Indian Journal of ChemistryVol. 16A, May 1978, pp. 383-387

Application of Extended Debye-Hiickel Theory in Deriving GibbsEquations for the Adsorption of Bolaform Electrolytes

D. K. CHATTORAJ & L. GHOSHDepartment of Food Technology & Biochemical Engineering, ]adavpur University, Calcutta 700032

Received 29 September 1977; accepted 24 January 1978

An appropriate expression for the calculation of the kT-coefficients (m) of the Gibbs equa-tion for the adsorption of an organic bolaform electrolyte in the pres ence and absence of a neutralsalt has been derived on the assumption that the Helmholtz type of double layer exists at theliquid interface. It has also been shown from the mathematical analysis that the activity para-meters l; and", in this expression may be calculated on the assumption that the extended Debye-Htickel theory for the bulk activity coefficients remains valid particularly when the ionic concen-trations are relatively high. The numerical values of m for various concentrations of the elec-trolytes have been graphically presented. In the absence of the neutral salt, values of m con-siderably decrease from the ideal value' three' when the bolatorrn ion concentrations are high.Using the appropriate values of m thus calculated, the pressure-area curve for bolaform anionhas been constructed from the published data [Pal, R. & Chattoraj, D. K., J. colloid interface Sci.,52 (1975)56; Menger, F. M. & Wrenn, S., J. phys. Chem., 78 (1974), 1387] showing the variationof the interfacial tension with increasing concentrations of the bolaform electrolyte in the bulk.

THERE exists considers ble theoretics 1 in terestfor the derivation of the correct form of theGibbs equation-:" for the adsorption of an

organic electrolyte in the presence and absence ofan inorganic salt. Many of these derived equationshave been found suitable for the calculation of theamount of un i-univalent electrolytes adsorbed perunit area of the liquid interfaces+", Recently,surface and interfacial tensions of the air-We terand oil-water systems have been measured as func-tions of the increasing concentra tions of the uni-bivalent types of organic bola form electrolytes9,lo.

Attempts have been mr de in a few ca ses to eva lua tethe surface excess con centra tions of the dibc sicorganic ions with the help of the Gibbs equationseither after neglecting the interionic a ttra ctioneffects in the bulk phase-s or using the Debye-Hiickellimiting law for the calculation of the bulk activitycoefficients of the bola form ions", Such incompleteassessment of the interionic effect in the bulkwiII be very much unsatisfactory when the con-centra tion of the organic ions is significantly high.An attempt has therefore been made in this paperto derive a suitable form of the Gibbs equationfor the adsorption of the bola form electrolytesusing the extended form of the Debye-Huckeltheory for the calculation of the bulk activity co-efficients. The application of this equation to theexisting experimental data has also been made andthe pressure-area curves thus constructed for thebola form ions examined critically.

Derivation of the Gibbs EquationLet RNaz stand for the organic surface active

electrolyte whose concentration in the aqueousmedium ma y be va ried. For the bola form electro-lytes such 2S disodium sebacate, the organic anion

is bivalent so that Z is equal to two in me.gnitudeand negative in sign. The aqueous medium mayalso contain a neutral salt such 2S N<.Cl whose con-centra tion during the experiments is usua lly keptconstant. Let CR and C stand for the bulk concen-trations of the organic ar.d the inorganic electrolytesRN2z and Na Cl respectively having a commoncation, The organic sa It is a Iso c. ssumed to becompletely dissociated in the bulk (Eq. 1)RNaz;;:R-z+ZNa+ ... (1)Gibbs equa tion for the 2 dsorption of electrolytesin the presence of s-types of differer.t ior s acquiresthe forrn4 (2)-dY=knr;dlnf,C; ... (2)Here T, stands for the surface excess of ith typeof ions per unit area of the liquid surfs ceo Thesurface excesses of the organic and sodium ions areshown to be positive whereas that for chloride ionsit is shown to be negatives, The mole r concentra-tion and corresponding activity coefficient of theith ion in the bulk are expressed by C' and Ii res-pectively. In Eq. (2), k and T are the Boltzmanconstants and the absolute temperature respectively.We may then write Eq. (2) in the more explicitform4 (3)-dY=kT(rRd In/RCR+rNa+d In fNa£Na+-rCI-

d lnfcl- CC1- ... (3)It has already been shown! that at constant C,rcl-dln/cl-Ccl- is negligibly small so that Eq. (3)assumes the form (4)-dy=rRkT m dIn CRwhere the kT-coefficient mEq. (5):

m=~ [1+4> rNa+d In CNa+]rRd In CR

... (4)ma y be expressed by

... (5)

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INDIAN J. CHEM., VOL. 16A, MAY 1978

The activity parameters ~ and ,p in Eq. (5) maybe given by relations (6) and (7) respectively.

