J. Phys. Chem. 1984, 88, 2155-2163 error. If more precise experimental data become available, perhaps over greater ranges of concentration and temperature, then a n extension using Pitzer’s equations might be justified. We point out, however, t h a t temperature and pressure effects on terms involving the “higher order electrostatic effects” will become significantly more complex.
Acknowledgment. Contribution No. 263 from the thermodynamics research laboratory at the Bartlesville Energy Technology Center, Department of Energy, where the research was funded by the Enhanced Oil Recovery Program, and in cooperation with Associated Western Universities, Inc. Glossary constant in Debye-Hiickel activity and osmotic expressions A, activity of the solvent a1 Bl,, Bny ion interaction parameters b ion-size parameter critical micelle concentration expressed in molality cmc g rational osmotic coefficient
I K
7
2155
ionic strength, mol kg-’ thermodynamic equilibrium constant based on molalities molar mass of the solvent total or stoichiometric molality of surfactant molality of species i aggregation number mole fraction of component i charge on ion i mole fraction of surfactant in micellar form fraction of counterions “bound” to the micelle activity coefficient of species i mean ionic activity coefficient for monomeric surfactant electrolyte mean stoichiometric activity coefficient for a surfactant solution shielding factor leading to effective micellar charge number of ions of species i produced from ionization of one molecule of electrolyte osmotic coefficient on a stoichiometric basis osmotic coefficient on a mixed electrolyte basis
Model for Thermodynamics of Ionic Surfactant Solutions. 2. Enthalpies, Heat Capacities, and Volumes Earl M. Woolley*+ Department of Chemistry, Brigham Young University, Provo, Utah 84602
and Thomas E. Burchfield* U S . Department of Energy, Bartlesville Energy Technology Center, Bartlesville, Oklahoma 74005 (Received: July 18, 1983; In Final Form: October 17, 1983)
Equations have been derived for apparent and partial molar quantities for ionic surfactant solutions. The equations are derived from expressions for osmotic and activity coefficients that are based on a mass-action model and a rigorous thermodynamic treatment for mixed electrolyte solutions. A maximum of two ion-ion interaction parameters is required to quantitatively describe the concentration dependence of experimental enthalpy, heat capacity, and volume data for a variety of aqueous solutions of anionic and cationic surfactants over ranges of concentration and temperature. Application of the equations to aqueous solutions of sodium dodecyl sulfate and octylamine hydrobromide is discussed. Changes in the thermodynamic quantities for the micellization reaction a t a standard state of infinite dilution and at the critical micelle concentration are obtained.
Introduction Models for thermodynamic properties of surfactants in solution have been summarized in recent Most models have been developed from molecular-level considerations of t h e hydrophobic and electrostatic interactions that lead to aggregation of surfactant molecules. Such models have usually been used t o derive equations for the change in the Gibbs o r Helmholtz free energy for formation of a micelle. In recent years a number of detailed studies of the concentration and temperature dependence of relative apparent molar enthalpies, I#JL, apparent molar heat capacities, I#Jc, and apparent molar volumes, d,, of aqueous surfactant solutions have been T h e available data on thermodynamic quantities for micelle formation have been summarized by Stenius et aL2’ Although a wealth of data on apparent molar properties now exists in t h e literature, relatively little has been published on quantitative thermodynamic models to represent the concentration dependence of these properties. Desnoyers and co-workers have applied a phase separation model to enthalpies, heat capacities, volumes, free energies, and entropies of ionic surfactantslo and a mass-action model to similar data for nonionic surfactants.22 Faculty Research Participant through the Associated Western Univer-
sities, Inc.
Recently Eriksson and co-workers4J have made model calculations for alkyl carboxylates. (1) Wennerstrom, H.; Lindman, B. Phys.Rep. 1979, 52, 1. (2) Hall, D. G.; Pethica, B. A. In “Nonionic Surfactants”; Shick, M. J., Ed.; Marcel Dekker: New York, 1967; p 516. (3) ,Tanford, C. “The Hydrophobic Effect: Formation of Micelles and Biological Membranes”; Wiley: New York, 1980. (4) Eriksson, F.; Eriksson, J. C.; Stenius, P. In “Solution Chemistry of Surfactants”; Mittal, K. L., Ed.; Plenum: New York, 1979; p 297. (5) Eriksson, F.; Eriksson, J. C.; Stenius, P. Colloids Surf.1981, 3, 339. (6) Hall, D. G. Colloids Surf.1982, 4 , 367. (7) Musbally, G. M.; Perron, G.; Desnoyers, J. E. J. Colloid Interface Sci. 1974, 48, 494. (8) De Lisi, R.; Perron, G.;Desnoyers, J. E. Can.J. Chem.1980, 58, 959. (9) De Lisi, R.; Ostiguy, C.; Perron, G.; Desnoyers, J. E. J . Colloid Interface Sci.1979, 71,147. (10) Desnoyers, J. E.; De Lisi, R.; Perron, G. Pure Appl. Chem.1980, 52, 433. (11) Desnoyers, J. E.; De Lisi, R.; Ostiguy, C.; Perron, G. In “Solution Chemistry of Surfactants”; Mittal, K. L., Ed.: Plenum: New York, 1979; Vol. 1, p 221. (12) Desnoyers, J. E.; Roberts, D.; De Lisi, R.; Perron, G. In “Solution Behavior of Surfactants”; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982; Vol. 1, p 343. (13) Rosenholm, J. B.; Grigg, R. B.; Hepler, L. G. In “Solution Behavior of Surfactants”; Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1982; VOl. 1, p 359. (14) Brun, T. S.; Hoiland, H.; Vikingstad, E. J . Colloid Interface Sci. 1978, 63, 89.
This article not subject to U S . Copyright. Published 1984 by the American Chemical Society
2156 The Journal of Physical Chemistry, Vola88, No. 10, 1984
In this paper, we present a model for the concentration dependence of apparent molar thermodynamic properties (relative apparent molar enthalpies, apparent molar heat capacities, and apparent molar volumes) of ionic surfactant solutions. Our app r ~ a c his ~based ~ on a mass-action model of micelle formation. It accounts for ionic interactions in solution by treating the micellar solution as a mixed electrolyte, using the Guggenheim equations for osmotic and activity coefficient^.^^-^' Equations for the concentration dependence of the apparent molar quantities are obtained from temperature and pressure derivatives of expressions for osmotic and activity coefficients. We report here the results of application of the model to published thermodynamic data at several temperatures for an anionic surfactant having a low critical micelle concentration (cmc), sodium dodecyl sulfate (0.008 m), and for a cationic surfactant having a relatively high cmc, octylamine hydrobromide (0.2 m). Our model provides a framework for examining the thermodynamic properties of surfactant solutions. It also helps to elucidate the contributions that the monomer and the micellar aggregates make to the observed thermodynamic quantities.
Description of the Model The aggregation of an anionic surfactant, MA, can be approximated by a simple mass-action equilibrium involving one aggregate species having an aggregation number, n, with a fraction of "bound" counterions, P.
