). CuSO•• ZnSO. and MgS04, 7HzO suggest that these electrolytes form ion-pairs in solution and that A dataare best explained by a model that assumes that these solutions are characterized by the following equilibria:

K.

A + + 8 - ¢ (A + .8 -) (uncharged pairs)

K;(A + .8-) ~ (A + ---8 -) (solvent separated ion-pairs)

The A0 and KA values evaluated from this model also compare very well with the corresponding values obtained by the usualmethods. The model also does not require the adjustment of the ion size parameter.

The interpretation of conductance behaviour ofelectrolytes has so far been based on the assumptionthat an ion is a rigid charged sphere that moves in ahydrodynamic and electrostatic continuum! -9. Suchan assumption, however. is valid for very dilutesolutions (C < 0.005M) and fails to hold forconcentrated solutions. The situation is all the moreacute!" -16 when it comes to an analysis of theconductance data of electrolytes that form ion-pairs inwater. This calls 17 for a suitable approach to expressthe conductance data of these electrolytes in water.According to Bockris and Reddyl7 the approach muststart from an altogether different model of ionicbehaviour. Singh 18 has recently proposed a model ofionic behaviour that assumes that at any concentrationeach ion has a definite probability to be in either theDebye or the lattice configuration and that theprobability that an ion in an m molal solution is in theDebye configuration is given 18bye -15 m. This model I K

has been quite successful in explaining 18- 21 a numberof thermochemical properties of simple 1 : 1 and 2: 1electrolytes in water in the concentration range C=0.0005 to 1.0 M. Further, this model has recentlybeen utilized to derive an expression+' for theequivalent conductance (A) of an electrolyte by takinginto consideration the effect of the external electricalfield on the behaviour of these ions. According to thisapproach. A for a C molar solution of an electrolyte isgiven by Eq. (I).

II (A+BAo)fl!2e-ISmA = /\ - - --1+-r,fJT;r-

_(I_('-I~[SICI/3+QIC] ... (I)

where

116

... (2)

... (3)

(AO)L = equivalent conductance at infinite dilution ifthe lattice configuration holds. and all the other termshave their usual significance22.23• This approach 22 ,

however, tacitly assumes that as long as an ion is in theDebye configuration, the equilibrium ionic andpotential distributions are given by the classicalDebye-Huckel solution!", and that of the tworetarding forces arising due to electrophoretic andrelaxation effects, the relaxation retardation is handledby the solution of the time-dependent (a.c.) or time-independent (d.c.) solution of the fourth orderassymetric Poisson-Boltzmann equatiorr'". This isstrictly valid for very dilute solutions only: forconcentrated solutions such a problem is insurmount-able. Since the probability that an ion is in the Debyeconfiguration has been shown 18 to be given by e -ISn,

and as it is very small for C> 0.02 M solutions weexpect that no significant uncertainties in thetheoretical A values would be introduced if it isassumed that these are given as a zerothapproaximation by the same classical results.However. in the lattice configuration the changed ionicand potential distributions around the central ionshould have been taken into consideration whileworking out the electrophoretic and relaxationalretarding forces. As the mathematical problemsassociated with such an approach are truly formidable.it has been circumvcntcd+' using an elementary

SI~GH et al.: ION-PAIRS IN WATER & TRANSITION MODEL OF ELECTROLYTIC BEHAVIOUR

treatment 1-. While such a simplistic treatrnent" of thevarious retardational forces has been quite successfulto predict:" A of a number of I : I electrolytes upto C= 1.0 M. it would be interesting to see if this approachcould be utilised to extract useful information aboutsuch electrolytes as RbClO... RbCl03• CsClO .•.CsC103• KCIO .•. KC103• KN03• AgN03• CuSO .•.ZnSO .•and MgSO .•which are known to form ion-parsin water+': I .•. 13. Although conductance equations dueto Fuoss' and Justicell are available to extract such aninformation about these electrolytes from theirmeasured A. there is a considerable controversyregarding the conductance equation that should beutilized for the purpose. Thus while Fuoss 10 maintainsthat the conductance equation due to Justicell is basedon a model that is physically unrealistic and that theparameters J 3: Z and K A obtained therefrom are mainlycurve fitting parameters without physical significance,Justice and Justice.'? maintain that the classical Fuoss-On sager expansion (Eq. 4)