~=1+ d In/Rdin CR

4>= !(1+ d InfNa+)~ dIn CNa+

The value of m also depends on the models of theelectrical double layer formed at the interface'when the organic ions are adsorbed gradually atthe air-water or the oil-water interfaces. Accordingto the picture given by Stern, the interfacial doublelayer is partly fixed (Helmholtz type) and partlydiffuse (Gouy type) so that'

rNa+d In CNa+ Z2rRdln CR = Z+ .£(1+ e

CR

••• (6)

•.. (7)

« 'l'a/2BT)

1+1... (8)

where

••• (9)

Here '1"a stands for the absolute magnitude of thepotential (without sign) in the diffuse double layerand E, the electronic charge. Z here stands for theabsolute magnitude of charge of organic ions (with-out sign). . .

If all the ionic sites of the adsorbed organic ionsbecome bound with Na+ counter-ions, the diffuselayer disappears ('1"a=O) and the charged interfacecontains only the fixed double layer of the Helm-holtz model type. In this situation, 1in Eq. (8)tends to infinity' so that

rNa+d In CNa+ Z2rRdinCR =Z+C •.• (10)

CR

Eq. (10) strictly valid for the H~lmh<:>ltz doublelayer may be applicable for any situation for theGibbs equation if 5% error is allowed' for t~e evah~a-tion of rR• This error may become associated WIthm due to the uncertainties in the real model of theinterfacial double layer.

The activity parameters ~ and ,p may be calculatedusing the extended theory of Debye and Huc~elll.According to this theory In /R and In fNa+ are grvenby Eqs. (11) and (12)

InfR= _ AZ2v'~ ••. (11)l+a_Bv'fL

Av'~In I. ••• (12)JNa+=- l+a+Bv'~

In Eqs. (11) and (12) A and 13 are ~onstants whosevalues can be calculated on the baSIS of the Debye-Huckel theoryw; :a: and a+ are the radii of. theorganic anions and inorganic ca tions respectlv~lyboth of which on replacement by an average radiusa± for the ions" in Eqs. (11) and (12) may lead torelation (13)In/R=Z2InfNa+ ... (13)

384

Differentiating Eq. (11) with respect to d In CR- andthen combining the resulting equation with relation(6), it may be shown that~=1-

2v'2Z(Z+1)CR+2C((I+a± ~ v'2Z(Z+I)CR+4C r•.. (14)

~ can thus be calculated from the known values ofCR and C. According to the Debye-Hiickel Iimitinglaw, 13 is equal to zero and the expression for ~will be converted into the same form previously de-rived by Chattoraj-. The expression of ~ only forthe special case of uni-univalent organic electrolytehas also been derived by Chattorajs using theextended Debye-Huckel theory. The same expres-sion may be obtained by putting Z equal to unityin Eq. (14).

Since CNa+ is equal to ZCR+C, d In CNa+ (ordCNa+/CNa+)in Eq. (7) maybe replaced by the expres-sion ZCR din CR/(ZCR+C). Now combining Eqs.(6) and (13) with Eq. (7), it may be shown that

,p= ~{1+ ~-;1 (z + ~)} ... (15)

From the known value of ~ at a given value of CR,therefore, value of ,p can easily be calculated withthe help of Eq. (15). Combining Eqs. (5), (10) and(15), we obtain relation (16)

m=~ [1+~{~~1 + Z~:~C}] ... (16)

At given values of CR and C therefore m may beestimated if the single activity parameter ~ is cal-culated with the help of Eq. (14).

ResultsCalculation of the kT-coeflicient (m) - In Eq. (14),

the value of A is equal to 2·303 ADH where ADH' theDebye-Htickel parametert-, for water is 0·509 at25°. The value of the other Debye-Htickel para-meter B for water is 0'330x 108• In the calcula-tion of m for the adsorption of a uni-univalentelectrolvtev", a± has been taken arbitrarily to be5 A. The exact value of a± remains uncertain forbolaform electrolyte. In the present case, m hasbeen calculated with a± equal to either 5 or 10 Arespectively.

