Nov., 1959
DIFFCSION COEFFICIENT OF DODECYLTRIMETHYLAMMONIUM CHLORIDE
1921
DIFFUSION COEFFICIENT OF DODECYLTRIMETHYLAMMOSIUM CHLORIDE IN AQUEOUS SOLUTIONS A T 23' BY REBECCA A. PARKER AND STANLEY P. WASIK Contribution from the National Bureau of Standards, Washington, D. C. Received M a y 22, 1060
The diffusion coefficient of the colloidal electrolyte dodecyltrimethylammonium chloride has been measured as a function of concentration in aqueous solutions a t 23' a t concentrations up to 1.5 g./100 ml. The diffusion coefficient is approximately a step function of the concentration; it decreases from a value of about 8.2 X 10-1 cm.2/sec. a t concentrations below the critical micelle concentration (crnc.) to about 2.2 X 10-6 cm.2/sec. above the cmc. This behavior is consistent with other properties of the solutions.
Introduction The diffusion of paraffin-chain salts in aqueous solution might be expected to be unusual. These salts are generally thought to dissolve in water in the form of two different species': a strong monomeric electrolyte and a micellar electrolyte containing highly charged colloidal aggregates of the paraffin-chain ions. Since the two species are in equilibrium, their relative concentrations vary as a function of the concentration of the solution. Hence, the diffusion coefficient of the salt would be expected to be dependent on concentration. The approximation generally made concerning the relative concentrations of the monomeric and micellar species is that the salt dissolves in water as a strong monomeric electrolyte until the concentration is equal to the critical micelle concentration (cmc.) ; a t this concentration, the solution is saturated with monomeric ions and any further added salt dissolves to form micelles. (This is equivalent to assuming that the micelles are composed of an infinite number of monomeric ions'). If this approximation is valid, the diffusion coefficient should be a step function of the concentration with the step from the higher value corresponding to the monomeric electrolyte to the lower value corresponding to the micellar electrolyte occurring a t the cmc. The general differential equation for a unidirectional diffusion process is Fick's Second Law2 "at= ; ( D g )
where c = concentration t = time x = distance in the direction of diffusion D = diffusion coefficient
If diffusion is studied a t a liquid junction between two solutions of different concentrations, the boundary conditions which are imposed upon the solution of the diffusion equation are that a t t = 0, for x < 0, c = c1 and for x > 0, c = c2 where c1 < c2. It is assumed that the concentrations a t large values of II: in the diffusion cell are unaltered during an experiment. With these restrictions, no general solution for equation 1 has been obtained. However, if it is assumed in addition that x/ (1) G. S. Hartley, "The Colloidal State 11. A q u e o u s Solutions of Paraffin-Chain Salts .4 Study in Micelle Formation," Herrnann et Cie., Paris, 1936. (2) A. Fick, Ann. Phuszk, 170, 59 (1855).
d = 0, where 0 is a function of c o11lyl3equation 1 can be transformed and integrated a t constant time to give D in this case is a differential diffusion coefficient corresponding t o the definite concentration c rather than an integral coefficient appropriate t o the range of concentrations, c1 to c. The integral in equation 2 must be evaluated from the experimental data. This calculation is valid only if (x/dQc is constant during the experiment. Dodecyltrimethylammonium chloride was chosen for these experiments because the properties of its aqueous solutions have been studied extensively. Measurements of the viscosity of the s o l ~ t i o n s 4 ~ ~ indicate that the micelles are approximately spherical. Other properties of the solutions which have been measured are turbidity14 electromotive force of concentration cells16 micellar electrophoretic mobility' and electrolytic conductivity.8 Experimental The quaternary salt, dodecyltrimethylammonium chloride, was obtained from Armour and Co. It was purified by recrvstallization from anhydrous mixtures of ethanol and diethylkther. A chemical analysis of the carbon, hydrogen, nitrogen and chlorine content of the salt gave the theoretical values for its composition within the error of the determinations. Dextrose, National Bureau of Standards Standard Sample No. 41, was dried and used without further purification. The cell used for the diffusion experiments and the experimental twhnique have been described in detail by Longsworth.9 A sharp, plane, horizontal boundary between two solutions of different concentrations was formed in a rectangular cell. The subsequent concentration distribution a t the boundary was observed with a Rayleigh interference optic system. The temperature of the water-bath in which the cell was immersed during the measurements waa controlled a t 23 & 0.003'. The water-bath and optic system were those of a modified Aminco Model B Electrophoresis Apparatus. Photographs of the interference patterns9 were taken on glass plates. With a two-dimensional comparator, about fifty measurements of the refractive index of the solution a8 a function of distance in the cell were made for each pattern. The relation between the refractive index and the concentration of the solutions was determined with a differential refractometer. For dext,rose, the rate of change of the refractive index with concentration (dnldc) -
(3) L. Boltz~nann,ibid.,289, 959 (1894). (4) L. n l . Kushner, W.D. Hubbard and R . A. Parker, J. Research Null. Bur. Standards. 59, 113 (1957) RP 2778. ( 5 ) 12. 4 . Parker and S. P. Wasik, THIBJOURNAL,62, 967 (1958). (6) L. BI. Kushner and R. 4.Parker, ibid., 61, 822 (1957). (7) R. .4. Parker and S. P. Wasik, manuscript in preparation. ( 8 ) Unpublished data, this Laboratory. (9) L. G. Longsworth, J . A m . Chem. floc., 74, 4155 (1952).
