Article pubs.acs.org/JPCB
Theory of Soret Coefficients in Binary Organic Solvents Semen Semenov*,† and Martin Schimpf‡ †
Institute of Biochemical Physics RAS, Kosygin St. 4, 119334 Moscow, Russia Department of Chemistry, Boise State University, Boise, Idaho 83725, United States
‡
ABSTRACT: Thermodiffusion in binary molecular liquids is examined using the nonequilibrium thermodynamic model, where the thermodynamic parameters are calculated using equations based on statistical mechanics. In this approach, thermodiffusion is quantified through the variation in binary chemical potential and its temperature and concentration dependence. The model is applied to solutions of organic solvents, in order to compare our theoretical results to experimental results from the literature. A measurable contribution of the orientation-dependent Keezom interaction is shown, where the possible orientations are averaged using the Boltzmann weighting factor. Calculations of enthalpies of evaporation from the model yield good agreement with experimental values from the literature. However, calculations of the associated energetic parameters were several times larger than those reported in the literature from numeric simulations of material transport.
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BACKGROUND Recent attempts to explain thermodiffusion (the Soret effect) in organic liquid mixtures using models based on nonequilibrium thermodynamics and equilibrium statistical thermodynamics have been a topic of discussion in the literature. Such models are based on the temperature and concentration dependence of the chemical potentials of the components μi, as outlined below. The material flux Ji⃗ in a nonhomogeneous and nonisothermal mixture can be defined as1,2 Ji ⃗ = −niLi∇
μi T
− niLiQ ∇
1 T
δμ* =
v2 μ v1 1
δμ* =
(1)
v2 osm δP ϕ
(4)
The same general approach is used by Morozov.6 The issue related to any approach that expresses parameters of mass and thermodiffusion through pressure becomes obvious from the statistical−mechanical expressions for pressure and chemical potential:7,8 P = −kT
(2)
where vi are the specific molecular volumes of the respective components. For ease of reading, we will subsequently refer to component 1 as the “solvent” and component 2 as the “solute”, although the model applies to the entire concentration range. The last term in eq 2 is the free energy of the solvent molecules displaced from the volume occupied by the solute. We refer to this parameter as the chemical potential of a virtual particle consisting of the solvent but having the same shape and dimensions as the solute. The final expressions2−5 use the Gibbs−Duhem equation to express the binary chemical potential through the (nonuniform) excess pressure Posm as follows: © 2014 American Chemical Society
(3)
Here δ indicates the difference in the parameter between the hot and cold reservoirs, μs and vs are the chemical potential and specific molecular volume of the solvent, respectively, P is the total pressure, and ϕ is the volume fraction of the solute. The parameter δμs/νs is interpreted as the difference in the puresolvent partial pressure between hot and cold reservoirs, and eq 3 is rewritten as2−5
where ni are the numeric volume concentrations of the components, Li and LiQ are the Onsager coefficients, and T is the temperature. The material transport parameters are expressed through the temperature and concentration dependence of the binary chemical potential:2−5 μ* = μ2 −
δμ ⎞ v2 ⎛ ⎜δP − s ⎟ ϕ⎝ vs ⎠
∂ ln Z ∂V
(5)
∂ ln Z ∂Ni
(6)
μi = −kT
Here V is the volume of the system, Ni is the number of particles of type i, and Z is the partition function. Statistical mechanical models typically begin with the chemical potential of a component as an ideal gas:7,8 Received: October 28, 2013 Revised: February 17, 2014 Published: February 18, 2014 3115
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⎡ 2πmikT ⎤ 3 kT ln⎢ ⎥ ⎣ h2 ⎦ 2
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J1⃗ = J2⃗ = 0 (7)
Note that eq 9 is valid only for a closed stationary system. In a subsequent article,14 Firoozabady and co-workers use the Gibbs−Duhem equation to calculate the nonzero pressure gradient. The resulting expression for the Soret coefficient is used by Hartmann and co-workers15 to parametrize their empirical data on the thermodiffusion of several pairs of organic liquids. The parameter commonly used for characterizing the steadystate concentration distribution in nonisothermal systems is the Soret coefficient:
Here mi is the molecular mass of the respective component, k is the Boltzmann constant, and h is Planck’s constant. Equation 7 applies to spherically symmetric particles and ignores the contribution of the internal degrees of freedom, like rotation. We note that the classical terms in the partition function are related to kinetic energy, which is a function of mass and its distribution through the particle body, not volume. Thus, the terms based on eq 7 which are not functions of volume are omitted from eq 4. In another approach,9 the mass effects are also ignored, leading to final expressions that ignore the isotope effects described by the classical partition function, thus considering only the much smaller quantum term of the partition function. The use of chemical potentials is preferable in statistical mechanical models because the basic expression for material flux Ji⃗ (eq 1), which can be defined by the nonequilibrium thermodynamics expression for entropy production, contains no excess pressure. Furthermore, the terms related to kinetic energy contained in the chemical potential of an ideal gas play a notable role in molecular thermodiffusion.10 Another group of authors based their research in thermodiffusion on the assumption of constant pressure, even in nonhomogeneous and nonisothermal systems. This assumption is rooted in classical monographs,1,2 where it is formulated axiomatically. By assuming constant pressure, we lose the important physical parameter and mathematical variable necessary to characterize the system. This issue is closely related to the fact that the material fluxes of the components determined by eq 1 should also satisfy the condition of material conservation: J1⃗ + J2⃗ = 0
(9)
ST =
Q2 − Q1 ∇ϕ = ϕ(1 − ϕ)∇T Tϕ(∂μ2 /∂ϕ)P , T
(10)
where Qi is the so-called heat of transport. There are different definitions in the literature for the heat of transport, but the most common definition is
Qi =
LiQ Li
(11)
Using experimental data for equimolar mixtures, authors that assume constant pressure15 concluded that the heat of transport is defined by that of a pure component, i.e., independent of the concentration of the binary mixture. They refer to the respective property of the pure component as the thermophobicity. According to eq 10, the concentration dependence of the Soret coefficient is determined by the concentration dependence of its chemical potential. For an ideal or quasi-ideal solution, μ2(ϕ) ≈ kT ln ϕ and the Soret coefficient should, therefore, be constant across the concentration range, which we know from experimental data to be incorrect. Thus, the assumption of constant pressure in nonisothermal systems is erroneous. We note that the assumption of constant pressure has been discussed in the literature for some time. Piazza17 indicates Wan’t Hoff18 as the first researcher whose approach using the assumption of constant pressure failed in his description of thermodiffusion. In calculations of the Soret coefficient, we follow an approach outlined previously,19 where a solution to the problem of inconsistency in the mass transport equations was proposed. The consistency was shown to be restored by putting the following restriction on the Onsager coefficients:
