J. Phys. Chem. 1983, 87,1397-1406
1397
Thermodynamic Properties of Alkyldimethylamine Oxides in Water. Application of a Mass-Action Model for Micellization Jacques E. Desnoyers, Gaston Caron, Rosarlo DeLisl,+ David Roberts,' Alaln ROUX,~ and Gerald Perron Department of Chemistry, Universl de Sherbrooke, Sherbrooke. Quebec, Canada J1K 2R 1 (Received: July 26, 7982; In Final Form: November 9, 1982)
The freezing points, enthalpies of dilution, volumetric heat capacities, densities, and sound velocities of the homologous series R(CH&NO, for R = butyl, hexyl, octyl, and decyl, were measured in water at 25 O C and as a function of temperature in the case of octyl. The osmotic coefficients and the apparent and partial molar relative enthalpies, heat capacities, volumes, compressibilities, and expansibilities were calculated. Isochoric heat capacities and isothermal compressibilities can also be derived from these data. There is a gradual change in the trends of these functions when going from the lower homologue, which behaves like a medium-size alcohol, to the higher one, which is a typical nonionic surfactant. The osmotic coefficients are positive in the premicellar region at the freezing temperature but become negative at higher temperatures. The concentration dependence of the various functions can be accounted for quantitatively with a simple mass-action model. Aggregation numbers and thermodynamic functions of micellization can be derived with this model.
Introduction Direct studies of the thermodynamic properties of surfactants as a function of concentration are ideal to test quantitatively the various theories and models for micellization. The simplest model for micellization is that of a pseudophase transition which can account semiquantitatively for the concentration dependence of apparent molar quantitie~.'-~This model has proved to be very useful in defining the critical micelle concentration (cmc) and in deriving thermodynamic functions of micellization through apparent1p2r4p5or ~ a r t i a l molar ~ ~ ~ ,quantities. ~ However, significant deviations are observed in the micellar region. For example, the phase-separation model predicts a sharp maximum in the apparent molar heat capacity $c in the transition region and a discoptinuity without maximum in the partial molar value Cp,2if the enthalpy of micellization is finite.3 Experimentally, maxima are observed with both functions. As we would expect, deviations from the phase-separation model are larger for shorter chain surfactants. The mass-action modelgll is probably preferable to account for the thermodynamic data of short-chain surfactants. This model also has the advantage that it can be used to derive aggregation numbers from the data. 1 2 7 1 3 To investigate the mass-action model fully, one should choose very simple surfactant systems as a starting point. Ionic surfactants are not convenient for this purpose since, in addition to the micellization process, the ionic dissociation of the micelle has to be accounted for. The interactions between the micelles and the counterions will affect the thermodynamic properties and, in particular, the entha1~ies.l~Nonionic surfactants avoid these complications, and among these the alkyldimethylamine oxides (C,DAO) are of particular interest. They behave as typical nonionic surfactants at pH above 7 and at the same time they have cmc's which are reasonably high.12J5-21 While Benjamin has measured the enthalpies12 and volumes1gof
some of the C,DAO, not enough data points were obtained in the transition region to test an association model. The free energies, enthalpies, heat capacities, volumes, and compressibilities were therefore investigated over a wide concentration range for the C,DAO where x = 4 , 6 , 8 , and 10. In the case of CSDAO the volumes and heat capacities were measured as a function of temperature. It is therefore possible to obtain also the expansibilities with this surfactant. The observed trends for the volumes and heat capacities have already been reported.22 These studies show that there is a smooth transition between C4DA0, which behaves like a medium-size alcohol in water, and C&AO, where a sharp cmc is observed. There is a strong
* Address correspondence t o this author at the following address: Institut National de Recherche Scientifique, C.P. 7500, Ste-Foy, Quebec, Canada G1V 4C7. Istituto di Chimica Fisica, Universita di Palermo, Palermo, Italy. J Koningklijke/Shell Exploratie en Produktie Laboratorium Volmerlaan 6, 2288GD Rijswijk, Holland. 8 Laboratoire d e Thermodynamique e t de Cinetique, Universit6 de Clermont 11, 63170 Aubiere, France.
