2260
J. Phys. Chem. 1996, 100, 2260-2268
Effect of Large Changes in Temperature and Pressure on the Thermodynamic Properties of Micellization and on the Distribution Constant of a Polar Solute in Micellar Solutions A. Inglese* Department of Chemistry, UniVersity of Bari, Via Orabona 4, 70126 Bari, Italy
R. De Lisi and S. Milioto Department of Physical Chemistry, UniVersity of Palermo, Via Archirafi 26, 90123 Palermo, Italy ReceiVed: August 7, 1995; In Final Form: October 30, 1995X
Density measurements of pentanol (PentOH)-dodecyltrimethylammonium bromide (DTAB)-water mixtures as functions of both alcohol and surfactant (mS) concentrations were carried out at 0.1 MPa from 45 to 75 °C and at 19 MPa from 25 to 130 °C. The standard (infinite dilution) partial molar volumes and expansibilities of DTAB in water and the corresponding properties in the micellar phase were calculated from the experimental data. As far as PentOH in DTAB micellar solutions is concerned, with the exception of the standard partial molar volume (V°R) data at 130 °C and 19 MPa, all the V°R Vs mS trends are monotonic curves with mS. The data of V°R as a function of mS were treated by means of an equation previously used for data obtained at ambient conditions. So it was possible to evaluate simultaneously the distribution constant (K) of PentOH between the aqueous and micellar phases and the standard partial molar volume of alcohol in the aqueous (V°R,f) and the micellar (V°R,m) phases. The standard free energy of transfer (∆G°t) of PentOH from the aqueous to the micellar phases was calculated from K values. By using experimental literature data for the present system at 25 °C, ∆G°t values as functions of temperature and pressure were calculated. The very good agreement between the simulated and experimental values indicates that the model used also holds at higher temperatures. Equations correlating the standard free energy, enthalpy, and entropy of transfer to temperatures up to 130 °C and pressures up to 19 MPa were proposed.
Introduction The effects of concentration, polar headgroup, chain length, and additives on the structural and thermodynamic properties of micellar systems were intensively studied in the last 2 decades. However, a large part of this work was performed at atmospheric pressure and temperatures not too far from ambient, although many surfactant solutions are used in numerous technical, industrial, and biological processes at high temperatures and pressures. Large changes in these parameters might produce strong effects on the molecular interactions responsible for the aggregation process and, consequently, on the structure of aggregates either in the presence and in the absence of additives. Fruitful information about these interactions can be derived from the change of thermodynamic properties of micellar solutions with temperature and pressure. So far, only the effect of large changes in pressure on the properties of micellization of water + surfactant binary systems was systematically studied with spectroscopic and thermodynamic techniques.1-14 In spite of some discordant conclusions, all these studies showed that the critical micelle concentration (cmc) and the thermodynamic properties of micellization are significantly affected by pressure. For alkyltrimethylammonium bromide aqueous solutions, the cmc Vs pressure profiles are convex curves with the maximum at 75 MPa.2 From these curves, it follows that the molar volume of micellization is positive for P < 75 MPa and negative for P > 75 MPa. Only a few studies examined the effect of large changes in temperature on the cmc,15 aggregation number, and volume of micellization.16-22 The results are consistent with the temperature effect on the structure of water. With increasing temper* To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-2260$12.00/0
ature, the peculiar structure of water, mostly affecting the selfassembly of amphyphilic molecules, is strongly altered and the properties of water become similar to those of normal polar liquids. Much less studied was the thermodynamics of additive solubilization in micellar solutions as functions of temperature and pressure. In recent years, water + surfactant + solute ternary systems have been intensively studied by means of different techniques at 25 °C and atmospheric pressure.23-34 As a result, the structure of the aggregate and the thermodynamic properties of solutes in aqueous and micellar phases are wellknown. It was observed that they depend on the nature of both the surfactant and the additive. To the best of our knowledge, with the exception of the effect of small changes in temperature at atmospheric pressure,35 no studies dealing with large changes of temperature and pressure are reported in the literature for the water-surfactant-additive ternary systems. In the present paper, we report the apparent molar volumes of dodecyltrimethylammonium bromide (DTAB) in water and of pentanol in water + DTAB solutions at 0.1 and 19 MPa from 25 to 130 °C as functions of both the additive and the surfactant concentration. This experimental effort is predominantly directed toward the study of the pressure and temperature dependence of the partial molar volume of the unmicellized and micellized surfactant in water, of the distribution constant of pentanol between the aqueous and the micellar phases, and of the standard (infinite dilution) partial molar volume of pentanol in the micellar phase. At the same time, the models used to fit the experimental data at ambient conditions will be tested at high temperatures and pressures. Since for the present ternary system a complete set of thermodynamic properties (volume, heat capacity, enthalpy, © 1996 American Chemical Society
Large T and P Changes in Micellar Solutions free energy, entropy, and compressibility) are known at 25 °C and 0.1 MPa,34,36 the present data can be also used to verify the reliability of the thermodynamic relationship to predict the thermodynamic properties in a large interval of temperature and pressure. Experimental Section Materials. Dodecyltrimethylammonium bromide (DTAB, Sigma, 99%) was twice recrystallized. The sample was dissolved in hot ethyl alcohol (Fluka, puriss p.a. > 99.5%) and then crystallized by adding cold acetone (Fluka, puriss p.a. > 99.5%). The crystallized surfactant was filtered and dried under vacuum at 60 °C for 2 days. 1-Pentanol (PentOH, Fluka, puriss p.a. > 99%) was fractionally distilled over calcium hydride and stored over molecular sieves. All solutions were prepared using degassed, deionized, and distilled water. The concentrated solutions (m > 0.1 mol kg-1) were prepared by weighing the components at (0.01 mg. The diluted solutions were obtained from a stock solution (0.5 mol kg-1) by careful mass dilution. The molalities of diluted and concentrated solutions were known with an estimated accuracy of (0.06% and (0.02%, respectively. Experimental data dealing with pentanol in surfactant solutions are not reported. At each temperature, pressure, and surfactant concentration, four density measurements were performed for alcohol concentration between 0.03 and 0.30.6 mol kg-1 depending on the alcohol solubility. Method. The equipment used was described elsewhere.37 Briefly, density measurements were carried out with a vibrating tube flow densimeter (Sodev, Model 03D). The temperature was controlled within 0.002 °C in the range 30-75 °C and (0.005 °C at 100 and 130 °C with a closed-loop programmable circulating thermostat. The estimated accuracy of temperature in the two ranges was (0.01 and (0.04 °C. The pressure was controlled by two back-pressure regulators (BPR; Circle Seal, Model 21U22512) connected in series at the outlet of the densimeter. The second BPR, set to regulate the pressure at a value of about 0.4 MPa lower than the first BPR, reduces the pressure noise in the first BPR and allows to pressurize, at a pressure close to the working one, the liquid in the sample loop before injection in the densimeter. The prepressurization is necessary to eliminate the change in the baseline produced by the pressure fluctuation observed when the liquid in the sample loop is injected in the densimeter at a pressure very different from the working one. The pressure stability of the instrument and the estimated pressure accuracy were (0.05 and (0.1 MPa, respectively. The base flow was supplied by a high-pressure liquid chromatography pump (stability: (0.02%). The densimeter was calibrated at each experimental temperature and pressure using as reference liquid distilled water and deuterium oxide (Sigma, 99.96 atom % D). The densities of the two reference fluids were calculated from the equation of state provided by Haar et al.38 and Hill et al.39 The accuracy on the density measurements was 3 ppm at low temperatures and pressures and 5 ppm at high temperatures and pressures. At 0.1 MPa, measurements were limited up to 75 °C, since at higher temperatures their reproducibility was not good because of the bubbles formed in the flowing fluid when preheated before entering in the vibrating tube. This problem was absent at 19 MPa, and consequently, it was possible to perform accurate measurements up to 130 °C.