A = A 0 - SC1 Z + EC log C

=J(a).C+J3!z(a).C3i2+0(C) ... (4)

is based on a restrictive primitive model that relies onthe use of the linearlized Boltzmann-Poisson equation -their conductance equation uses the more exactdistribution function of Meeron. On the other handPethybridge and Taba 13 have recently analysed theirconductance data of MgSO .• > HzO in water in termsof the conductance equation due to Pittsz6.1 .• and alsodue to Fuoss " and have concluded that whereas theFuoss conductance equation+' gives the best fit over amoderate concentration range, it requires very large d-values (which in turn depend on the concentrationrange analysed). Further Glueckauf?" maintains thatthe Pitts conductance equationZ6.1 .• is justified for m< O.OOS and that beyond this point species likeM(SO .•h. M2(SO .•)2+ and uncharged Mz(SO .•)z (M=metal) species arise in increasing proportions.Glueckauf"" further concludes that the success ofPethybridge and Taba to fit their conductance data 13

with the Pitts equatiorr'";' .•upto C=0.12 M must beregarded as a coincidence. In view of these the analysesof the A data of RbCIO .•, RbCI03, CsCIO .•. CsCI03•

KCIO .•. KCIO-,. KN03• AgN03• CuS04• ZnSO .• andMgSO .• in terms of this approach+ to extract someuseful information about these electrolytes thatform 1.'."".30 ion-pairs in water would be even moreinteresting.

Results and DiscussionIn Eq. (I). C refers to the concentration of ions that

are free and is equal to the analytical concentrationonly if every ion from the ionic lattice from which theelectrolyte has been obtained is stabilized as an

independent mobile charge carrier. However, as aresult of electrostatic attraction, ions come into thesphere of influence of ions of opposite sign to form acertain proportion of uncharged ion-pairs so that atany concentration only a fraction of the ions is free toconduct electricity. It would then mean that althoughsuch electrolytes are completely ionized they are notcompletely dissociated in solutiorr'", Consequently if IXis the fraction of the ions that are free and if C is theconcentration of the electrolytic solution. then theconcentration of the ions that responsible forconductance would be IXC.In that case Eq. (1) reducesto Eq. (5)

° (A +B Ao)(l: IXCiZ~)I/Z e -ISm

A = A - I + 1.4(0.Sl: IXCiZ~)1/2

-SI(IXC)I/3(l-e-l~

-QI(IXC)(1-e-l~ ... (S)

If KA is the association constant for the equilibriumbetween the free ions and the ion-pairs then KA is givenby ':' Eq. (6)

KA = (1 - IX)!(C(ZCiZ±) ... (6)

where it is the mean formal activity coefficient ofthe electrolyte.

Since the effect of ion-association on the measured Aof an electrolyte that forms ion-pairs in solutionshould appear gradually with increase in con-centration. Eq. (S) should express this fact clearly. ForRbCIO .•. RbC103• CsCIO a- CsCI03, KClO .•, KCI03,

KN03 and AgN03 since C((which depends mainly onKA (~ I) for these 1: 1 electrolytes) varies from 0.9998to 0.9843 and ~s it also varies gradually as theconcentration of these electrolytes changes from 0.001to 0.03 M Eq. (S) for these 1: 1 association electrolytesclearly shows that compared to Eq. (1) forunassociated electrolytes, the effect of association withincrease in concentration of these 1: 1 electrolytesappears gradually. The situation. however. is quitecomplicated for 2: 2 electrolytes (KA for which is about200 times that for these 1: 1 electrolytes and It alsochanges considerably with increase in concentration ofthese electrolytes). The increase in association with anincrease in the concentration of these 2: 2 electrolyteswould. as expected. not appear gradually. Furthersuch solutions are believed 16 to contain not onlyuncharged but charged species also. These speciesmake significant contributions 16 to the measured Abeyond a certain concentration of these electrolytes.