(8)
In the classical mongraphs,1,2 the gradient expression for material flux (eq 1) is written for only one component. The flux of the other component is taken from eq 8, assuming the gradient equation for the second component is fulfilled automatically. However, for modeling nondilute mixtures, e.g., binary mixtures across the concentration range, this practice leads to a result that depends on which of the two components is selected for calculating its flux from eq 1. In order for the approach to work across the entire concentration range, eq 1 must be written for both components, as well as satisfying conservation of mass (eq 8). However, that results in three equations (two eq 1 and eq 8) for two component concentrations, which are not necessarily self-consistent for any value of the kinetic coefficients. Consequently, the system of material transport equations based on nonequilibrium thermodynamics is valid only when certain restrictions are placed on the kinetic coefficients in the expressions for material flux. This issue has been addressed by introducing a drift velocity,11 which is common for components in a system at constant pressure. However, such an approach leads to the unacceptable assumption that the system will move as a whole in approaching steady state.12 In the pioneering work of Firoozabady and co-workers,13 restrictions are placed on the chemical potentials of the components, and the condition of mass conservation expressed by eq 8 is replaced by a softer condition:
Q i = −μi
(12)
The resulting expression for the Soret coefficient is ∂μP*
ST =
∂T ∂μ *
2ϕ(1 − ϕ) ∂ϕP
(13)
where the subscript P refers to constant pressure. Equation 13 is practically identical to the respective expression for the Soret coefficient obtained by others who replaced the osmotic pressure with the binary chemical potential using eq 4.3−6 The only difference is a factor of 2 in the denominator of eq 13. This factor arises from the incompressibility of the liquid, as expressed by n1v1 + n2v2 = 1
(14)
The factor 2, which was not accounted for in the previous work,3−6 is obtained under the standard rule of differentiation 3116
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of composite functions in calculating the gradient of chemical potential: 2
∑
∂μk [n2 , n1(n2)] ∂nl
l=1
=
∂μk
Since the pair correlation functions in liquids should approach unity at infinite distance, we will use that approximation:
∇nl
gij(r , λ) = 1
This assumption means that the local distribution of the solvent molecules is not disturbed by the considered particle. The approximation of constant local density is used in the study of liquids as regular solutions.24 Combining eqs 2, 7, 16, 17, 19, and 20 and accepting the standard form of the van der Waals intermolecular interaction defined by25
∂μk ∂n1 ∇n 2 ∂n1 ∂n2
∇n 2 +
∂n2 ∂μ = 2 k ∇ϕ ∂ϕ
(15)
The statistical mechanical expression for chemical potential at constant volume is defined as21 μiV = μ0i +
∫0
1
∑
dλ
j = 1,2
ϕj vj
∫V
Φij = −εij(σij/r )6
i out
(16)
where μ0i is the chemical potential of the ideal gas represented by eq 7 and parameter λ describes the gradual “switching on” of the intermolecular interaction. A detailed description of this representation can be found elsewhere.20,21 The parameter r is the distance between a molecule of the surrounding liquid and the center of the considered molecule; gij(r,⃗ λ) is the pair correlation function, which expresses the probability of finding a molecule of the surrounding liquid at r(⃗ r = |r|⃗ ) if the considered molecule is placed at r = 0; and Φij is the molecular interaction potential. Equation 16 is the obvious and exact generalization of the respective expressions for one-component systems, as outlined previously.10 The chemical potential at constant pressure can be related to the chemical potential at constant volume by ∇μip = ∇μiV +
∫V
i out
∇Πi dv
μP* =
vij =
j = 1,2
r
∫∞
2Φij(r′) r′
⎤ dr′⎥ ⎥⎦
(23)
vij = (1 − lij)(vii1/3 + vjj1/3)3
(24)
εij = (1 − kij) εiiεjj
(25)
Here 1 − lij is the parameter reflecting nonsphericity of the molecule and kij is a fitting parameter known as the coupling parameter. Equation 24 approximates the van der Waals volume for two different molecular particles of a complex nonspherical shape. The minimal value of the nonsphericity parameter is 1 − lij = 1/8, which corresponds to spherical molecules. Equations 24 and 25 allow the binary chemical potential to be expressed through the molecular enthalpy of vaporization, which can be obtained from the literature. Using eq 20 and the approximation of a regular solution (eq 20), the reduced molecular enthalpy of vaporization in kT units can be written as ∞ vjjεii 1 hi = Φii(r ) dv = i vkT vjkT Vout i (26)
(18)
The local pressure gradient is calculated using the condition for establishing local equilibrium in a thin spherical layer of thickness l and area S, concentric with the considered molecule.10,23 This condition expresses the local conservation of specific free energy F(r) = Πi(r) + ∑j=1.2 ϕjΦij(r)/vj in the isothermal system when the layer is shifted from position r to r + dr while maintaining spherical symmetry. The change in free energy is due to both changes in parameters within the layer and changes in the area dS and volume of the layer. The changes in volume and surface area are related by dV = 2r dS, and we obtain a modified equation for equilibrium in the closed spherical surface, leading to a definition of the pressure gradient that is related to the change in surface area. That definition has the same nature as the surface (Laplace) pressure gradient discussed elsewhere:2,7 ϕj ⎡ ⎢Φij(r ) − vj ⎢⎣
4π 3 σij 3
is the respective van der Waals volume and ρi is the component density. We note that the use of eq 21 for the van der Waals interaction potential assumes spherically symmetric molecules, which is an oversimplification in many cases. The ratio ρ1/ρ2 approximates the mass ratio of a virtual particle to that of the solute, assuming they occupy the same volume. In the integration over Viout in eqs 16 and 17, the lower limit is r = σij. There is the standard way to express parameter vij through the van der Waals molecular volume vii and expressing ε12 through the energetic parameters ε11 and ε22 of identical particles:26
Here Viout is the volume external to the molecule of the ith component and Πi is the local pressure distribution around this molecule. Equation 17 expresses the relation between the forces acting on a molecular particle at constant versus variable local pressure. This equation is the obvious generalization of the equation that connects the gradients of chemical potentials at constant volume and pressure:22
Πi = − ∑
⎞ v12 v ⎛ v (ε11 − ε12)(1 − ϕ) + 12 ⎜ε12 − 22 ε22⎟ϕ 3v1 3v2 ⎝ v12 ⎠ ρ ϕ 3 + kT ln + kT ln 1 1−ϕ 2 ρ2 (22)
Here
(17)
∇μiP = ∇μiV + vi∇P
(21)
where εij is the energy of interaction and σij is the minimal molecular approach distance, we obtain the following binary chemical potential at constant pressure:
∞
gij( r ⃗ , λ)Φij(r ) dv