(n@)M++ nA- = M,BA,"(l-p)
(1)
Although we formally derive equations for an anionic surfactant, A-, with counterions, M+, the resulting equations are identical with those derived for a cationic surfactant, A+, with counterions, M-. A surfactant solution having a concentration greater than the critical micelle concentration (cmc) can be considered as a mixed electrolyte solution containing mA mol kg-' of the 1:l electrolyte, [M+, A-1, and mB mol kg-' of the I:n(l - 8) electrolyte, [n( 1 - P)M+, MnBA,"('-@]. The equilibrium molalities mA and mB are related to the total or stoichiometric molality, m, by m = mA + nmB
(2)
where the equilibrium molalities of the species M+, A-, and [M$A;n(1-8'] are mM,mA, and mB, respectively. The mole fraction of surfactant micellized, a, is defined as a = nmB/m
(3)
The equilibrium constant for reaction 1 is given by
Woolley and Burchfield where the activity coefficients for the species M+, A-, and MnBA["('-@)are yM, yA,and 7 6 , respectively. The activity coefficient term in eq 4 can be expressed by using the Guggenheim equations2e27 for mixed electrolyte solutions.23
+
- P - 1]AyZ1/2/(l log (yB/yMn'yAn) = -n[a2n(1 b ~ ' / ~+) ~ , , [ n m ( 2 ~-a P - I ) ] + B,,[m(l - 2pa)I ( 5 ) In eq 5 , Z is the ionic strength of the solution, b is an ion-size parameter, B1, and B,, are ion-ion interaction parameters for monomeric and micellar surfactant, respectively, and 6 is a "screening factor" for the micellar charge that was introduced in our previous paper23 to account for the effective charge on the micelle. The Guggenheim equation^^"^' for y M , ?A, and yBlead to the expression for the mean stoichiometric activity coefficient for the surfactant, ya, given in log (y*) = -AyZ1/2/(1 + b z q (1/2) log (1 - Pa) + Bl,[m(2
+ (1/2)
log(1 - a )
+
- a - P a ) / 2 ] + B,,(ma/2n)
(6) and, via the Gibbs-Duhem relationship, to the expression for the stoichiometric osmotic coefficient for the solution, 4, given in23
4 = 1 - a(1
+
The function
ub) is defined by24-26
- 1 / n ) / 2 - (In 10)A,Z3/2a(bZ1/2)/3m + Bl,[(l - a ) ( l - pa)m(ln 10)/2] Bny[a(lPa)m(ln 10)/2nI ( 7 )
a b ) = 3[1
+
+ y - 1 / ( 1 + y) - 2 In (1 + y ) ] / y 3
(8)
The ionic strength used in eq 5-7 is calculated with the equation 1= Cm,z,'/2 = [2(1
- a ) + nh2(1 - P)Za+ (1 - P ) a ] m / 2 (9)
where the "effective charge" on the micelle is *n(@ We have shown previously that 6 < 1 in order for eq 1-9 to represent experimental osmotic and activity coefficient data for a variety of surfactants over wide ranges of c o n c ~ t r a t i o n .We ~ ~ have also shown that setting b = 1 in eq 5-7 is appropriate to properly represent the data, consistent with the suggestion by Guggenheim2' for simple electrolyte solutions. Equations 6 and 7 can be used to write the expression for the total free energy of the ionic surfactant solution (per mole of surfactant) relative to the conventional pure water and infinitely dilute solute standard states (XI = 1 and m = 0) in24-26
(C - G 0 ) / 2 m R T = In (yt) + (1/2) In (m) - 4
(10)
By straightforward application of the relationship in eq 1 124to [d(AG/T)dT], = - M / P = - m + L / p (1 5) Vikingstad, E.; Skauge, A.; Hoiland, H. J . Colloid Interface Sci. 1978, 66, 240.
(16) Kale, K. M.; Zana, R. J . Colloid Interface Sci. 1977, 61, 312. (17) Berg, R. L.; Noll, L. A.; Good, W. D. In "Chemistry of Oil Recovery"; Johansen, R. T., Berg, R. L., Eds.; American Chemical Society: Washington, DC; ACS Symp. Ser. No. 91, 1979, p 87. (18) Berg, R. L. BERC/TPR-77/3, Bartlesville Energy Research Center, Bartlesville, OK, 1977. (19) Mazer, N. A,; Olofsson, G. J . Phys. Chem. 1982, 86, 4584. (20) Pilcher, G.; Jones, M. N.; Espada, L.; Skinner, H. A. J . Chem. Thermodyn. 1969, 1, 381. (21) Stenius, P.; Backlund, S.; Ekwall, P. In "Thermodynamics and Transport Properties of Organic Salts"; Fanzosini, P., Sanensi, M. Eds.; IUPAC Chemical Data Series No. 28; Pergamon: Oxford, 1980; p 275. (22) Desnoyers, J. E.; Caron, G.; De Lisi, R.; Roberts, D.; Roux, A.; Perron, G. J . Phys. Chem. 1983, 87, 1397. (23) Burchfield, T. E.; Woolley, E. M. J . Phys. Chem. preceding article in this issue. (24) Pitzer, K. S.;Brewer, L. 'Thermodynamics" (Revision of G. N. Lewis and M. Randall, "Thermodynamics," 1st ed);McGraw-Hill: New York, 1961; 2nd ed. (25) Robinson, R. A.; Stokes, R. H. "Electrolyte Solutions";Butterworths: London, 1959; 2nd ed. H. S.; Owen, B. B. "The Physical Chemistry of Electrolytic (26) Solutions ; Reinhold: New York, 1958; 3rd ed. (27) Guggenheim, E. A. Philos. Mag. 1935, 19, 588.
wed,
(11)
eq 10, we obtain the expression for relative apparent molar enthalpy, &, of the surfactant solution given in
dL = -2RP(d[ln ( y d I / W ,
+ 2Rp(&/dT),
(12)
The first term on the right-hand side of eq 12 is the partial molar enthalpy of surfactant, Similarly, application of eq 13 and 1524 to eq 10 and 1 1 leads to the expressions in eq 14 and 16 for the apparent molar heat capacity, &, and the apparent molar volume, &, respectively, of the surfactant solution.
e2.
[d(AH)/dTI, = Acp =
-
cp02)
(13)
4, - q . 0 2 = - 4 R 7 W n ( 7 * ) 1 / m p 2 ~ P ( d ~ [ (yt)] l n /dP},
+ 4 ~ T ( a + / d T )+, 2RP(d24/dTz), (14)
[a(AG)/dP], = AV = m(& - P2)
(15)
4" - P2= 2RTP[ln ( r t ) l / W , - 2 R T ( d d ~ / d P h (16) The sum of the first two terms on the right-hand side of eq 14
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2157
Model for Enthalpies, Heat Capacities, and Volumes TABLE I: Debye-Huckel Parameters Used in Eq 5-7 and in Curve Fitting to Eq 20-22, Using Eq 12, 14, and 16
'*O0O
so00
1.529 22.52 5 0.4952 1515 1.696 25.84 15 0.5026 1729 1.780 27.40 20 0.5066 1847 1.865 28.95 25 0.51 08 1973 1.952 30.52 30 0.5150 21 04 2.040 32.13 35 0.5 196 224 1 45 0.5291 2545 2.223 35.76 39.39 2903 2.418 55 0.5393 a From ref 28. From ref 29 and 30, except A L at 55 "C, which is interpolated from values in ref 31. (aA =~AL/ 41n(10)RTZ. (aA /aP),=-3AV/4ln(1O)R? e(aZAy/ aT2)I, = 3(Ac - 2AL/?)/4 In (1O)RT'.
