Anal. Chem. 1999, 71, 196-200
Constant Chemical Potential Titration. Application to Determination of Nonionic Surfactant Concentrations Mireille Turmine,* Christelle Mace´, Franc¸ ois Millot, and Pierre Letellier
Laboratoire d’Energe´ tique et Re´ activite´ aux Interfaces, EA 1519, case 39, Universite´ Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
Titration is most often associated with the idea of a stoichiometric reaction. Generally, it is not considered possible to titrate a compound against a reagent unless the titration reaction is near total and if the result is not a product of well-defined stoichiometry. In this work, we illustrate that accurate titration is possible with compounds and reagents that interact to form an association of undefined stoichiometry. Our model is the potentiometric titration of nonionic surfactant with cationic surfactant using a cationic surfactant-selective electrode. The result of this reaction is mixed micelles, the composition of which depends on the concentrations of the two surfactants in solution. The use of titration, and especially of potentiometric titration, is most often based on the notion of a stoichiometric reaction. Generally it is difficult to imagine that a compound of unknown concentration can be titrated against a reagent of given concentration unless (i) the product is a compound of well-defined composition and (ii) the reaction proceeds to near completion. This is, however, a restricted definition of titration. Indeed, titration is possible with compounds that interact even weakly and that do not lead to an association of definite stoichiometry. Adsorption phenomena and aggregates of mixed constitution (including microemulsions and micelles) are examples of such nonstoichiometric associations. In this study, we show that it is possible to titrate a compound by potentiometry, providing that a conditional stoichiometry is created by fixing a number of parameters of the medium. So, in this case, if a nonionic compound interacts with an ionic reagent, the concentration of which can be followed with a selective electrode, a potentiometric titration can be envisaged. We used as a model system the titration of nonionic surfactants (Brij 35, Tween 80) with a cationic amphiphile, dodecyltrimethylammonium (DTA+), as titrant. These two surfactants form mixed micelles in aqueous solution. Using a DTA+-selective electrode, the nonionic surfactant can be titrated against the cationic surfactant. Generally, nonionic surfactant titrations involve analytical techniques, many of which are technically demanding (such as * Corresponding author: (e-mail)
[email protected] or
[email protected]; (fax) 33 1 44 27 30 35.
196 Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
colorimetry, chromatography, and spectroscopy).1 Although nonionic or zwitterionic surfactants have not net charge, various methods for potentiometric titration have been described. They are all based on the formation of complexes between the surfactant and a bivalent cation such as barium or calcium, the concentration of which is followed with selective electrodes.2-5 This potentiometric method differs from previously described methods in that the electrode is not used as end-point indicator but as a sensor which can identify a particular state of the solution. It may be valuable for routine titration as the process is easily automated. EXPERIMENTAL SECTION Products. Dodecyltrimethylammonium bromide (DTABr), dodecylpoly(oxyethylene glycol)23 (Brij 35) and poly(oxyethylene)20 sorbitan monooleate (Tween 80) (Aldrich) were used without further purification. All solutions of these products in pure water were prepared by gravimetry. The critical micelle concentrations (cmc’s) of these surfactants, in pure water at 298 K are6 Brij 35, 9.1 × 10-5 M; Tween 80, 2 × 10-5 M; and DTA+,Br-, 1.6 × 10-2 M. Electrochemical Measurements. The cationic surfactantselective electrode was prepared in our laboratory as described previously.7,8 This electrode was made with a plasticized poly(vinyl chloride) (PVC) membrane containing 20 wt % PVC, 80 wt % dinonyl phthalate (the plasticizer), and as carrier, dodecyltrimethylammonium tetraphenylborate salt. The electrochemical cell used was constituted by this selective electrode in conjunction with a bromide ion-selective electrode (Tacussel, XS220) because this study was made in 0.2 mol L-1 NaBr solution. The emf values of this electrochemical cell were steady within (0.2 mV and measured at 298.0 ( 0.1 K using a LPH530T, Tacussel millivoltmeter. The response of this cell is (1) Cullum, D. C.; Platt, P. Recent Developments in the Analysis of Surfactants; Porter, M. R., Ed.; Elsevier Applied Science: New York, 1991; Vol. 32, p 5. (2) Sugawara, M.; Nagasawa, S.; Ohashi, N. J. Electroanal. Chem. 1984, 176, 183. (3) Vytras, K. Ion-Sel. Electrode Rev. 1985, 7, 77. (4) Jones, D. L.; Moody, G. J.; Thomas, J. D. R.; Birch, B. J. Analyst 1981, 106, 974. (5) Moody, G. J.; Thomas, J. D. R. Nonionic surfactants-chemical analysis; Cross, J., Ed.; Marcel Dekker Inc.: New York, 1987; p 117. (6) Detergents for membrane research, Biochemicals Catalogue Bohringer Mannheim GmbH. (7) Jezequel, D.; Mayaffre, A.; Letellier, P. Can. J. Chem. 1991, 69, 1865. (8) Gloton, M. P.; Mayaffre, A.; Turmine, M.; Letellier, P.; Suquet, H. J. Colloid Interface Sci. 1995, 172, 56. 