In Fig. I, m for a bola form electrolyte has beenplotted as a function of -log CR at several fixedvalues of C. Each full line in this figure has beenconstructed with a± equal to 5 A, whereas dashedline corresponds to a±=10 A. Each dotted linein this figure is based on the ideal bulk behaviourof the electrolytes when /R and fNa+ are unity sothat ~=,p=1. At a given value of C, these threecurves differ from each other significantly onlywhen CR is relatively high. At a given value ofCR, m decreases with decrease in the average radiusfrom 10 to 5 A but both these values are signi-ficantly lower than that calcula ted on the basis of theideal bulk behaviour of the electroytes. It is alsoevident that m is equal to unity for a value of CR

CHATTORAJ & GHOSH: GIBBS EQUATION FOR ADSORPTION OF BOLAFORM ELECTROLYTES

-2 -3O·S·I-----.L------'--- -L... --'

-4log CR

-4

~_------c'OA

c. cfoOOI

-30·5'-'-----....&...-------1-----"""-------'

-I

Log '.

less than 0·01 molar provided the concentration ofC is as high as one molar (vide curves F and F/).However, when C becomes lower than one molarconcentration, values of m shown in curves A to E01" A'to E' become significantly greater than unityparticularly if the concentration of the organicelectrolyte is high and that of the inorganic saltlow. In the complete absence of the neutral salt(C equal to zero), m is equal to 3 at all values ofCR provided the interionic attraction effect isnegligible. However, m is observed to decreasesignificantly from this ideal value at relativelyhigher values of CR if appropriate value of ~ isincorporated in Eq. (16).

Pal and Chattorajt have recently used the Debye-Huckel limiting law (B equal to zero in Eqs. 11 and12) during the application of the Gibbs equationfor the calculation of the surface excess of thebolaform ions. In Fig. 2 the kT-coefficient m,calculated on the basis of the Debye-Huckel limit-

Fig. 1 - Plot of kT-coefficient (m) vs logof the concentration of the organicelectrolyte (CR) [Curves A, B, C, D, Eand F are drawn according to the ex-tended Debye-Huckel equation, [a± = 5A]; curves A', B', C', D', E' and F' aredrawn according to the extended Debye-Huckel equations, [a± = 10 A]; andcurves A', B", CH, DH, EN and F* are drawnaccording to the ideal behaviour without

interionic attractions (~ = .p = 1)]

-5

Fig. 2 - Plot of kT-coefficient (m) vs logof the concentration of the organicelectrolyte (CR) [Curves A, B, C, D, Eand F are drawn according to the

limiting Debye-Huckel equation)

-s

ing law, has been plotted against -log CR' Asin Fig. 1, the value of m (vide curves E and F)remains close to unity when the concentration ofthe organic electrolyte is less than 0·01 molar andC is equal to 1·0 and 0·1 molar concentrations.However, when CR exceeds 0·01 molar concentration,m becomes significgntly less than unity which isabnormal and unexpected. With the applicationof the extended Debye-Hiickel theory m is foundto be slightly greater than unity in this range ofconcentration. With decrease of C, the curvesB, C and D in Fig. 2 exhibit maxima. The values ofm on the right side of the maximum at a given valueof CR calculated on the basis of the limiting andextended theories are found to be close to each other.On the left side of the maximum, m decreases signi-ficantly with increase in CR (decrease in -log CR)

according to the limiting law. On the basis of theextended theory, however, m increases with increasein CR at a constant concentration of C so that

385

INDIAN J. CHEM., VOL. 16A, MAY 1978

,,~!

I••EC•••u ,t•Q..•••c,.~

1=:::'

St-i

'25 \0

It

II

300 400 500o 2

A(A) per Boloform ion

Fig. 3 - (1t-A) isotherms for the disodium sebacate in the absence of inorganic salt (C) [Curves: (A) ideal equation;(B) extended Debye-Huckel equation (a± = 5 A); (B') extended Debye-Hiickel equation (a± = 10 A); and (C) limiting

Debye-Hiickel equation]

maximum in the curve has not been exhibited. Inthe complete absence of the salt also, the featureof the corresponding curves in Figs. 1 and 2 aredistinctly different from each other.Pressure-area curves of the bolaform electrolytes-

Recently Pal and Chattoraj" have presented theirdata on the lowering of the boundary tension ofoil-water systems as functions of the concentrationsof disodium sebaca te both in the presence andabsence of one molar concentration of sodiumchloride. The lowering of tension represents thesurface pressure 7t. The corresponding area A peradsorbed molecule obtained from the value of 1/rRdirectly depends on the kT-coefficient (m). Theconcentrations of the sebacate ions in the presenceof excess (one molar) neutral salt were alwaysbelow 0·030 molar so that 'one kT form' of theGibbs equation was used for the calculation ofA and 7t-A curves were subsequently constructed",According to the extended Debye-Hiickel theory,this calculation seems to be justified.

Using the data of Pal and Chattoraj", the 7t-Acurves of sebacate in the absence of neutral salthas been constructed with the help of the limitinglaw, extended law and ideal behaviour for theelectrolytes in the bulk for the calculation of Inat various values of Ce (Fig. 3). For 10 dynes surfacepressure, values of A become 40, 56 and 70 Azrespectively if m is respectively calculated by limit-ing law, extended law and ideal behaviour forelectrolytes in the bulk. Extended law is perfectlysound in this respect and further the positions ofthe 7t-A curves in Fig. 3 based on the extended lawremain close to each other for a± equal to 5 and10 A respectively.

