J. Phys. Chem. B 2006, 110, 22207-22212
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Tricritical Phenomena in Quasi-ternary Mixtures of Water + n-Decane + n-Undecane + tert-Butanol† Yongsheng Xiang,‡ Xueqin An,§ Nong Wang,‡ and Weiguo Shen*,‡,| Department of Chemistry, Lanzhou UniVersity, Lanzhou, Gansu, 730000, China, Department of Chemistry, East China UniVersity of Science and Technology, Shanghai, 200237, China, and College of Chemistry and EnVironment Science, Nanjing Normal UniVersity, Nanjing, 210097, China ReceiVed: October 21, 2005; In Final Form: January 22, 2006
Coexistence curves for the quasi-ternary system of water + n-decane + n-undecane + tert-butanol have been determined by measurements of the refractive index in three coexisting liquid phases. The binary mixtures of n-decane + n-undecane constructed the quasi-pure components in which the mass fraction β of n-decane controls the approach to the tricritical point. The coexistence curves can be fitted to Scott’s extended theory and can be extrapolated to a tricritical point at (44.1 ( 0.3) °C and β ) 0.77 ( 0.02 corresponding to an average n-alkane-carbon number of 10.19 ( 0.02. The nonclassical critical amplitude ratio has been confirmed to be 4-5% smaller than the classical value, which is consistent with Fisher’s prediction.
1. Introduction In a multicomponent liquid mixture, a tricritical point is defined as a point at which three coexistence phases become simultaneously identical.1 According to the phase rule, at least three components are required for the existence of an isolated tricritical point in four-dimensional T-P-composition space. Griffiths proposed an asymptotic theory2 to describe the tricritical phase behaviors for systems very close to the tricritical point. Scott developed an extended theory,3 for binary mixtures or quasi-binary mixtures, which accounts for departure from this limiting theory. The tricritical phenomena of three-component mixtures with one being a low molecular mass hydrocarbon solvent and the others being two higher molar mass hydrocarbons, such as methane + 2,2-dimethyl butane + 2,3-dimethylbutane and ethane + n-heptadecane + n-octadecne, have been extensively studied.4-9 The experimental results demonstrated that the two higher molecular mass hydrocarbons in the ternary mixtures could be treated like a single component, and its thermodynamic properties varied continuously by changing the relative amounts of the two higher molecular mass hydrocarbons. Therefore, the extended theory for a binary mixture may be used to analyze the experimental results, and the relative amounts of the two higher molecular mass hydrocarbons controls the approach to the tricritical point. The tricritical phenomena of the ternary mixtures of a bimodal polymer in a solvent have also been studied.10 Because the weight-average molecular mass ratio of a longer chain polymer to a shorter chain polymer provided an extra degree of freedom, a tricritical point might be approached and the tricritical phenomena might be described by the extended classical theory in such systems either at the equilibrium vapor pressure or at constant pressure. †
Part of the special issue “Charles M. Knobler Festschrift”. * Corresponding author. Tel.: +86 21 64250047. Fax: +86 2164252510. E-mail address:
[email protected]. ‡ Lanzhou University. § Nanjing Normal University. | East China University of Science and Technology.
