Equation surfactant concentration

Furthermore, in a series of polyoxyethylene nonylphenol nonionic surfactants, the value of varied linearly with the HLB number of the surfactant. The value of K2 varied linearly with the log of the interfacial tension measured at the surfactant concentration that gives 90% soil removal. Carrying the correlations still further, it was found that from the detergency equation of a single surfactant with three different polar sods, was a function of the sod s dipole moment and a function of the sod s surface tension (81). 

The main equation of the model describes the dependence of retention factor, k, from surfactant concentration, c and modifier concentration, c ... 

Two other general ways of treating micellar kinetic data should be noted. Piszkiewicz (1977) used equations similar to the Hill equation of enzyme kinetics to fit variations of rate constants and surfactant concentration. This treatment differs from that of Menger and Portnoy (1967) in that it emphasizes cooperative effects due to substrate-micelle interactions. These interactions are probably very important at surfactant concentrations close to the cmc because solutes may promote micellization or bind to submicellar aggregates. Thus, eqn (1) and others like it do not fit the data for dilute surfactant, especially when reactants are hydrophobic and can promote micellization. 

By combining (1), (3) and (4), expressions (5) and (6) are obtained. These, or similar, equations readily explain why first-order rate constants of micelle-assisted bimolecular reactions typically go through maxima with increasing surfactant concentration if the overall reactant concentration is kept constant. Addition of surfactant leads to binding of both reactants to micelles, and this increased concentration increases the reaction rate. Eventually, however, increase in surfactant concentration dilutes the reactants in the micellar pseudophase and the rate falls. This behavior supports the original assumption that substrate in one micelle does not react with reactant in another, and that equilibrium is maintained between aqueous and micellar pseudophases. 

Key questions in these treatments are the constancy of a (or P) and the nature of the reaction site at the micellar surface. Other questions are less troubling for example the equations include a term for the concentration of monomeric surfactant which is assumed to be given by the cmc, but cmc values depend on added solutes and so will be affected by the reactants. In addition submicellar aggregates may form at surfactant concentrations near the cmc and may affect the reaction rate. But these uncertainties become less important when [surfactant] > cmc and kinetic analyses can be made under these conditions. In addition, perturbation of the micelle by substrate can be reduced by keeping surfactant in large excess over substrate. 

According to this equation, the plot of l/kw as function of C would show a minimum and the plot beyond Cmin would be linear with a positive slope and positive intercept. However, at the optimum surfactant concentration corresponding to the maximum in the plot of kv against [Surfactant], the following relationship is obtained. 

This model leads to Equation (2) describing the observed rate constant kohs as a function of surfactant concentration [S], cmc, and aggregation number N ([S]-cmc)/A corresponds to the concentration of micelles. 

The Menger-Portnoy model is closely related to the Berezin model employing partition coefficients instead of equilibrium constants.For the case where only two pseudophases (bulk water and micelle) are considered, the partitioning of the reactant is given by the partition coefficient P. This leads to Equation (4) describing observed rate constants as a function of surfactant concentration. 

Addition of further surfactant then dilutes the reactants by increasing the volume of the micellar pseudophase and the observed reaction rate (or reaction rate constant) decreases. This kinetic scheme results in Equation (7) describing the observed second-order rate constant obs,2 as a function of micellized surfactant concentration [8]. ... 

Equations 27 and 28 present the extension of the Szyszkowski-Langmuir model to the adsorption of one-surfactant systems with aggregation at the interface. For the formation of dimmers on the surface, n = 2 and Eqs. 27 and 28 can be expanded to obtain the Frumkin equation of adsorption state. In general, the surface aggregation model described by Eqs. 27 and 28 contains four free parameters, including coi, n, b and Fc, which can be obtained by regression analysis of the data for surface tension versus surfactant concentration in the solution. 

Azeotrope Micellization. Here the micelle mole fraction of 1 is the same as that of the monomers (10.111. For fixed a = the monomer and micelle compositions are equal to a for all total surfactant concentrations. If an azeotrope exists, this azeotropic condition and Equation 5 imply that... 

Azeotrope Micellization. Again, if azeotrope exists, x = a = a. Then for all total surfactant concentrations above the cmc this condition and Equations 15 and 16 imply that... 

This equation contains the counterion concentration Cj + which depends on the total surfactant concentration. It follows that x would depend on c j+ and hence would vary above the cmc. This contradiction implies that azeotrope micellization cannot occur if = J3(x). Of course, if c c, the C + would be constant and azeotropy can again occur. If d fdx = 0, azeotropy can be also possible. For ySj = / 2 = 0.7, Cj = 0, c /cj = 3.0, and w(x) = A + B ( 2x-l), which is the Redlich-Kbter expansion (12), with A = -3 and B = 0, one finds from Equation 21 that 0.8113. No value of Qj can be calculated if = 0.7, / 2 = 0.3, and / (x) = iX + /32(1 x). Figures 1 and 2 illustrate this point showing monomer and micelle concentrations (or inventories) for a = 0.8113. In the ionic/ionic case, the micelle composition x and the ratio Cj/c2 are constant above the cmc. In the ionic/nonionic case (Figure 2) the micelle composition varies with total surfactant concentration. Osborne-Lee and Schechter (22) have found evidence of azeotrope micellization for... 

