1396
J. Chem. Eng. Data 2005, 50, 1396-1403
Ternary Diffusion Coefficients of Glycerol + Acetone + Water by Taylor Dispersion Measurements at 298.15 K Thomas Grossmann and Jochen Winkelmann* Institut fu¨r Physikalische Chemie, Universita¨t Halle - Wittenberg, Geusaer Strasse, D-06217 Merseburg, Germany
The concentration dependence of the mutual diffusion coefficients in the ternary liquid mixture glycerol + acetone + water is determined at 298.15 K by the Taylor dispersion technique along two concentration paths of constant water mole fractions of 0.468 and 0.420, ranging from the binary subsystem toward the phase boundary in the vicinity of the critical solution point. The eigenvalues of Fick’s diffusion coefficient matrix are given, and the influence of the optical properties of the system on the diffusion coefficient determination is discussed. It was found that the determinant |D| continuously declines on approaching the phase boundary.
Introduction Diffusion is an important elementary process of mass transport in liquids and of mass transfer through fluid interfaces (e.g., in living cells and in technical extraction processes). Therefore, it is necessary to know the diffusion coefficients as a function of concentration and especially their behavior when approaching the phase boundary in multicomponent systems with a liquid-liquid phase separation. To investigate diffusion phenomena, different experimental methods are established: diaphragm cell,1-5 conductometric,6,7 and optical8-12 (e.g., Gouy, Rayleigh, and holographic interferometry and also dynamic light scattering for the measurement of mutual diffusion coefficients13-15). As a fast and simple method, the Taylor dispersion technique is well established in the case of binary mixtures of organic compounds and of electrolyte solutions.16-24 With this method, it is also possible to investigate diffusion processes in ternary systems.25-28 The fact, however, that in organic liquid systems one experimentally measurable quantity has to be used to extract two eigenvalues or four elements of Fick’s diffusion coefficient matrix demonstrates the considerably higher complexity of the ternary diffusion problem, and as a consequence, publications of ternary diffusion coefficients are rather rare. The aim of our work was to study the behavior of the diffusion coefficients in a ternary liquid mixture with a miscibility gap depending on the distance from the phase boundary and in the vicinity of the critical solution point. As a model system, we chose mixtures of glycerol + acetone + water. This system was previously investigated by Pertler33 and Rutten.39 In his holographic interferometry measurements, Pertler found that, approaching the phase boundary, the two main elements of the 2 × 2 matrix of Fick’s diffusion coefficients should coincide whereas the offdiagonal elements approach zero. We decided to perform dynamic light scattering (DLS) measurements in the vicinity of the critical solution point,29 and we found * Corresponding author. E-mail: jochen.winkelmann@ chemie.uni-halle.de. Fax: +49 3461 462129. Phone: +49 3461 462090.
different transport modes. To identify the character of these transport processes, we want to compare the DLS results with measurements of the classical Fick’s diffusion matrix. The critical slowing down of the diffusional transport processes is investigated with respect to the dependency of the concentration along two paths of constant water content where water is considered to be the solvent. The two concentration paths were chosen in connection with the DLS experiments. Because of the high scattering intensities near the critical solution point of such a ternary system, overlapping areas exist in which both methodss the Taylor dispersion technique and DLSscan be applied. This will enable us to investigate mass transport phenomena in a crossover region from the homogeneous phase to areas near or close to the phase boundary.29 Because ternary diffusion data from the literature are rather scarce and experimental uncertainties are much higher than in the binary case, we performed systematic diffusion measurements along two different concentration paths from the binary subsystem to the phase boundary. The eigenvalues of Fick’s matrix and the respective determinants were calculated to check the quality and internal consistency of the experimental data. Taylor Dispersion Method When describing mass transport, Fick30 found the following relation for the diffusion process between two components
J ) -D(grad c)
(1)
where J is the molar flux, grad c is the concentration gradient, and D is the diffusion coefficient. In the case of a ternary mixture, this single relation transforms into a system of two coupled mass fluxes, assuming the third component to be the solvent. The Taylor dispersion method is a rapid and simple technique used to determine mutual diffusion coefficients. A small volume of sample solution is injected into the laminar flow of a carrier stream of the same mixture at a slightly different concentration. At constant temperature, the liquid carrier flows with constant velocity through a capillary with an inner radius R. At the end of the capillary, a detector (e.g.,
10.1021/je050082c CCC: $30.25 © 2005 American Chemical Society Published on Web 05/28/2005
Journal of Chemical and Engineering Data, Vol. 50, No. 4, 2005 1397 differential refractometer, UV detector, conductometer, or other suitable flow-through detectors) monitors the change in concentration. The injected square pulse develops into a parabolic velocity profile, and the radial concentration gradient causes radial diffusion that changes the rectangular pulse shape into a Gaussian concentration profile.