Fisher and Sarbach11 suggested that the tricritical exponents should be essentially classical; however, the amplitude ratios should deviate from the classical values. They estimated that the ratios might be smaller than the classical values by 2-5% and predicted that the nonclassical tricritical behaviors might be observed in more precise measurements of three-phase coexistence equilibria near the tricritical point. Unfortunately, while the classical tricritical exponents have been well confirmed by experiments, the nonclassical amplitude behaviors still remain unobserved because of experimental uncertainties. According to the Bronsted principle of congruence,12 a mixture of two n-alkanes behaves as a quasi-pure n-alkane with the number of carbons being the average of that in the twoalkane mixture. This quasi-pure n-alkane may be mixed with water and tert-butanol to form quaternary solutions (or termed the quasi-ternary solution), and three liquid-phase equilibria may be observed at the equilibrium vapor pressure. Kahlweit13 reported the approach to the tricritical point of the quaternary system of water + n-decane + n-dodecane + tert-butanol by varying the relative amount of the two alkanes or the average carbon number. The closer the carbon number of the two n-alkanes is, the more valid the congruence principle is. Therefore, we proposed a study on the quaternary system of water + n-decane + n-undecane + tert-butanol. Refractive index measurements of coexisting phases are widely used in the study of critical phenomena. The refractive index may be converted to the molar fraction, volume fraction, or mass fraction in binary mixtures14 and bimodal polymer solutions.10 It also may be used as a good order parameter to describe the critical scaling behavior.14 Goh et al.6 pointed out that density might be a good order parameter, and the threephase-equilibrium curves plotted by using mass density as the order parameter were remarkably more symmetrical. The refractive index is related to the mass density. When the mass density or refractive index is used as a good order parameter, the coexistence curve may be simply illustrated in a twodimensional plot of temperature against the order parameter for a quasi-ternary solution. Shen et al.10 found that the total mass fraction of polymers in bimodal polystyrene mixtures was such
10.1021/jp0560415 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/23/2006
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TABLE 1: Mass Fractions of Decane in the Quasi-pure Component, Average Carbon Numbers, and the Overall Compositions of Eight Quasi-ternary Mixtures β
cn
tm (°C)
∆t (°C)
wH2O
wOil
wOH
0.092 0.185 0.281 0.379 0.416 0.477 0.526 0.577
10.91 10.80 10.70 10.60 10.56 10.50 10.45 10.40
53.875 52.540 51.185 49.760 49.269 48.257 47.736 46.920
0.784 0.579 0.432 0.346 0.295 0.219 0.189 0.104
0.351 0.332 0.320 0.319 0.319 0.302 0.320 0.318
0.521 0.528 0.536 0.540 0.550 0.549 0.548 0.555
0.128 0.140 0.145 0.140 0.130 0.150 0.132 0.127
a sufficiently good order parameter that the quasi-binary extended theory may be successfully applied to the study of the tricritical phenomena of these mixtures. In their studies, the total mass fraction of polymers was converted from the directly measured refractive index and the coexistence curves plotted by using refractive index were even more symmetrical. These indicate that the refractive index may be a good order parameter for analyzing the tricritical experimental data with the help of the quasi-binary extended theory. In this paper, we report the refractive indexes in the three coexisting liquid phases of the quasi-ternary system of water + n-decane + n-undecane + tert-butanol and discuss the tricritical phenomena through Scott’s extended theory. 2. Experimental Section 2.1. Materials. Decane (>99%) and undecane (>99%) were obtained from Merck-schucardt Co. and dried and stored over a 0.4-nm molecular sieve. tert-Butanol (99%) was supplied by Tianjin Chemical Co. The water was distillated and deionized, and its electrolytic conductivity was about 1.8 × 10-7 s‚cm-1. 2.2. Preparation of Mixtures. The quasi-pure components were prepared by mixing the proper amounts of decane and undecane in flasks to form the binary solutions with the desired mass fractions, β, of decane. The quasi-ternary mixtures were then prepared by weighing the quasi-pure component, water, and tert-butanol in glass tubes of 10-mm i.d. provided with Acethread connections, which allowed them to be sealed with Telflon caps. The three-phase equilibria may be illustrated by an “S”-type curve in a plot of the temperature against a density variable. However, only for a sample with a proper overall
Figure 1. Relative height of the two interfaces vs θ for the mixture with β ) 0.526.
Figure 2. Coexistence curves of temperature vs refractive index for quasi-ternary systems with various values of β.