J. Phys. Chem., Vol. 100, No. 6, 1996 2261 Results and Discussion Water-DTAB Binary System. Critical Micelle Concentration and Degree of Counterion Dissociation. As will be seen later, the knowledge of the critical micelle concentration (cmc) and the degree of counterion dissociation of the micelles (β) at the working temperatures and pressures is needed for the data analysis. Both parameters can be derived from the literature conductivity measurements as functions of concentration at the different temperatures15 and pressures.2 We reported elsewhere36 equations correlating the cmc and β to the pressure at 25 °C derived on the basis of the data of Tuddenham and Alexander.2 From these equations, by increasing the pressure from 0.1 to 19 MPa, the increase in the cmc is 2 × 10-3 mol kg-1 and that in β is about 0.02. While there are not thermodynamic relationships which permit us to predict the effect of temperature on (∂β/∂P)T, there are for (∂(cmc)/∂P)T. In fact, on the basis of the pseudophase transition model for micellization,40 by neglecting the pressure effect on β, the dependence of the cmc on P is given by
(
)
∂ln(cmc) ∂P
) T
∆Vm (2 - β)RT
(1)
where ∆Vm is the volume of micellization. From the data in Table 2, at 25 °C the quantity at the righthand side of eq 1 is 1.6 × 10-3 MPa-1, and therefore, by increasing the pressure from 0.1 to 19 MPa, the cmc should increase by about 5 × 10-4 mol kg-1, which is of the same order of magnitude as the cmc’s experimental uncertainty. Since, as will be seen later, the absolute ∆Vm value at higher temperatures is smaller than that at 25 °C, by increasing the temperature, the absolute (∂ln(cmc)/∂P)T value decreases. Therefore, it is reasonable to assume that β and cmc at 19 MPa are the same as those at 0.1 MPa. As far as the temperature effect is concerned, since in our pressure interval no appreciable difference is expected for β and cmc, from the literature data reported by Evans et al.15 in the temperature ranges 25-100 °C at 0.1 MPa and 100-160 °C at 0.5 MPa, the following equations were derived:
β ) 0.202 + 1.48 × 10-3t + 1.45 × 10-5t2
(2)
cmc ) 0.0186 - 2.30 × 10-4t + 4.3 × 10-6t2
(3)
where t is the temperature in °C. The values calculated through eqs 2 and 3 agree with those obtained from a different source of conductance data.41 Apparent Molar Volumes. The apparent molar volume (VΦ,S) of DTAB in water was calculated by using the following equation:
VΦ,S )
MS 103 (d - do) d mSddo
(4)
where MS and mS are the molecular weight and the molality of the surfactant, respectively, while d and do are the density of solution and water, respectively. The density difference, ∆d ) d - do, and the apparent molar volumes of DTAB in water at the experimental temperatures and pressures are reported in Table 1. Figures 1 and 2 show the apparent molar volumes of DTAB as functions of mS at 0.1 and 19 MPa, respectively, at the working temperatures. As Figure 1 shows, in the postmicellar region, at 45 and 65 °C, VΦ,S increases with concentration, tending to constant values at high mS. This significant change
2262 J. Phys. Chem., Vol. 100, No. 6, 1996
Inglese et al.