(20)
∫
Below, we use eq 26 to compare thermodynamic parameters obtained from the literature with those obtained from thermodiffusion data. In calculating the temperature derivative of the binary chemical potential, the temperature dependence of the specific
(19) 3117
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Table 1. Physical Parameters of the Solvents Examined molar volume Vm (cm3)
reduced molecular enthalpy hi in kT units
cyclohexane
106 (ref 29)
acetone
75 (ref 29)
tetralin isobutylbenzene
136 (ref 34) 158 (ref 36)
11.7 (ref 34) 11.7−12.9 (ref 32) 11.3 (ref 34) 11.0−11.9 (ref 32) 16.7 ± 0.3 (ref 34) 15.4 ± 0.3 (ref 34)
solvent
∂T
=−
We note that the Keezom interaction can be utilized in a wider sense beyond the angular dependence of the intermolecular interaction between hard dipoles. Thus, eq 27 is also valid for molecules of complex shape with strong anisotropy in the dielectric polarizability. Considering the atoms constituting a molecule as separate particles that interact with atoms constituting other molecules, we will also obtain the orientation-dependent potential of intermolecular interactions. This orientation-dependent potential can be averaged over the possible orientations using the approach by Keezom, which is based on the interaction of hard dipoles. In this case, eq 27 will also be valid for nonpolar molecules with no hard dipole moments. Using eqs 13, 22, and 27, the Soret coefficient can be written as 1 ST = 2
h2*K ϕ T
+
ϕ(1 − ϕ)(h1* + h2*) + 3
9 2T
ln
(28)
v12ε12 − v22ε22 v2kT
(30)
and h2* =
are defined as the binary reduced enthalpy of the first and second components, respectively, expressed in units of kT. The parameter
h2*K =
v22 ε22K v2 kT
0.0014 (refs 30 and 31)
0.79 (ref 29)
58 (ref 29)
0.0008 (ref 35) 0.0095 (ref 36)
0.97 (ref 34) 0.85 (ref 36)
132 (ref 34) 134 (ref 36)
(32)
ρ⎤ h *K 1⎡ * 9 ⎢α1h1 (1 − ϕ) − α2h2*ϕ + 2 ϕ + ln 1 ⎥ 6 ⎢⎣ T 2T ρ2 ⎥⎦ (33)
ρ1
(29)
84 (ref 29)
Equation 33 describes a linear concentration dependence of the Soret coefficient in ideal and quasi-ideal solutions, confirming the empirical observation of a near-linear concentration dependence of the Soret coefficient in certain nonpolar organic solvent pairs.16 However, the dependence was strongly nonlinear for solvent pairs having vastly different properties, such as cyclohexane and acetone. Additional examples are discussed below. We note that eq 33, as well as the empirical data, contradicts the theory subsequently developed by Hartmann et al.,15 which predicts a constant Soret coefficient across the concentration range in such systems. Note that in expressing the terms of the binary chemical potential given by eq 2 through the binary reduced enthalpies of the first and second component given by eqs 29 and 30 we introduce no new physical concept. The expressions are derived only to estimate the reduced molar enthalpies from thermodiffusion parameters, in order to compare the resulting values with literature data obtained by other methods to make some useful general conclusions. For example, eqs 29 and 30 indicate that, for miscible solvents that do not have a large difference in physicochemical and geometric parameters, i.e., ε11 ≈ ε22; v12/v1 ≈ v12/v2 ≈ v22/v2, where the energetic parameter ε12 by definition lies within the narrow interval between ε11 and ε22, the binary reduced enthalpies of the first and second components are related by the approximate relationship h2* ≈ −h1* . This result agrees with data summarized in Table 2, lending support to the proposed theory. In eq 28, we have ignored the concentration dependence of the specific molecular volumes. This is a reasonable approximation for organic solvents where the maximum change in the specific molecular volume across the concentration range
ρ2
v12(ε12 − ε11) v1kT
0.78 (ref 33)
ST =
where
h1* =
0.0011 (ref 32)
The parameter h*1 + h*2 indicates the difference in physical properties of the components. When |h1* + h2*| ≪ 12, the mixture can be considered as an ideal or quasi-ideal solution. The expression for the Soret coefficient in such binary mixtures can be written as
(27)
α1h1*(1 − ϕ) − α2h2*ϕ +
molar mass M (g)
⎛ v (ε − ε12) v ε − v12ε12 ⎞ Tc = ⎜ 12 11 + 22 22 ⎟ 12kv1 12kv2 ⎠ ⎝
εijK T
density ρ (g/cm3)
examine the experimental data of Wittko and Köhler,16 where one of the solvents is always the highly symmetric nonpolar molecule cyclohexane. An analysis of eq 28 without the Keezom term was made previously.10 The denominator in eq 28 can become zero at a temperature Tc when ϕ = 1/2 and h1* + h2* < 0, where Tc is defined as
molecular volume was the only factor taken into account in previous works.10,12,23 However, the interaction potential may also contain a temperature-dependent component involving the so-called Keezom or orientation dipole interaction parameter εijK(T).25 This component of the interaction potential is obtained by weight-averaging Φij(r)⃗ over all orientations between the hard dipoles using a Boltzmann factor. The specific expression for the orientation dipole−dipole interaction potential can be found elsewhere.25 For our calculations, the temperature derivative of the Keezom potential can be written as
∂εijK (T )
thermal expansion coefficient αT (K−1)
(31)
is the reduced Keezom enthalpy of the second component in kT units. In eq 28, we assume that only the second component has a polarity or anisotropy that contributes to the respective Keezom term. In the experiments, we will use the model to 3118
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is less than a couple of percent. However, in strongly polar liquids such as water and alcohols, this concentration dependence cannot be ignored and such systems require a separate study. Preliminary evaluations indicate that the Soret coefficient can still be calculated when the concentration dependence of the molecular volume is known and substituted directly into the equations. However, calculation of the dynamic parameters becomes a much more complex problem. Finally, we note that calculation of Soret coefficients in electrolytes was outlined in previous work.27,28
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RESULTS In this section, we compare the theoretical results based on eqs 28−31 with empirical data. Using eqs 29 and 30, we could calculate parameters h*1 and h*2 for substituting into eq 28, relying on the literature for values of the remaining parameters. However, there is significant variance in enthalpies of vaporization reported in the literature (see Table 1). Together with the fitting parameters in eqs 24 and 25, the wide variance in enthalpies of vaporization would allow nearly any values of h1* and h2* to be obtained, including different signs in those values. Consequently, we formulate and solve the inverse problem; that is, we obtain the experimental values of h*1 and h*2 and use them to calculate values of the enthalpies of vaporization for the components. The concentration dependences of the Soret coefficient calculated using eq 28 for three binary systems are illustrated in Figures 1−3. Calculated values
Figure 2. Concentration dependence of the Soret coefficient for the mixture cyclohexane−tetralin. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28.
Figure 3. Concentration dependence of the Soret coefficient for the mixture cyclohexane−isobutylbenzene. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28.