TABLE 2: Refractive Indexes of Three Phases in Quasi-ternary Mixtures at Midpoint and Critical-End-Point Temperatures ∆t (°C)
nmR
nmβ
nmγ
nUc
nUe
nLc
nLe
0.104 0.189 0.219 0.295 0.346 0.432 0.579 0.784
1.3736 1.3742 1.3743 1.3744 1.3745 1.3744 1.3746 1.3749
1.3692 1.369 1.3688 1.3683 1.3681 1.3675 1.3669 1.3665
1.3648 1.3637 1.3629 1.362 1.3612 1.3599 1.3583 1.357
1.3722 1.3722 1.3722 1.3721 1.3720 1.3718 1.37165 1.37165
1.3640 1.3624 1.3619 1.3607 1.3599 1.3586 1.3569 1.3554
1.3671 1.366 1.3657 1.3649 1.3644 1.3635 1.36245 1.3613
1.3746 1.3753 1.3755 1.3758 1.3760 1.3761 1.3764 1.3768
composition does an entire S-type curve appear, thereby allowing the refractive indexes of the three coexisting phases to be measured in the whole temperature range from the upper critical end point (UCEP) to the lower critical point (LCEP). We carefully searched for the proper composition using the method described by Shen et al.10 The accuracy in determinations of β and the compositions of the quasi-ternary mixture was better than 0.001. 2.3. Measurement of Refractive Indexes. The sample with the proper composition was prepared in a 10-mm path-length rectangular fluorimeter cell provided with an Ace-thread connection and sealed by a Teflon cap. No evaporation from the cell could be detected over the period of measurements of the whole coexisting curve. The refractive indexes n of the three coexisting phases in the rectangular cell were measured for various temperatures t by the method of minimum deviation. The accuracy in measurement of the refractive index was (0.0001. The apparatus and the experimental procedure for measurement of the refractive index have been described previously.10,14 After each change of the temperature, we waited for 1 h for thermal equilibrium, and then, the sample was mixed by end-over-end rotation of the cell for about 5 min in the water
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Figure 3. Plot of average carbon number cn vs (∆t)2/3. The line is the least-squares fit.
Figure 5. Plot of mass fraction β of decane in the quasi-pure component vs (∆t)2/3. The line is the least-squares fit.
Figure 4. Plot of midpoint temperature tm vs (∆t)2/3. The line is the least-squares fit.
Figure 6. Plot of ln(nUc - nLc) vs ln(∆t).
bath. The cell was set in the bath for an additional period of time for phase equilibrium to be reached after rotation. The time needed for phase equilibrium to be reached was determined by continuous measurements of the refractive indexes of coexisting phases up to 36 h. It was found that after 1 h the refractive indexes did not change with time at most temperatures. However, when the temperature was quite close to the critical point, it was necessary to wait for an additional 3 h for phase equilibrium. The temperature was measured with a Pt resistance thermometer connected to a multimeter/data acqustion system (Keithley model 2700). The accuracy and precision in the temperature measurement were about (0.02 K and (0.001 K, respectively. The temperatures of UCEP and LCEP were determined with a precision of (0.002 K. It was observed that mixtures nominally of the same composition had different values of upper and lower critical temperatures, differing by as much as 0.1 K. This may be attributed to the uncontrollable impurities introduced in the preparation of the samples. However, the difference ∆t ) tU - tL in temperatures between UCEP and LCEP, which describes the size of the three-phase region, was reproducible within (0.004 K.
3. Results and Discussions The mass fractions, β, of decane and the corresponding average carbon numbers, cn (cn ) x10cn10 + x11cn11, where x10 and x11 are mole fractions and cn10 and cn11 are the carbon numbers of decane and undecane), in the quasi-pure components used to prepare the quasi-ternary solutions are listed in columns 1 and 2 of Table 1. Table 1 also lists the overall compositions of the systems, where wH2O, wOH, and wOil are the mass fractions of water, tert-butanol, and quasi-pure oil in the quasi-ternary solutions, respectively. With these compositions, the whole three-phase region between the upper critical points and the lower critical points could be observed. The height fractions for each of the three phases, which approximately equal the volume fractions of the phases, were found to be approximate to 1/3 and insensitive to the relative temperature θ for those quasi-ternary mixtures with the overall compositions listed in Table 1, where θ is defined by (2t - tU - tL)/(tU - tL) and tU and tL are the UCEP and LCEP temperatures. A typical variation of the height fractions with θ is shown in Figure 1, where R, β, and γ represent the upper, middle, and lower phases, respectively.