10.1021/ac980921g CCC: $18.00
© 1998 American Chemical Society Published on Web 12/01/1998
Nernstian (59 mV per logarithmic unity) between 3 × 10-6 and ∼3 × 10-3 mol L-1 (it is the DTABr cmc in presence of 0.2 mol L-1 NaBr). Nonionic surfactants were added with a micropipet (Biohit France), and cationic surfactant was added with a burette of 5.00 ( 0.02 mL or a Metrohm microburette of 0.2500 ( 0.0005 mL. Principle of Titration. The basis of this titration is a theoretical study on the behavior of mixed micelles and interactions between nonionic and ionic surfactants.9 In this previous study, showed that when the chemical potential of the cationic surfactant is controlled then the composition of the mixed micelle and concentrations of the two surfactants free in solution are fixed. The result is that when the chemical potential of the cationic surfactant is maintained constant, a conditional stoichiometry is fixed. This is the basis of the constant chemical potential titration (CCPT) of nonionic surfactants (at DTABr constant chemical potential). This approach to titration is thermodynamically valid. The principles must be briefly defined to well-understand this titration. Thermodynamic Justification of Constant Chemical Potential Titration. Consider a monophasic system constituted of n1 mol of DTABr solution, n2 mol of nonionic surfactant solution, and n3 mol of solvent, at temperature T and pressure P. Solutes 1 or 2 can be compounds or salts. Once equilibrium is established, the free energy of this solution can be written
G ) n1µ1 + n2µ2 + n3µ3
Figure 1. Principle of the constant chemical potential titration (CCPT).
reacts as though there were a “stoichiometry” of the aggregate resulting from the association between 1 and 2.
(∂n1/∂n2)µ1,n3 ) νµ1,n3 The value of this “conditional stoichiometric coefficient” depends on the chosen experimental conditions (n3 and µ1). This is the basis of the constant chemical potential titration. The experimental process is the following. Consider a solution containing the three constituents (n1, n2, n3); µ1 is the chemical potential of 1. A small quantity (δn2) of 2 is added to the solution; 2 interacts with 1. The chemical potential of 1 decreases. A quantity (δn1) of 1 is then added to the solution to restore the initial chemical potential value of 1 (Figure 1). The chemical potential of 1 is thereby maintained constant; the number moles of water similarly remains constant. The ratio δn1/δn2 can be identified with the partial differential (∂n1/∂n2)µ1,n3 whose value is that of the conditional stoichiometric coefficient:
and therefore
dG ) V dP - S dT + µ1 dn1 + µ2 dn2 + µ3 dn3
where µ1, µ2, and µ3 are the chemical potentials of the two solutes and solvent, respectively. Their values depend on the medium composition at equilibrium and show the possible associations between solutes and between solutes and solvent. n1, n2, and n3 are total molar quantities of mixture constituents, whatever the envisaged interactions. The different magnitudes that intervene in the expression of G are linked together by cross-differentiation relations. At constant temperature and pressure,
( ) ∂µ2 ∂µ1
( )
∂n1 )∂n2 n2,n3
δn1/δn2 ) (∂n1/∂n2)µ1,n3 ) νµ1,n3
(1)
(2)
µ1,n3
For simplicity, temperature and pressure are not included in constant magnitudes associated with each partial differential. They have previously been applied to determine compositions of mixed micelles composed of Brij 35 and dodecyltrimethylammonium bromide.9 Thus, the right-hand term of eq 2 shows that different solutions possessing the same value of chemical potential of 1 and the same number of moles of solvent will react identically to perturbations of the numbers of moles of 1 and 2, the ratio (∂n1/ ∂n2)µ1,n3 being independent of the system content. If, in a series of experiments, the number of moles of solvent (n3) and the chemical potential of 1 (µ1) can be kept constant, then the system (9) Palous, J. L.; Turmine, M.; Letellier, P. J. Phys. Chem. B 1998, 102, 5886.