TABLE 1: Density Difference and Apparent Molar Volume of Dodecyltrimethylammonium Bromide at Experimental Temperatures (°C) and Pressures (MPa)a mS
103∆d
103∆d
VΦ,S
0.011 53 0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1
0.164 0.403 0.533 0.718 0.920 1.132 1.420
t ) 45.00; P ) 0.10 296.84 0.159 7 297.91 0.203 8 298.56 0.307 0 299.28 0.434 1 299.50 0.543 1 299.70 0.748 7 299.84 0.871 8
1.715 2.141 3.118 4.238 5.139 6.759 7.618
299.93 300.03 300.10 300.15 300.19 300.14 300.18
0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1 0.159 7
0.328 0.412 0.573 0.732 0.902 1.130 1.365
t ) 65.00; P ) 0.10 303.30 0.203 8 304.34 0.307 0 304.58 0.434 1 304.79 0.543 1 304.99 0.748 7 305.09 0.871 8 305.15
1.709 2.494 3.405 4.123 5.448 6.142
305.22 305.24 305.25 305.29 305.21 305.23
0.011 53 0.030 74 0.042 84 0.061 49 0.080 59 0.102 0 0.130 1
0.096 0.245 0.363 0.497 0.655 0.802 1.026
t ) 75.00; P ) 0.10 307.50 0.159 7 307.51 0.203 8 307.29 0.307 0 307.59 0.434 1 307.51 0.543 1 307.81 0.748 7 307.73 0.871 8
1.257 1.573 2.321 3.156 3.868 5.043 5.735
307.62 307.68 307.61 307.65 307.59 307.62 307.58
0.006 19 0.011 24 0.011 53 0.015 84 0.022 70 0.025 96 0.028 21 0.030 74 0.042 79 0.042 84 0.051 10
0.112 0.208 0.236 0.290 0.396 0.420 0.447 0.475 0.644 0.647 0.746
t ) 25.00; P ) 19.17 288.71 0.061 47 288.35 0.078 33 288.82 0.080 59 288.45 0.130 1 289.30 0.159 7 290.57 0.203 8 290.88 0.307 0 291.25 0.434 1 291.61 0.543 1 291.55 0.748 7 292.00 0.871 8
0.892 1.107 1.142 1.771 2.160 2.696 3.932 5.345 6.492 8.485 9.571
292.06 292.37 292.33 292.68 292.67 292.78 292.85 292.92 292.96 292.99 293.02
0.011 53 0.028 21 0.042 79 0.051 10 0.061 47 0.078 33 0.080 59 0.102 0
0.106 0.262 0.400 0.477 0.568 0.709 0.688 0.915
t ) 65.02; P ) 19.15 302.42 0.130 1 302.33 0.159 7 302.25 0.203 8 302.24 0.307 0 302.23 0.434 1 302.40 0.543 1 302.44 0.748 7 302.41 0.871 8
1.152 1.410 1.766 2.594 3.548 4.308 5.714 6.469
302.50 302.40 302.47 302.39 302.42 302.44 302.29 302.30
0.010 63 0.011 53 0.015 84 0.022 70 0.025 96 0.028 21 0.030 74 0.042 79 0.042 84 0.051 10 0.051 10
0.062 0.066 0.087 0.128 0.141 0.164 0.186 0.247 0.244 0.311 0.306
t ) 100.00; P ) 19.15 312.69 0.061 47 312.72 0.078 33 312.60 0.080 59 312.81 0.102 0 312.70 0.159 7 312.61 0.203 8 312.35 0.307 0 312.61 0.434 1 312.71 0.543 1 312.28 0.748 7 312.41 0.871 8
0.365 0.484 0.508 0.670 1.050 1.361 2.072 2.866 3.500 4.711 5.302
312.39 312.07 311.91 311.57 311.46 311.26 310.98 310.86 310.83 310.60 310.63
0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1 0.159 7
0.088 0.128 0.178 0.289 0.436 0.610 0.796
t ) 130.00; P ) 19.24 323.25 0.203 8 323.10 0.307 0 323.19 0.434 1 322.38 0.543 1 321.59 0.748 7 321.04 0.871 8 320.64
1.098 1.774 2.584 3.219 4.376 5.051
320.10 319.42 318.95 318.76 318.47 318.30
VΦ,S
mS
a Units are mol kg-1 for concentration, g cm-3 for density, and cm3 mol-1 for apparent molar volume.
is essentially due to the transfer of the surfactant from water to the micelles since the micelle-micelle interactions can be
Figure 1. Effect of temperature on the apparent molar volumes, corrected for those at infinite dilution, against concentration for dodecyltrimethylammonium bromide in water at 0.1 MPa.
Figure 2. Effect of temperature on the apparent molar volumes, corrected for those at infinite dilution, against concentration for dodecyltrimethylammonium bromide in water at 19 MPa.
neglected since the micelle concentration is very small. At 75 °C, the VΦ,S change at the cmc is very small. Interesting is the temperature effect on the VΦ,S Vs mS plots at 19 MPa. In fact, at 25 °C, VΦ,S is essentially constant up to the cmc, beyond which it increases monotonically with mS, reaching a constant value at 0.1m. At 65 °C, VΦ,S practically does not depend on the surfactant concentration, while at higher temperatures, it is practically constant in the premicellar region and decreases monotonically with mS in the postmicellar one. Moreover, at high temperatures, the break at the cmc in the VΦ,S Vs mS trends becomes broader since the micellization occurs in a larger concentration range. Standard Partial Molar Volume. In the premicellar region, at a given temperature and pressure, the VΦ,S data as a function of mS can be treated by means of the following equation:
VΦ,S ) V°S + AVxmS + BVmS
(5)
where V°S is the standard (infinite dilution) partial molar volume, AV is the Debye-Hu¨ckel parameter, and BV the surfactantsurfactant pair interaction parameter. The partial molar volume can be evaluated by introducing eq 5 into the following equation:
V2 )
∂(mSVΦ,S) ∂mS
(6)
At 0.1 MPa, the low cmc values15 (Table 2) did not permit us to study the premicellar region. So, the V°S values at the experimental temperatures were calculated by combining some literature data. In particular, the V°S value at 0.1 MPa and 45
Large T and P Changes in Micellar Solutions
J. Phys. Chem., Vol. 100, No. 6, 1996 2263
TABLE 2: Volumetric Properties of Dodecyltrimethylammonium Bromide in Water and Critical Micelle Concentration at Experimental Temperatures (°C) and Pressures (MPa)a T
P
15.00 25.00
0.10 0.10
35.00 45.00 65.00 75.00 25.00
0.10 0.10 0.10 0.10 19.17
65.02
19.15
100.00
19.15
130.00
19.24
V°S
Av
Bv
284.2d 288.4e 289.2g 292.2d 296.4g 303.5g 307.0g 288.5 ( 0.1 288.6g 302.3 ( 0.1 302.2g 312.4 ( 0.1 312.5g 322.7 ( 0.2 322.4g
cmcb
βc 0.23 0.25
5.3 3.7 1.7 0.5 4.7
0.0161 0.0154 0.0153f 0.0158 0.0172 0.0219 0.0263 0.0154
0.27 0.30 0.36 0.39 0.25
∆Vm
Vf
Vm
284.2 288.4
292.0d 294.9e
7.8 6.7
297.5d 300.1 305.2 307.5 293.0
1.994h
-13
292.2 296.4 303.5 307.0 288.3
2.685h
-12
302.3
302.4
0.2
0.022
0.36
3.716h
-14 ( 4
312.4
310.4
-2.0
0.038
0.50
5.133h
-13 ( 5
323.0
317.7
-5.3
0.061
0.64
a Units are cm3 mol-1 for volume, mol kg-1 for critical micelle concentration, cm3 kg1/2 mol-3/2 for the A interaction parameter, and cm3 kg v mol-2 for the Bv interaction parameter. b Calculated according to eq 3. c Calculated according to eq 2. d From ref 43. e From ref 26. f From ref 49. g See text. h From ref 44.