/an,
cp+
is the partial molar heat capacity of the surfactant, and the first term on the right-hand side of eq 16 is the partial molar volume of the surfactant, In applying eq 12, 14, and 16 to eq 6 and 7, we need to consider the effects of temperature and pressure on all the parameters in eq 4-9: K, a,n, /3, 6, A,, B!,, Bw. The temperature and pressure dependence of A, was obtained from the parameters summarized in Table I.28-31 Variation of a and K with temperature and pressure are handled in a straightforward fashion using with 4, 5, and 9 along with the definitions in24-26
v2.
(a In K / a T ) ,
=A H O / R ~
( a A H " / a T ) , = ACp"
(a In K / d P ) T
(17) (18)
=-AP/RT
(19)
Although it is possible to include temperature and pressure derivatives of n, /3, and 6, we will show in the Results section that it is unnecessary to do so in order to fit experimental data with eq 12, 14, and 16. Furthermore, one cannot be self-consistent in using eq 17-19 if n or /3 change with temperature or pressure, since, for the mass-action model with one aggregate species, the reaction stoichiometry would be changing. Explicitly taking the derivatives in eq 12, 14, and 16 while holding n, 0, and 6 constant, leads to equations of the form given in @L
=
4, =
+ ~LLAH+ " klL(aBly/aT)p + knL(aBn,/aT)p
~ O L
(20)
+ koc + k,ACpo + k l , ( d 2 B l , / d p ) p+ kn,(d2Bn,/r3p)p
~ p o z
(21)
4" =
P 2
+ kov
kwAP
klv(aBl,/aP)T
+ knv(aBn,/aP)T (22)
The coefficients kix in eq 20 and 22 are explicit functions of m , K , a,n, 0, 6 , A,, B,,, Bny,and T and are given in the Appendix. Values of CY are calculated from given values of all the rest of these parameters by an iterative method using eq 4, 5, and 9. A least-squares treatment of experimental q5L and 4, data with eq 20 and 22 leads to the thermodynamic parameters for the ionic surfactant solution. The value of K is varied in order to find the optimum value (minimum root-mean-square deviation) for each function, 4Land dv. The derivation of eq 21 shows that the coefficients k,, klc,and k,, depend on values of AHo, (aB,,/aT),, and (aBny,dT&,,in addition to m , K,a,n, /3, 6 , A,, B1,, By,, and T . Consequently, the parameters in eq 20 must be determined (or estimated) before (28) Bates, R. G. "Determination of pH"; Wiley: New York, 1964; p 406. (29) Perron, G.; Desrosiers, N.; Desnoyers, J. E. Can. J. Chem. 1976, 54, 2163. (30) De Lisi, R.; Ostiguy, C.; Perron, G.; Desnoyers, J. E. J . Colloid Interface Sci. 1979, 71, 147. (31) Clarke, E. C.; Glew, D. N. J . Chem. SOC.,Faraday Trans. 1 1980, 76, 1911.
-
" x
~
1
A
6000
3 \
i 3000
a
0
1
-3000
0
0.25
,
,
,
1
1
0.5
rn/
1
,
,
,
1
,
0.75
,
,
,
1
1
I
,
,
,
I .25
MOL K G - I
Figure 1. Relative apparent molar enthalpies of aqueous octylamine hydrobromide from ref 8: 0,V, and + at 5, 15, and 25 O C , respectively. The solid lines are our least-squares fit using the parameters in Table 11.
& data can be treated by eq 21. As before, the value of K is varied in order to find the optimum value (minimum root-mean-square deviation) for the & data. It should be noted that the value of K merely specifies the molality a t which the break (cmc) will occur in the dx vs. m curve. Only those parameters in eq 20-22 that are statistically significant a t or above the 95% confidence level are retained. Results To test the model, we used data for octylamine hydrobromide and sodium dodecyl sulfate. The hydrobromide, a cationic surfactant, has a rather high cmc of about 0.22 mol kg-' at 25 OC8 with a low aggregation number which we estimate to be about 20.32 The dodecyl sulfate has a rather low cmc of about 0.0082 mol kg-' a t 25 0C33and an aggregation number of about 64.34*35 W e have shown in a previous paperz3that the osmotic and activity coefficient data for these two surfactants are described quantitatively by setting B,, = 0, B,, = 0, b = 1 in eq 5-7 and by using 6 = 0.68 for the hydrobromide and 6 = 0.52 for the dodecyl sulfate in eq 5 and 9. Relative apparent molar enthalpies for octylamine hydrobromide are given at 5, 15, and 25 "C,* based on measured enthalpies of dilution at 25 "C and measured apparent molar heat capacities at 5, 15 and 25 "C. Apparent molar heat capacities and apparent molar volumes are also given a t 5, 15, and 25 OC,s and at 35,45, and 55 0C.36 W e have corrected the apparent molar heat capacities a t 35, 45, and 55 " C for a calorimetric error.37 Data at 5, 15, and 25 "C were smoothed by the original authors.8 Enthalpy of dilution data are given for sodium dodecyl sulfate a t 20, 25, 30, and 35 "C.lS Apparent molar heat capacity data are very difficult to obtain for this compound because of the low cmc, but some data have been reported at 25 "C.' Apparent molar volume data have been reported by Doughty38 for a specially purified sample. l 8 The value of K that corresponds to the minimum in the rootmean-square deviation for eq 20-22 is dependent upon the values of n and @ chosen for the particular surfactant. For sodium dodecyl sulfate, n = 6434,35and /3 = 0.7739are fairly well estab(32) Leibner, J. E.; Jacobus, J. J . Phys. Chem. 1977, 81, 130. (33) Moroi, Y.;Nishikido, N.; Uehara, H.; Matuura, R. J . Colloid Interface Sci. 1975, 50, 254. (34) Muller, N. In 'Micellization, Solubilization, and Microemulsions"; Mittal, K. L., Ed.; Plenum: New York, 1977; Vol. 1, p 229. (35) Lianos, P.; Zana, L. J . Colloid Interface Sci. 1981, 84, 100. (36) Leduc, P. A,; Fortier, J. L.; Desnoyers, J. E. J . Phys. Chem. 1974, 78, 1217. (37) Desnoyers, J. E.; de Visser, C.; Perron, G.; Picker, P. J . Solution Chem. 1976, 5, 605. (38) Doughty, D. A. J . Phys. Chem. 1981, 85, 3545. (39) Piercy, J.; Jones, M. N.; Ibbotson, G. J . Colloid Interface Sci. 1971, 37, 254.