REBECCA A. PARKER AND STANLEY P. WASIK
1922
Vol. 63
the initial boundary is not infinitely sharp when t = 0, was less than f 1 0 seconds. The evaluation of the diffusion coefficient with equation 2 requires a simultaneous determination of c, z and dc/dz. This may be done conveniently by measuring two of the quantities and calculating the third. In these experiments, data for c as a function of 2 were differentiated to obtain dc/dr as a function of c and r ; these calculations were also performed on the National Bureau of Standards IBM-704 computer. The integral L z d c in equation 2 was evaluated
/ 7
0 2 t
00
graphically for each of the experimental points. In order to estimate the error in the calculation, values for the diffusion coefficient of dextrose were calculated from the data used previously to check the calibration of the instrument. The error in the evaluation of the diffusion coefficient by equation 2 was found to be *0.5% except for concentrations near CI and c2. In these regions the error is much larger because of the relatively large error in the integration and because of a peculiarity in the differentiation program for the computer which results in less accurate values for dcldx. The value for t used in these calculations was the actual elapsed time from the start of the experiment. Since for all the experiments considered, t was greater than 6 hours, the error caused by neglecting the zero time correction was less than &lO sec/6 hours or &0.05%.
0 I IO1
lCm/sec
I/
Fig. 1.-A comparison between experimental and theoretical values for dc/dE as a function of e. e = x/di; c1 = 0; c2 = 1.027 g./100 ml. The points represent values for dc/dO calculated by differentiation of the experimental data. The line represents values for dc/dE obtained by a numerical solution of the diffusion equation, if the diffusion coefficient as a function of concentration (see Fig. 2) is approximated by three straight lines.
--
i 3
c2
04
26
08
I
,z
,o
I4
1
CONCENTRATION 1 ~ ~ 1 0 0 m l l
Fig. 2.-Diffusion coefficient of dodzcyltrimethylammonium chloride in aqueous solutions a t 23 The points represent the values for the diffusion coefficient calculated from four different experiments: CI = 0, cz = 0.560 g./lOO ml., t = 8 hours; c1 = 0.570 g./100 ml., c2 = 1.567 g./lOO ml., 1 = 24 hr.; c1 = 0, cz = 1.027 g./IOO ml., t = 15 hr.; c1 = 0, c2 = 1.027 g./100 ml., t = 20.25 hr. The value for the cmc. was obtained from measurements of the turbidity of the solutions.
.
was found to be 0.144 ml./g. For dodecyltrimethylammonium chloride a t concentrations below the cmc., dn/dc = 0.165 ml./g.; a t concentrations above the cmc., dnldc = 0.157 ml./g. From these data, the concentration distribution was calculated. In order to check the calibration of the apparatus, measurements were made of the diffusion coefficient of dextrose in aqueous solutions a t 25". The experiments performed by Longsworth9Jo were duplicated. Since the diffusion coefficient of this sugar is constant, it may be calculated from the properties of the probability function and data for c as a function of x a t a particular value for t . 9 ~ 1 1 These calculations were performed on the National Bureau of Standards IBM-704 electronic computer. The diffusion coefficient of dextrose in a solution of concentration 0.380 9.1 100 ml. obtained in six separate measurements was 6.729 (+0.006) X cm.Z/sec., compared with the value of 6.728 (&0.006) X 10-6 cm.2/sec. measured by Longsworth.lO The zero time correction12 in these experiments, i.e., the correction to the observed time due to the fact that ~
(10) L. G. Longsworth, THIS JOURNAL, 68, 770 (1954). (11) "Tables of the Error Function and Its Derivative. National Bureau of Standards Applied Mathematics Series. 41," U. 8. Government Printing Office, Washington, D. C., 1954. (12) L. G. Longsworth, J . A n . Chem. SOC.,69, 2510 (1947).
