Langmuir 1993,9, 96-100
96
Temperature Dependence of Critical Micelle Concentrations and Heat Capacities of Micellization for Ionic Surfactants Norbert Muller Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 Received August 28,1992. In Final Form: October 13,1992 When the critical micelle concentration of an ionic surfactant has a minimal value, cmc*, occurring at a temperature !P,the cmc at other temperatures depends on cmc*, T / P , and ACp/(l + a). ACp is the change in heat capacity for micellization, and a is the fractional degree of counterion binding. When ACp/(l + a) is treated as a constant, a simple equation for this relation is readily obtained. It ia tested on 20 sets of experimental data and found to fit with root mean square deviations lees than 1% for 19 of them and 1.6%for the other. The possibility that AC, varies with temperature is not ruled out, but the fits are not significantlyimprovedby assuminga linear dependence. Approximate best-fit parameters are presented and used to find values of ACp and other changesin thermodynamicvariablesof micellization for each surfactant. Most of the results are consistent with a suggestion of Evans and Ninham that (1 + a)RTIn cmc is equal to the contribution of hydrocarbon chain-water interactionsto the Gibbs energy of micellization, but there are several anomalies remaining to be resolved. Introduction Critical micelle concentrations (cmc) have been determined as a function of temperature for many ionic surfactants. Such results often define a U-shaped curve with a minimum not far from room temperature.' These have been used to evaluate changes in thermodynamic properties for micellization. For micelles that consist of m surfactant ions and p counterions, with a = p/m, the usual procedure is to write AGmo = (1 + a)RT In cmc
(1)
AHmo= -(1+ a)R'la d(ln cmc)/dT
(2)
under conditions of constant pressure. Attempts to obtain the corresponding heat capacity changes, ACp, from the temperature derivatives of AHmohave met with limited success, because the result at any particular temperature is very sensitive to possible small errors in the nearest data points. It sometimes appears that ACp varies with the temperature rather erratically.2-6 Quite recently some of these observations were reexamined by La Mesa, who introduced a very interesting 'reduced variables" treatment? When cmc* is the minimal value of cmc and P the temperature at the minimum, he found that plotting (cmc - cmc*)/cmc* against I(T- !P)/ PI for an assortment of surfactants gave points falling fairly close to a single curve represented by the empirical equation (cmc - cmc*)/cmc* = I(T - P)/!Plr (3) with y = 1.74 f 0.03. The exponent y has no obvious physical meaning, and it is not clear why eq 3 works as well as it does or what is the main source of the deviations of data points from the curve. These considerationssuggested taking a further look at these U-shaped curves to find whether a relation similar to eq 3 can be derived thermodynamically. The results, (1) Kresheck, G. C. In Water: A ComprehenuioeTreatise;Franks, F., Ed.;Plenum Press: New York, 1975; Vol. 4, p 95. (2) Adderson, J. E.;Taylor, H. J. Colloid Sci. 1964, 19,495. (3) Stead, J. A.; Taylor, H. J. Colloid Znterjace Sci. 1969, 30, 482. (4) Adderson, J. E.; Taylor, H. J. Pharm. Pharmacol. 1970,22, 523. (5) Adderson, J. E.; Taylor, H. J . Pharm. Pharmacol. 1971,23, 312. (6) La Mesa, C. J.Phys. Chem. 1990,94, 323.
presented here, appear to offer a more satisfying way to analyze this sort of data, eepecially when the goal is to find ACp.
Theory There has been much discussion regarding the applicability of eq 2, in light of the fact that the aggregation number, m, and the number of bound counterions,p, are not expected to be independent of the temperature.'^^ For ionic surfactants it is reassuring that good agreement is found when results from eq 2 can be compared with those from calorimetric measurements.1Js* In this paper, the validity of eq 2 is provisionallyassumed in order to explore more fully what it implies about the variation of cmc with temperature in the neighborhood of the minimum. When the temperature range is not toowide, areasonable first approximation is to treat ACp as a constant. Noting that eq 2 implies that AHmo vanishes at P,it follows that at other temperatures AHm'
ACp(T- P )
(4)
while ASmo = ASm'* + ACp In ( T / P ) (5) where A&"* is the entropy change at P and is equal to -AG,O*/!P. Then
ACmo ACp[(T- P )- T In ( T / P ) ] - TUm'* (6) and by eq 1
In cmc = [ACJ(l+ a)RI[(T - P ) / T + In ( P / T ) I-
ASmo*/(l+ a)R (7)
Since when T = !P
In cmc* = -ASmo*/(l + a)R
(8)
this gives
In (cmc/cmc*) = [AC,J(l+ a)RI[l- !P/T + I n P / T ) l (9) (7) Muller, N.In Micellization Solubilization, and Microemubiom; Mittal, K. L., Ed.; Plenum Press New York, 1977; Vol. 1, p 229. (8) Espada, L.; Jones, M. N.;Pilcher, G. J. Chem. Thermodyn. 1970, 2, 1.