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TABLE 3: Parameters of Equations 5, 6, 9, and 10 exponent of ∆t 0.63 ( 0.1 2/3a 2/3a a
ν1/2 (K)-1/3
nt 1.3708 ( 0.0001 1.3706 ( 0.0001
νb01 (K)-2/3
νb20 (K)-2/3
(-5.1 ( 0.2) × 10 (-5.0 ( 0.4) × 10-3 -3
(5.35 (0.06) × 10-3
(-1.4 ( 0.4) × 10 (-1.4 ( 0.4) × 10-4
ν3/2b11 (K)-1
ν3/2b30 (K)-1
(4 ( 13) × 10-5
(1.4 ( 0.3) × 10-4
-4
Value fixed at theoretic prediction.
The mass fraction, β, of decane in the quasi-pure component is also a choice for the measure of the distance from the tricritical point, which was found to be proportional to (∆t)2/3, as shown in Figure 5. A least-squares fitting gives a value of 0.77 ( 0.02 for β. The extended theory gives a general expression for the coexistence curve of the refractive index against the temperature
n ) nt + z0ν1/2(∆ζ)1/2 + (b01 + b20z02)ν(∆ζ) + (b11z0 + b30z03)ν3/2(∆ζ)3/2 + O[ν2(∆ζ)2] (3) where nt is the tricritical refractive index of the quasi-ternary mixture, ν and bij are the constants, and z0 is a solution of the reduced asymptotic equation
z03 - 3z0 - 2θ ) 0
(4)
When (∆t)2/3 is chosen as the measure of the distance from the tricritical point, i.e., ∆ζ ) (∆t)2/3, using eq 3 for the different combinations of seven characteristic points on the coexistence curves yields
Figure 7. Plot of ln(nLe - nUe) vs ln(∆t).
The refractive indexes of three coexisting phases at various temperatures were measured for eight quasi-ternary mixtures with different values of β. Figure 2 shows the (t, n) three-liquidphase curves for the eight mixtures. According to Scott’s theory,11 the tricritical behaviors can be quantified in terms of seven characteristic points on a coexistence curve: the refractive index nUc and nUe of the critical and noncritical phases at tU, the corresponding values of nLc and nLe at tL, and the values of nmR, nmβ, and nmγ for the upper, middle, and lower phases at the midpoint temperature tm ) (tU + tL)/2. These characteristic points are indicated on one of the coexistence curves in Figure 2. The values of the refractive indexes at tm were directly determined from the coexisting curves, while the values of nUc, nLc, nUe, and nLe were obtained by extrapolations of the diameters (nR + nβ)/2 and (nβ + nγ)/2 and the coexistence curves. Table 2 lists the values of the seven characteristic points for eight quasi-ternary mixtures. The extended theory11 predicts that ∆t and tm can be expressed as a power series:
∆t ) 2j2(∆ζ)3/2 + 2j4(∆ζ)5/2 + O(∆ζ)7/2
(1)
tm ) tt + j1(∆ζ) + j3(∆ζ)2 + O(∆ζ)3
(2)
where tt is the tricritical temperature, ji is the coefficient, and ∆ζ measures the distance from the tricritical point; for these systems, an appropriate choice for ∆ζ is ∆cn ) cn - cnt, where cnt is the average carbon number at the tricritical temperature. A plot of (∆t)2/3 against cn in Figure 3 yields a straight line and gives the tricritical value of cnt ) 10.19 ( 0.02, which is close to 10.27 reported by Kahlweit.13 Because ∆t was directly measured and more accurate than ∆cn, we prefer to use (∆t)2/3 instead of ∆cn as the measure of the distance from the tricritical point. Figure 4 shows a plot of tm against (∆t)2/3, which yields a straight line and gives the tricritical temperatrue tt of 44.1 ( 0.3 °C, in good agreement with 44.9 °C reported by Kahlweit.13