2158
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
Woolley and Burchfield
TABLE 11: Parametersa for Fitting Experimental @L Data for Octylamine Hydrobromide from Ref 8 to Eq 20, Using n = 20, p = 0.60, and 6 = 0.68 a (aB,y/a n p , rmsd, t, "C In K intercept, J mol-' AH"/n, J mol-' kgmol-' K-' kgmol-' K-' J mol-' 5 36.2 -202 t 26 12020 t 201 -(3.94 i 0.09) x 10-3 (5.8 t 0.6) x 10.' 47 -(3.67 f 0.08) X loT3 (5.2 2 0.4) x 38 -184 f 21 8684 c 140 15 38.6 -(3.35 i: 0.07) x 10-3 (4.7 c 0.3) x lo-, 35 -171 f 21 5476 t 111 25 41.2 (-3.0 x 10-3) (4.2 X IO-,) 0 (2620) 35 (41) 45 (41) (3.7 x 10-2) (0) (-2.7 x 1 0 - 3 ) 0 (3.2 X lo-') (-2400) (-2.4 x 1 0 - 3 ) 55 (41) 0 a The t values listed are the statistical root-mean-squaredeviations. Rmsd is the root-mean-squaredeviation of the fit to eq 20.
up,
TABLE 111: Parametersa for Fitting Experimental @v Data for Octylamine Hydrobromide from Ref 8 and 36 to Eq 22, Using n = 20, p = 0.60, and 6 = 0.68 w,,/am-, rmsd,b t, "C In K v",, cm3 mol-' AV"/n, cm3 mol-' kg mol-' atm-' cm3 mol-'
a
8.07 i 0.03 4 3 . 9 t 0.2) x 10-5 0.05 5 36.4 169.52 i: 0.02 7.45 i 0.03 -(2.6 t 0.2) x 10-5 0.04 15 38.8 171.65 t 0.02 6.93 i: 0.02 -(I .7 ?: 0.2) x 10-5 0.03 25 40.6 173.74 C 0.02 6.46 t 0.10 0.09 35 41.0 175.26 f 0.03 45 41.4 177.63 i. 0.02 5.12 t 0.08 0.07 55 42.0 179.10 t 0.07 5.0 t 0.2 0.1 8 The t values listed are the statistical root-mean-squaredeviations. Rmsd is the root-mean-square deviation of the fit to eq 22.
TABLE IV: Parametersa for Fitting Experimental oc Data for Octylamine Hydrobromide from Ref 8 and 36 to Eq 21, Using n = 20,
p = 0.60, and 6 = 0.6823
rmsd.c In K C,",, J K-' mol-' ACp"/n, J IC'mol-' (8ZBly/aTz)p,kgmol-' K-* J K-* mol-' 5 37.0 607.1 r 1.6 -327.1 C 2.2 (2.4 f 0.5) x 10-5 2.9 619.8 t 0.7 -316.3 i 0.9 (1.6 i: 0.2) x 10-5 1.3 15 39.2 -297.5 C 0.5 (2.0 i. 0.1) x 10-5 0.8 25 39.6 621.9 f 0.4 -273 t 3 (3.5 f 0.6) x 10-5 2.0 356 39.8 612.6 r 2.0 611.7 f 1.5 -259 i: 2 (6.7 ?: 0.5) X 1.7 45'7 37.8 (5.8 i 1.0) x 10-5 4.4 589.0 * 3.9 -224 * 5 55b 36.7 a The values listed are the statistical root-mean-squaredeviations. Based on calculations using projected enthalpy parameters as discussed in the Results section and given in Table 11. Apparent molar heat capacities were corrected as described in ref 37. Rmsd is the root-meansquare deviation of the fit to eq 21. t, O C
lished. However, for octylamine hydrobromide, the values of n = 2032and /3 = O.6O4O that we have chosen are estimates. The data can be adequately represented by using n = 64 and 6 = 0.17 for sodium dodecyl sulfate and n = 20 and /3 = 0.60 for octylamine hydrobromide a t all temperatures. In Tables 11-IV, we give the least-squares parameters of the fit of experimental &, &, and 4cdata8-36for octylamine hydrobromide to eq 20-22. W e will now briefly review each property in turn, but we leave consideration of the values of AHo, A V , and ACpo until the Discussion section. In Figure 1, we show plots of the smoothed relative apparent molar enthalpies of octylamine hydrobromide solutions a t 5, 15, and 25 O C . 8 The lines in this figure correspond to our least-squares fit of the data to eq 20 involving (AH'), (dBl,/dT)p, (B,,/aT),, and an intercept to rescale all the smoothed 4Lvalues. The values of these four parameters are given in Table 11. The intercept should be zero if the extrapolation of experimental $L data to m = 0 is performed properly. This is often very difficult to do, since the experimental data become more uncertain a t low concentrations and the form of the extrapolation function can change the ~ value of the intercept. W e conclude that experimental $ J data for octylamine hydrobromide can be adequately represented up to 1 mol kg-' a t 5-25 OC by eq 20 using the parameters given in Table 11. In Figure 2, we show plots of the apparent molar volumes for octylamine hydrobromide solutions a t 5 to 5 5 O C . The lines in this figure correspond to our least-squares fit of these data to a ( A T ) , and (BB1,/BP)Tat 5, 15, and form of eq 22 involving Pz, 25 O C , and Pozand (AVO) at 35,45, and 55 O C . Numerical values of these parameters are given in Table 111. The deviations of our values of Pozfrom those reported ear lie?^^^ are 0.02, 0.03, 0.02, (40) Anacker, E. W.; Underwood, A. L. J . Phys. Chem. 1981,85, 2463.
I85
I
& z
180
m 0 z
, 175 >
e
1
170
1
I65
0
0 25
,
~
,
~
0 5
m/
1
~
~
0 75
,
,
1
,
1
,
,
,
1
I 25
MOL K G - 1
Figure 2. Apparent molar volumes of aqueous octylamine hydrobromide from ref 8 and 36: 0,V, +, A, 0, and 0 at 5, 15, 25, 35, 45, and 55 O C , respectively. The solid lines are our least-squares fit using the parameters in Table 111.
0.08, 0.09, and 0.26 cm3 mol-' a t 5, 15, 25, 35, 45, and 5 5 OC, respectively. These differences are within experimental uncertainty. W e conclude that experimental dVdata for octylamine hydrobromide a t 5-55 O C and up to 1 mol kg-' are represented adequately by eq 22 using the parameters given in Table 111. In Figure 3, we show plots of the apparent molar heat capacities for octylamine hydrobromide solutions a t 5, 25, and 45 OC. The data at 5 and 25 O C were smoothed by the original authorss while the data a t 45 O C were not.36 Values of AHo, (aBl,/aT)p,and (aB,,/aT), used in eq 21 for 35,45, and 55 OC were estimated by extrapolation of the values given in Table I1 to these tem-
,
~
,
~
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2159
Model for Enthalpies, Heat Capacities, and Volumes
TABLE V: Parametersa for Fitting Experimental @L Data for Sodium Dodecyl Sulfate from Ref 18 to Eq 20, Using n = 64,
p = 0.77, and 6 = 0.52 intercept, &in, w1,y/anp, OB,y/ a n p , rmsd,b J mol-' J mol-' kg mol-' K-' kg mol-' K-' J mol-' 2.87 i 0.08 29 3294 i 39 -(5.8 i 0.3) X -68 i 36 20 (530)' 109 i 46 913 i 50 -(3.4 5 0.3) X lo-' 3.28 i 0.10 33 25 (530)' -2020 i 58 -(2.2 i 0.3) X IO-' 2.62 + 0.1 1 56 250 5 40 30 530 2.45 i 0.12 54 -4492 i 69 -(2.2 i 0.4) X lo'* 299 i 52 35 530 Rmsd is the root-mean-square deviation of the fit to eq 20. The a The i. values listed are the statistical root-mean-square deviations. values here do not correspond to the minimum in the rmsd. Instead, values were taken at a K value consistent with the other K values. The @L data show very little break near the cmc. See Figure 4. In K
t, "C
TABLE VI: Parametersa for Fitting Experimental Qv Data from Ref 38 and Qc Data from Ref 7 for Sodium Dodecyl Sulfate at 25 "C to Eq 22 and 21, Respectively, Using n = 64, p = 0.77, and 6 = 0.52 -
data
In K
cm3 mol-'
cm3 mol-'
rmsd,b cm3 mol"
@"
537
235.68 i 0.04
10.68 i 0.08
0.09
data
AT/?!,
v"2,
-
2000
-
i 0
0
7
rmsd,b
ACpo/n, J K-' mol-'
J gi;bl-'
In K
4000
\
J K-'
eJ-2000
mol-'
aC (530)' 1 0 1 6 i 24 -485 i 27 15 The r values listed are the statistical root-mean-square deviations. Rmsd is the root-mean-square deviation of the fit to eq 21 and 22. The values here do not correspond to the minimum in the rmsd. Instead, values were taken at a K value consistent with the other K values. Since there are no QC data at concentrations less than the cmc, K cannot be determined unambiguously.