Results A number of diffusion experiments were performed a t 23" with aqueous solutions of dodecyltrimethylammonium chloride of various concentrations up to 1.5 g./100 ml. Regardless of the values of c1 and cz chosen, 0 was a function of c only. Therefore, equation 2 could be used to calculate the diffusion coefficient. The circles in Fig. 1 represent a typical set of values for dc/d0 calculated from the data for an experiment in which c1 = 0 and cz = 1.027 g./100 ml. If the diffusion coefficient were constant, the shape of this curve would be Gaussian. The experimental curve is highly asymmetrical and bears slight resemblance to a Gaussian curve. T'alues calculated for the diffusion coefficient of the salt are shown in Fig. 2 . The data, the results of four different experiments, agree within & 0.05 X cm.2/sec. The value for the cmc. shown in Fig. 2 was obtained from measurements of the turbidlty of the solution^.^ In Fig. 2 the diffusion coefficient is seen to be approximately a step function of the concentration. If the diffusion coefficient a t concentrations below the cmc. is assumed to be constant, the best value for the integral diffusion coefficient, calculated by the method described by Longsworthg from data for repeated experiments in which CI = 0 and c2 = 0.560 g./100 ml., is 8.12 (hO.01) X cm.2/sec. This is close to the value of 8.22 X low6cm.2/sec. calculated with equation 2 for c = 0.560/2 g./100 ml. If the diffusion coefficient a t concentrations above the cmc. is assumed to be constant, the best value for the integral diffueion coefficient, calculated from data for repeated experiments in which c1 = 0.570 g./100 ml. and cz = 1.567 g./100 ml., is 2.22 (hO.01) X cm.2/sec. This is close to the value of 2.24 X lop6 cm.2/sec. calculated with equation 2 for c = (0.570 1.567)/2 g./100 ml. The discrepancies may be attributed to the fact that the calculation described by Longsworth is strictly applicable only when D is a constant. h further check on the calculation of the diffusion coefficient from equation 2 was obtained by solving
+
Nov., 1959
DIFFUSION COEFFICIENT OF DODECYLTRIMETHYLAMMONIUM CHLORIDE
1923
the non-linear diffusion equation (equation 1) diffusion coefficient of a cationic colloid of connumerically for given values for D as a function of centration c in the presence of a strong supporting c. The values for the diffusion coefficient used in electrolyte of concentration c’ these calculations were obtained by approximating the data shown in Fig. 2 by three straight lines. The results of a typical cal~ulation’~ are shown in Fig. 1 as the solid line. I n both the experiment and the calculation compared in Fig. 1, the values where chosen for c1 and cp were 0 and 1.027 g./IOO ml. U = equiv. conductance of colloidaI cations The agreement between the experimental data n+ = their valence in solution and the numerical solution is seen to be very good. V = equiv. conductance of negatively charged gegenions n- = their valence in solution Discussion X = equiv. conductance of supporting electrolyte According to a theory proposed by Hartleyl to explain the behavior of colloidal electrolytes, If the supporting electrolyte is considered to be the the diffusion coefficient a t infinite dilution would be monomeric salt and if the diffusion coefficient of the expected to correspond to that of the monomeric micelles is extrapolated to infinite dilution of electrolyte. A value for the diffusion coefficient micelles (ie., the crnc.), the second term in the of the monomeric ions, D,, a t infinite dilution brackets is zero. DM may then be calculated from may be calculated from the Nernst e q ~ a t i o n ’ ~the electrophoretic mobility of the micelles extrapcm.2/volt sec.), olated to the cmc. (32.5 X as extended by Hartley16 the charge (14) and degree of ionization (36Yo) of the micelles in solution,7 and values for the concentration potential.6 The value for D M calcuwhere lated from these data is 1.86 X cm.2/sec., compared with 1.80 X cm.2/sec. obtained by a u = equiv. conductance of the cation v = equiv. conductance of the anion linear extrapolation of the data in Fig. 2. Also, f = mean activity coefficient of the ions since V/n- > U/n+, this equation predicts an The limiting equivalent conductance of the elec- increase in the diffusion coefficient with increasing trolyte is approximately 95.8 ml. (mole ohm concentration, as is observed. cm.) - l a 8 Assuming independent ionic conduction, Thus, the magnitude of the diffusion coefficient v = 73.316 and u = 22.5. The quantity d In of the salt is consistent with other properties of the f / d In c = 0 when c = 0. The calculated value for solutions. The agreement between theoretical cm.2/sec. and experimental values for the diffusion coefDm at infinite dilution is 9.05 X The value obtained by extrapolation of D as a ficient is very good considering the assumptions function of c in Fig. 2 is 8.45 X cm.2/sec. made in the derivation of the equations and the This discrepancy is not surprising since the lowest approximations made in the calculations. The change in the diffusion coefficient