Application of the Compensated Arrhenius Formalism to Self-Diffusion

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J. Phys. Chem. B 2010, 114, 8600–8605

Application of the Compensated Arrhenius Formalism to Self-Diffusion: Implications for Ionic Conductivity and Dielectric Relaxation Matt Petrowsky and Roger Frech* Department of Chemistry and Biochemistry, UniVersity of Oklahoma, Norman, Oklahoma 73019 ReceiVed: March 5, 2010; ReVised Manuscript ReceiVed: May 27, 2010

Self-diffusion coefficients are measured from -5 to 80 °C in a series of linear alcohols using pulsed field gradient NMR. The temperature dependence of these data is studied using a compensated Arrhenius formalism that assumes an Arrhenius-like expression for the diffusion coefficient; however, this expression includes a dielectric constant dependence in the exponential prefactor. Scaling temperature-dependent diffusion coefficients to isothermal diffusion coefficients so that the exponential prefactors cancel results in calculated energies of activation Ea. The exponential prefactor is determined by dividing the temperature-dependent diffusion coefficients by the Boltzmann term exp(-Ea/RT). Plotting the prefactors versus the dielectric constant places the data on a single master curve. This procedure is identical to that previously used to study the temperature dependence of ionic conductivities and dielectric relaxation rate constants. The energies of activation determined from self-diffusion coefficients in the series of alcohols are strikingly similar to those calculated for the same series of alcohols from both dielectric relaxation rate constants and ionic conductivities of dilute electrolytes. The experimental results are described in terms of an activated transport mechanism that is mediated by relaxation of the solution molecules. This microscopic picture of transport is postulated to be common to diffusion, dielectric relaxation, and ionic transport. 1. Introduction The transport of mass or charge in liquids is usually viewed in hydrodynamic terms where the translational movement of a molecule (diffusion) or ion (ionic conductivity) is hindered by the resistive drag exerted by the solvent. Closely related is rotational motion of a dipole in solution (dielectric relaxation), which is also hindered by a resistive drag. In these examples, the resistive force is proportional to the solvent viscosity η through application of Stokes’ law and results in the following hydrodynamic expressions1–3

Ionic conductivity:

Λ)

q 6πηr

(1)

Dielectric relaxation:

τ)

4πηr3 kBT

(2)

Diffusion:

D)

kBT 6πηr

(3)

where D is the diffusion coefficient, Λ is the molar conductivity, that is, Λ ) σ/c (σ is ionic conductivity and c is total salt concentration), q is the charge on an ion, r is the radius of the moving ion (or particle), τ is the dielectric relaxation time, kB is Boltzmann’s constant, and T is temperature. Equations 1-3 often produce significant errors in the calculated quantities,1,3–10 but they are still commonly used because there are no better alternatives. These equations are inaccurate because each is based on a hydrodynamic model to describe microscopic transport. Hydrodynamic theory provides a good description of * To whom correspondence should be addressed.

macroscopic motion, for example, a steel ball bearing falling in a 10 gallon vat of oil. However, the application of hydrodynamic theory to molecules is much less straightforward for a number of reasons;1 (1) molecular movement is discontinuous (not straight-line motion), (2) ions and molecules are not necessarily spherical, (3) the proper value to use for the radius r is usually not clear, and (4) the macroscopic viscosity is used for η, but it is the viscosity in the immediate vicinity of the molecule that is important. On a microscopic level, motion of a particle (i.e., a molecule or ion) is governed by the intermolecular forces between the surrounding molecules in addition to the forces between the particle and the surrounding molecules. The magnitude of these forces depends on the positions and, in the case of dipolar interactions, the orientations of the interacting species. In solids, where molecules occupy equilibrium positions with relatively fixed orientations, transport is usually described very well by the Arrhenius equation. For example, the temperature-dependent ionic conductivity for a solid or polymer electrolyte below the glass transition temperature is conventionally represented as11

σ ) σoe-Ea/RT

(4)