Equation 34 allows the fitting parameter kij characterizing the interaction between different kinds of molecules to be removed from consideration. Next we calculate the function h2(h1) and compare the results with experimental values from the literature. Plots of the function h2(h1), calculated from eq 34 using l12 values summarized in Table 2, are plotted in Figure 4. The vertical lines A and B indicate the two extremes of h1 for cyclohexane reported in the literature. According to Figure 4, the values of the reduced molecular enthalpy for acetone, tetralin, and isobutylbenzene that match literature values can be obtained using a reduced molecular enthalpy for cyclohexane that lies within the range reported in the literature. In addition to the parameters h*1 and h*2 , Table 2 contains the Keezom component of the reduced molecular enthalpy h2*K obtained in fitting the experimental data to eq 28. For comparison, Figure 5 represents the best-fit curve for cyclohexane−acetone obtained under the assumption h2*K = 0. A comparison of Figures 1 and 5 illustrates that including the
Figure 1. Concentration dependence of the Soret coefficient for the mixture cyclohexane−acetone. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28.
of parameters h1*, h2*, and h2*K are summarized in Table 2. As shown below, the Keezom interaction parameters are necessary for obtaining accurate theoretical curves. Combining eqs 29 and 30 and accounting for eq 26, we obtain an equation that expresses the theoretical heats of vaporization of the components as a function of experimental values of the reduced enthalpies of the components: h1* −
3 ⎛ v2 v v 1/3 ⎞ h2* = 2 h2 − (1 − l12)⎜⎜1 + 21/3 ⎟⎟ h1 v1 v1 v1 ⎠ ⎝
(34) 3119
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Table 2. Theoretical Values of Parameters Calculated from Experimental Data solvent couple
h*1
h*2
h2*K
1/3 3 (v22/v11) = [1 + (v1/3 2 /v1 )]
l12
cyclohexane−acetone cyclohexane−tetralin cyclohexane−isobutylbenzene
−27.6 −11.0 −8.6
20.4 11.5 9.0
6.5 3.3 2.8
6.7629 9.0851 9.8321
0.41 0.59 0.63
Keezom term significantly improves the agreement with empirical data.25 In the absence of the Keezom term, it is impossible to obtain reasonable values of ST in dilute solutions of the considered solvents. The literature values of the solvent dipole moments other than apolar cyclohexane decrease with the values of h2*K in Table 2. This result confirms qualitatively the contribution of the Keezom interaction in classical expressions of ST. Next we evaluate the energetic parameters εii using values of the reduced molecular enthalpy for the components and eqs 26 and 31. In calculating the ratio vii/vi, we assume vii ranges from vii = 2vi for the hindered closest possible approach possible only at the predetermined mutual orientation, to vii = 8vi for spherical particles with a random isotropic mutual interaction. Using these estimates and eq 26, we obtain the following range for the energetic parameter εii = from
Figure 4. Dependence of the reduced enthalpies of vaporization for the mixed solvents h2 on the respective parameter of cyclohexane h1 according to eq 34. Acetone, solid line; tetralin, dashed line; isobutylbenzene, dotted line. The vertical lines A and B designate the limits of the literature values of h1 for cyclohexane.
hi h to i 8 2
(35)
Calculated values of the energetic parameters are summarized in Table 3. Note that these estimates are related to the enthalpy of vaporization as defined by eq 26 and to estimates of the van der Waals volume defined by eq 35. These first-order estimates, which yield the lowest possible values for the energetic parameters, are several times larger than values used in simulations of transport processes.37,38 In one such work,37 a new version of NEMD was used and the energetic parameter εii was assumed to be equal to kT, while, in the other work,38 the approximation εii = kT/2 is used in calculations by the same method. The latter value is typical for the NEMD simulations.
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CONCLUSIONS A theory of thermodiffusion based on nonequilibrium thermodynamic and statistical mechanics is applicable to material transport in nonisothermal binary mixtures of organic liquids. A relatively small Keezom interaction can contribute significantly to thermodiffusion because of its strong temperature dependence. Besides good agreement with experimental data, the theory yields values of the component thermodynamic parameters that agree with experimental data using fewer fitting parameters than the number used in standard thermodynamic calculations, which rely on modeling the potentials of intermolecular interactions. However, calculations of the associated energetic parameters are several times larger than those reported in the literature from simulations of transport.
Figure 5. Concentration dependence of the Soret coefficient for the mixture cyclohexane−acetone with zero Keezom interaction. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28, where h 2*K = 0.
Table 3. Comparison of Literature and Calculated Values of Reduced Molecular Enthalpy solvent cyclohexane acetone tetralin isobutylbenzene
reduced molecular enthalpy hi in kT units, literature data 11.7 11.3 16.7 15.4
reduced molecular enthalpy hi in kT units, eq 34 and Table 2
reduced energetic parameter εii in kT units, eq 35
12.7 11.9 16.8 16.2
1.59−6.35 1.48−5.95 2.1−8.4 2.0−8.1
(ref 34); 11.7−12.9 (ref 32) (ref 34); 11.0−11.9 (ref 32) ± 0.3 (ref 34) ± 0.3 (ref 34) 3120
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AUTHOR INFORMATION
Corresponding Author
*Phone: +7(495)939-74-39. E-mail: [email protected]. Author Contributions
The authors thank W. Koehler for sharing the tabulated data. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) De Groot, S. R.; Mazur, P. Non-equilibrium Thermodynamics; North-Holland: Amsterdam, The Netherlands, 1962. (2) Kondepudi, D.; Prigogine, I. Modern Thermodynamics; Wiley: New York, 1999. (3) Dhont, J. K. G.; Wiegand, S.; Duhr, S.; Braun, D. Thermodiffusion of Charged Colloids: Single-Particle Diffusion. Langmuir 2007, 23, 1674−1683. (4) Dhont, J. K. G. Thermodiffusion of Interacting Colloids. I. A Statistical Thermodynamics Approach. J. Chem. Phys. 2004, 120, 1632−1641. (5) Dhont, J. K. G. Thermodiffusion of Interacting Colloids. II. A Microscopic Approach. J. Chem. Phys. 2004, 120, 1642−1653. (6) Morozov, K. I. Soret Effect in Molecular Mixtures. Phys. Rev. E 2009, 79, 031204−1−11. (7) Landau, L. D.; Lifshitz, E. M. Statistical Physics, Part 1 (English translation from Lifshitz, E. M.; Pitaevskii, L. P.); Pergamon Press: Oxford, U.K., 1980. (8) Mayer, J. E.; Goeppert-Mayer, M. Statistical Mechanics, 2nd ed.; Wiley: New York, 1977; Chapter 7. (9) Hartmann, S.; Koehler, W.; Morozov, K. I. The Isotope Soret Effect in Molecular Liquids: A Quantum Effect at Room Temperatures. Soft Matter 2012, 8, 1355−1360. (10) Semenov, S. N. Statistical Thermodynamics of Material Transport in Non-Isothermal Binary Systems. EPL 2012, 97, 66003. (11) Leroyer, Y.; Würger, A. Soret Motion in Non-Ionic Binary Molecular Mixtures. J. Chem. Phys. 2011, 135, 054102−054110. (12) Semenov, S. N.; Schimpf, M. E. Comment on “Soret Motion in Non-Ionic Binary Molecular Mixtures” [J. Chem. Phys. 135, 054102 (2011)]. J. Chem. Phys. 2012, 137, 127101-1−127101-2. (13) Shukla, K.; Firoozabadi, A. A New Model of Thermal Diffusion Coefficients in Binary Hydrocarbon Mixtures. Ind. Eng. Chem. Res. 1998, 37, 3331−3339. (14) Ghorayeb, K.; Firoozabadi, A. Molecular Pressure, and Thermal Diffusion in Nonideal Multicomponent Mixtures. AIChE J. 2000, 46, 883−891. (15) Hartmann, S.; Wittko, G.; Köhler, W.; Morozov, K. I.; Albers, K.; Sadowski, G. Thermophobicity of Liquids: Heats of Transport in Mixtures as Pure Component Properties. Phys. Rev. Lett. 2012, 109, 065901-1−065901-5. (16) Wittko, G.; Köhler, W. Universal Isotope Effect in Thermal Diffusion of Mixtures Containing Cyclohexane and Cyclohexane-D12. J. Chem. Phys. 2005, 123, 014506-1−014506-6. (17) Piazza, R. Thermal Diffusion of Ionic Micellar Solutions. Philos. Mag. 2003, 83, 2067−2085. (18) Wan’t Hoff, J. H. Z. Z. Phys. Chem. 1887, 1, 481. (19) Semenov, S. N.; Schimpf, M. E. Mass Transport Thermodynamics in Nonisothermal Molecular Liquid Mixtures. Phys.-Usp. 2009, 52, 1045−1054. (20) Kirkwood, I.; Boggs, E. The Radial Distribution Function of Liquids. J. Chem. Phys. 1942, 10, 394−402. (21) Fisher, Z. Statistical Theory of Liquids; Chicago University: Chicago, IL, 1964. (22) Haase, R. Thermodynamics of Irreversible Processes; AddisonWesley: Reading, MA, 1969. (23) Semenov, S. N. Statistical Thermodynamic Expression for the Soret Coefficient. EPL 2010, 90, 56002-1−56002-8. (24) Kirkwood, J. G. Order and Disorder in Liquid Solutions. J. Phys. Chem. 1939, 43, 97−107. 3121