°C was obtained from the V°S of nonyltrimethylammonium bromide42 (245.95 cm3 mol-1) and the methylene group contribution evaluated by using the following equation:43
VCH2(cm3 mol-1) ) 14.66 + 0.046t
(7)
where t is the temperature (°C). The V°S value at 75 °C was obtained from the V°S values at 0.9 and 32 MPa21 by assuming that the compressibility is pressure independent. Indeed, a large dependence of isothermal compressibility on pressure was observed for sodium decanoate in water.10 Neverthless, by taking for DTAB the same dependence as for sodium decanoate, the correction is negligible. Therefore, the V°S value at 65 °C was interpolated from the linear plot of V°S as a function of temperature where the present and literature data43 were used. Whenever comparisons are possible, it is observed that the experimental VΦ,S value at 0.011m is close to V°S, indicating that the surfactant-surfactant interactions are small. At 19 MPa, it was possible to investigate the premicellar region. Then, the V°S values were calculated through eq 5 by using the AV values we interpolated from those reported as a function of temperature at 20 MPa44 for each temperature. They are reported in Table 2 together with the V°S values calculated at our experimental conditions from the data reported by Archer et al.21 using a different procedure for the representation of the thermodynamic properties of the surfactant solution.45 As can be seen, the two series of data are in good agreement. As Figure 3 shows, at both 0.1 and 19 MPa, V°S increases linearly with temperature, with slopes corresponding to the standard partial molar expansibility, which are 0.379 ( 0.006 and 0.322 ( 0.008 cm3 mol-1 K-1 at 0.1 and 19 MPa, respectively. The two straight lines cross each other at about 32 °C, and consequently, the standard partial molar isothermal compressibility, K°T,S ) -(∂V°S/∂P)T, is negative for t < 32 °C and positive for t > 32 °C. These findings agree with literature results dealing with the standard partial molar isoentropic compressibility, K°S,S, obtained from speed-of-sound measurements.36 Remember that K°S,S is generally smaller than K°T,S by about 0.01 cm3 mol-1 MPa-1.36,42 From the two straight lines, the dependence of K°T,S (cm3 mol-1 MPa-1) on temperature is given by
K°T,S ) -0.095 + 0.0030t
Figure 3. Partial molar volumes of dodecyltrimethylammonium bromide in the standard state and in the micellized state as functions of temperature at 0.1 and 19 MPa. For references, see Table 2.
calculations, values very close to those reported in the literature36 are obtained for K°T,S at 25 °C and for the thermal coefficient of K°T,S. Partial Molar Volume of Micellized Surfactant. On the basis of the pseudophase transition model for micellization,46 the partial molar volume of the surfactant in the micellized form (Vm) can be evaluated by extrapolating at the cmc the trend of the partial molar volume as a function of molality in the postmicellar region. In the present case, since at high mS VΦ,S is constant, the partial and apparent molar volumes are equal; therefore, Vm ) VΦ,S. The Vm values are reported in Table 2 and shown in Figure 3. Regardless of the pressure value, Vm increases with temperature in a linear manner, indicating that the expansibility of the surfactant in the micellar phase (Em) is constant despite the large excursion in temperature. The Em values are 0.258 ( 0.004 and 0.234 ( 0.002 cm3 mol-1 K-1 at 0.1 and 19 MPa, respectively. The expansibility of DTAB and the pressure effect on it are larger in the aqueous phase than in the micellar phase due to the hydrophobic nature of the micelle. From the best fits to the Vm Vs t plots, the following equation correlating the partial molar isothermal compressibility of the surfactant in the micellized form, KT,m (cm3 mol-1 MPa-1), to temperature was obtained:
KT,m ) 0.068 + 0.0013t
(9)
(8)
from which, within the uncertainty involved in the present
which gives values very close to those reported in the literature36 for KT,m at 25 °C and for its thermal coefficient.
2264 J. Phys. Chem., Vol. 100, No. 6, 1996
Inglese et al.
Figure 4. Volume of micellization of dodecyltrimethylammonium bromide as a function of temperature at 0.1 and 19 MPa. For references, see Table 2.
Figure 5. Effect of temperature on the standard partial molar volumes of pentanol in dodecyltrimethylammonium bromide aqueous solutions as functions of the surfactant concentration at 0.1 MPa. Lines, best fits according to eq 12.
Volume of Micellization. The volume of micellization of the surfactant is given by46
∆Vm ) Vm - Vf
(10)
where Vf is the partial molar volume of the dispersed surfactant. At 0.1 MPa,V°S was used instead of Vf because of the lack of experimental data. This means that the interaction contributions between the dispersed surfactants are considered negligible according to the low cmc values. At 19 MPa, the Vf values were calculated by taking the V°S, AV, and BV values reported in Table 2 and eqs 5 and 6. The ∆Vm and Vf values are collected in Table 2. As Figure 4 shows, the volume of micellization decreases with temperature because the expansibility of the surfactant in the micellar phase is smaller than that in water (or aqueous phase). The stronger temperature dependence for ∆Vm at 0.1 MPa with respect to that at 19 MPa is a consequence of the smaller expansibility of the surfactant in both the aqueous and micellar phases at 19 MPa. The temperature at which the volume of micellization changes from positive to negative values depends on pressure. At 0.1 and 19 MPa, it is 78 (extrapolated value) and 73 °C, respectively. This behavior is consistent with the increment of the degree of polydispersity of micelles with temperature determined by the increasing population of aggregates with smaller aggregation number. In fact, a correlation between the volume of micellization and the aggregation number (ni) of the micelles through the cmc dependence on pressure was reported elsewhere.21 For instance, at 25 °C and 0.1 MPa, the volume of micellization of sodium dodecyl sulfate changes from -27.5 to 28.9 cm3 mol-1 as ni increases from 40 to 80. Since at the selected temperature and pressure Vf does not depend on ni, the negative ∆Vm values for small ni indicate that the void space in the micelles and, hence, the compressibility of the surfactant in micellar phase increase by increasing ni. The present results agree with the above findings. By increasing the temperature, the contribution of small or negative volumes of micellization due to the formation of the small aggregation number micelles becomes more important and ∆Vm decreases and changes sign at a temperature which is pressure dependent. Since pressure and temperature have opposite effects on ni, it is supported that the higher the pressure is, the lower the temperature is at which the ∆Vm sign inversion occurs (Figure 4). Water-DTAB-Pentanol Ternary Systems. At constant surfactant concentration, the apparent molar volume of PentOH in the DTAB solutions (VΦ,R) was calculated by means of eq 4 by replacing mS and MS with the molality (mR) and the molecular
Figure 6. Effect of temperature on the standard partial molar volumes of pentanol in dodecyltrimethylammonium bromide aqueous solutions as functions of the surfactant concentration at 19 MPa. Lines, best fits according to eq 12.