1993 American Chemical Society
Temperature Dependence of Cmc’s
Langmuir, Vol. 9, No. 1, 1993 97
Table I. Data Sets Used To Test Equations 9 and 12 and Numerical Rerults data set 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
compound dodecyl pyridinium bromide2
l-dodecyl-4-methoxypyridiniumbromide3 1-dodecyl-4-methoxypyridiniumchloride3 1-methyl-4-dodecyloxypyridinium bromide3 1-tetradecyl-dmethoxypyridiniumbromide3 decyl a-picolinium bromide4 dodecyl a-picolinium bromide‘ tetradecyl a-picolinium bromide‘ decyltrimethylammonium bromide5 dodecyltrimethylammonium bromide5 dodecylbenzyldimethylammonium bromide5 sodium n-octyl sulfate15 sodium n-decyl sulfate13 sodium n-decyl sulfate15 sodium 2-decyl sulfate13 sodium n-dodecyl sulfate13 sodium n-dodecyl sulfatelS sodium n-tetradecyl sulfate13 sodium 2-tetradecyl sulfate13 sodium 4-tetradecyl sulfate13
temp range PC) 5-70 5-55 5-60
15-60 20-60 5-70 10-70 10-70 5-70 5-70
10-70 10-55 0-65 10-55 10-65 0-65 10-55 25-75 10-65 25-75
no. of pointa 14 11 12 10 9 14 120 13 14 14 13 10 14 10 12 14 10 11 12 11
10Scmc* (mole fraction) 20.14b 15.14b 22.05b 6.136 3.566 83.23 18.35 4.60 113.0 26.4 9.86 238b 58.76 59.66 80.56 14.86 15.1b 3.67b 5.8g6 9.096
T+ (“C) -ACp/(l 13 17 28.6 12 4 23 19 13, 25 20 18 28.5 30 27 38 25 24 20 27 32.5
+ a)
241 323 219 291 256 236 267 332 168 241 328 272 253 291 259 308 308 325 333 350
rma dev (%) 1.55 0.41 0.50 0.40 0.84
0.75 0.69 0.72 0.70 0.39 0.77 0.78 0.24 0.40 0.43 0.42 0.52 0.25 0.26 0.23
a A datum at 35 “C was discarded because of an apparent typographical error. Cmc’s converted to mole fractions by dividing molalities or molarities by 55.5.
or
Numerical Trials with Equation 9 cmc = cmc* exp[AC$(T,P)/(l+ a)RI
(10)
where F ( T , P ) is an abbreviation for the function [l P / T + In(T*/ll?l. Equationsgand loareindeedreduced variables equations in that the ratio cmc/cmc* depends only on T / P if ACp and a are fixed. They differ from eq 3 because each surfactant is allowed to have its own characteristicvalue of ACp/(l+ a). Moreover,the function F(T,P),like the experimental curves, is not symmetrical about the point T = P.It is significant that eq 9 shows that the shape of each curve is almost exclusively determined by the value of AC,. This was noted earlier in papers where equations somewhat akin to eq 9 were also presented?JObut these relate cmc(T)to the values of cmc, mm0, and ACpat 26 O C rather than to cmc*, P, and ACp. A possible way to obtain an expression superior to eq 9 is to remove the assumption that ACp is constant, replacing it by the slightly more realistic but still simple assumption of a linear dependence, or ACp ACp*[l - B(T - P)] (11) where B is a constant which needs to be determined empirically and is probably no larger than 0.01.10 Then using the reasoning described above, it is easily shown that
In (cmc/cmc*) = [ACp*/(l + a)RI [F(T,P)BF’CT,P)I (12) where Er(T,P) stands for the function [PIn (T/T*) + (!P2- T2)/2TJ. Unlike F ( T , P ) , which is negative whenever T # P, the functionFt(T,P), is positive when T < P and negative when T > P. Closely similar reasoning has been used to derive equations analogous to eqs 6 and 12 which describe the temperature dependence of the Gibbs energy of denaturation for a single domain globular protein assumed to undergo two-state unfolding.11J2 (9) Muebally, G. M.; Perron, G.; Desnoyers, J. E. J. Colloid Interface Sci. 1974, 48, 494. (10) M ~ e b d yG. , M.;Perron, G.; Desnoyers, J. E. J. Colloid Interface Sci. 1976,54,80.