The activation energy Ea and exponential prefactor σo are both temperature-independent variables, although σo can exhibit a very weak temperature dependence. From elementary statistical mechanics, an Arrhenius equation implies that the mechanism of transport is a thermally activated process, which is quite different from hydrodynamic motion. In an activated process, an energy barrier (represented by Ea) must be surmounted for an ion or molecule to move from one site or orientation to another. In liquids, the molecules can readily change both orientation and position, and ion transport no longer obeys the Arrhenius

10.1021/jp1020142  2010 American Chemical Society Published on Web 06/16/2010

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equation. Consequently, the temperature dependence is most often described by empirical functions; for example, for ion transport, the Williams-Landel-Ferry (WLF) equation12 or the Vogel-Tamman-Fulcher (VTF) equation13–15 is commonly used.16–19 However, the resulting empirical fitting parameters provide neither physical significance nor insight into the fundamental mechanism(s) governing ion transport. We take a completely different approach to transport phenomena in liquids by discarding the hydrodynamic model in favor of a picture in which the positions and orientations of the solution species (molecules and, in the case of conductivity, ions) are dynamically coupled to each other through intermolecular forces. One can view the resistive force on the molecules as resulting from this coupled motion, but the accumulated effect of the intermolecular forces is not adequately represented on a macroscopic scale by the viscosity. Instead, we must search for a solvent or solution property that better accounts for these molecular-level interactions. A small positional/orientational change of the species in a small locally stable configuration during a transport process results in a small shift from local equilibrium. This shift is followed by the relaxation of the species to new local equilibrium positions and orientations via their coupled motions. This picture is described in more detail in the Summary and Conclusions section. Since the intermolecular forces coupling the molecules are electrostatic in nature, we postulate that the static dielectric constant, εs, characterizes the dynamical response of solvent or solution molecules to the small changes in their local electric fields. We also postulate that transport in liquids is Arrhenius-like except, that the positional/orientational contribution from the species is represented by a dielectric constant dependence in the exponential prefactor. Throughout this article, all references to the dielectric constant will mean the static value of the real part of the dielectric constant. We have previously demonstrated that the temperaturedependent ionic conductivity, σ(T,εs), of dilute, liquid electrolytes and the temperature-dependent dielectric rate constant, k(T,εs), of pure liquids can be expressed in the following form20,21

σ(T, εs) ) σo(εs(T))e-Ea/RT

(5)

k(T, εs) ) ko(εs(T))e-Ea/RT

(6)

The dielectric rate constant is defined as the inverse of the dielectric relaxation time τ. The implicit dependence of both σ and k on the static dielectric constant εs is noted in eqs 5 and 6. The quantities σo(εs(T)) and ko(εs(T)) are the exponential prefactors for σ and k, respectively, and Ea is an energy of activation. Temperature-dependent conductivities or dielectric rate constants of these electrolytes or solvents deviate from Arrhenius behavior due to the temperature dependence of the static dielectric constant contained in the exponential prefactor. Scaling the temperature-dependent conductivities/rate constants to conductivities/rate constants at a reference temperature removes the dielectric constant dependence and allows the energy of activation to be calculated. The exponential prefactor can then be determined by dividing the conductivity/rate constant by the Boltzmann factor exp(-Ea/RT). When the temperature-dependent prefactors are plotted against the temperature-dependent static dielectric constants, the prefactors lie on a single master curve. Since the temperature dependence in the exponential prefactor must be scaled out to elucidate