weight (MR) of alcohol and by using for do the density of the corresponding water-surfactant binary system. For a given temperature, pressure, and surfactant concentration, VΦ,R shows a linear dependence from mR according to the following equation:
VΦ,R ) V°R + BVmR
(11)
Therefore, from linear fits, the standard (infinite dilution) partial molar volume of pentanol (V°R) and the alcohol-alcohol interaction parameter (BV) were obtained. Their values together with the corresponding uncertainties are collected in Table 3, while the dependence of V°R on mS at 0.1 and 19 MPa is shown in Figures 5 and 6, respectively. At both working pressures, with the exception of V°R data at 130 °C and 19 MPa, in the postmicellar region, the V°R Vs mS trends are monotonic curves which increase with the surfactant concentration and tend to a constant value. This behavior reflects the progressive transferring of PentOH from the aqueous to the micellar phases due to the increase of the amount of the latter phase with mS. At high surfactant concentration, the bulk V°R tends to the value in the micellar phase. On the basis of the pseudophase transition model for micellization and a mass action model for the additive distribution, we reported elsewhere26 an equation correlating V°R to mS in terms of the distribution constant (K) of the additive between the aqueous and the micellar phases and the partial molar volume of the additive in the aqueous (V°R,f) and micellar (V°R,m) phases
Large T and P Changes in Micellar Solutions
J. Phys. Chem., Vol. 100, No. 6, 1996 2265
TABLE 3: Standard (Infinite Dilution) Partial Molar Volumes and Limiting Slope for Pentanol in Water + DTAB Solutions at Different Temperatures (°C) and Pressures (MPa) mS
V°R
Bv
mS
V°R
Bv
0.011 53 0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1
104.09(0.09) 105.36(0.06) 105.93(0.04) 106.36(0.04) 106.88(0.07) 107.17(0.05) 107.47(0.02)
t ) 45.00; P ) 0.10 5.9(0.5) 0.159 7 107.62(0.00) 3.6(0.3) 0.203 8 107.97(0.04) 2.2(0.3) 0.307 0 108.38(0.02) 2.1(0.1) 0.434 1 108.73(0.01) 1.2(0.3) 0.543 1 108.85(0.01) 1.0(0.2) 0.748 7 109.08(0.02) 0.69(0.07) 0.871 8 109.11(0.03)
0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1 0.159 7
107.80(0.07) 108.09(0.05) 108.66(0.02) 109.15(0.04) 109.48(0.09) 109.70(0.02) 109.84(0.04)
t ) 65.00; P ) 0.10 3.5(0.3) 0.203 8 110.08(0.11) 0.8(0.3) 3.3(0.3) 0.307 0 110.52(0.05) 0.5(0.2) 2.20(0.08) 0.434 1 110.87(0.01) 0.19(0.04) 1.6(0.2) 0.543 1 110.91(0.03) 0.3(0.1) 1.1(0.4) 0.748 7 111.19(0.03) -0.04 (0.09) 1.06(0.09) 0.871 8 111.15(0.02) 0.19(0.06) 1.1(0.1)
0.011 53 0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1
108.38(0.00) 108.81(0.12) 109.14(0.02) 109.24(0.08) 109.98(0.04) 110.28(0.00) 110.51(0.03)
t ) 75.00; P ) 0.10 2.50(0.01) 0.159 7 110.83(0.02) 3.1(0.5) 0.203 8 110.98(0.03) 3.26(0.09) 0.307 0 111.36(0.06) 3.8(0.4) 0.434 1 111.65(0.05) 2.1(0.2) 0.543 1 111.79(0.05) 1.39(0.02) 0.748 7 111.85(0.02) 1.6(0.1) 0.871 8 111.90(0.02)
0.011 53 0.030 74 0.042 84 0.061 47 0.080 59 0.130 1 0.159 7
t ) 25.00 ; P ) 19.17 102.23(0.01) 1.27(0.07) 0.203 8 104.93(0.09) 103.10(0.03) 0.5(0.2) 0.307 0 105.29(0.02) 103.47(0.00) 0.41(0.02) 0.434 1 105.51(0.00) 103.80(0.02) 0.19(0.08) 0.543 1 105.75(0.00) 104.06(0.03) 0.0(0.1) 0.748 7 105.82(0.05) 104.49(0.05) -0.2(0.2) 0.871 8 105.86(0.03) 104.82(0.09) -0.6(0.4)
0.011 53 0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.130 1
106.14(0.05) 106.65(0.06) 106.90(0.06) 107.23(0.03) 107.39(0.10) 107.69(0.05) 108.09(0.02)
t ) 65.02; P) 19.15 3.1(0.2) 0.159 7 108.32(0.04) 3.0(0.4) 0.203 8 108.53(0.05) 3.1(0.3) 0.307 0 108.86(0.01) 2.1(0.1) 0.434 1 109.08(0.02) 1.4(0.4) 0.543 1 109.14(0.02) 2.4(0.2) 0.748 7 109.24(0.02) 1.05(0.09) 0.871 8 109.29(0.03)
0.011 53 0.030 74 0.042 84 0.061 47 0.080 59 0.102 0 0.159 7
111.15(0.04) 111.03(0.04) 110.9(0.2) 111.28(0.04) 111.49(0.04) 111.83(0.03) 111.91(0.03)
t ) 100.00; P ) 19.15 1.4(0.2) 0.203 8 112.23(0.05) 1.9(0.2) 0.307 0 112.65(0.06) 2.5(0.9) 0.434 1 112.76(0.04) 2.1(0.2) 0.543 1 112.98(0.03) 2.0(0.1) 0.748 7 113.07(0.01) 1.4(0.1) 0.871 8 112.98(0.02) 1.6(0.1)
0.030 74 0.042 84 0.061 47 0.080 59 0.130 1 0.159 7