67, 336 (1978); 70, 448 (1979). (18) D. G. Kolp, R. G. Laughlin, F. P. Krause, and R. E. Zimmerer, J . Phys. Chem., 67, 51 (1963). (19) L. Benjamin, J . Phys. Chem., 70, 3790 (1966). (20) L. Benjamin, J . Colloid Interface Sci., 22, 389 (1966). (21) J. M. Corkill and K. W. Hermann, J . Phys. Chem., 67,934 (1963). (22) J. E. Desnoyers, D. Roberts, R. DeLisi, and G. Perron in "Solution
0022-3654/83/2087-1397$0 1.50/0
(1) K. M. Kale and R. Zana, J. Colloid Interface Sci., 61, 312 (1977). (2) E. Vikingstad,A. Skange, and H. Hailand, J. Colloid Interface Sci., 66, 240 (1978). (3) J. E. Desnoyers, R. DeLisi, and G. Perron, Pure Appl. Chem., 52, 433 (1980). (4) G. Douheret and A. Viallard, J. Chim. Phys. Phys.-Chim. Biol., 78, 85 (1981). (5) J. B. Rosenholm. Colloid Polvm. Sci., 259, 1116 (1981). (6) G. M. Musbally, G. Perron, and J. E. Desnoyers, J . Colloid Interface Sci., 54, 80 (1976). (7) J. E. Desnoyers, R. DeLisi, C. Ostiguy, and G. Perron in "Solution Chemistrv of Surfactants". Vol. 1. K. L. Mittal. Ed., Plenum Press, New York, 19?9, p 221. (8) K. Shinoda and E. Hutchinson, J . Phys. Chem., 66, 577 (1962). (9) D. G. Hall and B. A. Pethica in 'Nonionic Surfactants", M. J. Schick, Ed., Marcel Dekker, New York, 1967, Chapter 16. (10) P. Mukerjee in "Physical Chemistry: Enriching Topics from
Colloid and Surface Science", H. Van Olfen and K. J. Mysels, Eds., Theorex, La Jolla, CA, 1975, Chapter 9. (11) J. B. Rosenholm, T. E. Burchfield, and L. G. Hepler, J . Colloid Interface Sci., 78, 1981 (1980). (12) L. Benjamin, J. Phys. Chem., 68, 3575 (1964). (13) B.-0. Persson, T. Drakenberg, and B. Lindman, J . Phys. Chem., 83, 3011 (1979).
(14) R. DeLisi, C. Ostiguy, G. Perron, and J. E. Desnoyers, J . Colloid Interface Sci., 71, 147 (1979). (15) K. W. Hermann, J. Phys. Chem., 66, 295 (1962); 68, 1540 (1964). (16) W. L. Courchene, J. Phys. Chem., 68, 1870 (1964). (17) S. Ikeda, M. Tsunoda, and H. Moeda, J . Colloid Interface Sci.,
Behavior of Surfactants", Vol. I, K. L. Mittal and E. J. Fendler, Eds., Plenum Press, New York, 1982, p 343.
0 1983 American Chemical Society
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Desnoyers et al.
The Journal of Physical Chemistry, Vo/. 87, No. 8, 1983
relaxational or equilibrium displacement contribution to the heat capacity and little indication of interactions between the micelles beyond the cmc. This series should therefore be ideal for a quantitative investigation of the mass-action model for micellization. Self-consistent parameters should be obtained for all properties if the theoretical approach is good.
Experimental Section The techniques used for these investigations have all been described previously and are the same as in previous studies on s u r f a c t a n t ~ . ~ JThe ~ * ~freezing ~ * ~ ~ points were obtained with the equilibrium technique.25 Some difficulties arose with the analysis of the solutions since there was some ambiguity about the exact density of the solutions, as will be discussed later. The concentrations of the solutions were therefore checked through a conductivity titration with HC1. Even though the amine oxides are weak bases, the analyses give concentrations good to f 0 . 1 % . The freezing points were also checked with a Model 3R Advanced Instruments osmometer and the agreement was comparable to the experimental error. The enthalpies of dilution were measured with a Sodev flow mixing microcalorimeter and at high molalities with a Parr solution calorimeter, the heat capacities per unit volume were measured with a Sodev heat capacity microcalorimeter, the density was measured with a Sodev vibrating tube densimeter, and the compressibilities were measured through sound velocities by using a Nusonic solution monitor. All these techniques can in principle give reliable data down to about 0.01 mol kg-', which allows studies of the C,DAO up to Clo. The preparation and purification of the C,DAO samples used for most experiments have been described before.22 A major difficulty arose since the densities from different stock solutions were systematically shifted. This was attributed to the highly hygroscopic character of the amine oxides. It was difficult to correct for the water content of the solid surfactants by Karl Fischer titration and conductivity titrations could not be reproduced to better than 0.5%. Such an error was sufficient to account for the discrepancies between the data from different stock solutions. Although the absolute apparent molar volumes can be shifted by as much as 0.5 cm3 mol-' from one set to another, the concentration dependence seemed unaffected. Since each property was studied with the same stock solution, these systematic errors should not affect too much our data analysis through the mass-action model. However, a t the latter part of this study, when we were determining osmotic coefficients, it occurred to us that the origin of our difficulties could be the presence of traces of sodium sulfite in our amine oxide solutions. Sodium sulfite is used in the preparation process to eliminate the excess peroxide but it could be possible that it was not completely eliminated by the acetone recrystallization. A test with BaCl, definitely showed the presence of such traces. New amine oxide samples were prepared in which MnOz was used instead in catalytic amount to eliminate the excess peroxide. The products were all quite insoluble in the aqueous solution. The initial conductivities of the amine oxide solutions were now all very low at all concentrations and the reproducibility of the conductivity titration was much better (0.1%). (23) R. DeLisi, G. Perron, and J. E. Desnoyers, Can. J. Chem., 58,959 (1980). (24) R. DeLisi, G. Perron, J. Paquette, and J. E. Desnoyers, Can. J . Chem., 59, 1865 (1981). ( 2 5 ) J. E. Desnoyers, G. Perron, and C. Ostiguy, J. Chem. Educ , 5 5 , 137 11978)
FVI
kp'
Flaure 1. Excess amarent and Dartial molar isentrooic comoressibikes of amine oxides C,DAO in water at 25 OC: q 5 s t= 4i,s and = Ks,2- $ O * .