Arrhenius behavior, the temperature-dependent conductivity/ rate constant described by eqs 5 and 6 is said to obey a “compensated” Arrhenius equation. In this article, we extend the scope of our earlier work to consider mass transport of uncharged molecules. Specifically, we apply the compensated Arrhenius formalism to temperature-dependent self-diffusion coefficients of pure n-alcohols. 2. Experimental Section Methanol, ethanol, 1-propanol, 1-butanol, 1-hexanol, and 1-octanol (g99% pure) were obtained from Aldrich and used as received. All chemicals were stored, and all samples were prepared in a glovebox (e1 ppm H2O) under a nitrogen atmosphere. For the diffusion measurements, the samples were contained in a 5 mm OD and 20 cm long glass NMR tube sealed with parafilm. The sample height in the NMR tube was constricted to 0.8 cm. Proton self-diffusion coefficients were measured using a Varian VNMRS-400 MHz NMR spectrometer equipped with an Auto-X-Dual broad-band 5 mm probe. The corresponding Larmor frequency of 1H in this field is 399.965 MHz. An FTS XR401 air-jet regulator was used to control the temperature in the measurement range from -5 to 80 °C. The gradient field strength was calibrated at each temperature from literature diffusion coefficient values for either methanol,22 ethanol,22 or water.23 There is about a 15% difference between the lowest and highest gradient calibration factor over the temperature range of -5 to 80 °C. The diffusion measurements were made using the Stejskal-Tanner pulsed field gradient NMR spin-echo technique.24 For a given gradient duration and interval, the attenuation of the signal intensity was recorded as a function of the gradient field strength over the range of 0.07-0.75 T m-1. Plotting the logarithm of intensity versus the square of the gradient field strength produced a linear relationship whose slope yielded the diffusion coefficient.25 The static dielectric constant was obtained using a HP 4192A impedance analyzer to measure the capacitance from 1 kHz to 13 MHz. The sample holder was an Agilent 16452A liquid test fixture. All measurements were taken in the aforementioned glovebox. A Huber ministat 125 bath was used to regulate the temperature to (0.1 °C from -5 to 80 °C. The static dielectric constant εs was calculated from the measured capacitance C through the equation εs ) RC/Co, where R accounts for stray capacitance and Co is the atmospheric capacitance.26 3. Results and Discussion As mentioned in the Introduction, diffusion coefficients are usually described in hydrodynamic terms,1,2,4,27–29 and their temperature dependence is often expressed with empirical equations that provide no physical insight into the transport process.23,30,31 Here, the temperature-dependent diffusion coefficient is written as a compensated Arrhenius expression, analogously to eqs 5 and 6

D(T, εs) ) Do(εs(T))e-Ea/RT

(7)

where Do(εs(T)) is the exponential prefactor. The diffusion coefficient and the static dielectric constant of a chosen alcohol are first measured over some temperature interval of interest. Next, an isothermal reference curve is constructed using a family of solvents that are chemically and structurally similar to the chosen alcohol. The reference curve is comprised of the diffusion coefficients plotted as a function of the static dielectric

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Figure 1. Self-diffusion coefficients for pure n-alcohols plotted against the alcohol static dielectric constant at both 25 and 45 °C; 1 ) octanol, 2 ) hexanol, 3 ) butanol, 4 ) propanol, 5 ) ethanol.

Figure 2. Compensated Arrhenius plot (Tr ) 25 °C) (squares) and conventional Arrhenius plot (triangles) for the temperature-dependent diffusion coefficients of butanol measured over the temperature range of -5 to 80 °C.