-4000
a
700
-6000 0
-
+,
tively. The solid lines are our least-squares fit using the parameters in Table V.
248
_I I
?
-
0 . 125
KG-l
Figure 4. Relative apparent molar enthalpies of aqueous sodium dodecyl sulfate from ref 18: 0, 0, and A at 20, 25, 30, and 35 OC,respec-
7
45J
0.I
0.075
rn/ MOL
I " " l " " l " " I " ' '
620
0.05
0.025
540
I
245
Y 7
' $
1
/-----
, 460 s" t
0
380
l
i l
300 0
0 25
,
,
,
,
i
,
,
,
0 75
0 5
m
,
/
MOL
l
,
,
l
I
,
l
~
l
I
~
,
25
K G - ~
Figure 3. Apparent molar heat capacities of aqueous octylamine hydrobromide from ref 8 and 36: 0 , +, and 0 at 5 , 25, and 45 'C,
respectively. The solid lines are our least-squares tit using the parameters in Table IV.
peratures. The lines in Figure 3 correspond to our least-squares fit of these data to a form of eq 21 using only ~ p o z ,(AC,"), and (aB,,/aP), a t each temperature. Numerical values of these parameters are given in Table IV. The deviations of our values of ~ " p O zfrom those reported earlier8,36are 1.1, -0.2, -0.2,0.6, -2.3, and 0.0 J K-' mol-' a t 5, 15, 25, 35, 45, and 55 "C, respectively. These differences are also within experimental uncertainty. W e conclude that experimental +c data for octylamine hydrobromide solutions a t 5-55 " C and up to 1 mol kg-' are adequately represented by eq 21 using only the parameters given in Table IV. In Tables V and VI, we give the parameters of the least-squares fit of experimental dL,18&38 and 4: data for sodium dodecyl sulfate to eq 20-22. In Figure 4, we show plots of the relative apparent molar enthalpies of sodium dodecyl sulfate a t 20, 25, 30, and 35 "C.18 The solid lines in this figure correspond to our least-squares fit of the data to eq 20 involving ( W )( ,8 B I 7 / ~ T ) , , ( C ~ B , , ~ / and ~ T )an ~ ,intercept to rescale all the 4L values. As discussed earlier in connection with the octylamine hydrobromide
1
233 0
0 02
,
,
,
,
1
,
0 04
m
,
1
,
1
1
0 06
,
,
,
1
,
0 08
/
,
,
0 1
/ MOL K G - 1
Figure 5. Apparent molar volume of aqueous sodium dodecyl sulfate at 25 'C from ref 38. The solid line is our least-squares fit using the
parameters in Table VI. data, this intercept merely rescales the relative apparent molar enthalpies. These experimental 4Ldata are adequately represented by eq 20 with the parameters given in Table V at 20-35 "C at concentrations up to about 0.1 mol kg-'. In Figure 5, we show a plot of apparent molar volume of sodium dodecyl sulfate solutions at 25 0C.38 The solid line corresponds to our least-squares fit of these data to a form of eq 22 involving only P,and ( A T ) . The numerical values of these parameters are given in Table VI. In Figure 6, we show a plot of the apparent molar heat capacity of sodium dodecyl sulfate solutions at 25 OC.' The solid line corresponds to our least-squares fit of these data to a form of eq 21 involving only the c?poz and (ACPo)values in Table VI and the ( A H o ) ,(dB,,/aT),, and (aB,,/aT), parameters from Table V a t 25 OC. Our c?,O2 = 1016 J K-' mol-' is in reasonable
2160
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
Woolley and Burchfield
TABLE VII: Effects of Changing n , p , and 6 on Parameters for Fitting Experimental Data for Sodium Dodecyl Sulfate at 25 "C to Eq 6 and 20-22 Activity Coefficienta
P 0.77 0.77 0.75 0.77
n
64 67 64 64
P 0.77 0.77 0.75 0.77
n
64 67 64 64
6
In K
intercept
0.52 0.52 0.52 0.60
539 564 532 53s
-0.0015
0.52 0.52 0.52 0.60
P
64 67 64 64
0.77 0.77 0.75 0.77
In Kd
6'
I os
107 Heat Capacity
0.52 0.52 0.52 0.60
(530) (555) (523) (528) -
Aflin, J mol"
(aB, y/a7-)p,
(aBny/aT,lp,
rmsd,b
kg mol-' K-'
kg mol-' K-'
J mol-'
91 3 91 2 827 899
-0.034 -0.034 -0.035 -0.035
3.28 3.4s 3.03 3.40
32.8 33.0 33.3 33.8
AC "In
(a2B,,1aT1),,
J K-"rn~f-' 1016 -485 1016 -485 1019 -488 1016 -486 Volume
kg mol-' K-2
J $'Pn'bl-'
rmsdb 0.0072 0.0074 0.0073 0.01 10
intercept, J mol-' 109 109
n
Bny,
kg mol-'
-0.0020 t0.0003 -0.0030 Enthalpy
In K d (530) (555) (523) (528)
6C
Bl y' kg mol-'
(a2Bny/aT ~ ) ? , rmsd,b kg mol-' KJ K-' mol-' 14.8 14.9 15.1
14.9
V",, AV"ln, OB, y/ap)T, (aBn,laP)T, rmsd,b cm2 mol-' cm3 mol'' kg mol-' atm-' kg mol-' atm-' cm3 mol-' 64 0.77 0.5 2 537 235.68 10.68 0.093 67 0.77 0.52 56 3 235.67 10.65 0.092 0.52 530 235.67 10.59 64 0.75 0.094 64 0.77 0.60 536 235.67 1OS8 0.091 a Activity coefficient data were fit to eq 6 as described in ref 23. The parameter 6 was optimized to give the minimum rmsd for the first three entries. The fourth entry of 6 = 0.60 was chosen in order to show the effects of changing 6 . Rmsd is the root-mean-square deviation of the fit to the applicable equation. These values of 6 are those chosen as described in footnote a . These values of In K are chosen to be consistent with those at 30 and 35 "C as explained in footnote c in Tables V and VI. P
n
In K
6'
l i t e r a t ~ r e .W ~ ~e are unaware of any estimates in the literature of (dn/dT) or (d@/dT) for octylamine hydrobromide. The effects of using different values of n, p, and 6 on ( A H o / n ) , (AC,'/n), (AVOln), P2,and Cpo2for sodium dodecyl sulfate, as seen from Table VII, are nearly negligible. This is probably due in part to the rather low cmc for this compound. For octylamine hydrobromide, the situation is not so clearly resolved, as is evident from the fact that the standard state parameters are clearly more sensitive to the choices of n, p, and 6 , as seen in Table VIII. This is as expected, since the extrapolation to infinite dilution is so much further when the cmc is high. In part, we may resolve this difficulty by showing that values of ( A H / n ) , (AC,/n), and (AV/n)valid at the crnc are more nearly independent of the choice of the values of n, p, and 6 . I
450
0
0 06
,
,
,
,
/
,
0 12
m/
I
I
I
I
0 18
,
,
,
,
I
/
,
,
0 24
j
0 3
MOL K G - 1
Figure 6. Apparent molar heat capacity of aqueous sodium dodecyl sulfate at 25 OC from ref 7. The solid line is our least-squares fit using the parameters in Table VII.