RyZ
It was therefore apparent that the origin of the differences in densities for solutions of different stock solutions arose from different amounts of traces of sulfites. Since we did not know what the concentrations of the impurities were, it was unfortunately not possible to correct our earlier measurementsz2and our relative enthalpies. In all cases the temperature was controlled to f10-3 K and the absolute value determined to 0.01 K with a Hewlett-Packard quartz thermometer. Results Heat Capacities and Volumes. The densities and heat capacities per unit volume of the four C,DAO and the derived apparent molar volumes & and heat capacities & are given in the Appendix. The data in the premicellar region were analyzed with the equation d y = Yo2+ Aym (1) where Y stands for V or Cp, m is the molality, and P,is the standard infinite-dilution partial molar quantity. The partial molar quantities Tzwere obtained from a plot of A($w)/Am against the mean molality. The heat capacity data for CIDAO and Cl,,DAO were obtained at 25 OC, those for C6DA0 a t 5 and 25 "C, and those for C8DA0 at 0.5, 2,5,15,25,35, and 55 OC. In the case of CGDAO,CBDAO, and C1,,DAO, the flow densimeter was placed in series with the flow microcalorimeter and as a result the temperatures of the volumetric data were systematically lower than those of heat capacities by 0.38 "C. This difference causes a negligible error in the calculation of r$c. The observed trends for the concentration and temperature dependences for 4v, (?p,z, &, and I&, the apparent molar expansibilities, and the parameters of eq 1have been given elsewhere.22 New parameters based on the massaction model will be discussed later. Isentropic Compressibilities. The solutions for sound velocities u were placed in a dilution cell and the dilutions were made by the addition of known masses of solvent. The isentropic compressibilities PS were derived from u in cm sdl and the densities d in g cm-3 of the same solutions by using the relation ps = l 0 2 / ( d u 2 )
12)
and the apparent molar isentropic compressibilities 4K,S were calculated from the difference between /3s and pSo. the value for pure water at 25 "C (44.77 X lo-* bar-') (3) +K,S = P S 4 V + 1000(pS - @ S o ) / m d O The original data and derived +K,S are given in the Appendix. The concentration dependences of +K,S and K s , ~ (from a plot of A(4K,sm)/hm vs. mean m ) are shown in Figure 1. As in the case of d Vand &-, &S of C,DAO varies
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983
Properties of Alkyldimethylamine Oxides in Water
TABLE I: Parameters of Eq 5 for the Calculation of
2ol
Apparent Molar Relative Enthalpies
@ L= A L m
solute
A I>
C,DAO C,DAO C,DAO C,,DAO
4780 11160 2.5 x 104
1710
+ BLm2
213 825 9550 4.25 X lo5
- _ -- - - - -
-4
m
BL
/(cm3 mol-')
oL/(kJ mol-')
oc/(J K-' mol-')
1040Ks/(cm3bar-' mol-' ) oE/(cm3K-' mol-')
T/"C T< 5' 15 25 35 55 0.12 1.62 4.62 14.62 25 34.62 54.62 0.5 2 5 15 25 35 55 0.5 2 5 15 25 35 55 25 25 2.4 9.6
y",
1 1 1 1 1
1 178.293 178.4 79 178.992 181.001 183.071 184.729 (188.50) 0 0 0 0 0 0 0 (773) (776) (794) (802) (808) (785) 811.67 -23.54 0.1974 (0.160) 0.1974
BY
1.200 -0.0580 -0.2886 -0.4169 -0.5873 -1.042 -6.359 -5.698 -4.706 -3.879 -2.369 -0.396 (+ 0.0) 11.80 11.84 11.90 11.88 11.48 12.92 13.00 (0) (0) (0) (0)
(0) (0)
-614.549 51.94 0.9442 0.5448 0.1304
mIi
YM
(mol k g - ' )
n
183.809 184.021 184.625 186.649 188.60 190.374 193.301 28.94 28.03 26.42 21.09 16.70 12.54 5.27 130.871 123.03 200.13 271.22 379.9 395.42 4 14.186 68.40 0.1958 0.1728 0.1825
0.4513 0.3549 0.2676 0.2069 0.1787 0.1678 0.3902 0.3672 0.3419 0.2669 0.21 26 0.1955 0.1750 0.3785 0.3628 0.3357 0.2591 0.21 0.1890 0.1576 0.3840 0.3644 0.3322 0.2638 0.2096 0.1844 0.1874 0.2776 0.2270 0.3683 0.3089