116.22(0.09) 116.07(0.04) 116.07(0.03) 116.06(0.08) 116.15(0.03) 116.15(0.04)
t ) 130.00; P ) 19.24 0.3(0.4) 0.203 8 116.23(0.04) 0.9(0.2) 0.307 0 116.25(0.05) 0.9(0.1) 0.434 1 116.30(0.03) 0.7(0.3) 0.543 1 116.31(0.01) 1.0(0.1) 0.748 7 116.28(0.03) 0.8(0.2) 0.871 8 116.26(0.01)
0.84(0.01) 0.5(0.1) 0.23(0.09) -0.02(0.04) -0.03(0.02) -0.25(0.06) -0.14(0.07)
1.21(0.07) 1.1(0.1) 0.8(0.2) 0.5(0.1) 0.5(0.2) 0.24(0.07) 0.43(0.04) -0.3(0.3) -0.49(0.08) -0.56(0.01) -0.92(0.00) -1.2(0.2) -1.2(0.1)
0.5(0.2) 0.6(0.2) 0.89(0.04) 0.11(0.06) 0.12(0.07) 0.17(0.08) 0.56(0.06) 1.0(0.2) 0.2(0.2) 0.5(0.2) 0.0(0.1) 0.12(0.03) 0.26(0.05)
1.0(0.1) 0.6(0.2) 0.68(0.09) 0.99(0.02) 0.67(0.08) 0.79(0.03)
a Units are mol kg-1 for concentration, cm3 mol-1 for volume, and cm3 mol-2 kg for the Bv interaction parameter.
V°R ) V°R,m + [(V°R,f - V°R,m) + Acdc∆Vm]Nf
(12)
where Acdc∆Vm indicates the displacement of the micellization equilibrium due to the presence of the additive. Nf is the mole fraction of PentOH in the aqueous phase with respect to the total amount of PentOH, which can be expressed as30
Nf )
1 1 + K(mS - cmc)
For a 1:1 ionic surfactant, the Acdc term is given by26
(13)
Acdc ) {2.3KS + (1 + β)K}cmc/2
(14)
where Ks is the Setchenov constant and the factor 2 takes into account the complete dissociation of the unmicellized surfactant. As mentioned above, at 130 °C and 19 MPa, the V°R Vs mS curve does not display the profile predicted by eq 12, showing a maximum at about 0.6 mol kg-1. This behavior is reminiscent of that observed for other systems47,48 and, generally, is observed whenever the property of transfer of the additive from the aqueous to the micellar phase is small, which is the case here for V°R,m - V°R,f. For this reason, eq 12 was fitted to the experimental points at 130 °C and 19 MPa in the increasing region only. At each temperature and pressure, eq 12 was solved by means of a nonlinear least-squares method from which K, V°R,m, and the term in brackets are obtained simultaneously. From these quantities and eq 14, the V°R,f values were derived by using the cmc, ∆Vm, and β values collected in Table 2. As far as Ks is concerned, it is available only at 25 °C and 0.1 MPa.49 Since Ks is related to the alcohol-surfactant pair interaction parameter for free energy (GRS) by50
KS ) 2GRS/RT
(15)
its derivative with respect to temperature and pressure is given by
( ) ∂KS ∂T
)-
2HRS
P
2
(HRS)298 + (CRS)298(T - 298)
) -2
RT2
RT
( ) ∂KS ∂P
) T
(16)
2VRS (VRS)298 + (ERS)298(T - 298) )2 (17) RT RT
where HRS, CRS, VRS, and ERS indicate the pair interaction parameters for enthalpy, heat capacity, volume, and expansibility, respectively. The above equations assume a linear dependence of the interaction parameters with temperature. The interaction parameters can be calculated from the corresponding standard partial molar properties of alcohol in the premicellar surfactant solutions and in water as
YRS )
Y°R - Y°R,W m
(18)
where m indicates the premicellar surfactant concentration at which Y°R is known. Literature data at 25 °C for HRS,51 C°PR,52 C°PR,W,52 V°R,36 V°R,W,36 E°R36 and E°R,W36 show that within the experimental uncertainties, the VRS, ERS, and CRS parameters are zero, while the HRS parameter is not. Consequently, pressure does not affect Ks, while temperature does. From eq 15 and the literature values for HRS (9.00 kJ kg mol-2) and for Ks (-0.34 kg mol-1)49 at 25 °C, the following equation correlating Ks to temperature was obtained:
Ks ) 6.93 -
2166 273 + t
(19)
The best fits to the experimental points in the postmicellar region are shown in Figures 5 and 6. The K, V°R,m, and V°R,f values are collected in Table 4 where V°R,W literature36,37 values are also reported. With the exception of the results at 130 °C and 19 MPa, the V°R,f values are essentially equal to those of V°R,W according, as mentioned above, to the negligible alcohol-surfactant pair interaction parameters. So the V°R,f Vs temperature curves, shown in Figure 7, are representative of those in pure water. To fit
2266 J. Phys. Chem., Vol. 100, No. 6, 1996
Inglese et al.