To provide a suitable test of the usefulness of eq 9 or 12, a data set should consist of no fewer than about 10 points covering a moderately large temperature range and preferably presented in numerical rather than graphical form. Twenty such data seta were used and are described in Table I. Eleven of these represent a variety of cationic surfactante studied in the same laboratory using conductance and the other nine pertain to seven linear and branched sodium alkyl sulfate^.^"^^ For sodium decyl and sodium dodecyl sulfate, two data sets of similar quality were found and used separately to test the consistency of the results. According to eq 9, In cmc is linearly dependent on F(T,P), the intercept being In cmc* and the slope ACJ(1 + a)R. Therefore, each set of data was treated in the following way: With a fixed trial value of P, F(T , P ) was evaluated for each data point, and a plot of In cmc versus F ( T , P ) was prepared. If the plot showed appreciable curvature, a new trial value of !P was chosen and the procedure repeated until the curvature was reduced as much as possible. The slope and intercept of the most nearly linear plot were used to obtain cmc* and ACJ(l+ a), and then a calculated cmc was obtained at each temperature by means of eq 10. The parameters thus found for each of the surfactants, and the root mean square (rms) values of the percent differences between the observed and calculated cmc’s, are also presented in Table I. It is evident that eq 9 fits the data surprisingly well. The only set which has an rms deviation larger than 0.85% is the first; there the three largest deviations are 2.05, 2.09, and 3.09%, and the rms deviation is 1.65%. As it happens, this system is the only one for which error limits are explicitly given in the original papers;2-sthey are said (11) Becktel, W. J.; ScheUman, J. A. Biopolymers 1987,261869. (12) Franks, F.; Hatley, R. H. M.; Friedman, H. L. Biophys. Chem. 1988, 31, 307. (13) Flockhart, B. D. J. Colloid Sei. 1961,16,484. Data are p m n t e d in numerical form in ref 14. (14) Critical Micelle Concentrationa of Aqueoue Surfactant S y s t e m ; Mukerjee, P., Mysels, K. J., Eds.; NSRDS-NBS W,Superintendent of Documents, Washington, DC, 1971; pp 61-54. (16) Goddard, E. D.; Benson,,G. C. Can. J. Chem. 1967, 36, 986. Numerical results are presented in ref 14.
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90 Longmuit, Vol. 9, No.1, 1993
to be f0.2 mm, which amounts to just under 2%. Thus, the discrepancies between calculated and observed values are probablywithinthe error limits in this case, and almost certainly so for the other surfactants. A consequence of the excellent fits obtained with eq 9 is that it is not possible to do significantlybetter by using eq 12. The data do not prove that AC, is constant, but they do not allow meaningful nonzero values of the parameter B in eq 12 to be evaluated. Similarly, these results cannot be said to prove the correctness of eq 2, but by themselves they give no indication that it is incorrect. The procedure presented here probably provides the best values of ACp/(l + a)that can be obtained from this sort of measurement. Its main strength is that each value is determined collectively by the whole set of data and hence much less sensitive to possible errors in a few points. The uncertainty of the final values of AC,/(l + a) is difficult to specify precisely and depends on which set of data is taken. When the rms deviation is relatively large, or when T* lies outside the range spanned by the experimental points, it is found that T* may be changed by 1,2, or possibly even 3 "C, with concomitant changes of up to about 10% in the slopes of the plots, without making the fit very much worse. This suggestsuncertainty limits of about *lo%, or somewhat less when the rms deviation is small. In this connection, it may be noted that data sets 13 and 14 give two values of ACp/(l+ a)for sodium n-decyl sulfate, each differing from their average by 7 % . There is no obvious reason to expect one set of results to be better than the other. Sets 16 and 17, for sodium n-dodecyl sulfate, each give the same value of AC,/ (1
+ a).