13.64 14.24 12.92 13.18 15.27 21.64 17.03 18.28 19.49 17.05 12.73 15.85 15.29 19.64 19.69 17.82 14.67 12.7 12.79 6.33 18.81 18.04 18.32 15.09 13.36 16.16 13.18 13.63 15.90 12.17 15.47
EQUIL
Ll
586.57 575.0 500.46 336.67 184.28 108.35 57.49 14.06 0.09715 0.2741 0.1841
0.015 0.023 0.024 0.019 0.021 0.027 0.064 0.037 0.050 0.022 0.024 0.048 0.098 0.044 0.049 0.053 0.038 0.036 0.044 0.009 8.8 10.4 6.6 5.6 2.2 3.8 10.7 0.24 0.0036 0.013 0.0057
a In view of the large error in the data in the premicellar region at high temperature, the uncertainty in these parameters may be larger.
The osmotic coefficient predicts an aggregation number of 15.1 f 2.2, essentially independent of temperature below 35 "C since no systematic trend can be observed. This value is in excellent agreement with that measured by CourchenelGand Corkill and HermannZ1from light scattering, i.e., 15. Also, as expected, mIdecreases regularly with T. The parameters for 4" are all quite reasonable. For example, for the data at 25 "C, the values that we had obtained graphically22were = 183.09 cm3 mol-', V M = 188.50 cm3 mol-l, and A v = -2.64 cm3 mol-2 kg, which are all in excellent agreement with those derived from the mass-action model. Also, the mean deviation between the model calculations and the experimental data is 0.024 cm3 mol-' is essentially the experimental uncertainty of 4". Therefore, volumes like osmotic coefficients can be accounted for quantitatively with the model if sufficient data points are available. The mean aggregation number is 16.7 but n seems to have a small temperature dependence. The relative enthalpies are all as expected. The values of LM from the model, e.g., 16.7 k J mol-' at 25 "C, are in excellent agreement with those derived from the partial molar relative enthalpies in the postmicellar region, 16.1 kJ mol-' at 25 "C. The A H M values decrease from a value of 25 kJ mol-' at 0.5 "C to 3.5 a t 55 "C. The mean aggregation number is 16.2 and, as in the case of volumes,
seems to depend slightly on temperature. An example of the agreement between the mass-action model and the experimental apparent molar heat capacities is shown in Figure 6. The modeling is excellent over the whole concentration range including the region of the characteristic hump, showing unambiguously that the origin of this hump is caused by the equilibrium displacement with a change in temperature. The program could be used at all temperatures. However, the initial slope Bc did not vary in any regular way. Also, as a result of the paucity of data points at low molalities, these initial slopes appeared to be generally too positive. More regular and ml as a function of trends were obtained for temperature if the initial slope was arbitrarily fixed at zero and G P O 2 thus determined graphically. In general, this decreased n from 1 to 2 units. These n are of the same magnitude as with other functions and again show a very slight temperature dependence. The equilibrium displacement contribution is positive and decreases with temperature as expected from the magnitude of AH,. However, the predicted EQUIL terms would vary from 846 J K-l mol-' at 0.5 "C to 16 J K-' mol-' while the experimental values vary from 587 to 57 J K-' mol-l. This difference could be due in part to the neglect of activity coefficients and their derivative in the calculation of these terms and in part to the oversimplified way
cpoz
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983
Properties of Alkyldimethylamine Oxides in Water
h. sroF
1403
I
C8
ClO
CMC
0.38
0.036
n
5
17
I
I 40 2
h o
a
-
Experimental Predicted @e Difference
80 4
120
l/m
6
160
8
200 10
240 12
mol'lkg
Figure 7. Calculation of aggregatlon numbers from osmotic coefficients using a pseudophase model.