[ (Rr ) ] 2
u(r) ) 2u j 1-
(2)
where u is the velocity, u j is the average velocity of the carrier flow, r is the radial coordinate, and R is the radius of the capillary. In the case of binary mixtures, a differential equation results that was solved by Taylor17 with some simplifying assumptions to describe the mass balance
∂2ci ∂ci )K 2 ∂t ∂z
height B3 at tR is given by
2x3h B3 ) γ u j Rxπ tR )
u j 2 R2 K) 48D
(4)
z)x-u jt
(5)
SN(t) )
x ∑[ tR
2
Wi
t i)1 W1 + W2
(
exp -
W1 )
[(
D22 -
t
R2
) (
)
D1
[(
) (
)
2
D2
)
∂t
∑K
k)1
∂2ck ik
with dispersion coefficients
Kii )
Dkk u2R2 48 DiiDkk - DikDki
(7)
2
(8)
Here Dkk denotes the main and Dik denotes the corresponding cross diffusion coefficients. Solving the respective differential equations for the binary and for the ternary case leads to the final working equations that can be used to estimate the diffusion coefficients from the detector signal vs flow-time curve. In the case of a binary mixture, the detector signal S(t) is described by
[
xD 12D (tR - t) S(t) ) B1 + B2t + B3 exp - 2 t R xt
[ [
x x
D1 )
1 D + D22 + (D11 - D22) 2 11
1+
D2 )
1 D + D22 - (D11 - D22) 2 11
1+
]
2
(9)
where S denotes the detector output signal, t is the time, tR is the retention time, and R is the inner radius of the capillary. B1 and B2 are baseline parameters. The peak
]x
D2 (13)
4D12D21
] ]
(D11 - D22)2 4D12D21
(D11 - D22)2
(14)
(15)
and the parameter R1 is given by
R1 )
Dik uR Kik ) 48 DiiDkk - DikDki 2
D1 (12)
where Di represents the eigenvalues of the matrix of the ternary diffusion coefficients
(6)
∂z2
]x
R2 R1 D21 R1 + D11 - D12 (1 - R1) R1 R2
J2 ) -D21(grad c1) - D22(grad c2)
∂ci
(11)
R2 R1 D R + D11 - D12 (1 - R1) R1 21 1 R2
W2 ) - D22 -
where Ji is the molar flux of component i in the volume fixed frame of reference. To obtain the four diffusion coefficients Dik, Price25 solved the corresponding differential equation (eq 6) for a ternary mixture
)]
12Di (t - tR)2
with the Wi as the normalized weights of the two exponential terms. These weights are given by
where D is the mutual diffusion coefficient. In a ternary mixture, the diffusion processes are described by a coupled set of Fick’s equations
J1 ) -D11(grad c1) - D12(grad c2)
(10)
where L is the length of the capillary, h is the length of the injected sample, and γ is the detector sensitivity. In the case of ternary mixtures, we inject a small sample of composition c1 + ∆c1, c2 + ∆c2 into a laminar flow of a carrier with c1, c2. From the corresponding fluxes J1 and J2, there exist two overlapping profiles from where the diffusion coefficients can be extracted. After introducing a normalized peak signal SN(t) according to Leaist,27 we obtain
(3)
with a transformed length coordinate z and the dispersion coefficient K as
L u j
R1∆c1 R1∆c1 + R2∆c2
(16)
To calculate the parameters R1, a linear dependency of the refractive index-concentration change is assumed, supposing small concentration jumps between the sample and carrier composition
∆n ) R1∆c1 + R2∆c2
(17)
The Ri are the concentration derivatives of the refractive index at the carrier composition. They account for the optical properties of the mixture; their ratio contributes substantially to the accuracy of a Taylor measurement in a given system. Finally, the following working equation results
S(t) ) B1 + B2t + B3SN(t)
(18)
where S(t) denotes the detector signal, B1 and B2 are the
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Journal of Chemical and Engineering Data, Vol. 50, No. 4, 2005
Table 1. Diffusion Coefficients and Standard Deviations of Binary Mixtures of Glycerol (1) + Water (3), Acetone (2) + Water (3), and Glycerol (1) + Acetone (2) at 298.15 K glycerol (1) + water (3)
acetone (2) + water (3)
glycerol (1) + acetone (2)
x1
109D/m2 s-1
109σ /m2 s-1
x2
109D/m2 s-1
109σ/m2 s-1
x1
109D/m2 s-1
109σ/m2 s-1
0.0005 0.0007 0.0493 0.0509 0.0986 0.1012 0.1974 0.2018 0.2930 0.2930 0.3046 0.3894 0.3894
0.8684 0.8648 0.6501 0.6499 0.5129 0.4964 0.2981 0.2982 0.1602 0.1602 0.1509 0.0966 0.0959
0.0013 0.0010 0.0008 0.0011 0.0006 0.0007 0.0008 0.0015 0.0009 0.0008 0.0010 0.0004 0.0003
0.0018 0.0018 0.0041 0.0043 0.0045 0.0979 0.1061 0.2916 0.3068 0.4818 0.5161 0.5234 0.5375 0.6695 0.6698 0.7099 0.8547 0.9774
1.2940 1.2384 1.2060 1.2259 1.2132 0.7544 0.7566 0.5956 0.5956 0.6152 0.8943 0.9048 0.9380 1.5651 1.6316 1.7530 3.4159 4.6505
0.0047 0.0024 0.0013 0.0013 0.0010 0.0011 0.0007 0.0008 0.0011 0.0081 0.0021 0.0021 0.0023 0.0035 0.0050 0.0024 0.0182 0.0255
0.0039 0.0107 0.0198 0.0286
2.3726 2.1833 1.9229 1.5458
0.0322 0.0612 0.0245 0.0099
baseline parameters, B3 is the peak height relative to the baseline, and SN is the normalized detector signal as given by eq 11. Experimental Section For our measurements, the acetone (ECD-tested) with a purity of 99.9% and a water content of