Discussion 1. Heat Capacities of Micellization. The results in Table I provide a partial explanation for the degree of success of La Mesa's treatment! which gave eq 3. The new eq 9 would itself defiie a "universal* curve if AC,/(l + a)were the same for all the surfactants, and with only two exceptions the actual values differ from a mean of -276 J/(mol K)by no more than about 20%. It seems likely that those surfactants considered by La Mesa that were not included here have similar heat capacities of micellization. Like all previous treatments of these data, the present one providesunambiguousthermodynamicquantities only if the degree of counterion binding, a,is known. This quantity itself depends somewhat on temperature and rather more on the nature of the surfactant headgroup, the length of the alkyl chain, and the counterion.' Values of a at 26 O C were found in the literature for about half of the surfactants considered here, and these were used aa a guide for estimating the missing ones. All are listed, together with the resulting values of ACp, in the first two columns of Table II. For materials hawng a given chain length, the experimental values typically fall in a rather narrow range, and therefore it seems unlikely that any of the estimated values will be in error by more than 0.05, leading to uncertainties not exceeding 3% in 1 + a. Calorimetric results for comparison are available for only six of the surfactants. For dodecylpyridinium bromide, data set 1, a thermometric titration methodle gave -386 i 20 J/(mol K),about 10% smaller than the value obtained here. For decyltrimethylammonium bromide, data set 9, two treatments of the same data9JO gave -302 f 20 and -365 J/(mol K);the smaller value is very near (16) Krmheck, G. C.; Hargraves, W. A. J. Colloid Interjace Sci. 1974, 48,481.
Table 11. Thermodynamic Parameters for Midlieation at 28 OC of Surfactants, Listed as in Table I data set 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-ACp (J/(mol 1+ a 1.77O 1.7Bb 1.7Bb 1.7Bb 1.83b 1.7V 1.7BC 1.W 1.72d 1.82d 1.7Bd 1.64b 1.700 1.700 1.70b 1.78" 1.78" 1.87" 1.87b 1.87b
K)) 427 575 390 518 468 401 475 598 289 439 584 446 430 495 440 548 548 608 623 655
a m o
-AGmo AHm'' (J/(mol (kJ/mol) (kJ/mol) K)) -ASmo/ACp 37.3 107.7 0.252 -5.2 38.8 -4.6 114.5 0.199 37.2 +1.4 129.3 0.332 42.6 -6.7 120.3 0.232 46.1 -9.9 121.6 0.260 -0.8 97.5 0.243 29.9 37.9 -2.9 117.7 0.248 124.9 0.209 44.5 -7.2 29.0 0.0 97.0 0.336 37.2 -2.2 117.3 0.267 40.7 -4.1 122.6 0.210 24.6 +1.6 87.6 0.196 31.3 +2.1 112.3 0.261 31.3 +LO 108.3 0.219 29.9 +5.7 119.5 0.272 38.9 0.0 130.5 0.238 -0.6 128.3 0.234 38.8 47.3 -3.0 148.5 0.244 155.6 0.250 45.1 +1.2 43.1 +4.9 170.0 0.246
Reference 1. Estimated. Reference 4. Reference 5.
that in Table 11. For dodecyltrimethylammonium bromide? data set 10, calorimetric measurements at three temperatures yield-423 f 20 J/(mol K), in good agreement with the present value. Each of the sodium alkyl sulfates has been studied several times.sJ0J6J7 Values reported for the octyl sulfate, data set 12, are -280 f 30,9 -326,'O and -310 f 3816J/(mol K),all considerably smaller than the present results. For the decyl sulfate, data seta 13 and 14, they are -394 f 10: -426,lO and -402 f 42ls J/(mol K), close to the lesser of the two values in Table 11. For the dodecyl sulfate, data sets 16 and 17, they are -616 f 10: -530,lO -561 f 42,lS and -673 f 20 J/(mol K),and the value found here from both sets of data is very close to the average of these. 2. Thermodynamics of Micellization. The quantities AGm", AHm", and AS," have previously been calculated for most of these surfactants using eqs 1 and 2. It is now possible instead to use eqs 1,4,6, and 8. The results at 26 "C,also presented in Table 11,are based on cmc's converted to mole fraction units, to provide unitary entropy and Gibbs energy changes. They are quite similar to values reported earlier,2-sJ3Js except for differences arising because the earlier researchers used eqs 1 and 2 either with the assumption a = 0 or with a = 1. Moreover, quantities calculated at other temperature by the present approach vary with the temperature smoothly and regularly, which was not always found before. This reflects the fact that each value now depends on parameters adjusted to fit the whole data set and hence is much lese severely affected by a few possible defective data points. Table II also includesvalues at 25 "Cfor the ratio AS,"/ AC,, which is discussed below. This quantity is independent of the value adopted for a. A number of researchers have reported attempts to separate changes in thermodynamic variables for micellization into additive contributions representing, on the one hand, interactions between hydrocarbon chains and water (subscript hc) and, on the other, headgroup, counterion, and surface interactions (subscript s).l&N The ~~
~
~
(17) Pilcher, G.; Jones, M. N.; Eepada, L.; Skinner, H. A. J. Chem. Thermodyn. 1969, I , 381. (18) Mukerjee, P. Adu. Colloid Interface Sci. 1967, 1, 248. (19) Emerson, M. F.; Holtzer, A. J. Phyu. Chem. 1987, 71,1898. (20) Evans,D. F.; Ninham, B. W. J. Phys. Chem. 1988,87, 6026.