Figure 6. Apparent molar heat capackles of C,DAO at 25 "C. Comparison wlth mass-action model.
provided there are no interactions between micelles. Under these conditions the value of YM is the same as that obtained from partial molar q ~ a n t i t i e s . A ~ plot of qjy against l / m is also useful in defining a cmc. It could be mentioned a t this stage that osmotic data can also be interpreted readily by using a pseudophase approach. Essentially, we assume that beyond the cmc the concentration of monomers remains constant and that the monomers coexist with micelles of aggregation number n. The osmotic coefficient is thus given by mqj = cmc + (m - cmc)/n (31) qj = l / n + (1 - l/n)(cmc/m)
of calculating A H M from eq 23. The ACMvalues obtained from the model are all in good agreement with those derived from the phase-transition modeLZ2 The compressibilities were measured at 25 "C only. The values of mI and n are all normal and the calculated ( A V M ) 2 / R Tis 14.4 X cm3 bar-' mol-', in excellent agreement with the experimental value 14.1 X cm3 bar-' mol-'. The value of KsMis 68.4 X lo4 cm3bar-' mol-' compared with K S ,in~ the micellar region, 69.9 X lo-' cm3 bar-' mol-'. The expansibilities were derived from the temperature dependence of qjv. The program was tested at three mean temperatures where it was possible to obtain reliable data. The convergence did not present any problems at 25 and 9.6 "C. At 2.4 "C it was necessary to fix Eo2by graphical extrapolation. The n and mI values are of the same order as with other properties. The predicted EQUIL term at 25 OC was 0.118 cm3 K-' mol-' compared with the experimental value of 0.097 cm3 K-l mol-'. Cl,$AO. The cmc of C&AO is of the order of 0.02 mol kg-' a t 25 "C. It is therefore difficult to obtain reliable thermodynamic data in the premicellar region with the present techniques. Also, it turns out that in many cases not enough data points were taken in the transition region to define with sufficient precision the inflection point. As a result the program did not always converge properly and, if it did, some of the derived parameters, e.g., the EQUIL term, were obviously wrong. It as therefore necessary to fix some of the parameters to get a meaningful fit. Properties at infinite dilution can be derived from the lower homologues assuming group additivity. Properties of the surfactant in the micellar form YM can also be derived from a phase-separation m ~ d e l . ' - ~ The J ~ apparent molar quantities are given by 4 y = YM + (cmc/m)AYM (30) and YM can then be derived from a plot of $ y vs. l / m
A plot of qj against l l m would then give n and the cmc (see, for example, ref 4 and 5). This type of plot is shown for C8DA0 and C&AO in Figure 7. The n values for the two surfactants are respectively 5 and 17, which are too low compared with those from light ~cattering.'~?~' On the other hand, the cmc's obtained are quite close to the mI values for CBDAO,indicating that our operational definition of the cmc is reasonable. The property of CloDAO which worked best with the mass-action model was the volume. If no parameters were fixed, an aggregation number of 29.4 was obtained, which is in good agreement with the value of -35 obtained by Herrmann.15 However, the value of PoAwas too low and the initial slope positive, which is improbable. PAwas therefore fixed at 215 cm3 m o F , a value consistent with the graphical extrapolation22and with the value estimated from group additivity by using the volumes of the lower members of the series. Although the aggregation number is now decreases to about 22, somewhat on the low side, all the other parameters appear reasonable. These parameters are given in Table V. The value of AVM is 7.4 cm3 mol-', in good agreement with the phase-equilibrium value, 7.85 cm3 mol-'.22 As a final test V Mare also fixed at the value 221.5 cm3 mol-' obtained from the phaseequilibrium model. n now goes up to 48 while the other parameters appear reasonable. These parameters are also given in Table 11. The osmotic coefficient would not converge a t all. The origin of the nonconvergence was probably the low-concentration data where the uncertainty was high. The program was then run after fixing mI at 0.038 mol kg-', the value of the cmc obtained at the freezing temperature from the pseudophase model (see above). The aggregation number a t the freezing temperature, 23, was not in good agreement with one of the values obtained from volumes. The relative enthalpies with no fixed parameters gave an aggregation number of 13.5. Fixing the initial slope at
\
x
\
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983
1404
Desnoyers et al.
TABLE V : Thermodynamic Parameters of C,,DAO at 25 ' C a property
AY
_.I_-__
@(atT f ) ov/tcm3 mol-')
or,/(k J mol-') pC/(JK
mol-')
10'c)ii,s/(cm3 bar-' mol-')
(-26)
2.976 -34.97 -29.47 10.88 45.36 (0) (0) 280
YM
m I/( mol kg-' )
n
221.16 (221.5) 16.33 (13.55) 461.5 (482) 84.85
(0.0380) 0.02049 0.02014 (0.0205) 0.0215 0.0219 0.0215 (0.0220)
22.13 21.70 48.04 15.5 18.5 15.3 20.0 22.02
EQUIL
L7
139.1 127.6 0.2672
0.050 0.15 0.18 0.078 0.1 5 7.2 9.5 7.9
Values in parentheses were fixed.