In order to evaluate KA, A0 and S I for an electrolytethat forms ion-pairs in solution from its A data via Eqs(5 and 6). it is first assumed that :i= I and so from the(A + M /(1 - G) versus cl -'(I - e -ISn,/( I - G) plot onegets approximate values of A0 and SI' With these

117

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Pres

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Pres

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(Lit)

(Lit)

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RbC

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Table I -Comparison of A0 (in international mhos) and KA and of A values (in international mhos) from Eqs (5), (9) and (16)with Corresponding Literature Values at

2SOC for Some 1: 1 and 2: 2 Electrolytes that Form Ion-pairs in Water --( ContdiVl

zA at concentration (C), (in mol litre -I) o

Electrolyte 110 KA y fJ S~ :I:Present Present ~

(Lit) (Lit) 0.001 0.005 0.01 0.02 0.03 0.05 ":-KNO) 144.80 0.60 60.0 Calc 141.92 138.38 135.56 130.92 126.56 117.92 0

( 145.03)23b (0.597)30 (46.26) Expl23b 141.80 138.50 134.80 132.40 126.30 120.40 z145.0 0.60 0.10 90.0 Calc' 141.80 138.04 134.98 130.20 126.21 122.41 ~

(46.27) ;:0Vl

144.50 0.60 0.005 84.0 Calc" 142.24 138.24 134.93 132.08 125.93 120.91(46.21) Z

~AgNO) 133.20 1.0 49.0 Calc. 130.42 127.16 124.76 121.33 118.58 113.82 »( 133.36)'·lb (1.0jl0 (44.7)23b Expl23b 130.50 126.82 123.97 119.10 115.20 108.95 ...;

tT1133.20 1.0 0.0 49.2 Calc" 130.50 126.83 124.00 119.74 116.27 110.59 ;:0

(44.7) Ro

133.30 1.0 0.001 44.75 Calc" 130.54 127.00 124.33 120.78 118.03 112.D3...;;:0

(44.70) »ZZnSO. 130.00 190.0 600.0 Calc. 115.99 99.32 85.65 63.13 7.07 Vl

(132.S0) (186.2)31 (I 77.44)23b Expl23b 115.53 95.49 84.91 71.24 61.20 :;131.0 190.0 0.5536 250.0 Calc' 111.88 94.61 84.91 74.20 58.69 0z

(177.44) s:131.0 190.0 0.06 177.96 Calc." 116.07 96.03 86.06 76.01 62.12 0(177.96) 0

tT1CuSO .• 129.5 160.0 583.99 Calc. 115.52 98.93 85.35 63.30 8.16 r

(133.60)2)b (199.50jl' (l77.18)23b Expl+" 115.26 94.07 83.12 72.20 59.Q5 0'TI

129.5 160.0 0.6263 194.91 Calx.' 109.66 93.33 82.88 76.55 45.54 tT1

(177.44) rtT1

130.0 160.0 0.5 177.34 Calc. *. 115.14 93.95 83.61 72.51 58.88 o...;(177.44) CXI0" 1.37 16.4 52.0 79.0 115.3 217.0 4200 ;:0

0MgSO. 130.50 156.0 895.0 Calc. 125.01 111.88 95.35 85.58 73.16 31.31 -17.50 r7H2O (132.67)13 (156.5) (177.00) Bxpl '? 125.10 104.02 88.60 81.896 76.508 67.85 59.51 -<...;

131.00 156.0 1.00 328.32 Calc. • 125.02 104.04 88.60 80.65 67.83 58.71 37.81 n(J 17.96) 0;,

tT1132.00 156.0 0.0076 211.0 Calc." 125.21 104.93 91.41 84.51 79.02 69.62 58.80 :I:

(177.96) »<

(a)S I Values in parentheses are the values evaluated from Eq. (2) with I. = 1.0 0c'From Eq. (9) ;:0

"From Eq. (16)--_.,._-, ...__ .... - ..._---

---c

INDIAN 1. CHEM" VOL. 25A, FEBRUARY t986

approximate values of A0 and 51' a series of KA valuesare next assumed and A °,51 and QI' so varied that thevalues calculated from Eq. (5) are as close to theliterature values as possible. Such AO, 51' KA and QIvalues for the various electrolytes are recorded inTable I and are also compared with the correspondingliterature valuesI3.23b.2S. Table 1 also contains the Avalues so calculated for the various electrolytes andthese values are also compared with the correspondingexperimental values.