Recently Menger and Wrenn-" have determinedthe surface tension of water as a function of theincreasing concentration of a series of cationic

386

types of bolaform electrolytes [RaN+ -(CH2)n-N+R3J; here R stands for methyl or n-butyl groupsand n may be 4, 8 or 12. They have also calculatedthe molecular a reas, co-a res s, crit ice l micelle con-centra tion (CMC) and sta r.da rd free energies ofadsorption of these electrolytes from the calculationof A using '3kT form' of the Gibbs equations validonly for the ideal electrolytes. Although Eqs.(14) and (16) have been deduced in the previoussection for bola form anion, these may also beshown to remain valid for the adsorption of bola-form cation of charge (Z) equal to +2. In Te ble 1,Y for these electrolytes measured by Mer:ger et al.IOa t severa 1 va lues of CRare quoted. Va lues of min all these cases calculated on the basis of Eq. (16)are observed to significantly deviate from idealvalve 'three' when the number of hydrocarbongroups in the compound is low. The calculationof A and other interfacial pars meters by Mergeret apo with assumption that m=3 may becomeerrOnEOUS in this circumstance. With increase ofhydrocarbon groups in the ions, the value of CRdecreases sharply so that the difference betweenthe calculated value of m and its ideal value isreduced considers bly ar.d consequently the errorin the calculation of these parameters is decreasedconsiderably. In Fig. 4, 1t for BusN+-(CHz)12-N+Bu3 has been plotted agair st A which in itsturn have been calculated using limitir g, extendedor ideal theories for bola form electrolytes. Thevalues of A at a given 7t are quite close to each othersince CR in these ce ses are r.ot high.

DiscussionTwo similarly chs rged ionic sites of a bola form

electrolyte are separated by a hydrocarbon chainof varying lengths. The physico-chemical beha-viour of such an electrolyte in the bulk wa ter is

CHATTORAJ & GHOSH: GIBBS EQUATION FOR ADSORPTION OF BOLAFORM ELECTROLYTES

20;;•.~c•v;;Q...• 15c•.....,

1:::'

Fig. 4- (IT-A) isotherms for the bolaforrnelectrolyte, Bu3N+(CH2)12 = N+Bu3 inthe absence of inorganic salt (G) [Curves:(A) ideal equation; (B) extended Debye-Huckel equation; and (C) limiting

Dc hye-Huckcl equation]

50 75 100.2

AlA) per BolofOlfft ;01\

TABLE 1 - CALCULATED kT-COEFFICIENT (m) VALUESUSING THE EXTENDED DEBYE-HDCKEL EQUATION

Bolaforrn electrolytew y Concen- er-«-tration efficient

in (m)molarity

Me3N+-(CH.).-N+Me. No surface 0·005 2·70activity

Bu3K'-(CH.l.-N+ BU3 57'5 0·100 2·46CIV[C value 0'505 2·52

Me3N+-(CH')8-N+Me3 63·4 0·100 2'46CMC value 0'505 2·52

Bu3N+-(CH.)s-N+Bu3 53·7 0·100 2·46

important from the biological standpoint since itshows some kind of physiological activities'< andalso it binds to ordinary and biological polymers.Compared to the extensive physico-chemical studiesof the bola form electrolytes in solution in bulk12-14,only very few studies9,lo have been made for theinterfacial behaviour of the bola form electrolytesfrom surface and interfacial tension measurements.Such interfacial studv mav become useful-'' for thebasic understanding of "the cellular membranebecause of the structural resemblance of the adsorb-ed bolaform electrolytes with paired phospholipidsuniquely in the interfacial membrane phase. Itis therefore expected that more studies of theinterfacial behaviour of these electrolytes will bemade in the near future.

From the present analysis, it ha s been shown thatthe calculation of the surface excess of the bola formelectrolyte with the help of the Gibbs equation willbe useful and accurate only when the interionica ttra ction between the electrolytes in the bulk is

taken into account on the basis of the extendedtheory of Debye and Hiic~el. T~le neglect o.f sucheffect may lead to a senous discrepancy III ifeconstruction of n-A curves. The discrepancy Withthe corrected and ideal values of A is reducedconsiderably with an increase in hydrocarbon groupsin the bola form electrolyte.

AcknowledgementThe authors are grateful to the Research and

Development Organization, Minis~ry of Defen~e,Government of India, New Delhi, for financialassistance during the investigation.

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387