25 kJ mol-2 kg (see Table I) made things slightly worse since n went down to 12. The next step was to fix mI instead at the value obtained with volumes 0.0205 mol kg-'. The value of n now went up to 15.5. The value of t2in the postmicellar region is 13.55 kJ mol-'. The value from the model with no fixed parameters was 13.81 kJ mol-', in good agreement. On the other hand, if mIis fixed at 0.0205 mol kg-', the value of LM goes up to 16.3 kJ mol-', which is obviously too high. A further test was therefore made by fixing LM at 13.55 kJ mol-l. A slightly better value was then obtained for n, 18.5. The program could not converge properly with heat capacities either if no parameters were fixed. The initial slope Ac was obviously too positive, 110 J K-' moF2 kg, n too low at 16, and the EQUIL term too low, 126.7, while the predicted value of (AHM)2/RT"?was 217 J K-' mor2 kg. Fixing CpoA at 988 J K-' mol-' and Ac a t 0 decreased n slightly and increased the EQUIL term to 139 J K-' mol-2. The value of ACM,527 J K-' mol-', agrees reasonably with the phase-transition If CP,Mis also fixed a t 482 J K-' mol-', the value from the phase-separation model, the aggregation number went up to 20 and mI was now in better agreement with other values but the EQUIL term was decreased slightly. The same difficulties were encountered with r $ ~ , ; . If KsoAwas fixed at the value obtained by group additivity, -26 X cm3 bar-' mol-', and mI a t the value of 0.022, good convergence was obtained and a value of n = 22 was obtained in agreement with other properties and KS,Mwas as expected. However, the EQUIL term was 2 orders of magnitude too low, 0.26 X lo-* compared with the predicted value of 22 X cm3 bar-' mol-'. If the EQUIL term and Ksoz were fixed, the program would not converge. If KS,Mwas fixed also a t 87.1 X cm3 bar-I mol-', the program converged slowly and the values of n oscillated between 32 and 38 and the m Iwas too high. I t would appear that the difficulties of convergence and of deriving meaningful aggregation numbers primarily results from the paucity of data points in the transition. Since the functions are changing rapidly over a small concentration region, it is difficult to find an inflection point. Data at more closely spaced concentrations would be required. Discussion The simple mass-action model for micellization appears to fit quantitatively the thermodynamic properties of most amine oxides in water despite the numerous assumptions and approximations made, namely, a single-step association process, no interactions between micelles, only pair interactions between monomers, neglect of activity coefficient in the equilibrium equation, and a temperature- and pressure-independent aggregation number. The minimum number of parameters necessary to define a function varies from three with osmotic coefficients to six with the second From the parameters derivative function $,-, &, and @Ka in Tables 11-V, the functions can be calculated at any
molality by using eq 28, eq 29, and the appropriate relation for the apparent molar quantity. While the model is ideal for medium- or short-chain surfactants, it can possibly be applied to longer chain ones such as Cl&AO provided the thermodynamic properties in the premicellar region can be predicted by group additivity and enough data points are generated in the transition region. For more complex systems, such as ionic surfactants, the model can be refined by the introduction of a Debye-Huckel term for the monomers and the degree of dissociation of the micelles. However, it must be borne in mind that the more parameters that are involved, the more data points that would be required to extract the parameters. The mass-action model is also quite useful to derive the thermodynamic functions of micellization from AYM = YM - Yo*- AYaImI
(32)
where aI is calculated from eq 28. These AYM are, in general, in excellent agreement with those derived from the phase-separation The enthalpies of micellization at different temperatures can be derived from two approaches. The $ J ~data at various temperatures were derived from the data at 25 "C and from the heat capacities and AHMextracted by using eq 32 or else it can be calculated from
where AHoMis the enthalpy of micellization at To. If ACM is given by ACM = AC"M + B(T - To) (33) then AHM = AHOM
+ (AC"M - BTo)(T - To) + ( B / 2 ) ( F - To2) (34)
For C6DA0, B = 9.1 J K-2 mol-' and AC"M is -428 J K-' mol-'. The predicted values for A H M a t 0.5 and 55 "C using eq 34 are respectively 27.8 and 5.8 kJ mol-' while those derived from Table IV are respectively 25 and 3.5 kJ mol-'. If the temperature dependence of ACM is neglected, the values from eq 34 at 0.5 and 55 "C are now 25 and 1.8 kJ mol-' in better agreement with those from Table IV. These self-consistency tests support reliability of the AHMderived from the mass-action model. The temperature dependence of AVM can also be used to derived AEw At 10 and 25 "C AEM would be -0.06 and -0.004 cm3 K-' mol-', respectively. The corresponding values derived from the data in Table IV are -0.05 and -0.19 cm3 K-' mol-'. This latter values appears to be erroneous. The variation of mI and n of C6DA0 are shown for different properties as a function of temperature in Figure 8. The values of ml from different properties are generally in excellent agreement. Those from osmotic coefficients appear to be slightly out of line at low temperatures. The
The Journal of Physical Chemistry, Voi. 87,
Properties of Alkyldimethylamine Oxides in Water
No. 8, 1983 1405
?.