agreement with a predicted 1024 J K-' mol-' based on group contribution method^.^' We now discuss briefly the effects of changing n, p, and 6 on the parameters in Tables 11-VI, with special emphasis on the effects on the thermodynamic quantities (AH'), (AVO), (AC,'), P,, and ~ p o 2 In . Tables VI1 and VIII, we give examples of these parameters obtained by using different n, @, and 6 . Values of (dn/dT), -0.64 and (dp/dT), N -0.001 for sodium dodecyl sulfate, based on theoretical estimates have been reported in the (41)
Perron, G.; Desnoyers, J. E. Fluid Phase Equil. 1979, 2, 239.
Discussion Although based on a simplistic view of the equilibria and energetics of micelle formation, our model fits all the experimental data as a function of concentration at all temperatures within the experimental error. Only two ion interaction parameters are required to fit the relative apparent molar enthalpy of octylamine hydrobromide a t concentrations up to 1 m (4.6 times the cmc). For the heat capacities and volumes, a maximum of one interaction parameter is required. For sodium dodecyl sulfate, two interaction parameters are required to fit the enthalpy data up to 0.1 m (12 times the cmc), and no interaction parameters are required to fit the heat capacity and volume data. Application of the model to thermodynamic data for other surfactants will be described in a separate paper.42 Some calculations were done to account for deviations from the limiting law in the premicellar region by including formation (42) Woolley, E. M.; Burchfield, T. E., manuscript in preparation.
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2161
Model for Enthalpies, Heat Capacities, and Volumes
TABLE VIII: Effects of Changing n , p, and S on Parameters for Fitting Experimental Data for Octylamine Hydrobromide at 5 "C to Eq 7 and 20-22
Osmotic Coefficient'"
Bn,,
517,
In K 36.9 47.8 37.4 39.8
n
P
6
20 25 20 20
0.60 0.60 0.62 0.60
0.68 0.60 0.72 0.60
n
P
6 C
In K
20 25 20 20
0.60 0.60 0.62 0.60
0.68 0.60 0.72 0.60
36.2 47.0 36.6 38.6
intercept -0.0106 -0.0109 -0.0105 -0.01 02 Enthalpy
-
intercept, &In, J mol-' J mol-' - 202 12020 -214 11718 -21 I 12316 -210 12505 Heat Capacity
P
6C
In K
J K-'mol-'
20 25 20 20
0.60 0.60 0.62 0.60
0.68
37.0 48.0 37.6 39.7
607.1 606.7 605.2 608.7
0.72 0.60
n
P
6C
In K
ACpa/n, J K-'mol-'
CPOW
n
0.60
v",,
kg mol-'
cm3 mol''
Av"/n,
cm3 mol-'
(aB,,/aT),, kg mol-' K" -0.00394 -0.00402 -0.00400 -0.003 98
rmsdb 0.0032 0.0032 0.0032 0.0045
(a~,,/an,,
J mol-'
0.058 0.063 0.066 0.063
47 46 48 48
(a25,,/a T )f,
kg mol" K-g'
kg mol-' K-
x x x x
rmsd,b J K-' mol-' 2.9 2.0 2.9 3.5
10-5
10.~ 10-5 10-5
( a 5 1y/ap)T, kg mol'' atm-'
rmsd,b
kg mol-' K-'
(a ' B ,,/a T2) 2.45 2.21 2.57 3.61
-327.1 -322.3 -328.8 -330.0 Volume
kg mol-'
(~B,,/WT, kg mol-' atm-'
rmsd,b cm3 mol-'
0.68 36.4 169.52 20 0.60 8.07 -3.90 x 10-5 0.045 0.60 47.4 169.53 25 0.60 7.97 -3.96 x 100.043 20 0.62 0.72 37.0 169.53 8.07 -3.97 x 10-5 0.045 20 0.60 0.60 39.0 169.53 8.36 -4.01 x 1 0 4 0.046 '" Osmotic coefficient data were fit to eq 7 as described in ref 23. The parameter 6 was optimized to give the minimum rmsd for the first three entries. The fourth entry of 6 = 0.60 was chosen in order to show the effects of changing 6 . Rmsd is the root-mean-square deviation of the fit to the applicable equation. These values of 6 are those chosen as described in footnote a. of dimers. Results of the calculations indicated, as has been pointed out by Birch and Hall43for enthalpies of dilution of alkyl sulfates and alkyltrimethylammonium bromides, that deviations from Debye-Huckel behavior a t concentrations below the cmc can be accounted for equally well by including an ion interaction term or by dimerization. W e chose to account for nonideality a t concentrations below the cmc with an interaction coefficient [Le., (aB,,/aT),] rather than by assuming the formation of a dimer because the latter would involve the addition of two parameters that are difficult to determine, the equilibrium constant and the enthalpy change for dimer formation. In the literature, the quantity most frequently calculated from experimental thermodynamic data for micellar solutions is the change in a property at the critical micelle concentration. Various methods1~14~'5~19-21~44-47 have been used to derive the change in enthalpy, heat capacity, and volume for micelle formation from direct thermodynamic data on these quantities. In order to compare the results of our model calculations with values in the literature, the standard state (infinite dilution) quantities (Win), (ACpo/n),and ( A P / n ) must be related to values a t finite concentrations ( A H / n ) , (AC,/n), and ( A V / n ) . The change in a thermodynamic property ( A X / n ) ,where X = H , C,, or V, may be related to the value a t infinite dilution ( A X o / n )by consideration of the temperature and pressure de-
(43) Birch, B. J.; Hall, D. G . J . Chem. SOC., Faraday Trans. l 1972,68, 2350. (44) Musbally, G. M.; Perron, G.; Desnoyers, J. E. J . Colloid Interface Sci. 1976, 54, 80. (45) Rosenholm, J. B. Colloid Polym. Sci. 1981, 259, 1 1 16. (46) Paredes, S.; Tribout, M.; Ferreira, J.; Leonis, J. Colloid Polym. Sci. 1976, 254, 631. (47) Eatough, D. J.; Rehfeld, S . J. Thermochim. Acta 1971, 2, 443.