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Theory of Soret Coefficients in Binary Organic Solvents Semen Semenov*,† and Martin Schimpf‡ †
Institute of Biochemical Physics RAS, Kosygin St. 4, 119334 Moscow, Russia Department of Chemistry, Boise State University, Boise, Idaho 83725, United States
‡
ABSTRACT: Thermodiffusion in binary molecular liquids is examined using the nonequilibrium thermodynamic model, where the thermodynamic parameters are calculated using equations based on statistical mechanics. In this approach, thermodiffusion is quantified through the variation in binary chemical potential and its temperature and concentration dependence. The model is applied to solutions of organic solvents, in order to compare our theoretical results to experimental results from the literature. A measurable contribution of the orientation-dependent Keezom interaction is shown, where the possible orientations are averaged using the Boltzmann weighting factor. Calculations of enthalpies of evaporation from the model yield good agreement with experimental values from the literature. However, calculations of the associated energetic parameters were several times larger than those reported in the literature from numeric simulations of material transport.
■
BACKGROUND Recent attempts to explain thermodiffusion (the Soret effect) in organic liquid mixtures using models based on nonequilibrium thermodynamics and equilibrium statistical thermodynamics have been a topic of discussion in the literature. Such models are based on the temperature and concentration dependence of the chemical potentials of the components μi, as outlined below. The material flux Ji⃗ in a nonhomogeneous and nonisothermal mixture can be defined as1,2 Ji ⃗ = −niLi∇
μi T
− niLiQ ∇
1 T
δμ* =
v2 μ v1 1
δμ* =
(1)
v2 osm δP ϕ
(4)
The same general approach is used by Morozov.6 The issue related to any approach that expresses parameters of mass and thermodiffusion through pressure becomes obvious from the statistical−mechanical expressions for pressure and chemical potential:7,8 P = −kT
(2)
where vi are the specific molecular volumes of the respective components. For ease of reading, we will subsequently refer to component 1 as the “solvent” and component 2 as the “solute”, although the model applies to the entire concentration range. The last term in eq 2 is the free energy of the solvent molecules displaced from the volume occupied by the solute. We refer to this parameter as the chemical potential of a virtual particle consisting of the solvent but having the same shape and dimensions as the solute. The final expressions2−5 use the Gibbs−Duhem equation to express the binary chemical potential through the (nonuniform) excess pressure Posm as follows: © 2014 American Chemical Society
(3)
Here δ indicates the difference in the parameter between the hot and cold reservoirs, μs and vs are the chemical potential and specific molecular volume of the solvent, respectively, P is the total pressure, and ϕ is the volume fraction of the solute. The parameter δμs/νs is interpreted as the difference in the puresolvent partial pressure between hot and cold reservoirs, and eq 3 is rewritten as2−5
where ni are the numeric volume concentrations of the components, Li and LiQ are the Onsager coefficients, and T is the temperature. The material transport parameters are expressed through the temperature and concentration dependence of the binary chemical potential:2−5 μ* = μ2 −
δμ ⎞ v2 ⎛ ⎜δP − s ⎟ ϕ⎝ vs ⎠
∂ ln Z ∂V
(5)
∂ ln Z ∂Ni
(6)
μi = −kT
Here V is the volume of the system, Ni is the number of particles of type i, and Z is the partition function. Statistical mechanical models typically begin with the chemical potential of a component as an ideal gas:7,8 Received: October 28, 2013 Revised: February 17, 2014 Published: February 18, 2014 3115
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The Journal of Physical Chemistry B μ0i = kT ln ϕivi −
⎡ 2πmikT ⎤ 3 kT ln⎢ ⎥ ⎣ h2 ⎦ 2
Article
J1⃗ = J2⃗ = 0 (7)
Note that eq 9 is valid only for a closed stationary system. In a subsequent article,14 Firoozabady and co-workers use the Gibbs−Duhem equation to calculate the nonzero pressure gradient. The resulting expression for the Soret coefficient is used by Hartmann and co-workers15 to parametrize their empirical data on the thermodiffusion of several pairs of organic liquids. The parameter commonly used for characterizing the steadystate concentration distribution in nonisothermal systems is the Soret coefficient:
Here mi is the molecular mass of the respective component, k is the Boltzmann constant, and h is Planck’s constant. Equation 7 applies to spherically symmetric particles and ignores the contribution of the internal degrees of freedom, like rotation. We note that the classical terms in the partition function are related to kinetic energy, which is a function of mass and its distribution through the particle body, not volume. Thus, the terms based on eq 7 which are not functions of volume are omitted from eq 4. In another approach,9 the mass effects are also ignored, leading to final expressions that ignore the isotope effects described by the classical partition function, thus considering only the much smaller quantum term of the partition function. The use of chemical potentials is preferable in statistical mechanical models because the basic expression for material flux Ji⃗ (eq 1), which can be defined by the nonequilibrium thermodynamics expression for entropy production, contains no excess pressure. Furthermore, the terms related to kinetic energy contained in the chemical potential of an ideal gas play a notable role in molecular thermodiffusion.10 Another group of authors based their research in thermodiffusion on the assumption of constant pressure, even in nonhomogeneous and nonisothermal systems. This assumption is rooted in classical monographs,1,2 where it is formulated axiomatically. By assuming constant pressure, we lose the important physical parameter and mathematical variable necessary to characterize the system. This issue is closely related to the fact that the material fluxes of the components determined by eq 1 should also satisfy the condition of material conservation: J1⃗ + J2⃗ = 0
(9)
ST =
Q2 − Q1 ∇ϕ = ϕ(1 − ϕ)∇T Tϕ(∂μ2 /∂ϕ)P , T
(10)
where Qi is the so-called heat of transport. There are different definitions in the literature for the heat of transport, but the most common definition is
Qi =
LiQ Li
(11)