0.45
I
L 0
10
20
30 T
40
50
( O C )
Flgwe 8. Variation of m , and n of C8DA0 with temperature: (0)from 4; (0) from 4"; (+) from 4'; (V)from &; (0)from 4K;(-) predicted cmc with eq 35; (---) predicted cmc assuming a constant AH,.,.
predicted temperature dependence of the cmc is calculated from6 R In (cmc) = R In (cmc)(To)+ ( A H O M - TOAC'M + T$/2)(1/T l/T,J - (AC"M - 7'8)In ( T / T J - (B/2)(T - 7'0)(35) where B is defined by eq 33. The predicted cmc's follow closely the m Ifrom osmotic data. On the other hand, if a constant AHMis assumed, then a better agreement is observed with the mI from other properties. The aggregation numbers for C8DA0 are essentially the same for all properties investigated with the exception of those from osmotic coefficients which are smaller a t low temperatures and larger a t high temperatures. These aggregation numbers appear to go through a slight minimum as a function of temperature but this effect is probably not real and probably results from some of the approximations made. It is therefore probably valid to assume that n is independent of temperature. It is difficult to decide how real these aggregation numbers are in view of the approximations and assumptions made in the model. Especially when n turns around 6, there is probably a broad distribution even if the estimated error in n is less than 1. More elaborate models, involving, for example, a distribution of micellar sizes or corrections for activity coefficients in the equilibrium equation, may yield somewhat different average n. However, this would also result in more parameters needed to fit the data and there is little advantage in doing so with the present systems. The model can also be used to generate data for the calculation of heat capacities a t constant volume and isothermal compressibilities. When the parameters in Table IV were used, the coefficient of thermal expansion a, the density d, the isobaric specific heat c p , and the isentropic compressibility /3s were generated at fixed molalities from the apparent molar quantities. Then isothermal compressibilities /3 and specific heat capacities at constant volume c v were calculated from /3 = /3s + a2T/dCp (36) cv = CPPS/P (37) Taking Po and cv,oas 45.235 X lo* bar-' and 4.1363 J K-' g-', we calculated dK and 4c,vof CsDAO at 25 "C in the usual way, and the results are compared with 4 K 3 and 4cp in Figure 9. The general trends are the same for c $ ~and 4K,S, @K being systematically more positive by 10 x -15 X cm3 bar-' mol-' as generally observed with alcohols and surfactants. The mass-action model can also
'g *1
c //
0.0
" -2.0
0
0.4
0.a
I .2 mol kg-l
I. 6'
Flgure 9. Apparent molar heat capacities at constant volume and isothermal compressibilities of C8DA0 at 25 OC. Comparison with 4c,p and 4 K . S .
be used to derive an aggregation number from q~~ and the value 12.27 is comparable with that from c # I ~and , ~ so are the parameters mI and EQUIL. The c $ ~are , ~less positive than bCs by about 125 J K-' mol-', the main difference being in the premicellar region. The negative initial slope of $c,v in the premicellar region is more in line with the expected trends in the monomer-monomer interactions if the main effect of the interaction is a reduction of the hydrophobic hydration. The aggregation number derived from 4c,v is 13.38, in excellent agreement with that obtained from $cs, 13.56. Excellent agreement is also observed with the parameters mI and EQUIL. Acknowledgment. We are grateful to the Natural Science and Engineering Council of Canada, to the Quebec Ministry of Education, and to the France-Quebec Exchange program for financial assistance. We also thank Dr. R. Palepu for his help in establishing the purification procedure of the amine oxides.
List of Main Symbols Y thermodynamic function, stands for G, H, V ,E , K , Yo,
4Y AY
Yi
K
cP,0r-L standard partial molar quantity of solute apparent molar quantity interaction parameter in premicellar region osmotic coefficient isentropic compressibility molality of solute or of species i where i can be water (W), monomers (A), or micelles (M) activity of species i activity coefficient of species i equilibrium constant for the micellization process aggregation number fraction of surfactant in the monomeric state value of a at the inflection point d2a/dm2= 0 molality of the inflection point or cmc thermodynamic function of micellization
1408
AY’M d , do
Ps, pso U
0
AHID cp, c y a in eq
J. Phys. Chem. 1983, 87,1406-1408
value of AYM at 25 ‘C density of the solution and of the solvent isentropic compressibility of the solution and the solvent in bar-l sound velocitv freezing point depression integral heat of dilution specific heat capacity at constant pressure and at constant volume coefficient of thermal expansion
P
36
isothermal compressibility Registry NO. C~DAO,20762-86-1;C ~ A O3, u i w - 7 ; C ~ A O ,
2605-78-9; CIODAO, 2605-79-0.