from the Concentration at which D could be determined in these experiments was 0.01 g./lOO ml. Below value corresponding to the monomeric electrolyte this concentration, u, v and d In f/d In c would be to that for the micellar electrolyte would be exexpected to be strongly dependent on concentra- pected to occur a t the cmc. If the micelles were tion. The value for D, when c = 0.01 g./100 very much larger than the monomeric ions, the ml. is 8.3 (*0.2) X cm.2/sec. The quantity diffusion coefficient would be an infinitely sharp d In f/d In c a t this concentration was calculated step function of the concentration. The step in from values for the concentration potential.6 Fig. 2, however, occurs over a change in concentraThe equivaIent conductance a t this concentration tion of about 0.3 g./lOO ml. Thus, even though a is 93.ti8 If it is assumed that this decrease in the micelle of this salt is an aggregate of about 40 equivalent conductance from the value a t infinite paraffin-chain ions4 the assumption that the micelle dilution may be attributed to a 2.3% decrease in is composed of an infinite number of monomeric the equivalent conductance of each ion, D, is ions is a very rough approximation in considering calculated to be 8.35 X cm.2/sec., in good the behavior of the salt in solution. It may be agreement with the experimentally determined noted in this connection that the region above the crnc., in which the diffusion coefficient decreases value at this concentration. If the approximation generally made concerning rapidly with increasing concentration, is the the relative concentrations of the monomeric and region in which the viscosity and turbidity of the micellar species is valid, the diffusion coefficient a t solution deviate from the behavior expected, if concentrations above the cmc. would be expected all of the salt added a t concentrations above the to correspond to that of the micellar electrolyte. cmc. dissolved in the form of micelles of constant A value for this diffusion coefficient, DM, may size and ~ h a r g e . ~ If the cmc. is defined as the concentration a t be estimated from Hartley’s equation17 for the which dD/dc is a maximum, the data for the dif(13) Private communication, W. C. Rheinboldt, Applied Mathefusion coefficient indicate a cmc. of 0.583 g./100 matics Division, National Bureau of Standards, 1958. ml. f 0.01 compared with the value of 0.570 (14) W. Nernst, 2. physik. Chem., 2, 613 (1888). (1.5)G. S. Hartley, Phil. Mag., 12, 473 (1931). g./100 ml. f 0.01 determined by measurements of (16) B. E. Conway, “Electrochemical Data,” Elsevier Publ. Co., the tubidity of the solution^.^ Houston, Texas, 1952. The radius of large spherical particles may be (17) G. 8. Hartley and C . Robineon, h o c . Roy. SOC.(London), A134, 20 (1931). calculated from the value for their diffusion coef-
1924
S. LINDENBAUM, C. F. JUMPER AND G. E. BOYD
Vol. 63
ficient in solution by means of the Stokes-Einstein relation
presence of 0.04 A I NaCI8 give a value of 19 f 1 A. for the micellar radius, compared with 19.1 calculated from the micellar eight.^ D = kT/Bvr Conclusions where The diffusion coefficient of dodecyltrimethylk = Boltzmann's constant ammonium chloride a t 23" in aqueous solutions a t 7 = viscosity of the solution concentrations up to 1.5 g./lOO ml. is approximately r = radius of the particle a step function of the concentration. The difThis equation is applicable to uncharged spheres or fusion coefficient decreases from a value of about to charged spheres in the presence of swamping 8.2 X loW6cm.2/sec. at concentrations below the concentrations of electrolyte. From the value cmc. to 2.2 x cm.2/sec. a t concentrations for the diffusion coefficient of the micelles, extrap- above the cmc. These values are consistent olatgd to the cmc., the cadius is calculated to be with other properties of the solutions. The be13 A. A value of 18.6 A. is calculated from the havior of the electrolyte may be explained by the micellar weight4 by assuming that the micellar usually accepted theory of colloidal electrolytes. density is 0.75 g./mL6 and that a monomolecular Acknowledgment.-The authors thank Dr. Werlayer of water is bound to the surface of the micelle. ner C. Rheinboldt and John P. Menard of the If the concentration of monomeric electrolyte a t the Applied Mathematics Division for writing the cmc. is not sufficient t,o swamp the effects of the programs for the NBS IBM-704 electronic comhigh micellar charge, the radius calculated from the puter used in the calculations. The authors also diffusion coefficient would be expected to be the thank Mrs. Rita Y. Cowan of the Chemistry Dilower of the two, as is observed. The agreement vision, who performed the graphical integrations between the two values would then be expected to and many other essential calculations, and Erle R. improve as the concentration of supporting elec- Deardorff of the Chemistry Division, who pertrolyte is increased. Preliminary measurements of formed the chemical analysis of dodecyltrimethylthe diffusion coefficient of the micelles in the ammonium chloride.