Using experimental data for equimolar mixtures, authors that assume constant pressure15 concluded that the heat of transport is defined by that of a pure component, i.e., independent of the concentration of the binary mixture. They refer to the respective property of the pure component as the thermophobicity. According to eq 10, the concentration dependence of the Soret coefficient is determined by the concentration dependence of its chemical potential. For an ideal or quasi-ideal solution, μ2(ϕ) ≈ kT ln ϕ and the Soret coefficient should, therefore, be constant across the concentration range, which we know from experimental data to be incorrect. Thus, the assumption of constant pressure in nonisothermal systems is erroneous. We note that the assumption of constant pressure has been discussed in the literature for some time. Piazza17 indicates Wan’t Hoff18 as the first researcher whose approach using the assumption of constant pressure failed in his description of thermodiffusion. In calculations of the Soret coefficient, we follow an approach outlined previously,19 where a solution to the problem of inconsistency in the mass transport equations was proposed. The consistency was shown to be restored by putting the following restriction on the Onsager coefficients:
(8)
In the classical mongraphs,1,2 the gradient expression for material flux (eq 1) is written for only one component. The flux of the other component is taken from eq 8, assuming the gradient equation for the second component is fulfilled automatically. However, for modeling nondilute mixtures, e.g., binary mixtures across the concentration range, this practice leads to a result that depends on which of the two components is selected for calculating its flux from eq 1. In order for the approach to work across the entire concentration range, eq 1 must be written for both components, as well as satisfying conservation of mass (eq 8). However, that results in three equations (two eq 1 and eq 8) for two component concentrations, which are not necessarily self-consistent for any value of the kinetic coefficients. Consequently, the system of material transport equations based on nonequilibrium thermodynamics is valid only when certain restrictions are placed on the kinetic coefficients in the expressions for material flux. This issue has been addressed by introducing a drift velocity,11 which is common for components in a system at constant pressure. However, such an approach leads to the unacceptable assumption that the system will move as a whole in approaching steady state.12 In the pioneering work of Firoozabady and co-workers,13 restrictions are placed on the chemical potentials of the components, and the condition of mass conservation expressed by eq 8 is replaced by a softer condition:
Q i = −μi
(12)
The resulting expression for the Soret coefficient is ∂μP*
ST =
∂T ∂μ *
2ϕ(1 − ϕ) ∂ϕP
(13)
where the subscript P refers to constant pressure. Equation 13 is practically identical to the respective expression for the Soret coefficient obtained by others who replaced the osmotic pressure with the binary chemical potential using eq 4.3−6 The only difference is a factor of 2 in the denominator of eq 13. This factor arises from the incompressibility of the liquid, as expressed by n1v1 + n2v2 = 1
(14)
The factor 2, which was not accounted for in the previous work,3−6 is obtained under the standard rule of differentiation 3116
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of composite functions in calculating the gradient of chemical potential: 2
∑
∂μk [n2 , n1(n2)] ∂nl
l=1
=
∂μk
Since the pair correlation functions in liquids should approach unity at infinite distance, we will use that approximation:
∇nl
gij(r , λ) = 1
This assumption means that the local distribution of the solvent molecules is not disturbed by the considered particle. The approximation of constant local density is used in the study of liquids as regular solutions.24 Combining eqs 2, 7, 16, 17, 19, and 20 and accepting the standard form of the van der Waals intermolecular interaction defined by25
∂μk ∂n1 ∇n 2 ∂n1 ∂n2
∇n 2 +
∂n2 ∂μ = 2 k ∇ϕ ∂ϕ
(15)
The statistical mechanical expression for chemical potential at constant volume is defined as21 μiV = μ0i +
∫0
1
∑
dλ
j = 1,2
ϕj vj
∫V
Φij = −εij(σij/r )6
i out
(16)
where μ0i is the chemical potential of the ideal gas represented by eq 7 and parameter λ describes the gradual “switching on” of the intermolecular interaction. A detailed description of this representation can be found elsewhere.20,21 The parameter r is the distance between a molecule of the surrounding liquid and the center of the considered molecule; gij(r,⃗ λ) is the pair correlation function, which expresses the probability of finding a molecule of the surrounding liquid at r(⃗ r = |r|⃗ ) if the considered molecule is placed at r = 0; and Φij is the molecular interaction potential. Equation 16 is the obvious and exact generalization of the respective expressions for one-component systems, as outlined previously.10 The chemical potential at constant pressure can be related to the chemical potential at constant volume by ∇μip = ∇μiV +
∫V
i out
∇Πi dv
μP* =
vij =
j = 1,2
r
∫∞
2Φij(r′) r′
⎤ dr′⎥ ⎥⎦
(23)
vij = (1 − lij)(vii1/3 + vjj1/3)3
(24)
εij = (1 − kij) εiiεjj
(25)
Here 1 − lij is the parameter reflecting nonsphericity of the molecule and kij is a fitting parameter known as the coupling parameter. Equation 24 approximates the van der Waals volume for two different molecular particles of a complex nonspherical shape. The minimal value of the nonsphericity parameter is 1 − lij = 1/8, which corresponds to spherical molecules. Equations 24 and 25 allow the binary chemical potential to be expressed through the molecular enthalpy of vaporization, which can be obtained from the literature. Using eq 20 and the approximation of a regular solution (eq 20), the reduced molecular enthalpy of vaporization in kT units can be written as ∞ vjjεii 1 hi = Φii(r ) dv = i vkT vjkT Vout i (26)
(18)
The local pressure gradient is calculated using the condition for establishing local equilibrium in a thin spherical layer of thickness l and area S, concentric with the considered molecule.10,23 This condition expresses the local conservation of specific free energy F(r) = Πi(r) + ∑j=1.2 ϕjΦij(r)/vj in the isothermal system when the layer is shifted from position r to r + dr while maintaining spherical symmetry. The change in free energy is due to both changes in parameters within the layer and changes in the area dS and volume of the layer. The changes in volume and surface area are related by dV = 2r dS, and we obtain a modified equation for equilibrium in the closed spherical surface, leading to a definition of the pressure gradient that is related to the change in surface area. That definition has the same nature as the surface (Laplace) pressure gradient discussed elsewhere:2,7 ϕj ⎡ ⎢Φij(r ) − vj ⎢⎣
4π 3 σij 3
is the respective van der Waals volume and ρi is the component density. We note that the use of eq 21 for the van der Waals interaction potential assumes spherically symmetric molecules, which is an oversimplification in many cases. The ratio ρ1/ρ2 approximates the mass ratio of a virtual particle to that of the solute, assuming they occupy the same volume. In the integration over Viout in eqs 16 and 17, the lower limit is r = σij. There is the standard way to express parameter vij through the van der Waals molecular volume vii and expressing ε12 through the energetic parameters ε11 and ε22 of identical particles:26
Here Viout is the volume external to the molecule of the ith component and Πi is the local pressure distribution around this molecule. Equation 17 expresses the relation between the forces acting on a molecular particle at constant versus variable local pressure. This equation is the obvious generalization of the equation that connects the gradients of chemical potentials at constant volume and pressure:22