constant for each member of the solvent family measured at a chosen reference temperature Tr. The values of εs used in the reference curve must span the range of εs measured for the chosen alcohol over the temperature interval of interest. Two reference curves for the family of n-alcohols, at reference temperatures Tr ) 25 and 45 °C, are shown in Figure 1. Each curve gives the isothermal dependence of the diffusion coefficient on the static dielectric constant for the group of n-alcohols. The diffusion coefficients in Figure 1 increase with dielectric constant, a trend that is qualitatively similar to that for ionic conductivities and dielectric rate constants previously studied.20,21,32 A compensated Arrhenius plot for the diffusion coefficient data is constructed by scaling the temperaturedependent diffusion coefficients for a chosen alcohol to reference diffusion coefficient data at temperature Tr so that the exponential prefactors cancel. This procedure is similar to that used to construct compensated Arrhenius plots for both conductivity and dielectric rate constant data.20,21,32 The reference diffusion coefficient curve is empirically fit to a function of the form D ) x1 exp(εs/x2) + x3 (x1, x2, and x3 are fitting parameters) so that the diffusion coefficient corresponding to a particular temperature T for the chosen alcohol member, D(T,εs), can be divided by the reference diffusion coefficient at Tr that has the same value for the static dielectric constant, D(Tr,εs). This procedure ensures that the exponential prefactors in the numerator and denominator cancel because both depend exclusively on the temperature-dependent static dielectric constant. There is no theoretical justification for using an exponential functional form to fit the reference curve; it was chosen because it very accurately fits the reference curve data. If the compensated Arrhenius equation describes the temperature dependence of the diffusion coefficient, a plot of ln[D(T,εs)/D(Tr,εs)] versus T-1 should be linear with slope ) -Ea/R and y-intercept ) Ea/(RTr). Figure 2 shows a compensated Arrhenius plot created by scaling temperature-dependent diffusion coefficients for butanol over the temperature range of -5 to 80 °C to diffusion

Petrowsky and Frech

Figure 3. Compensated Arrhenius plot (Tr ) 25 °C) (squares) and conventional Arrhenius plot (triangles) for the temperature-dependent diffusion coefficients of butanol measured over the temperature range of -5 to 80 °C.

coefficients at a reference temperature of 25 °C. For comparison, Figure 2 also shows a conventional Arrhenius plot of the butanol diffusion coefficient data. Both plots are linear to a high degree of accuracy. When fit to a linear function, the compensated Arrhenius plot has a correlation coefficient of 0.9992, while that for the conventional Arrhenius plot is 0.9980. Diffusion coefficient data are often much more Arrhenius-like than conductivity data, and therefore, diffusion data are sometimes fit with the conventional Arrhenius equation instead of an empirical function.23,33 However, even though both plots may be linear, the energy of activation calculated from the compensated Arrhenius equation is substantially different than that calculated from the conventional Arrhenius equation. For example, the data in Figure 2 give an activation energy of 37.2 kJ/mol for the compensated Arrhenius plot (calculated from the slope), while the Ea value is 23.5 kJ/mol for the conventional Arrhenius plot. The disparity in the magnitude of the Ea value between compensated Arrhenius and conventional Arrhenius plots is best explained by considering Figure 3. Figure 3 shows the same data that are depicted in Figure 2, except that the conventional Arrhenius data have been plotted on a secondary y-axis. The compensated Arrhenius plot and conventional Arrhenius plot are approximately parallel; the difference in slope between the two lines is apparent from the disparate scales of the y-axes for both plots. The effect of the compensated Arrhenius procedure is to shift a conventional Arrhenius plot to a different location on the y-axis with a slightly different slope. The activation energies calculated from both the slope and y-intercept for the compensated Arrhenius data in Figure 2 as well as Ea values calculated from temperature-dependent diffusion data for other alcohols using the compensated Arrhenius procedure are summarized in Table 1. The uncertainty in each Ea value is also given in Table 1. The procedure by which this uncertainty is calculated has been previously described.34 In addition to the linear fit of the data using the compensated Arrhenius equation, the internal consistency of Ea values obtained from the slope and intercept is also a good measure of the appropriateness of this procedure. Table 1 shows the consistency between Ea values obtained from the slope and intercept and also demonstrates that changing the reference temperature or alcohol member does not substantially affect the value of Ea. Table 1 also gives activation energy values obtained from conventional Arrhenius plots for the diffusion data of several different alcohols. There is an appreciable increase in the Arrhenius activation energy as the alcohol chain length increases. Once the energy of activation is determined from the compensated Arrhenius plot, the exponential prefactor can be calculated from eq 7 by dividing the temperature-dependent

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TABLE 1: Activation Energies of Several Alcohols Calculated from Both Compensated Arrhenius Plots and Conventional Arrhenius Plots for Self-Diffusion Data compensated Arrhenius data system