rivatives of the concentration quotient, (K/I'), where yMn0yAn)is defined by eq 5, as in
r = (ye/
AH = R P [ a In ( K / r ) / a T ] , = AHO - R P ( a In r / a T ) , (23) AC, = [ a ( A m / a q , = AC; - 2 m ( a In AV = - m [ a In
r/aT), - R P [ a 2In ( r ) / a P ] ,(24)
( ~ / r ) / a=~A ]P , + m ( a In r / a P ) , (25)
Equations 23-25 can be applied to eq 5 by using the parameters in Tables I-VIII. Values of thermodynamic parameters valid at the cmc, (AX*/n),calculated from eq 23-25 by using the parameters listed in Tables 11-VII, are compared with other values in the literature for octylamine hydrobromide and sodium dodecyl sulfate in Table IX. Similar application of eq 23-25 with the parameters in Tables VI1 and VI11 shows that changes in n and /3 do not significantly affect the values of (hX*/n), as illustrated in Table X. A number of values for the enthalpy, heat capacity, and volume of micellization of sodium dodecyl sulfate derived from direct thermodynamic property measurements have appeared in the literature. Values for the enthalpy of micellization have been calculated from titration cal~rimetry,"~*~* enthalpies of s o l ~ t i o n , ' ~ * ~ ~ enthalpies of mixing?O and enthalpies of d i l ~ t i o n . ~ A ~ Jvariety *~~~ of calculational methods have been used to derive enthalpies of micellization from the experimental calorimetric data. Values for the enthalpy of micellization of sodium dodecyl sulfate a t 25 (48) Kresheck, G. C.; Hargraves, W. A. J . Colloid Interface Sci. 1974, 48, 48 1.
(49) Benjamin, L. J . Phys. Chem. 1964, 68, 3575.
2162
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
Woolley and Burchfield
TABLE IX: Comparison of Values of aH*/n, ACp*/n, and AV*/n Valid at the cmc, Calculated from Eq 23-25 with Values in the Literature Obtained from Direct Thermodynamic Property Measurement AH*/n, J mol-' ACu*/n, J K-' mol-' A V * / n , cm' mol-' r, "C cmc, mol kg-' model lit. model lit. model lit. Octylamine Hydrobromide 5 0.24' 8720 87008 -330 -3498 8.81 9.88 15 0.22a 5690 5280' -326 -3488 7.90 8.8' 25 0.21' 2690 1975' -304 -332' 7.20 8.1' 35 0.22' (-1 70)b -270 (-310)' 6.40 6.9' 45 0.22' (- 2670)b -231 (-225)' 5.06 6.16 55 0.23a (-4990)b -198 4.9 Sodium Dodecyl Sulfate 20 0.00828 1290 3800,33 1 loo4' 25 0.00820 -754 -2100,48 -1250,49 130,'' -497 -516? 10.65 10.8? 360," 680,46 217047 30 0.0083 -3300 -4000,48 -3300,'' -2500,20 -260019 35 0.00845 -5790 -5000,'' -710048 -47819 'From ref 8. Values based on an extrapolation as described in the text. TABLE X: Effects of Changing n , p , and 6 on the Values of AH*/iz, AC,*/n, and AV*/n Calculated by Using the Parameters in Tables VI1 and VI11 and Eq 23-25
n
20 25 20 20 64 67 64 64
P
aH*/n, J mol-'
AC, * I n , J K-'
mol-' For Octylamine Hydrobromide at 5 "C 0.60 0.68 8718 -330 0.60 0.60 8590 -329 0.62 0.72 8800 -332 0.60 0.60 9083 -325 For Sodium Dodecyl Sulfate at 25 "C 0.77 0.52 -754 -497 0.77 0.52 -754 -497 0.75 0.52 -763 -499 0.77 0.60 -762 -498 6
A V*/n,
cm' mol-' 8.81 8.71 8.83 9.01
10.65 10.61 10.55 10.54
"C range from -2100 to 2170 J mol-', a difference of 4000 J mol-'. This discrepancy can in part be attributed to the difficulty of making calorimetric measurements a t concentrations below and near the cmc (0.0082 m),differences in purity of the sodium dodecyl sulfate used in the studies:' and the different calculational methods used. For sodium dodecyl sulfate, our values of ( A H * / n )agree well with values in the literature determined by direct calorimetric measurement^'^^^^*^"^ and calculated from the temperature dependence of the critical micelle c ~ n c e n t r a t i o n . Our ~ ~ value of ( A v * / n ) is in agreement with the value of Musbally et al.' The value of (AC,*/n) for sodium dodecyl sulfate is tentative because there are no data a t concentrations below the cmc. For octylamine hydrobromide, thermodynamic quantities for the micellization reaction derived from our model differ somewhat from those calculated by the method of De Lisi et al.s The graphical method of De Lisi et a1.* involves equating the change in a given thermodynamic quantity at the cmc to the difference in the extrapolated trends in the partial molar quantity above and below the cmc. This calculational method is consistent with the phase-separation model of micelle formation. As De Lisi et alSs have pointed out, the phase separation model may not be a good approximation for shorter-chain surfactants, such as octylamine hydrobromide, with relatively high critical micelle concentrations for which ion-ion interactions may significantly affect the concentration dependence of the partial molar quantities. The crnc of sodium dodecyl sulfate is only one-thirtieth that of octylamine hydrobromide, and therefore ion-ion interactions are perhaps less likely to influence the partial molar quantities. In fitting 4Land data to eq 20 and 21, we have used B1, = 0 and B,, = 0 a t each temperature. This gives rise to an apparent inconsistemy, since (aB,,/aT), # 0 and (aB,,/aT), # 0. Using values of B,, and B,, that are consistent with their temperature derivatives given in Tables 11, IV, and V changes the values of the parameters in eq 20 and 22 by less than the uncertainty in those parameters. For example, using B,, = -0.22
and B,, = 25 for sodium dodecyl sulfate a t 35 "C leads to values of AHo/n, (aB,,/dT),, (aB,,,/aT),, and AH*/n that differ from those in Tables V and IX by -18 J mol-', 0.0014 kg mol-' K-I, -0.02 kg mol-' K-', and 23 J mol-', respectively.
Conclusions Equations describing the concentration dependence of directly measurable thermodynamic properties (h,+c, and 4v)of micellar solutions have been derived from expressions for osmotic and activity coefficients based on a mass-action model of a g g r e g a t i ~ n . ~ ~ Activity coefficients are thus included rigorously, based on the Guggenheim equations for mixed electrolytes. The micelles are assumed to be monodisperse and the aggregation number, n, and the fraction of counterions "bound" to the micelle, /3, are assumed to be independent of temperature and pressure. The effects of ion-ion interactions on apparent molar thermodynamic quantities are included. The model has been applied to octylamine hydrobromide and sodium dodecyl sulfate, two ionic surfactants with cmc's that differ by a factor of 30. The model quantitatively fits the concentration , and dVdata for these surfactants over wide dependence of q ! ~ ~ q5,, ranges of concentration and temperature. The change in thermodynamic quantities a t the conventional standard state of infinite dilution and at the cmc are determined from the model calculations. The model provides a framework that can be used to correlate and understand the concentration dependence of experimentally measurable apparent molar thermodynamic quantities for ionic surfactant solutions.