Examination of Table 1shows that while the A° andK•. values of RbC104, RbCl03, CsC104, CsC103,

KC103, KN03, AgN03 and KCI04 in water at 25°Ccompare well with the corresponding A° and KAvaluesZ3b.Z5,30.31 evaluated from their A data by themore involved method II , the same is not true of the A0

values of ZnS04, CuS04 and MgS04, 7H20. Againwhile the calculated A values for RbCl04, RbC103,

CsCl04, CsC103, KCI04 and KCl03 in water at 25°C,compare fairly well with the correspondingexperimental values." in the range 0.001 M < C< 0.05 M, the values so calculated for ZnS04, CuS04

and MgS04. 7HzO in water do not compare well withthe experimental A values+". The sloeps SI soevaluated for the various 1 : 1electrolytes also comparewell with those obtained using Eq. (2), This, however,is not true ofCuS04, ZnS04 and MgS04. 7H20. In allthese calculations I± for the 2: 2 electrolytes areassumed to be the same as that for ZnS04 and havebeen taken from the literature+"; the I± values for theremaining I : 1 electrolytes in the concentration rangeof interest here have been computed from the Debye-H uckel equation (since the relevant information tocompute their Ii values from the transition model!"are not available, and as the transition functione -15 m ~ 1 in the manner described elsewhere+". Thedata necessary to convert molarity were taken from theliterature.P".

The failure of Eq. (5) to satisfactorily describe theA data, in particular of ZnS04' CUS04 andMgS04. 7H20, may be traced to the assumption thatit is the concentration of the unpaired ions thatdetermine the A of these electrolytes. It may sohappen31 that in the solutions of these electrolytes ananion (B) diffuses into the sphere of influence (Gurneycosphere) of a cation (A +) and forms a solvent-separated ion-pair (A + --- B-) having atleast onesolvent molecule between it and the cation. The solventseparated ion-pair then forms uncharged ion-pair(A'" B -). This implies that the solutions of theseelectrolytes are characterized by the equilibria (7)

... (7)

and that the free A -r and B - and (A T + B ) species arethe only species+' that contribute to the measured A of

120

these electrolytes in solution, Consequently if C(I-CI()is the concentration of (A + --- B -) ion-pairs and if i' isthe fraction of these which are uncharged contactpairs, then the concentration of (A + . B -) = Cy(l - C1().The concentration of the solute species that contributetoward A is then C = C[l - y(l-CI()] and as the A ofthese solutions is expressed by Eq. (8f8

A = P AJPC) ... (8)where P is the fraction of the solution that contributesto the transport current, A is then expressed by Eq. (9)

A = [1 - y(1-CI()]

[° (A+BAO)r'I;2e-ISm'

x A - 1+1.41'1/2

-(1-e-ISm){SIC'I/3+QIC'}] ... (9)

where r = 21 = L c:Z; ... (10)

In order to employ Eq. (9) to express the A of theseelectrolytes in water, it is first simplified by assumingthat 0( = I = y and then from the intercept and slope ofthe (A + M)/(I- G) versus CI/3 (1 - e -I~/(l - G) plotone obtains approximate value of A° and SI of theseelectrolytes in solution. A series of KA values are nextassumed and A0, S I and ~' then so varied that thecalculated A values are as close to the experimentalvalues as possible. Such values (denoted by *) for theaqueous solutions of these electrolytes are recorded inTable 1 and are also compared with the correspondingexperimental values. Table I also records KA, SI and}'values of these electrolytes.

