Indian Journal of Chemistry Vol. 16A, May 1978, pp. 383-387
Application of Extended Debye-Hiickel Theory in Deriving Gibbs Equations for the Adsorption of Bolaform Electrolytes
D. K. CHATTORAJ & L. GHOSH
Department of Food Technology & Biochemical Engineering, ]adavpur University, Calcutta 700032
Received 29September 1977;accepted 24January 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 neutral salt has been derived on the assumption that the Helmholtz type of double layer exists at the liquid 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 anion has 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 variation of the interfacial tension with increasing concentrations of the bolaform electrolyte in the bulk.
T
HERE exists considers ble theoretics 1 in terest for the derivation of the correct form of the Gibbs equation-:" for the adsorption of an organic electrolyte in the presence and absence of an inorganic salt. Many of these derived equations have been found suitable for the calculation of the amount of un i-univalent electrolytes adsorbed per unit area of the liquid interfaces+", Recently, surface and interfacial tensions of the air-We ter and 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 mrde in a few ca ses to eva lua te the surface excess con centra tions of the dibc sic organic ions with the help of the Gibbs equations either after neglecting the interionic a ttra ction effects in the bulk phase-s or using the Debye-Hiickel limiting law for the calculation of the bulk activity coefficients of the bola form ions", Such incomplete assessment of the interionic effect in the bulk wiII be very much unsatisfactory when the con- centra tion of the organic ions is significantly high.
An attempt has therefore been made in this paper to derive a suitable form of the Gibbs equation for the adsorption of the bola form electrolytes using the extended form of the Debye-Huckel theory for the calculation of the bulk activity co- efficients. The application of this equation to the existing experimental data has also been made and the pressure-area curves thus constructed for the bola form ions examined critically.
Derivation of the Gibbs Equation
Let RNaz stand for the organic surface active electrolyte whose concentration in the aqueous medium 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.gnitude and negative in sign. The aqueous medium may also contain a neutral salt such 2S N<.Cl whose con- centra tion during the experiments is usua lly kept constant. Let CR and C stand for the bulk concen- trations of the organic ar.d the inorganic electrolytes RN2z and Na Cl respectively having a common cation, The organic sa It is a Iso c.ssumed to be completely dissociated in the bulk (Eq. 1)
RNaz;;:R-z+ZNa+ ... (1)
Gibbs equa tion for the 2dsorption of electrolytes in the presence of s-types of differer.t ior s acquires the forrn4 (2)
-dY=knr;dlnf,C; ... (2)
Here T,stands for the surface excess of ith type of ions per unit area of the liquid surfs ceo The surface excesses of the organic and sodium ions are shown to be positive whereas that for chloride ions it is shown to be negatives, The mole r concentra- tion and corresponding activity coefficient of the ith ion in the bulk are expressed by C' and
Ii
res-pectively. In Eq. (2), k and T are the Boltzman constants and the absolute temperature respectively.
We may then write Eq. (2) in the more explicit form4 (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 mdIn CR where the kT-coefficient m Eq. (5):
m=~ [1+4> rNa+d In CNa+]
rRd In CR
... (4)
ma y be expressed by... (5)
383
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+
dInfNa+)
~ dIn CNa+
The value of m also depends on the models of the electrical double layer formed at the interface' when the organic ions are adsorbed gradually at the air-water or the oil-water interfaces. According to the picture given by Stern, the interfacial double layer is partly fixed (Helmholtz type) and partly diffuse (Gouy type) so that'
rNa+d In CNa+ Z2
rRdln CR
=
Z+.£(1+
eCR
••• (6)
•.. (7)
«'l'a/2BT)
1+1
... (8)
where
••• (9) Here
'1"a
stands for the absolute magnitude of the potential (without sign) in the diffuse double layer and E, the electronic charge. Z here stands for the absolute magnitude of charge of organic ions (with-out sign). . .
If all the ionic sites of the adsorbed organic ions become bound with Na+ counter-ions, the diffuse layer disappears
('1"a=O)
and the charged interface contains only the fixed double layer of the Helm- holtz model type. In this situation,1
in Eq. (8) tends to infinity' so thatrNa+d In CNa+ Z2
rRdinCR =Z+C •.• (10)
CR
Eq. (10) strictly valid for the H~lmh<:>ltz double layer may be applicable for any
situation
for the Gibbs equation if5%
error is allowed' for t~e evah~a- tion ofr
R• This error may become associated WIth m due to the uncertainties in the real model of the interfacial double layer.The activity parameters ~ and
,p
may be calculated using the extended theory of Debye and Huc~elll.According to this theory In
/R
and In fNa+ are grven by Eqs. (11) and (12)InfR= _ AZ2v'~ ••.(11)
l+a_Bv'fL
InI. Av'~ •••(12)
JNa+=- l+a+Bv'~
In Eqs. (11) and (12)
A
and13
are ~onstants whose values can be calculated on the baSIS of the Debye- Huckel theoryw; :a: and a+ are the radii of. the organic anions and inorganic ca tions respectlv~ly both of which on replacement by an average radius a± for the ions" in Eqs. (11) and (12) may lead to relation (13)In/R=Z2InfNa+ ... (13)
384
Differentiating Eq. (11) with respect to d In CR- and then 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 of CRand C. According to the Debye-Hiickel Iimiting law,
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 for the special case of uni-univalent organic electrolyte has also been derived by Chattorajs using the extended Debye-Huckel theory. The same expres- sion may be obtained by putting Z equal to unity in Eq. (14).Since CNa+ is equal to ZCR+C, d In CNa+ (or dCNa+/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 with the help of Eq. (15). CombiningEqs.