At high surfactant concentrations the surface pressure - area curves tend towards the surface pressures of the pure surfactant i(AII -r 0). Thus the integrals in equation 15 appear to be zero for > 1.4 nm2 molecule" and the adsorptions are then equal to the adsorptions for the monolayer-free system. In contrast, the Pethica equation at this area still imposes a significant correction factor on the adsorption the slope (3II/3 Inmg) for Am = 1.4 nm molecule" equals that for the monolayer-free system but (2m-2m)/... 

Here km and kw are the second-order rate constants in the micellar pseudo-phase and the aqueous phase, respectively, Phrp and Prfc are the partition coefficients for HRP and RFc, respectively, between the micellar and aqueous phases (PA= [AJm/fA],, A = HRP or RFc), C is the total surfactant concentration without cmc (C = [surfactant] t-cmc), and V is the molar volume of micelles. Equation (39) simplifies assuming Phrp <3C 1 and PRFc 1. In fact, the hydrophilic enzyme molecule is expected to be in the aqueous phase, while hydrophobic, water-insoluble ferrocenes have a higher affinity to the micellar pseudo-phase. Taking also into account that relatively low surfactant concentrations are used, i.e., CV <5iC 1, Eq. (39) transforms into Eq. (40). 

Recall that kexp and k0 are rate constants with and without surfactant, respectively, for the reaction in question, and that c and cCMC are surfactant concentrations in the reaction mixture and at the CMC, respectively. Therefore kexp> k0, c, and cCMC are experimentally accessible. Equation (28) predicts that a plot of (kexp — k0) -l versus (c — cCMC) -l is a straight line with a slope and intercept that have the following significance ... 

Figure D3.5.5 Equilibrium surface tension of sodium dodecyl sulfate (SDS) at the air-water interface as a function of surfactant concentration. The corresponding surface coverage was calculated using the Gibbs absorption equation (Eq. D3.5.26).

Equations D3.5.30 and D3.5.32 are both very valuable. They state that the rate of adsorption can be obtained from plots of the interfacial tension versus either tA- (for t—>0) or lth (for the long-term solution f— >). With these two equations the tool to extract the adsorption rate from experimentally obtained surface tension-time curves is at hand. It should be noted that instead of the Gibbs model, one could use one of the previously mentioned adsorption isotherms such as the Langmuir adsorption isotherm to convert interfacial tension to interfacial coverage data. The adsorption isotherms may be obtained by fitting equilibrium surface tension data versus surfactant concentration. 

In Section 3.2 it was shown that surfactants can influence the magnitude of Kp. To use equation (3) to simulate changes in Kp, the following information is required 1) the aqueous surfactant concentration in the CFSTR as a function of time 2) the sorbed phase surfactant concentration in each of the NK sites 3) the magnitudes of Kmn and Kmc for each surfactant and, 4) the magnitude of Ks. Unfortunately, the required information to incorporate equation (3) into the distributed-rate model was not determined for this study. As a result, the influence of the surfactants on the distribution coefficient was not considered. 

Also shown in Figure 7 is the theoretical curve based upon Equation 16, and taking into account particle swelling by monomer and growth by polymerization. In contrast to it, both experimental curves rise much more steeply. The experiment run in the absence of surfactant, sodium dodecyl sulfate (SDS), has the steeper slope. This pronounced difference due to surfactant concentration, even at levels as low as the 6 X 10 molar used here, has always been found in the many experiments which have been run (33). 

This is an equation for a straight line, the slope of which is 2R /t, and describes the plots in Figure 8. From the experimental slopes one may calculate the half-life, T, and the Fuchs stability factor, W. These values are given in Table III. They confirm that coagulation is an important process in determining particle size and number from the outset. With no surfactant present the particles apparently have a net attraction for each other (W < 1), and disappear with a half-life on the order of four milliseconds. As surfactant concentration is increased it... 

The adsorption of surfactant on a mineral surface is generally an exothermic process and at a constant surfactant concentration in the solution its adsorbed amount decreases with increasing temperature. Using the partial derivation 31n c/3T for a constant number of CH2 groups n of an hydrocarbon chain of a surfactant and constant values of the surface coverage a, we arrive to an equation for calculation of the total adsorption energy E. From the relation E = Ep + Ea, Richter and Schneider74 derived an equation which is valid for all isotherms ... 

Equation (9) allows one to obtain the surface excess from the variation of surface or interfacial tension with surfactant concentration. T can be obtained from the slope of the linear portion of the y - log C curve as illustrated in Fig. 6 for A/L and L/L interfaces. 

For the mixture of non-ionic and ionic surfactants in water with no electrolyte, the coefficient decreases from 2 to 1 with a decrease in the ionic surfactant concentration at the interface [ 12]. For the ionic surfactant solution in the presence of electrolyte such as NaCl, KC1, NaBr and KBr [ 13-15] the Gibbs adsorption equation is ... 

Near the erne, the aggregation number corresponding to the maximum in the size distribution curve is lower than the value given by the above equation, but as the total surfactant concentration increases it approaches that value. Using the Ben-Naim-Stillinger expression for the free energy, we ohtain... 

II.2. Charged Lyotropic Lamellar Liquid Crystals. In the case of a charged system, two changes in the system of eqs 17 must be made.80 81 First y should be replaced in eq 17a, for the surfactant, by y + fy=mole fraction of the surfactant in water, Xsw, should be replaced in the same equation by the mole fraction of the surfactant in water, in the vicinity of the interface CX SW). Assuming that the surfactant in the aqueous phase is totally dissociated, the concentration of surfactant near the interface (Xsw) can be related to the average surfactant concentration in the water phase, Xsw, by using the equilibrium and mass balance relations... 

---