Temperature Dependence of Cmc's
treatment of Evans and Ninhammsuggeststhat the Gibbs energy change given by eq 1should be identified with the alkyl chain contribution, A&,', and that AGOo= -aRT In cmc, 80 that RT In cmc = AGhco + AG,O. It would then be coneiatent to suppose that the changes in enthalpy and entropy given by eqs 2 and 4, as well as the corresponding heat capacity changes, should also be determined mainly by hydrocarbon chain-water interactions. In the following paragraphs, it will be shown that the data, broadly speaking,support thisinterpretation. However, a number of discordant results that cannot readily be rationalized on thia baais are listed at the end of the Discussion. The hydrocarbon chain terms should show significant similarities to the changes of thermodynamic variables for the transfer of alkanes or alkyl groups from water into nonpolar solvents.ls Such transfers have the following characteristics: (1)AC,,, is negative and proportional to the surface area of the hydrocarbon molecule or group.21 (2) The unitary entropy change, AS,", is positive, and at 26 O C the ratio AStro/ACp,, is approximately -0).26.22-24 (3) AHk" is positive at low temperatures, passing through zero near 22 O C and becoming negativeon further heating.= It may then be expected that the data in Table I1 should show similar characteristics, and to a large extent they do so.
Looking first at the heat capacity changes, they are indeed invariably negative. The table includes several pairs or groups of surfactants belonging to the same homologous series. For most of these (but not those in data sets 2 and 6), the magnitude of ACp increases with increasinglength of the alkyl chain, by an average of about 60 J/(mol K) per added methylene group. Data for the transfer of pentane and hexane from water solution to the pure liquid state2sor for transfer of various hydrocarbon gases from aqueous solution to vapor26give increments of 50 f 10 J/(mol K) per added methylene group, a very satisfactory agreement. Turning to the ratio ASm0/ACp, the average of the values in Table I1 is -0.247, quite near the Sturtevant valueF2 and 12 of the 18 surfactants give values that differ from the average by 10% or less. Sturtevant's discussion suggests that this would not be expected if the entropy and heat capacity changes included major contributions not attributable to hydrophobic hydration. Finally, transfer enthalpies of six hydrocarbons from aqueous solution to the liquid state are zero at an average temperature of 22 O C , but with some variation from compound to compound.23 For the surfactants, AHmo= 0 at P,and according to Table I the average value of Ts is also 22 OC. None of the values deviate by more than f18 O C , which corresponds to &6% when Kelvin temperatures are used. The micelliition enthalpies at 26 O C in Table I1are always much smaller than the Gibbs energy changes and seldom much larger than values of AH,' reported for hydrocarbon liquids.23 These considerations provide considerablesupport for the hypothesis that the changes in thermodynamic variables appearing in Table I1 arise almost exclusively from interactions of the hydrocarbon chains with water. However, a number of difficulties remain which suggest that this view cannot be entirely correct. (21) Gill, 5.J.; Dec, 5.F.; Olofseon, G.; Wad&, I. J. Phys. Chem. 1985, 89,3768. (22) Sturtevant, J. M.R o c . Natl. Acad. Sci. U.S.A. 1977, 74, 2236. (23) Baldwin, R. L.R o c . Natl. Acad. Sci. U.S.A. 1988,83,8069.
(24)Muller,N. Acc. Chem. Res. ISSO, 23,23. (26) Gill, 5.J.; Nichols, N. F.; Wadsa, I. J . Chem. Thermodyn. 1976,
8,445. (26) Dec, S.F.;Gill, S. J. J . Solution Chem. 1985, 14, 827.