Tr (°C)

propanol

25 45 25 45 25 45

butanol hexanol

Ea,slope (kJ mol-1) 38.6 ( 0.7 37.2 ( 0.9 37.2 ( 0.4 36.5 ( 0.5 35.4 ( 0.7 36.7 ( 0.6

diffusion coefficient by the Boltzmann factor exp(-Ea/RT). For the purpose of calculating the exponential prefactor, the energy of activation for the alcohols is taken as the average of the slope and intercept activation energy values for the data in Table 1. Therefore, the average Ea value for the alcohols is given as 37.0 kJ mol-1. Figure 4a plots the temperature-dependent diffusion coefficients against the temperature-dependent static dielectric constants, resulting in a series of well-separated curves. However, Figure 4b shows that when the temperature-dependent exponential prefactors are plotted against the temperaturedependent static dielectric constants, the data fall on a single master curve, analogous to behavior observed for both conductivity and dielectric relaxation data.20,21 Figure 5 plots the exponential prefactors against the dielectric constant for prefactors that are calculated using the Ea value obtained from the conventional Arrhenius plot of butanol. A master curve is not observed for these data, and it is clear that the magnitude of the Ea value affects the qualitative shape of the resulting graph. For the alcohol diffusion data presented in Figure 4a, a master curve is only observed for Ea values in the range approximately from 33 to 47 kJ/mol. Table 1 shows that conventional Arrhenius plots of the data yield Ea values that do not lie within the range needed to form a master curve, while compensated Arrhenius plots give activation energies that do form a master curve. Temperature-dependent diffusion coefficients for methanol were also measured, but these data are omitted from the figures and tables because they are not entirely consistent with the compensated Arrhenius behavior of the other alcohols. If the methanol data are included in Figure 1, the reference curves are no longer sharply increasing at high dielectric constants, but rather, these graphs appear to start leveling off. Furthermore, if the methanol exponential prefactors are included in Figure 4b, these prefactors do increase with dielectric constant similarly to the other alcohol data, but the methanol data are shifted to the right of the master curve. Two points must be made

conventional Arrhenius data Ea,intercept (kJ mol-1) 38.6 ( 0.7 37.5 ( 0.9 37.1 ( 0.4 36.6 ( 0.6 35.4 ( 0.7 36.8 ( 0.6

system ethanol propanol butanol hexanol octanol

Ea (kJ mol-1) 19.1 ( 0.2 24.2 ( 0.6 23.5 ( 0.4 27.1 ( 0.7 27.8 ( 0.5

regarding the anomalous methanol data. First, IR studies of n-alcohol-carbon tetrachloride mixtures have shown that the OH stretching frequency of methanol is different than that for alcohols C3 through C18, whose spectra are practically identical.35 This implies that the hydrogen bonding network of methanol is somewhat different from that of the other alcohols, and this result may explain why the methanol diffusion coefficient data is not entirely consistent with that of the other alcohols. Second, previous work has shown that the compensated Arrhenius formalism is not very effective for systems where the static dielectric constant is large, that is, approximately εs > 30.21 It is well-known that the dielectric constant for methanol is much higher than that for any other alcohol, and this may be a contributing factor as to why the methanol prefactors are not consistent with the master curve. The temperature dependence of ionic conductivities and of dielectric relaxation rate constants has also been studied using a compensated Arrhenius analysis.20,21,32 It is particularly interesting to compare the activation energies found in this study with activation energies obtained from conductivity and dielectric relaxation measurements in comparable systems. These data are listed in Table 2. A comparison of the Ea values in Table 2 with the compensated Arrhenius Ea values in Table 1 shows that at least in the family of alcohols, the energy barriers for transport in each of these three processes are approximately equivalent. These activation energy data provide compelling evidence linking together the mechanisms of transport for diffusion, ionic conductivity, and dielectric relaxation. 4. Summary and Conclusions Application of the compensated Arrhenius formalism to the temperature dependence of self-diffusion coefficients begins with the hypothesis that deviations from a conventional Arrhenius expression are due to the intrinsic temperature dependence of the dielectric constant in the exponential prefactor. Analysis of

Figure 4. (a) Diffusion coefficients versus static dielectric constants for ethanol, propanol, butanol, hexanol, and octanol over the temperature range of -5 to 80 °C. (b) Exponential prefactors versus the static dielectric constant for the data of (a) using the Ea value 37.0 kJ/mol.