(5), (10) and(15), we obtain relation (16)
m=~
[1+~
{~~1+
Z~:~C}] ...(16)At given values of CR and C therefore m may be estimated if the single activity parameter ~ is cal- culated with the help of Eq. (14).
Results
Calculation of the kT-coeflicient (m) - In Eq. (14), the value of A is equal to 2·303 ADH where ADH' the Debye-Htickel parametert-, for water is 0·509 at 25°. 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-univalent electrolvtev", a± has been taken arbitrarily to be
5 A.
The exact value of a± remains uncertain forbolaform electrolyte. In the present case, m has been calculated with a± equal to either 5 or 10
A
respectively.
In Fig.
I,
m for a bola form electrolyte has been plotted as a function of -log CR at several fixed values of C. Each full line in this figure has been constructed with a± equal to 5A,
whereas dashed line corresponds to a±=10A.
Each dotted line in this figure is based on the ideal bulk behaviour of the electrolytes when/R
and fNa+ are unity so that ~=,p=1. At a given value of C, these three curves differ from each other significantly only when CR is relatively high. At a given value of CR, m decreases with decrease in the average radius from 10 to 5A
but both these values are signi- ficantly lower than that calcula ted on the basis of the ideal bulk behaviour of the electroytes. It is also evident that m is equal to unity for a value of CRCHATTORAJ & GHOSH: GIBBS EQUATION FOR ADSORPTION OF BOLAFORM ELECTROLYTES
-2 -3
O·S·I---.L---'---
-L... --'-4
log CR
-4
~_---c'O
Ac.cfoOOI
-3
0·5'-'---....&...---1---"""---'
-I
Log '.
less than 0·01 molar provided the concentration of C is as high as one molar (vide curves F and F/).
However, when C becomes lower than one molar concentration, values of m shown in curves A to E
01" A'to E' become significantly greater than unity particularly if the concentration of the organic electrolyte is high and that of the inorganic salt low. In the complete absence of the neutral salt (C equal to zero), m is equal to 3 at all values of CR provided the interionic attraction effect is negligible. However, m is observed to decrease significantly from this ideal value at relatively higher values of CR if appropriate value of ~ is incorporated in Eq. (16).
Pal and Chattorajt have recently used the Debye- Huckel limiting law
(B
equal to zero in Eqs. 11 and 12) during the application of the Gibbs equation for the calculation of the surface excess of the bolaform ions. In Fig. 2 the kT-coefficient m,calculated on the basis of the Debye-Huckel limit-
Fig. 1 - Plot ofkT-coefficient (m)vs log of the concentration of the organic electrolyte (CR) [Curves A, B, C, D, E and F are drawn according to the ex- tended Debye-Huckel equation, [a±
=
5 A]; curves A', B', C', D', E' and F' are drawn according to the extended Debye- Huckel equations, [a± =10 A]; and curvesA', B", CH, DH, EN and F* are drawn according to the ideal behaviour withoutinterionic attractions (~
= .p =
1)]-5
Fig. 2 - Plot ofkT-coefficient (m)vs log of the concentration of the organic electrolyte (CR) [Curves A, B, C, D, E and F are drawn according to the
limiting Debye-Huckel equation)
-s
ing law, has been plotted against -log CR' As in Fig. 1, the value of m (vide curves E and F) remains close to unity when the concentration of the organic electrolyte is less than 0·01 molar and C 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 is abnormal and unexpected. With the application of the extended Debye-Hiickel theory m is found to be slightly greater than unity in this range of concentration. With decrease of C, the curves B, C and D in Fig. 2 exhibit maxima. The values of mon the right side of the maximum at a given value of CR calculated on the basis of the limiting and extended 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 the extended theory, however, m increases with increase
in
CRat 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
I
t
I
I300 400 500
o 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. In the complete absence of the salt also, the feature of the corresponding curves in Figs. 1 and 2 are distinctly different from each other.