Πi = − ∑
⎞ v12 v ⎛ v (ε11 − ε12)(1 − ϕ) + 12 ⎜ε12 − 22 ε22⎟ϕ 3v1 3v2 ⎝ v12 ⎠ ρ ϕ 3 + kT ln + kT ln 1 1−ϕ 2 ρ2 (22)
Here
(17)
∇μiP = ∇μiV + vi∇P
(21)
where εij is the energy of interaction and σij is the minimal molecular approach distance, we obtain the following binary chemical potential at constant pressure:
∞
gij( r ⃗ , λ)Φij(r ) dv
(20)
∫
Below, we use eq 26 to compare thermodynamic parameters obtained from the literature with those obtained from thermodiffusion data. In calculating the temperature derivative of the binary chemical potential, the temperature dependence of the specific
(19) 3117
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Table 1. Physical Parameters of the Solvents Examined molar volume Vm (cm3)
reduced molecular enthalpy hi in kT units
cyclohexane
106 (ref 29)
acetone
75 (ref 29)
tetralin isobutylbenzene
136 (ref 34) 158 (ref 36)
11.7 (ref 34) 11.7−12.9 (ref 32) 11.3 (ref 34) 11.0−11.9 (ref 32) 16.7 ± 0.3 (ref 34) 15.4 ± 0.3 (ref 34)
solvent
∂T
=−
We note that the Keezom interaction can be utilized in a wider sense beyond the angular dependence of the intermolecular interaction between hard dipoles. Thus, eq 27 is also valid for molecules of complex shape with strong anisotropy in the dielectric polarizability. Considering the atoms constituting a molecule as separate particles that interact with atoms constituting other molecules, we will also obtain the orientation-dependent potential of intermolecular interactions. This orientation-dependent potential can be averaged over the possible orientations using the approach by Keezom, which is based on the interaction of hard dipoles. In this case, eq 27 will also be valid for nonpolar molecules with no hard dipole moments. Using eqs 13, 22, and 27, the Soret coefficient can be written as 1 ST = 2
h2*K ϕ T
+
ϕ(1 − ϕ)(h1* + h2*) + 3
9 2T
ln
(28)
v12ε12 − v22ε22 v2kT
(30)
and h2* =
are defined as the binary reduced enthalpy of the first and second components, respectively, expressed in units of kT. The parameter
h2*K =
v22 ε22K v2 kT
0.0014 (refs 30 and 31)
0.79 (ref 29)
58 (ref 29)
0.0008 (ref 35) 0.0095 (ref 36)
0.97 (ref 34) 0.85 (ref 36)
132 (ref 34) 134 (ref 36)
(32)
ρ⎤ h *K 1⎡ * 9 ⎢α1h1 (1 − ϕ) − α2h2*ϕ + 2 ϕ + ln 1 ⎥ 6 ⎢⎣ T 2T ρ2 ⎥⎦ (33)
ρ1
(29)
84 (ref 29)
Equation 33 describes a linear concentration dependence of the Soret coefficient in ideal and quasi-ideal solutions, confirming the empirical observation of a near-linear concentration dependence of the Soret coefficient in certain nonpolar organic solvent pairs.16 However, the dependence was strongly nonlinear for solvent pairs having vastly different properties, such as cyclohexane and acetone. Additional examples are discussed below. We note that eq 33, as well as the empirical data, contradicts the theory subsequently developed by Hartmann et al.,15 which predicts a constant Soret coefficient across the concentration range in such systems. Note that in expressing the terms of the binary chemical potential given by eq 2 through the binary reduced enthalpies of the first and second component given by eqs 29 and 30 we introduce no new physical concept. The expressions are derived only to estimate the reduced molar enthalpies from thermodiffusion parameters, in order to compare the resulting values with literature data obtained by other methods to make some useful general conclusions. For example, eqs 29 and 30 indicate that, for miscible solvents that do not have a large difference in physicochemical and geometric parameters, i.e., ε11 ≈ ε22; v12/v1 ≈ v12/v2 ≈ v22/v2, where the energetic parameter ε12 by definition lies within the narrow interval between ε11 and ε22, the binary reduced enthalpies of the first and second components are related by the approximate relationship h2* ≈ −h1* . This result agrees with data summarized in Table 2, lending support to the proposed theory. In eq 28, we have ignored the concentration dependence of the specific molecular volumes. This is a reasonable approximation for organic solvents where the maximum change in the specific molecular volume across the concentration range
ρ2
v12(ε12 − ε11) v1kT
0.78 (ref 33)
ST =
where
h1* =
0.0011 (ref 32)
The parameter h*1 + h*2 indicates the difference in physical properties of the components. When |h1* + h2*| ≪ 12, the mixture can be considered as an ideal or quasi-ideal solution. The expression for the Soret coefficient in such binary mixtures can be written as
(27)
α1h1*(1 − ϕ) − α2h2*ϕ +
molar mass M (g)
⎛ v (ε − ε12) v ε − v12ε12 ⎞ Tc = ⎜ 12 11 + 22 22 ⎟ 12kv1 12kv2 ⎠ ⎝
εijK T
density ρ (g/cm3)
examine the experimental data of Wittko and Köhler,16 where one of the solvents is always the highly symmetric nonpolar molecule cyclohexane. An analysis of eq 28 without the Keezom term was made previously.10 The denominator in eq 28 can become zero at a temperature Tc when ϕ = 1/2 and h1* + h2* < 0, where Tc is defined as
molecular volume was the only factor taken into account in previous works.10,12,23 However, the interaction potential may also contain a temperature-dependent component involving the so-called Keezom or orientation dipole interaction parameter εijK(T).25 This component of the interaction potential is obtained by weight-averaging Φij(r)⃗ over all orientations between the hard dipoles using a Boltzmann factor. The specific expression for the orientation dipole−dipole interaction potential can be found elsewhere.25 For our calculations, the temperature derivative of the Keezom potential can be written as
∂εijK (T )
thermal expansion coefficient αT (K−1)
(31)
is the reduced Keezom enthalpy of the second component in kT units. In eq 28, we assume that only the second component has a polarity or anisotropy that contributes to the respective Keezom term. In the experiments, we will use the model to 3118
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is less than a couple of percent. However, in strongly polar liquids such as water and alcohols, this concentration dependence cannot be ignored and such systems require a separate study. Preliminary evaluations indicate that the Soret coefficient can still be calculated when the concentration dependence of the molecular volume is known and substituted directly into the equations. However, calculation of the dynamic parameters becomes a much more complex problem. Finally, we note that calculation of Soret coefficients in electrolytes was outlined in previous work.27,28
■
RESULTS In this section, we compare the theoretical results based on eqs 28−31 with empirical data. Using eqs 29 and 30, we could calculate parameters h*1 and h*2 for substituting into eq 28, relying on the literature for values of the remaining parameters. However, there is significant variance in enthalpies of vaporization reported in the literature (see Table 1). Together with the fitting parameters in eqs 24 and 25, the wide variance in enthalpies of vaporization would allow nearly any values of h1* and h2* to be obtained, including different signs in those values. Consequently, we formulate and solve the inverse problem; that is, we obtain the experimental values of h*1 and h*2 and use them to calculate values of the enthalpies of vaporization for the components. The concentration dependences of the Soret coefficient calculated using eq 28 for three binary systems are illustrated in Figures 1−3. Calculated values
Figure 2. Concentration dependence of the Soret coefficient for the mixture cyclohexane−tetralin. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28.