(3)
This operation of alternate additions can be repeated several times in the same reaction medium, such that the composition of the solution is modified at constant µ1 and n3 (n1 and n2 increase). The conditional stoichiometric coefficient value is constant, and additions of 1 and 2 can be taken into account
∑δn /∑δn 1
2
) δn1/δn2 ) νµ1,n3
so
∑δn
1
) νµ1,n3
∑δn
2
This proportionality can be exploited in the analytic plan. In given experimental conditions (µ1 and n3 constant), the conditional stoichiometric coefficient value can be determined by using solutions of compounds 1 and 2 of known titer, following the procedure described above. The titer of an unknown solution of 2 can be then determined from ν(µ1,n3) using a known titer solution of 1, by the same procedure. We tested whether this approach was experimentally possible, for nonionic surfactant titration by the dodecyltrimethylammonium ion (DTA+). To apply the relations demonstrated above to an ionic surfactant, DTABr, hereafter named 1, a nonionic surfactant (Brij 35 or Tween 80) named 2, and solvent, named 3, eq 3 must always be verified. Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
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unknown nonionic surfactant solution was then added, causing the emf to decrease. The same rC1 DTABr solution was added until the initial emf value, Er, was restored, and the volume added, δV1, was recorded. As previously, a certain number of alternate additions were made, and the volumes of 1 and 2 were calculated to determine the gradient, Px. We can write
δn1 rC1δV1 rC1 ) ) x Px δn2 xC δV C2 2 2
νµ1,n3 )
(6)
Figure 2. Description of the graph obtained in CCPT.
In this case, the chemical potential of DTABr can be expressed according to DTA+ and Br- activities
µ1 ) µθDTABr + RT ln aDTABr )
x
µθDTABr
C2 ) rC2(Px/Pc)
µ1 ) SµθDTABr + RT ln aDTA+ with S θ µDTABr
) µθDTABr + RT ln aBr-
Use of dilute surfactant solutions permits use of concentrations instead activities. Thus, µ1 is fixed if CDTA+ ) eC1 is maintained constant. A volume V of DTABr solution of a known concentration, eC , was placed in a beaker. A calibrated DTA+ ion-selective 1 electrode was placed in the solution in conjunction with a bromideselective electrode (Tacussel XS220) used as reference electrode. The emf of this electrochemical cell, Er, served as the reference to determine the DTA+ chemical potential in solution and then a conditional stoichiometry. Calibration. A volume, δV2, of a nonionic surfactant standard solution of concentration rC2 was added. Concentrations were such that the nonionic surfactant was micellized in the reaction medium. This led to a decrease of the emf value. Then, the initial emf value, Er, was restored by adding a volume δV1 of a solution of DTABr of known titer rC1 as described on Figure 1. This same operation was repeated several times. Total volumes of 1 and 2 were recorded. These values were reported on a graph as described in Figure 2 and lead to a straight-line whose the gradient Pc can be linked to the conditional stoichiometric coefficient defined in eq 3. Indeed, one can be written
δn1 rC1δV1 rC1 ) ) r Pc δn2 rC δV C 2
2
(5)
2
Titration. After the calibration operation, without changing the reaction medium, the standard nonionic surfactant solution was replaced with the solution of unknown titer (xC2). A volume of 198
(7)
+ RT ln(aDTA+aBr-) (4)
Determination of the value of the stoichiometric coefficient ν(µ1,n3) involves alternate additions of DTABr and nonionic surfactant to the reaction medium and monitoring the DTABr chemical potential value. Titration Procedure. The study used 0.2 mol L-1 NaBr; aBrcan be considered as constant, and eq 4 becomes