TABLE 4: Derived Quantities from the Fits of Equation 12 for Pentanol in Aqueous Dodecyltrimethylammonium Bromide Solutions at Experimental Temperatures and Pressuresa t, °C 15 Ksc K Kc V°R,m V°R,f V°R,w ∆V°t ∆G°t Ksc K Kc V°R,m V°R,f V°R,w ∆V°t ∆G°t
b
-0.59 10.3 35.3 106.31 ( 0.07 102.0 102.0 4.3 -8.5 ( 0.2
25b -0.34c,d 10.6 35.9 107.39 ( 0.07 102.5 102.5 4.9 -8.9 ( 0.2 -0.34 9.3 ( 0.7 32 ( 3 106.27 ( 0.05 102.4 ( 0.1 102.5e 4.0 -8.6 ( 0.2
35b -0.10 11.6 39.0 108.35 ( 0.07 103.4 103.4 5.0 -9.4 ( 0.2
45
65
75
P ) 0.10 MPa 0.12 0.52 11.3 ( 0.8 11.2 ( 1 38 ( 2 37 ( 3 109.54 ( 0.06 111.56 ( 0.07 104.3 ( 0.1 107.1 ( 0.1 104.7e 106.9e 5.2 4.5 -9.6 ( 0.2 -10.1 ( 0.2 P ) 19.2 MPa 0.52 10.3 ( 0.9 34 ( 3 109.69 ( 0.07 106.2 ( 0.1 106.2e 3.5 -9.9 ( 0.3
100
130
1.12 7.3 ( 1.4 24 ( 4 113.4 ( 0.1 111.4 ( 0.1 111.2e 2.0 -9.8 ( 0.5
1.56 4(2 13 ( 6 116.44 ( 0.06 117.7 ( 0.1 116.8e 1.3 8.6 ( 1.3
0.71 10.2 ( 1.3 33 ( 4 112.35 ( 0.11 108.5 ( 0.2 108.9e 3.9 -10.1 ( 0.3
a Units are cm3 mol-1 for volume, kg mol-1 for K and K , and kJ mol-1 for standard free energy. b Excluding K , data at 15, 25, and 35 °C and s s 0.1 MPa from ref 36. c Calculated according to eq 19. d From ref 26. c From ref 49. e From ref 37.
(∂K°R,f/∂T)P ) -(∂E°R,f/∂P)T
(20)
From analytical equations correlating E°R,f to temperature at the two pressures and eq 20, we obtain the dependence of the thermal coefficient of K°R,f on temperature. By combining this quantity ((∂K°R,f/∂T)P ) 2.1 × 10-5t) and the K°R,f value at 65 °C (0.047 cm3 mol-1 MPa-1), calculated as -∆V°R,f/∆P, the following equation was obtained:
K°R,f ) 0.003 + 1.05 × 10-5t2
Figure 7. Standard partial molar volumes of pentanol in dodecyltrimethylammonium bromide aqueous and micellar phases as functions of temperature at 0.1 and 19 MPa.
Figure 8. Standard partial molar expansibility of pentanol in dodecyltrimethylammonium bromide aqueous and micellar phases as functions of temperature at 0.1 and 19 MPa.
the experimental points, a second-order polynomial equation was used. From the best fits, the standard partial molar expansibilities of PentOH in the aqueous phase (E°R,f), shown in Figure 8 as a function of temperature, were obtained. As can be seen, the pressure effect on E°R,f increases with temperature. This effect corresponds to that of temperature on the standard partial molar isothermal compressibility of PentOH in the aqueous phase (K°R,f) since
(21)
The K°R,f value (0.010 cm3 mol-1 MPa-1) obtained from this equation at 25 °C can be considered to be in agreement with that reported in the literature36 (0.0115 cm3 mol-1 MPa-1) obtained by the more sensitive ultrasound measurements. However, it is to be noted that, although the difference between the calculated and the experimental values is about 10%, the accuracy using the indirect method is of the order of 100%. Figure 7 shows the V°R,m Vs t plots at 0.1 and 19 MPa. The present results are consistent with the values at 15, 25, and 35 °C and 0.1 MPa obtained in a previous work.36 At both pressures, V°R,m changes linearly with temperature, and by increasing the temperature, the volume of transfer of PentOH from the aqueous to the micellar phase (∆V°t ) V°R,m - V°R,f) goes through a maximum at about 40-50 °C, and the two curves tend to cross each other at about 90 °C (Figure 9). This is clearly evidenced by the standard partial molar isothermal compressibility of transfer, ∆K°t ) -(∂∆V°t/∂P)T, calculated from analytical equations correlating ∆V°t to temperature at 0.1 and 19 MPa (Figure 9). The linear increase of V°R,m with temperature indicates that the standard partial molar expansibility of PentOH in the micellar phase E°R,m is temperature independent. The E°R,m values at 0.1 and 19 MPa are 0.102 ( 0.002 and 0.098 ( 0.003 cm3 mol-1 K-1, respectively, indicating that pressure does not play a relevant role in the expansibility of PentOH in the micellar phase. Also, these values are equal to the molar expansibility of pure liquid PentOH (0.10 cm3 mol-1 K-1) and between those in pure water23 (0.03 cm3 mol-1 K-1) and in n-octane23 (0.15 cm3 mol-1 K-1) according to the solubilization of PentOH in the palisade layer of the micelle. From the E°R,m values at 0.1 and 19 MPa, the value of (2 ( 3) × 10-4 cm3 mol-1 MPa-1 K-1 was obtained for (∂K°R,m/∂T)P,
Large T and P Changes in Micellar Solutions
J. Phys. Chem., Vol. 100, No. 6, 1996 2267
Figure 9. Standard partial molar volume (∆V°t) at 0.1 and 19 MPa and isothermal compressibility (∆K°t) of transfer of pentanol from the aqueous to the micellar phases of dodecyltrimethylammonium bromide as a function of temperature.