Figure 3. Concentration dependence of the Soret coefficient for the mixture cyclohexane−isobutylbenzene. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28.
Equation 34 allows the fitting parameter kij characterizing the interaction between different kinds of molecules to be removed from consideration. Next we calculate the function h2(h1) and compare the results with experimental values from the literature. Plots of the function h2(h1), calculated from eq 34 using l12 values summarized in Table 2, are plotted in Figure 4. The vertical lines A and B indicate the two extremes of h1 for cyclohexane reported in the literature. According to Figure 4, the values of the reduced molecular enthalpy for acetone, tetralin, and isobutylbenzene that match literature values can be obtained using a reduced molecular enthalpy for cyclohexane that lies within the range reported in the literature. In addition to the parameters h*1 and h*2 , Table 2 contains the Keezom component of the reduced molecular enthalpy h2*K obtained in fitting the experimental data to eq 28. For comparison, Figure 5 represents the best-fit curve for cyclohexane−acetone obtained under the assumption h2*K = 0. A comparison of Figures 1 and 5 illustrates that including the
Figure 1. Concentration dependence of the Soret coefficient for the mixture cyclohexane−acetone. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28.
of parameters h1*, h2*, and h2*K are summarized in Table 2. As shown below, the Keezom interaction parameters are necessary for obtaining accurate theoretical curves. Combining eqs 29 and 30 and accounting for eq 26, we obtain an equation that expresses the theoretical heats of vaporization of the components as a function of experimental values of the reduced enthalpies of the components: h1* −
3 ⎛ v2 v v 1/3 ⎞ h2* = 2 h2 − (1 − l12)⎜⎜1 + 21/3 ⎟⎟ h1 v1 v1 v1 ⎠ ⎝
(34) 3119
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Table 2. Theoretical Values of Parameters Calculated from Experimental Data solvent couple
h*1
h*2
h2*K
1/3 3 (v22/v11) = [1 + (v1/3 2 /v1 )]
l12
cyclohexane−acetone cyclohexane−tetralin cyclohexane−isobutylbenzene
−27.6 −11.0 −8.6
20.4 11.5 9.0
6.5 3.3 2.8
6.7629 9.0851 9.8321
0.41 0.59 0.63
Keezom term significantly improves the agreement with empirical data.25 In the absence of the Keezom term, it is impossible to obtain reasonable values of ST in dilute solutions of the considered solvents. The literature values of the solvent dipole moments other than apolar cyclohexane decrease with the values of h2*K in Table 2. This result confirms qualitatively the contribution of the Keezom interaction in classical expressions of ST. Next we evaluate the energetic parameters εii using values of the reduced molecular enthalpy for the components and eqs 26 and 31. In calculating the ratio vii/vi, we assume vii ranges from vii = 2vi for the hindered closest possible approach possible only at the predetermined mutual orientation, to vii = 8vi for spherical particles with a random isotropic mutual interaction. Using these estimates and eq 26, we obtain the following range for the energetic parameter εii = from
Figure 4. Dependence of the reduced enthalpies of vaporization for the mixed solvents h2 on the respective parameter of cyclohexane h1 according to eq 34. Acetone, solid line; tetralin, dashed line; isobutylbenzene, dotted line. The vertical lines A and B designate the limits of the literature values of h1 for cyclohexane.
hi h to i 8 2
(35)
Calculated values of the energetic parameters are summarized in Table 3. Note that these estimates are related to the enthalpy of vaporization as defined by eq 26 and to estimates of the van der Waals volume defined by eq 35. These first-order estimates, which yield the lowest possible values for the energetic parameters, are several times larger than values used in simulations of transport processes.37,38 In one such work,37 a new version of NEMD was used and the energetic parameter εii was assumed to be equal to kT, while, in the other work,38 the approximation εii = kT/2 is used in calculations by the same method. The latter value is typical for the NEMD simulations.
■
CONCLUSIONS A theory of thermodiffusion based on nonequilibrium thermodynamic and statistical mechanics is applicable to material transport in nonisothermal binary mixtures of organic liquids. A relatively small Keezom interaction can contribute significantly to thermodiffusion because of its strong temperature dependence. Besides good agreement with experimental data, the theory yields values of the component thermodynamic parameters that agree with experimental data using fewer fitting parameters than the number used in standard thermodynamic calculations, which rely on modeling the potentials of intermolecular interactions. However, calculations of the associated energetic parameters are several times larger than those reported in the literature from simulations of transport.
Figure 5. Concentration dependence of the Soret coefficient for the mixture cyclohexane−acetone with zero Keezom interaction. The points are the experimental data from ref 16, and the solid line is the concentration dependence calculated using eq 28, where h 2*K = 0.
Table 3. Comparison of Literature and Calculated Values of Reduced Molecular Enthalpy solvent cyclohexane acetone tetralin isobutylbenzene
reduced molecular enthalpy hi in kT units, literature data 11.7 11.3 16.7 15.4
reduced molecular enthalpy hi in kT units, eq 34 and Table 2
reduced energetic parameter εii in kT units, eq 35
12.7 11.9 16.8 16.2
1.59−6.35 1.48−5.95 2.1−8.4 2.0−8.1
(ref 34); 11.7−12.9 (ref 32) (ref 34); 11.0−11.9 (ref 32) ± 0.3 (ref 34) ± 0.3 (ref 34) 3120
dx.doi.org/10.1021/jp410634v | J. Phys. Chem. B 2014, 118, 3115−3121
The Journal of Physical Chemistry B
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AUTHOR INFORMATION
Corresponding Author
*Phone: +7(495)939-74-39. E-mail: [email protected]. Author Contributions
The authors thank W. Koehler for sharing the tabulated data. Notes
The authors declare no competing financial interest.
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REFERENCES
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