νµ1,n3 )
Considering eq 5 and 6, the unknown concentration of the nonionic surfactant can be easily calculated from the two gradients determined graphically:
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
Calculation Procedure. All the equations given above require that n3, the number of moles of solvent, is maintained constant. But in the experimental process adopted, n3 varies. There are two aspects to this problem. 1. The first concerns the conditional stoichiometric coefficient value. This value is susceptible to change when the solvent content in the solution varies. In the various experiments, we found that ν(µ1,n3) variations are not very large for modifications of n3 (see Table 1). This is probably because the mixed-micelle composition depends mainly on the ratio between the molar fraction of 1 and 2 once the system is aggregated and not on the number of aggregated species. The ratio X1/X2 does not depend on n3. 2. The second problem concerns the calculation of the conditional stoichiometric coefficient from experimental measurements. Indeed, when the initial DTA+ concentration is restored, a part of the added cationic amphiphile serves to compensate the dilution effect caused by addition of the two surfactant solutions. This effect can be formalized by considering that the total Brconcentration was fixed and DTA+ activity was followed with cationic amphiphile-selective electrode. Consider a small volume (δV2) of nonionic surfactant solution (concentration rC2) added at any stage of the procedure to the initial solution whose free cationic concentration (eC1) is maintained constant. In the reaction medium, the concentration of 2 is higher than its critical micelle concentration in the reaction medium.
δn2 ) rC2δV2
Some of the cationic amphiphiles are committed to interaction with the nonionic surfactant. Then, a volume δV1 of a DTABr solution (concentration rC1) is added to restore the initial value of the potential of the DTA+-selective electrode.
δn1 ) rC1δV1 Some of the DTA+ ions (δndil) added serve to compensate for the dilution which results from the additions. If these volumes of dilute
Table 1 δV1 δV2 δy ∑δy ∑δV2 (mL) (mL) (10-6 mol) (10-6 mol) (mL) ν(µ1,n3) (a) Titration of Brij 35 in 0.2 M NaBr Solutiona calibration 1 0.0075 0.150 0.73 0.73 0.150 2 0.0078 0.120 0.77 1.50 0.270 3 0.0093 0.135 0.92 2.42 0.405 4 0.0079 0.115 0.77 3.19 0.520 5 0.0070 0.100 0.69 3.88 0.620 titration 1 0.0052 0.200 0.50 4.38 0.82 2 0.0046 0.190 0.44 4.82 1.01 3 0.0057 0.220 0.55 5.37 1.23 4 0.0057 0.220 0.55 5.92 1.45
0.100 0.106 0.106 0.108 0.104 0.097 0.104 0.104
(b) Titration of Tween 80 in 0.2 M NaBr Solutionb calibration 1 0.18 0.225 0.86 0.86 0.225 2 0.12 0.155 0.57 1.43 0.380 3 0.16 0.170 0.77 2.20 0.550 4 0.10 0.150 0.47 2.67 0.700 5 0.17 0.200 0.81 3.48 0.900 titration 1 0.14 0.350 0.65 4.13 1.250 2 0.17 0.450 0.79 4.92 1.700 3 0.14 0.400 0.65 5.57 2.100 4 0.15 0.300 0.71 6.28 2.400 5 0.11 0.350 0.50 6.78 2.750
0.072 0.088 0.062 0.079 0.078 0.073 0.068 0.098 0.060
a Initial DTA+ solution volume, 10 mL; rC ) [Brij 35] 2 in std soln ) 6.37 × 10-2 M; xC2 )[Brij 35]in unknown soln ) 2.39 × 10-2 M; eC1 ) -4 r + [DTA]in init soln ) 10 M; C1 ) [DTA ]in titration soln ) 0.1 M; reference emf of the electrochemical cell, -66.5 ( 0.2 mV. b Initial solution volume of DTA+, 10 mL; rC2 ) [Tween 80]in std soln ) 0.0513 M; xC2 ) [Tween 80]in unknown soln ) 0.0239 M; eC1 ) [DTA+]in initial soln ) 10-4 M; rC1 ) [DTA+]in titration soln ) 5 × 10-3 M; reference emf of the electrochemical cell, -46.7 ( 0.2 mV.