Figure 10. Distribution constant of pentanol between the aqueous and the micellar phases of dodecyltrimethylammonium bromide as a function of temperature at 0.1 and 19 MPa.
which is, obviously, temperature independent. Within the experimental uncertainties, this value is close to that reported in the temperature range 15-35 °C for the standard isoentropic compressibility of PentOH in the DTAB micellar phase (6.7 × 10-4 cm3 mol-1 MPa-1 K-1). So temperature and pressure affect the properties of PentOH in the aqueous phase more than those in the micellar phase. Inspection of Table 4 and Figure 10 show that the profiles of the distribution constant K Vs t at 0.1 and 19 MPa are similar; as occurs for ∆V°t, they increase up to 40-50 °C and then decrease. Also, the pressure dependence of K is small and negative. The uncertainty in K values is of the order of 10% up to 75 °C, and it increases to 20% and 50% at 100 and 130 °C, respectively. The K value was converted into the partition constant (Kc) on the molarity scale as23
Kc ) K/Vm
(22)
where Vm is the partial molar volume of the micellized surfactant. The basic thermodynamic relationships
∆G°t ) -RT ln Kc
(
)
∂(∆G°t/T) ∂T
( )
P
( ) ( )
) -R
∂ln Kc ∂T
P
(23) )-
∆H°t T2
(24)
∂∆G°t ∂ln Kc ) -RT ) ∆V°t (25) ∂P T ∂P T correlate the temperature and pressure dependence of the
Figure 11. Standard free energy of transfer of pentanol from the aqueous to the micellar phases of dodecyltrimethylammonium bromide as a function of temperature at 0.1 MPa (filled symbols) and 19 MPa (open symbols). Solid line, simulation at 0.1 MPa; broken line, simulation at 19 MPa.
partition constant to the standard partial molar enthalpy (∆H°t) and volume (∆V°t) for PentOH transfer from the aqueous to the micellar phases, respectively. Since Vm increases linearly with temperature, the profiles of Kc Vs t at 0.1 and 19 MPa are similar to those of K Vs t, shown in Figure 10. Therefore, from the plots in Figure 10 and from eq 24, by increasing the temperature, a decrease of ∆H°t is expected. Also, it should change from positive to negative values at about 40-50 °C, where the maximum is present in the K Vs t plot. As far as the pressure effect on the K Vs t profile is concerned, the plots indicate positive ∆V°t values which decrease with increasing temperature, as experimentally observed. From a quantitative point of view, the use of eq 23 to calculate ∆G°t is straightforward since the uncertainty on Kc gives an uncertainty on ∆G°t of the order of 5%. On the contrary, the uncertainty on the derived properties, i.e., ∆H°t and ∆V°t, calculated by means of eqs 24 and 25 is, often, of the same order of magnitude as the property value because temperature and pressure slightly affect ∆G°t. Therefore, as expected, eqs 24 and 25 cannot be used to evaluate more or less accurately ∆H°t and ∆V°t. However, they are useful to predict ∆G°t Vs T and P starting from experimental data at lower temperatures. The comparison between experimental and simulated data is not trivial since it can give information on the reliability of the model used also in the more drastic experimental conditions of the present work. By taking literature data at 25 °C and 0.1 MPa for ∆G°t (-8.9 kJ mol-1),51 ∆H°t (6.4 kJ mol-1),51 ∆V°t (4.9 cm3 mol-1),26 ∆K°t (0.062 cm3 mol-1 MPa-1),36 the standard partial molar heat capacity of transfer ∆Cp°t (-0.346 kJ mol-1),29 and the present values for ∆E°t (0.032 and 0.038 cm3 mol-1 K-1 at 0.1 and 19 MPa, respectively), ∆G°t as a function of P and T was calculated by means of eqs 24 and 25. As can be seen in Figure 11, simulated and experimental data agree very well. So despite the fact that the large change in P and T strongly affects the water and micellar structure, the model used also holds at high pressures and temperatures. Also, it is indirectly confirmed that the fit at 130 °C and 19 MPa is reliable since all three derived properties (V°R,m, V°R,f, and K) agree with those predicted from their plots as functions of temperature. The good agreement between simulated and experimental ∆G°t values as a function of temperature and the negligible dependence of ∆G°t on pressure support the following equations correlating ∆G°t, ∆H°t, and the entropy of transfer (∆S°t) to temperature in the range 15-130 °C and up to 19 MPa:
2268 J. Phys. Chem., Vol. 100, No. 6, 1996
Inglese et al.
Figure 12. Calculated standard free energy, enthalpy, and entropy of transfer of pentanol from the aqueous to the micellar phases of dodecyltrimethylammonium bromide as a function of temperature.
∆G°t ) 109.56 - 2.3687T + 0.346T ln T
(26)
∆H°t ) 109.56 - 0.346T
(27)
T∆S°t ) 2.0227T - 0.346T ln T
(28)
Remember that eqs 26-28 were obtained through experimental data at 25 °C and 0.1 MPa. The calculated ∆H°t and T∆S°t values, together with ∆G°t, as functions of temperature are shown in Figure 12. We can distinguish three regions: up to 45 °C, the transfer of PentOH from the aqueous to the micellar phases is entropy controlled; from 45 to 75 °C, both properties contribute to the negative ∆G°t values; from 75 to 130 °C, the process is enthalpy driven. Acknowledgment. We are grateful to the National Research Council of Italy (CNR) and to the Ministry of University and of Scientific and Technological Research (MURST) for financial support. References and Notes (1) Hamann, S. D. J. Phys. Chem. 1962, 66, 1359. (2) Tuddenham, R. F.; Alexander, A. E. J. Phys. Chem. 1962, 66, 1839. (3) Osugi, J.; Sato, M.; Ifuku, N. ReV. Phys. Chem. Jpn. 1965, 35, 32. (4) Tanaka, M.; Kaneshina, S.; Shin-no, S.; Okajima, T.; Tomida, T. J. Colloid Interface Sci. 1974, 46, 132. (5) Kaneshina, S.; Tanaka, M.; Tomida, T.; Matuura, R. J. Colloid Interface Sci. 1974, 48, 450. (6) Ueno, M.; Nakahara, M.; Osugi, J. ReV. Phys. Chem. Jpn 1977, 47, 25. (7) Rodriguez, S.; Offen, H. J. Phys. Chem. 1977, 81, 47. (8) Brun, T. S.; Høiland, H.; Vikingstad, E. J. Colloid Interface Sci. 1978, 63, 89. (9) Vikingstad, E.; Skauge, A.; Høiland, H. J. Colloid Interface Sci. 1978, 66, 240. (10) Brun, T. S.; Høiland, H.; Vikingstad, E. J. Colloid Interface Sci. 1979, 72, 59. (11) Nishikido, N.; Yoshimura, N.; Tanaka, M. J. Phys. Chem. 1980, 84, 558.
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