Acknowledgment. Contribution No. 264 from the thermodynamics research laboratory at the Bartlesville Energy Technology Center, Department of Energy, where the research was funded by the Enhanced Oil Recovery Program, and in cooperation with the Associated Western Universities, Inc. We thank Mrs. June Forbes for carefully typing the manuscript.
Nomenclature A,
El,, Bny b
Go*
cmc I
K k,, m mi n P p2
AXo/n
AX*/n 21
a
constant in Debye-Huckel expressions ion interaction parameters ion-size parameter standard partial molar heat capacity of solute critical micelle concentration expressed in molality ionic strength (mol kg-') thermodynamic equilibrium constant based on molalities coefficients in eq 20-22 defined in the Appendix total or stoichiometric molality of surfactant molality of species i aggregation number pressure standard partial molar volume of solute change in a thermodynamic property X ( X = H,C, V) for forming micelles at infinite dilution per mole of monomer change in a thermodynamic property X (X= H,C, V) for forming micelles at the cmc per mole of monomer charge on ion i mole fraction of surfactant in micellar form
J. Phys. Chem. 1984, 88, 2163-2167 fraction of counterions "bound" to the micelle shielding factor to give effective micellar charge activity coefficient product activity coefficient of species i mean stoichiometric activity coefficient osmotic coefficient on a stoichiometric basis apparent molar heat capacity relative apparent molar enthalpy
4"
2163
apparent molar volume
Registry No. Sodium dodecyl sulfate, 151-21-3; octylamine hydrobromide, 14846-47-0,
Supplementary Material Available: The coefficients k,, in eq 20-22 are given in the Appendix (4 pages). Ordering information is available on any current masthead page.
Viscosities and Excess Volumes of Binary Mixtures Formed by the Liquids Acetonitrile, Pentyl Acetate, 1-Chlorobutane, and Carbon Tetrachloride at 25 OC M. G. Prolongo, R. M. Masegosa, I. Herniindez-Fuentes, Departamento de Quimica Fhica, Facultad de Ciencias Quimicas, Universidad Complutense. Madrid-3, Spain
and A. Horta* Departamento de Quimica General y MacromolPculas, Facultad de Ciencias, Universidad a Distancia (U.N.E.D.),Madrid-3, Spain (Received: July 26, 1983; In Final Form: October 12, 1983)
Excess volumes (p) for the binary mixtures formed by acetonitrile or 1-chlorobutane with pentyl acetate and dynamic viscosities (7)for the binary mixtures formed by acetonitrile with 1-chlorobutane, pentyl acetate, or carbon tetrachloride and for the mixture 1-chlorobutane+ pentyl acetate have been determined at 25 O C . These mixtures have interesting properties as mixed solvents of polymers. Their study should help in understanding the phenomenon of cosolvency in polymer systems. The results of P are discussed in terms of the influence of interactions between components, order and degree of packing in the mixtures, and of free volume differences. The viscosities are well correlated with the volume properties of the mixtures but not with their free energies of mixing. The method of Bloomfield and Dewan is used to predict 7 theoretically from known mixing functions.
Introduction
results will be analyzed in terms of both kinds of contributions.
The binary systems in this work have interesting characteristics with respect to the solubility of polymers. A t 25 O C the pure liquids acetonitrile (MeCN), 1-chlorobutane (ClBu), carbon tetrachloride (CC14), and pentyl acetate (PAC) are bad solvents of poly(methy1 methacrylate), while the mixtures of MeCN with any one of the other liquids act as very good solvents for such a polymer (they are powerful cosolvent mixtures).' The mixture ClBu PAC shows the opposite behavior, its solvent power being lower than that of the pure components.2 The solution properties of a polymer in a mixed liquid depend not only on the interactions between the polymer and each one of the liquids but also on the interactions between the liquids themselves. In fact, these interactions between liquids are decisive in determining the solubilization of the polymer and the expansion of the macromolecular coils in solution. In order to adequately interpret the thermodynamic behavior of polymers in mixed solvents, it is therefore necessary to know the properties of the binary liquid mixtures acting as solvents. In the present paper we report measurements of excess volume, p,a t 25 OC for the mixtures M e C N + PAC and ClBu + PAC and of the dynamic viscosity, 11, also a t 25 O C for the following mixtures: M e C N + PAC, M e C N + ClBu, ClBu PAC, and M e C N CC14. The dynamic viscosity is an integral property of the liquid mixture which has been related to the interactions Our between liquids3+ and also to the structure of the
Experimental Section Density. The densities, D, of the system MeCN
+ PAC have been measured in a digital densimeter (Anton Paar) provided with a measuring cell DMA 601 and an electronic unit DMA 60. The pure components have been used as calibrating substances. The temperature in the measuring cell was regulated to 25.00 f 0.005 "C. The estimated error in the density is f5 X g.~m-~. The densities of the system ClBu + PAC have been determined by pycnometry. The pycnometers, made with precision capillary tube, were calibrated with twice-distilled water. The height reached by the meniscus in the capillary was measured with a cathetometer of precision 0.01 mm. The temperature control was 25.00 f 0.005 OC and the precision in D f 2 X g~cm-~. Viscosity. The kinematic viscosities, Y, of the pure liquids and their mixtures have been determined with two modified Ubbelohde viscometers having a capillary radius of 0.45 mm. Both were calibrated a t 25 'C by using benzene (C6H6),toluene (PhMe), cyclohexane (C6I-I 12), n-hexane (n-C6Hi4), n-heptane (n-C7H16), chloroform (CHCl,), ClBu, PAC, MeCN, and CC14 as calibrating substances of known 7 and D. The values of 11 and D taken from the literature for these liquids are shown in Table I. The conversion of Y into 11 by use of D is well-known: 9 = v/D. The densities of the systems MeCN + ClBu and MeCN CC14 have been taken from the l i t e r a t ~ r e . ~ J ~
(1) I. Fernlndez-PiBrola and A. Horta, Makromol. Chem., 182, 1705 (1981). (2) I. Fernlndez-PiBrola and A. Horta, Polym. Bull., 3, 273 (1980). (3) R. J. Fort and W. R. Moore, Trans. Faraday Soc., 62, 11 12 (1966). (4) P. Skubla, Colkcr. Czech. Chem. Commun., 46, 303 (1981). ( 5 ) V. A. Bloomfield and R. K. Dewan, J . Phys. Chem., 75, 31 13 (1971). ( 6 ) J. Nath and S. N. Dubey, J . Phys. Chem., 85, 886 (1981). (7) H. Vogel, and A. Weiss, Ber. Bunsenges. Phys. Chem., 86, 193 (1982).
(8) C. Jambon and G. Delmas, Can. J . Chem., 55, 1360 (1977). (9) I. Fernlndez-PiBrola and A. Horta, J . Chim. Phys. Phys.-Chim.Biol., 77, 27 (1980). (10) J. A. Riddik and W. B. Bunger, "Techniques of Chemistry", Vol. 11, Wiley-Interscience, New York, 1970. (1 1) J. Timmermans, "Physico-Chemical Constants of Pure Organic Compounds", Elsevier, Amsterdam, 1950 and 1965. (12) I. Brown and F. Smith, Aust. J . Chem., 7, 269 (1954).
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0 1984 American Chemical Society