SELECTIVITY COEFFICIENT MEASUREMENTS WITH VARIABLE CAPACITY CATION AND ANION EXCHANGERS BY
s. LINDENBAUM, c. F. JUMPER AND G. E. BOYD
Contribution from the Oak Ridge National Laboratory, Oak Ridge, Tennessee Received May 88,1959
Measurements a t % 5 O of equilibrium selectivity coefficients for the exchange of N a + with H+, with " I + , Cs+ and Z n + + ions, and for Cs+ with NH4+ ions in dilute aqueous mixed electrolyte solutions were conducted with specially synthesized variable cxchange capacity polystyrenesulfonic acid type cation exchangers. Coefficients also were measured for the exchange of Br- with C1- ions using variable capacity quaternary ammonium type strong-base anion exchangers. I n all exchanges the selectivity coefficient for the preferred ion decreased as the capacity per gram dry increased when the crosslinking was held constant. This behavior does not agree with the predictions made by a recent statistical thermodynamic theory of ion-exchange gels which states that increased exchange capacity increases the ion binding and therewith the selectivity.
A renewed interest in the capacity dependence of the selective uptake of different ions by organic ion exchangers has arisen as a consequence of the recent appearance of an attractive statistical thermodynamic theory of ion binding and exchange in polyelectrolyte solutions and gels. A molecular model for linear and cross-linked polyelectrolytes is proposed in this theory, and it has been argued,a "Since ion pairing only occurs when necessary to reduce strong electrostatic fields, we may understand why resins of high exchange capacity should be more selective than resins of lower capacity." A limited amount of data is available for comparison with this prediction. With variable capacity, divinylbenzene (DVB) cross-linked poly(1) Presented before t h e Division of Colloid Chemistry, 134th National iMeeting, American Chemical Society, Chicago, Illinois, September 7-12, 1958. (2) 6 . A. Rice and F. E. Harris, 2. physik. Chem., N. F., 8, 207 (1956). (3) F. E. Harris and S. A. Rice, THIS JOURNAL, 61, 1360 (1857).
styrene sulfonates, for example, it was found4" that the selectivity coefficientdb for the exchange of sodium and hydrogen ions, D$+, increased with exchange capacity, E (meq. g.-l dry), but, in disagreement with the theory, the coefficient for the exchange of silver and sodium ions, D%:, decreased with increasing E . Independent researches6 with nominal 14-15y0 DVB cross-linked monofunctional, variable capacity cation exchangers likewise showed that Dg?' increased with E and that a selectivity inversion occurred when the exchange capacity diminished below approximately three meq. g.-l. An increase of with E , and (4) (a) G. E. Boyd, B. A. Soldano and 0. D. Bonner, ibid., 58, 540 (1954). (b) The usual definition of selectivity coefficient is followed; thus, for the sodium-hydrogen ion exchange: 0 : ; ' = ! m ~ s + / m ~ + ) ~ / ( m ~ . + /where m ~ + t)h~e mNs+ and mH+ are the molalities of these ions in the exchanger, r, and in t h e equilibrium mixed electrolyte phase, w, respectively. ( 5 ) I. H. Spinner, J. Ciric and W. F. Graydon, Can. J . Chem., 32. 143 (1954).
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