Supplementary Material Available: Full-sized photocopies of the Appendix giving all the original data are available (8 pages). Ordering information is available on any current masthead page.
Pressure-Volume-Temperature Relations and Isotropic-Nematic Phase Transitions for a 4’-n-Alkyl-4-cyanobiphenyl Homologous Series Torhlakl Shlrakawa, Takao Hayakawa, and Tanekl Tokuda Department of Chemistry, Facuity of Science, Tokyo Metropol/tan University, Setagaya-ku, Tokyo 758 Japan (Received: August 13, 1982; In Final Form: November 18, 1982)
The P-V-T relations were measured for a 4’-n-alkyl-4cyanobiphenyl homologous series near the phase-transition point from the nematic to the isotropic phase. Log-log plots of the transition temperature vs. molar volume at the transition points gave linear relations for three homologues. The slopes d In T,/d In V, are found to be -7.62, -6.10, and -5.15 for 5CB, 6CB and 7CB, respectively. The first numerals in the abbreviated codes refer to the carbon number in the alkyl chain. Agreement was not good either qualitatively or quantitatively between the observed values of d In T,/d In V, and those calculated by the extended hard-rod theory hitherto presented, which suggests that the “softness” of the intermolecular potential should be taken into account for these systems.
Introduction The molecular shape asymmetry is an important factor which determines the properties of a particular substance whether it exhibits the liquid crystalline phase or n0t.l The effect of molecular shape on the potential function has attracted considerable interest from several investigators. The theory of the liquid crystalline phase is divided into three classifications: (1)order-disorder theory, (2) hard-rod theory, (3) mean-field theory. We have previously shown2 that the experimentally observed value of -d In T,/d In V, agreed with that calculated by the Pople-Karasz t h e ~ r ywhich , ~ was extended order-disorder theory, where T,is the transition temperature and V, is the molar volume at the transition points. The PopleKarasz theory, however, is not the one which takes into account the molecular shape factor, while those theories which are based on hard-rod models account for the effect of the length and the width of molecules. The theory of fluid structure based on hard-rod models was first discussed by Onsager.* Zwanzig6 proposed a version of Onsager’s theory which enabled the estimation of the virial coefficients to a much higher order. Alben,G Kimura,’ and Savithramma and Madhusudana8 also pro(1)Gray, G. w. ’Advances in Liquid Crystals”; Brown, G. H., Ed.; Academic Press: New York, 1976;Vol. 2, p 1. (2)Shirakawa, T.; Inoue, T.; Tokuda, T. J. Phys. C k m . 1982,86,1700. (3)Pople, J. A.;Karasz, F. E. J . Phys. Chem. Solids 1961, 18, 28. (4)Onsager, L. Ann. N. Y. Acad. Sci. 1949,51,627. (5) Zwanzig, R. J. Chem. Phys. 1963,39, 1714. (6)Alben, R.Mol. Cryst. Liq. Cryst. 1971,13,193. (7)Kimura, H.J.Phys. SOC. Jpn. 1974,36, 1280.
posed an extension of the hard-rod-model theory. They introduced the spherocylinder system which involves attractive forces as well as repulsive ones. With the hard-rod systems, the length-to-width ratio of a molecule determines the thermodynamic properties of the liquid crystalline phase. In a previous paper,2 we have suggested that the value d In T,/d In V, is a useful measure for testing theories. The value d In T,/d In V, is one of the calculable properties in hard-rod-model theory for the nematic liquid crystalline phase. If we measure the d In T,/d In V, value for compounds varying in length-to-width ratio, then we can check the theories easily. The homologous series of 4’-n-alkylcyanobiphenyl has four nematogenics. They are 5CB, 6CB, 7CB, and 8CB. In this paper, we have measured d In TJd In V, for three homologues of 4’-n-alkylcyanobiphenyl,and we compare these values with the theoretical values and discuss the applicabilities of the theories to the present system.
Experimental Section The compounds of the 4’-n-alkyl-4-cyanobiphenylhomologous series (alkyl = pentyl (5CB), hexyl (6CB), and heptyl (7CB)) were obtained from Tokyo Oka Industry, Ltd., and were used without further purification. The clear points of 5CB, 6CB, and 7CB were measured with a differential scanning calorimeter. The density of the materials was measured with a Lipkin-Devison type picnometer. (8)Savithramma, K.L.; Madhusudana, N. V. Mol. Cryst. Lzq. Cryst. 1980,62,63.
0022-3654/83/2087-1406$01.50/0 63 1983 American Chemical Society