S1 ) (nLc + nUc)/2 ) nt + (b01 + b20)ν(∆t)2/3 + O[ν2(∆t)4/3] (5) S2 ) (nLe + nUe)/2 ) nt + (b01 + 4b20)ν(∆t)2/3 + O[ν2(∆t)4/3] (6) S3 ) (nUc - nLc)/[4(∆t)2/3]1/2 ) ν1/2 + (b11 + b30)ν3/2(∆t)2/3 + O[ν5/2(∆t)4/3] (7) S4 ) (nLe - nUe)/[16(∆t)2/3]1/2 ) ν1/2 + (b11 + 4b30)ν3/2(∆t)2/3 + O[ν5/2(∆t)4/3] (8) S5 ) nmβ ) nt + b01ν(∆t)2/3 + O[ν2(∆t)4/3]
(9)
S6 ) (nmβ + nmγ)/2 ) nt + (b01 + 3b20)ν(∆t)2/3 + O[ν2(∆t)4/3] (10) S7 ) (nmR - nmγ)/[12(∆t)2/3]1/2 ) ν1/2 + (b11 + 3b30)ν3/2(∆t)2/3 + Oν5/2[(∆t)4/3] (11) To test the reliability of the classical theory, the variations of nUc - nLc, nLe - nUe, and nmR - nmγ with (∆t) are log-log plotted in Figures 6-8. The least-squares fits give the exponents of 0.35 ( 0.01, 0.35 ( 0.01, and 0.36 ( 0.01, respectively. These values are slightly lager than the theoretical value of 1/3, which may be attributed to ignoring the higher order corrections. The values of S1, S2, S5, and S6 were fitted to eqs 5, 6, 9, and 10 simultaneously by the least-squares method with the exponent of (∆t) being freely optimized. The values of the parameters from the fitting are listed in row 1 of Table 3. The optimized value of the exponent is 0.63 ( 0.1, which is well consistent
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J. Phys. Chem. B, Vol. 110, No. 44, 2006 22211
Figure 8. Plot of ln(nmR - nmγ) vs ln(∆t). Figure 10. Plots of S3, S4, and S7 vs (∆t)2/3: (O) S3; (4) S4; (0) S7. The lines are individual least-squares fits.
Figure 9. Plots of S1, S2, S5, and S6 vs (∆t)2/3: (O) S1; (4) S2; (0) S5; (]) S6. The lines are constructed by using the parameters listed in row 2 of Table 3, representing S5, S1, S6, and S2 from the top down.
with the theoretical value of 2/3. Therefore, the correctness of the classical exponents is conformed. With the exponent of (∆t) being fixed at the classical value of 2/3, eqs 5, 6, 9, and 10 were used to fit the values of S1, S2, S5, and S6 again. The results are listed in row 2 of Table 3. Figure 9 shows the plots of S1, S2, S5, and S6 against (∆t)2/3, where the lines represent values calculated by using eqs 5, 6, 9, and 10 and the parameters listed in row 2 of Table 3. A similar procedure was used for sets S3, S4, and S7 to obtain the values of ν1/2, ν3/2b11, and ν3/2 b30, which are listed in row 3 of Table 3. In the simultaneous least-squares fits, the values of ν1/2 were forced to be the same for S3, S4, and S7. However, the plots of S3, S4, and S7 against (∆t)2/3 in Figure 10 clearly demonstrate that the limiting value of ν1/2 for S4 is significantly larger than that for S3 and S7, while the limiting values of ν1/2 for S3 and S7 are possibly the same within the experimental uncertainties. Therefore, the classical theory fails to predict the tricritical amplitudes. As has been proposed by Goh et al.,8 the limiting values of the ratios (S3/S4) and (S7/S4) may be used to examine the extent of nonclassical tricritical behavior. It has been estimated that the values of these amplitude ratios might be smaller than the
Figure 11. Plots of S3/S4 and S7/S4 vs (∆t)2/3: (O) S3/S4; (4) S7/S4. The lines are individual least-squares fits.