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Figure 5. Exponential prefactors versus static dielectric constant for the data in Figure 4a that are calculated using the Ea value from the conventional Arrhenius plot of butanol (Ea ) 23.5 kJ/mol).

TABLE 2: Activation Energies Calculated from Compensated Arrhenius Plots for Temperature-Dependent Dielectric Relaxation and Ionic Conductivity Dataa system

-1

-1

Tr (°C)

Ea,slope (kJ mol )

Ea,intercept (kJ mol )

butanol hexanol octanol

40 40 40

Dielectric Relaxation 35.3 38.6 41.2

35.6 38.7 41.1

propanol butanol hexanol hexanol

25 25 25 50

Ionic Conductivity 39.4 40.7 39.9 41.5

39.6 40.9 39.5 41.2

a The dielectric relaxation data are for pure liquids, and the ionic conductivity data are for 0.0055 M solutions of tetrabutylammonium trifluoromethanesulfonate in alcohol.

temperature-dependent diffusion coefficient data (Figures 2 and 3) for a series of alcohols as described in the text clearly supports this hypothesis. In addition, when the exponential prefactors for the alcohols are plotted as a function of εs (Figure 4), all data fall on a single master curve, further supporting this hypothesis. We have previously shown that the temperature-dependent behavior shown in Figures 2-4 is also observed in the ionic conductivity of a number of dilute electrolyte solutions and in the dielectric relaxation of several pure liquids. The similarity of the compensated Arrhenius activation energies for diffusion, ionic conductivity, and dielectric relaxation in the alcohols and alcohol-based electrolytes (Tables 1 and 2) provides evidence linking the mechanisms of transport for these three processes. Additional support for this conclusion comes from the observation that each of these processes shows a similar dependence on the dielectric constant contained in the exponential prefactor. Further, qualitatively similar master curves for the three transport processes result when their exponential prefactors are plotted as a function of the static dielectric constant. A master curve would not be observed if the exponential prefactor was not a function of the solution static dielectric constant. Therefore, qualitatively similar master curves for the three processes further supports the hypothesis that it is the dielectric constant, not the viscosity, that characterizes mass and charge transport in the systems studied to date. What is the mechanism common to diffusion, dielectric relaxation, and ionic conductivity that involves some form of activation? Although this paper focuses on diffusion, it is easier to begin this discussion with dielectric relaxation, which has been studied by numerous authors for many decades.3,17,36–47 Eyring first pointed out that a relaxation rate might be viewed as a chemical rate constant and treated by transition-state

Figure 6. Illustration of the application of transition-state theory to a jump in a liquid composed of dipolar molecules.