Pressure-area curves of the bolaform electrolytes- Recently Pal and Chattoraj" have presented their data on the lowering of the boundary tension of oil-water systems as functions of the concentrations of disodium sebaca te both in the presence and absence of one molar concentration of sodium chloride. The lowering of tension represents the surface pressure 7t. The corresponding area A per adsorbed molecule obtained from the value of
1/r
Rdirectly depends on the kT-coefficient (m). The concentrations of the sebacate ions in the presence of excess (one molar) neutral salt were always below 0·030 molar so that 'one kT form' of the Gibbs equation was used for the calculation of A 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-A
curves of sebacate in the absence of neutral
salt
has been constructed with the help of the limiting law, extended law and ideal behaviour for the electrolytes in the bulk for the calculation of Inat various values of Ce (Fig. 3). For 10 dynes surface pressure, values of A become 40, 56 and 70
Az
respectively if m is respectively calculated by limit- ing law, extended law and ideal behaviour for electrolytes in the bulk. Extended law is perfectly sound in this respect and further the positions of the 7t-A curves in Fig. 3 based on the extended law remain close to each other for a± equal to 5 and
10
A
respectively.Recently Menger and Wrenn-" have determined the surface tension of water as a function of the increasing concentration of a series of cationic
types of bolaform electrolytes [RaN+ -(CH2)n- N+R3
J;
here R stands for methyl or n-butyl groups and nmay be 4, 8 or 12. They have also calculated the molecular a reas, co-a res s, crit ice l micelle con- centra tion (CMC) and sta r.da rd free energies of adsorption of these electrolytes from the calculation of A using '3kT form' of the Gibbs equations valid only for the ideal electrolytes. Although Eqs.(14) and (16) have been deduced in the previous section for bolaform anion, these may also be shown to remain valid for the adsorption of bola- form cation of charge (Z) equal to +2. In Teble 1, Y for these electrolytes measured by Mer:ger et al.IO
a t severa 1va lues of CRare quoted. Va lues of m in all these cases calculated on the basis of Eq. (16) are observed to significantly deviate from ideal valve 'three' when the number of hydrocarbon groups in the compound is low. The calculation of A and other interfacial pars meters by Merger et apo with assumption that m=3 may become errOnEOUS in this circumstance. With increase of hydrocarbon groups in the ions, the value of CR
decreases sharply so that the difference between the calculated value of
m
and its ideal value is reduced considers bly ar.d consequently the error in the calculation of these parameters is decreased considerably. In Fig. 4, 1t for BusN+-(CHz)12-N+Bu3 has been plotted agair st A which in its turn have been calculated using limitir g, extended or ideal theories for bola form electrolytes. The values of A at a given 7tare quite close to each other since CR in these ce ses are r.ot high.
Discussion
Two similarly chs rged ionic sites of a bola form electrolyte are separated by a hydrocarbon chain of 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.
..
•
15 c•..
...,
1:::'
Fig. 4- (IT-A) isotherms for the bolaforrn electrolyte, 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) VALUES USING THE EXTENDED DEBYE-HDCKEL EQUATION Bolaforrn electrolytew y Concen-
er-«-
tration efficient
in (m)
molarity
Me3N+-(CH.).-N+Me. No surface 0·005 2·70 activity
Bu3K'-(CH.l.-N+ BU3 57'5 0·100 2·46
CIV[C value 0'505 2·52
Me3N+-(CH')8-N+Me3 63·4 0·100 2'46
CMC value 0'505 2·52
Bu3N+-(CH.)s-N+Bu3 53·7 0·100 2·46
important from the biological standpoint since it shows some kind of physiological activities'< and also it binds to ordinary and biological polymers.
Compared to the extensive physico-chemical studies of the bola form electrolytes in solution in bulk12-14, only very few studies9,lo have been made for the interfacial behaviour of the bolaform electrolytes from surface and interfacial tension measurements.
Such interfacial studv mav become useful-'' for the basic understanding of "the cellular membrane because of the structural resemblance of the adsorb- ed bolaform electrolytes with paired phospholipids uniquely in the interfacial membrane phase. It is therefore expected that more studies of the interfacial behaviour of these electrolytes will be made in the near future.
From the present analysis, it has been shown that the calculation of the surface excess of the bola form electrolyte with the help of the Gibbs equation will be useful and accurate only when the interionic a ttra ction between the electrolytes in the bulk is
taken into account on the basis of the extended theory of Debye and Hiic~el. T~leneglect o.f such effect may lead to a senous discrepancy III ife construction of n-A curves. The discrepancy With the corrected and ideal values of A is reduced considerably with an increase in hydrocarbon groups in the bola form electrolyte.
Acknowledgement
The authors are grateful to the Research and Development Organization, Minis~ry of Defen~e, Government of India, New Delhi, for financial assistance during the investigation.
References
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sn..
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