Figure 3. (a) Titration of Brij 35 by DTABr. Determination of the two gradients leads to the concentration of the nonionic surfactant as described in eq 7. (b) Second example of titration of nonionic surfactant. In this experiment we titrate a solution of Tween 80.
solutions can be supposed to be additive, then
δndil ) eC1(δV1 + δV2)
Table 2 Pc (M)
The quantity of therefore
DTA+
associated with nonionic surfactant is
Brij 35 Tween 80
Px (M)
10-3
10-3
6.71 × 3.88 × 10-3
2.44 × 1.80 × 10-3
xC 2
(exp) (M)
error (%)
10-2
2.32 × 2.38 × 10-2
2.9 0.4
δy ) rC1δV1 - eC1(δV1 + δV2) be calculated from the ratio of these two gradients. It is this quantity that is taken into account in the calculation of ν(µ1,n3).
ν(µ1,n3) ) δy/δn2
x
C2 ) (Px/Pc)rC2
(8)
This calculation supposes that the conditional stoichiometric coefficient value is not very dependent on the solvent content of the solution. RESULTS AND DISCUSSIONS Table 1 detail two examples of titration of Brij 35 and Tween 80 solutions (xC2) against the DTA+ ions as described. Graphic Exploitation of Results. Determination of the Unknown Concentration. All the results were exploited graphically by plotting ∑δy against ∑δV2 (Figure 3). Two portions of straight line were observed on the plot, which correspond respectively to the calibration and the titration. The gradient of the calibration curve is Pc ) rC2ν(µ1,n3). The titration curve begins at the last point of the calibration curve; its gradient is Px ) xC2 ν(µ1,n3). The concentration of nonionic surfactant in solution can
For the experiments presented in Table 2, the error (%) for this titration is equal to =xC2 (real) - xC2(exp)|/xC2 (real) × 100. These error values are entirely satisfactory considering the ease and rapidity of the method. The accuracy of the titration of the chosen nonionic surfactant depends on experimental conditions, i.e., the nature and initial concentration of ionic surfactant, the concentrations of nonionic and ionic surfactants used in the calibration, and the concentration of the background salt. Therefore, accurate titration of a new compound requires a preliminary study to establish optimum conditions. This method allows titration of nonionic surfactants at concentration up or equal to the cmc. CONCLUSION Potentiometric titrations can be applied where there is a weak interaction between a solute to titrate and an ionic reagent which can be followed with a selective electrode. Thus, the development Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
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of selective electrodes facilitates the titration of many compounds that were not previously amenable to potentiometric titration, particularly as this method does not require that the electrode has a Nernstian behavior. The electrode only needs to be able to indicate the same potential value in same system states. Thus, the electrode is used as a sensor to identify an equilibrium state of the system. In the case of nonionic surfactants, we have shown that their titration in an aqueous or hydroorganic solution is possible if they form mixed micelles with ions whose chemical potential can be followed with a selective electrode. Brij 35 and Tween 80 could both be titrated with anionic surfactants such as sodium dodecyl sulfate in the presence of NaBr by following the chemical potential of dodecylsulfate. The proposed method is quick and can be automated for routine operation by associating successive operations of calibra-
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Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
tion and titration. This technique can be generalized to titration leading to stoichiometric compounds (acid-base or redox equilibria), but its accuracy is inferior to that obtained with the classic method. However, we used this technique to determine equilibrium constants with good precision.
ACKNOWLEDGMENT We thank Radiometer Analytical (Villeurbanne, France) for its technical support.
Received for review August 13, 1998. Accepted October 8, 1998. AC980921G