classical ones by 2-5%.13 Individual least-squares fits for the linear relations between Si (i ) 3, 4, 7) and (∆t)2/3 give the values of ν1/2 as (5.29 ( 0.04) × 10-3 (K)-1/3, (5.53 ( 0.04) × 10-3 (K)-1/3, and (5.25 ( 0.06) × 10-3 (K)-1/3 for S3, S4, and S7, respectively. The results from the fits are shown as the lines in Figure 10. It yields the limiting values of 0.96 ( 0.01 and 0.95 ( 0.01 for (S3/S4) and (S7/S4), which are smaller than 1 by 4-5% and are consistent with Fisher’s theoretical prediction.11 Another way to obtain the limiting values of the two ratios is to plot (S3/S4) and (S7/S4) against (∆t)2/3 and to extrapolate the straight lines to the tricritical point, as shown in Figure 11. The limiting values from the least-squares fits are 0.96 ( 0.01 and 0.95 ( 0.01 for (S3/S4) and (S7/S4), exactly the same as reported above. Therefore, we conclude that nonclassical tricritical behavior is confirmed. Acknowledgment. This work was supported by the National Natural Science Foundation of China (Projects 20173024, 20273032, and 20473035) and the Key Project of the Chinese Ministry of Education (No. 105074).
22212 J. Phys. Chem. B, Vol. 110, No. 44, 2006 References and Notes (1) Widom, B. J. Phys. Chem. 1973, 77, 2196. (2) Griffiths, R. B. J. Chem. Phys. 1974, 60, 195. (3) Scott, R. L. J. Chem. Phys. 1987, 86, 4106. (4) Creek, J. L.; Knobler, C. M.; Scott, R. L. J. Chem. Phys. 1981, 74, 3489. (5) Specovius, J.; Leiva, M.; Scott, R. L.; Knobler, C. M. J. Phys. Chem. 1981, 85, 2313. (6) Goh, M. C.; Specovius, J.; Scott, R. L.; Knobler, C. M. J. Chem. Phys. 1987, 86, 4120. (7) Ferandez-Fassanacht, E.; Williamson, A. G.; Sivaraman, A.; Scott, R. L.; Knobler, C. M. J. Chem. Phys. 1987, 86, 4133.
Xiang et al. (8) Goh, M. C.; Scott, R. L.; Knobler, C. M. J. Chem. Phys. 1988, 89, 2281. (9) Kumar, A.; Cannell, D. S.; Scott, R. L.; Knobler, C. M. J. Chem. Phys. 1988, 89, 3760. (10) Shen, W.; Gareth, R. S.; Knobler, C. M.; Scott, R. L. J. Phys. Chem. 1990, 94, 7943. (11) Fisher, M. E.; Sarbach, S. Phys. ReV. Lett. 1978. 41, 1127. (12) Bronsted, J. N.; Koefoed, K. D. Vidensk. Selsk. Mater. Fys. Skr. 1946, 22, No.17. (13) Kahlweit, M.; Strey, R.; Aratono, M.; Busse, G.; Jen, J.; Schubert, K. V. J. Chem. Phys. 1991, 95, 2842. (14) An, X.; Shen, W.; Wang, H.; Zheng, G. J. Chem. Thermodyn. 1993, 25, 1373.