Examination of Table 1 reveals that the A values socalculated for various I: I electrolytes compare wellwith the literature values in the concentration range C< 0,02 M only. For C> 0.02 M, while the calculatedvalues are lower than the literature values for RbCI04,RbCI03, CsC104, CsCl03, KCI04 and KC103, thereverse is the case for KNOJ and AgN03. The presentstudy has further revealed that the solutions ofCuS04,ZnS04 and MgS04· 7H 20 do not provide ideal sys-tems for analysis of their A data in terms of Eq. (9).Nevertheless the analysis of the A data of the various1 : I and 2: 2 electrolytes in terms of Eq. (9) clearlysuggests that the postulated 32(A + ---B -) entities whichmake meaningful and significant contributions to themeasured A of these electrolytes, add considerably toour understanding of the nature of these electrolytes inwater.

The failure of Eq. (9) to satisfactorily describe the Aof these electrolytes in water over the entireconcentration range suggests that the formation of(A + ---B -) and their subsequent dissociation intouncharged (A + . B ) ion-pairs is not the prime process

SI~GH et al.: ION-PAIRS IN WATER & TRANSITION MODEL OF ELECTROLYTIC BEHAVIOUR

that characterizes the solution of these electrolytes inwater.

It may so happen that when B - diffuses into thesphere of influence of AT. it forms instead anuncharged ion-pairs (A T . B -), which are notpermanent entities. Some of these somehow (perhapsdue to the diffusion of the solvent) form (A + ---B -),which contribute to the measured A of theseelectrolytes. In other words the solutions of theseelectrolytes in water are characterized by the equilibria(II and 12)

· .. (11)

A(A+ B-)¢ (A+---B-) · .. (12)

Consequently if 'X is the fraction of the ions that are freethen as the ionic concentration is C" = «C, A of suchsolutions due to equilibrium (11) would be given by Eq.(13) (from Eq. 1)

= [ o_(A+BAO)r"1;2(e-ISm)

A (X A 1+1.41"1/2

-(I-e-ISn')(SIC'I,3+QIC)] · .. (13)

Again as C( I - rx)is the concentration of (A + . B -) ion-pairs and if /3 is the fraction of these that yields (A + ---

B -) ion-pairs, then the concentration of the (A + --- B -)would be Cf3(l-rx). KA and K, characterizingequilibria (11) and (12) would then be given by Eqs (14)and (15) respectively.

· .. (14)

· .. (15)

Since •.•.'e ex pect /f to be very small, the concentration of(A T---B ) species would be very small so that they areat almost infinite dilution. The contributions that (A + -

--8 ) ion-pairs make towards the measured A ofaqueous solutions of these electrolytes (due toequilibria 12) would then be (in view of Eq. 7) given byA = {fA;(/10::::: /f/I, ''. Consequently A of the solutionsof electrolytes that are characterized by equilibria (IIand 12) would be given by Eq. (16)

A = 7.[AO _ 0~_BA°L!:~'-'s","1+ 1.411

-(I-e-ISm")(SIC"1 J+Q1C")l+{i!\O ... (16)I

In order 10 see the effectiveness of this model (i.c. Eq.16) to describe the ,\ of the present electrolytes in water

at 25e, a series of KA, K, and A° values for theseelectrolytes are assumed. SI values for theseelectrolytes are then computed from Eq. (2) and KA, A°and K, values (and sometimes the SI values also) sovaried that the A values calculated for theseelectrolytes from Eq. (16) are as close to theexperimental values as possible. Such KA, A0, «, (f3here) values, marked as (**) are recorded in Table I.Table 1 also gives the A values so evaluated for theseelectrolytes. The data so recorded reveal that thismodel (Eq. 16) best reproduces the A of theseelectrolytes in water. Further in almost all these casesSI values are very close to the S1 values evaluated fromEq. (2) (with X = 1). Again Table 1 shows that asf3 ::::;0 f 0, the basic arguments used in deriving Eq. (16)are justified. Further f3A ° term make an importantcontribution to the measured A of these electrolytes.The KA values evaluated in the present manner forthese electrolytes also compare well with thecorresponding KA values evaluated for theseelectrolytes from their A data by the more involvedmethod I I. The present approach thus does not requirethe adjustment of the ion-size parameter (whichsometimes yields unreasonable values 13) and yet yieldsA0, KA etc. values (for these electrolytes) that comparewell with those obtained from the more involvedmethods5.6.25•

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4 Fuoss R M & Accascina F, Electrolytic conductance(Interscience, New York) 1959.

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