Langmuir, Vol. 9, No. 1, 1993 99
For the homologous compounds in data sets 2 and 6, adding two methylene groups to the chain decreasesIACA by 107 J/(mol K), where an increase of about this magnitude would have been expected. Surfactants in data sets 2 and 3 have the same cation with different counterions, and IAC,l is 186 J/(mol K) smaller for the chloride than the bromide. For the dodecylpyridiniumsalts, it was reported that IACpl is 100 J/(mol K) larger for the chloride than for the bromide.16 Sincethe degreeof counterionbindingshould depend only slightly on the nature of the ion,' neither finding is easy to rationalize. Four of the surfactants give values of ASmo/ACp near -0.20, and two others give values near -0.33, rather far from the expected -0.26. None of these is otherwise abnormal in any obvious way, which makes it hard to suggest an explanation for these deviations. Finally, when compared with the hydrocarbon transfer data, most of the heat capacities of micellization are unexpectedlysmall. For example, ACP,* for hexane is -440 J/(mol K),%and the correspondingentries in Table I1 for surfactants in data sets 1 and 10, each with a dodecyl chain, are both smaller than this. It is possible, of course, that micellization removes only a portion, rather than the entire hydrocarbon chain, from contact with water.'* Sincethe materials consideredhere have a great variety of headgroups, it is hardly surprising that some of them show deviant behavior. However, it should be noted that almostevery familyof surfactants represented in the tables includes at least one member that behaves "normally". Thus, it is impossible to identify any molecular characteristic that leads to an anomalous heat capacity of micellization. Apart from the present results,there are aspects of Evans and Ninham's treatmentm that are enigmatic. For example, it requires that, for each material, in the absence of added electrolybe, AGhCo/AG," = -(1 + cr)/a,and the degree of counterion binding is so nearly the same for different surfactants that this ratio can probably vary only within the rather narrow range -2.27 f 0.16. Moreover, there is no reason why this equality should be required only at 26 OC,and if it is to hold over an extended range of temperatures, then it is also necessary to have both A S h c o / h s , o and AHhco/AH80 equalto+ + a)/a. (Strictly speaking, one should allow for the fact that a decreases on heating, but da/dT is so smallthat this has little effect.) It is hard to see why all this should be expected for diverse ionic surfactants irrespective of chain length, headgroup structure, or counterion. It is particularly puzzling that these conditions would imply that at temperatures close to Ts both and AG,Omust be almost entirely of entropic origin. This has of course long been recognized for Ai&,", but no reason has been proposed for believing that it should be true also for AG,0.27 A possible alternative conjecture that fits the present AGmO (1+ results about equally well is that A&" a)RT In cmc and AG," = 0, when there is no added electrolyte. One hesitates to propose thisbecause it would require rejecting the model calculations used by Evans and Ninham to evaluate AG.O, but it should be noted that AG,O includes opposing terms representing attractive and repulsive interactions, which allows at least the possibility of nearly complete cancellation, and the researchers themselves describe these terms as "extraordinarily difficult to quantify".m Smallnonzero values of AGOoarising (27) See, however, a dmueeion by Lindman, B.;Wennemt&m,H. Top. Curr. Chem. 1980, 87, 1, Section 6.3, which doea indicate that the electrostatic contribution to AG,O is predominantly entropic.
100 Longmuir, Vol. 9, No. 1, 1993
from slightly unequal opposingterms might then partially account for the mall and irregularly varying values of AHm* at 26 O C reported in Table 11.
Conclusions With the assumption that AC,/(l + a)is temperature independent, an equation is derived showing that the cmc at any temperature is a function of this quantity, cmc*, and T / P . For 20 seta of data taken from the literature, it fita 19 with mu deviations less than 1 % and the other with a 1.6% deviation. An altemative equation is also presented, assuming that AC, is a linear function of T, but it does not provide appreciably better fib. Although this treatment does not prove that AC,/(l + a)is actually constant, it gives a value for each surfactant that is coneistent with all points in the data set, relatively little
Muller
affected by the poeeible presence of a few deviant points, and hence probably the best that can be obtained from temperature-dependent cmc measurements. Using experimental or estimated values of 1 + a,the data yieM micellizationheat capacitiesin fairly good agreementwith the rather few available calorimetric resulta. The calculated values of AC, make it possible to teet the suggestion20that the quantity (1 + a)RTlncmc, often taken as the totalGibbs energy of micellization,is actually to be regarded as the hydrocarbon chain contribution to this Gibbs energy. Many of them are coneistent with tbie hypothesis, but there are some unexpected and puzzling exceptions. Although the estimated uncertainties of the heat capacity changes may be as large as lo%, these discrepancies seem too large to be attributed to experimental errors.