theory.48 However, the most detailed description of an application of transition-state theory to dielectric relaxation in pure liquids and solids was provided by Kauzmann,49 who began with a general picture of a dipole undergoing reorientational motion by a series of thermally activated discrete jumps rather than a continuous reorientation. He suggested that a reorienting dipole plus its immediate surrounding dipoles could be viewed as an activated complex during the transition from one locally stable configuration to another as it passed over an activation energy barrier. Kauzmann postulated that the dipoles became “less rigid” during the passage over the energy barrier, resulting in the large entropy of activation that is customarily observed in dielectric relaxation studies.49 In this paper, we modify the general picture of molecular relaxation introduced by Kauzmann to describe dielectric relaxation and extend this picture to diffusion and ionic conductivity. In all three cases, transport (which now includes both translational and reorientational motion) is assumed to occur through a series of thermally activated discrete jumps. However, it is important to define a jump as it is used in this paper. A jump is the process by which a group of dipolar molecules (plus an ion in the case of ionic conduction) undergoes a transition from one locally stable configuration to another. Figure 6 schematically depicts the “initial” state as a transient, locally stable configuration of a group of dipolar molecules. For the sake of simplicity, the figure illustrates six molecules, although there are probably more nearby molecules that should be considered. The activation energy Ea is the energy necessary to transform this local configuration to a higher-energy transition state indicated in the figure. The magnitude of the activation energy is governed by the intermolecular forces dynamically coupling the positions and orientations of the species with each other. The transition state then relaxes to a “final” state, where the dipoles have slightly different positions and orientations than in the initial state. In the case of ionic conduction, an external field would impose a preferred direction for the final position of the ion, whereas an external field imposes a preferred orientation for dipolar molecules undergoing dielectric relaxation. Presumably, the final state is another transiently stable local configuration with energy similar to that of the initial state. The transition state in Figure 6 is drawn as a slightly more disordered open structure that creates low-energy relaxation paths for the group of molecules. The creation of an unstable configuration with a lower-energy relaxation path is reminiscent of the formation of a transient “void” in free volume theory,

Compensated Arrhenius Formalism and Self-Diffusion which was introduced by Cohen and Turnbull50 to describe diffusion and later used by others to characterize ionic conduction.51 However, contrary to the arguments of Cohen and Turnbull, the formation of a local configuration analogous to a void is an activated process. Ratner showed that free volume theory is consistent with the Vogel-Tamman-Fulcher equation that is used to describe the curvature in plots of the diffusion or ionic conductivity data versus reciprocal temperature.52 However, we account for the curvature in a more direct way by considering the temperature dependence of the dielectric constant in the exponential prefactor of an Arrhenius-like expression (the “compensated” Arrhenius equation). In this paper, we utilize a compensated Arrhenius expression to describe the temperature dependence of the diffusion coefficient as we have previously done for dielectric relaxation21 and ionic conductivity.20,32 The similarity of the compensated Arrhenius activation energies for self-diffusion and dielectric relaxation to those for ionic conductivity (Tables 1 and 2) requires some further discussion. These similarities have interesting implications for the mechanism of ion transport in a liquid electrolyte because of our earlier postulate that the position and orientation of a mobile ion is dynamically coupled to the position and orientation of surrounding solvent molecules through intermolecular forces that also couple the motions of the solvent molecules to each other. In the cases of self-diffusion and dielectric relaxation of pure liquids, the intermolecular forces coupling the motion of the molecules to each other are the same. However, in the case of ion transport, the intermolecular forces coupling an ion to the solvent molecules will be primarily ion-dipole forces in the case of a polar solvent, while the forces coupling the motion of the solvent molecules with each other will be primarily dipole-dipole forces. Consequently, it might be expected that the activation energies for ionic conductivity would necessarily be quite different than those obtained from diffusion and dielectric relaxation studies. However, the striking similarity of the compensated Arrhenius activation energy value for ionic conductivity in a particular solvent with the values obtained from self-diffusion and dielectric relaxation studies of the same solvent can be rationalized by noting that the rearrangement of N solvent molecules during the transition from the initial to the final state is governed by N(N - 1)/2 dipole-dipole intermolecular interactions and N ion-dipole interactions between an ion and the surrounding solvent molecules. Since the rearrangement of solvent molecules presumably involves more than just nearest-neighbor solvent molecules, it seems reasonable that the value of the activation energy is governed primarily by dipole-dipole interactions with only a modest contribution from ion-dipole interactions, at least in the dilute, monovalent electrolyte solutions studied to date. Acknowledgment. We thank Dr. Susan Nimmo for her assistance with the setup for the NMR diffusion experiments. We also thank the National Science Foundation for funding the NMR equipment (Grant CHE#0639199). We gratefully acknowledge support for M.P. provided by C-SPIN, the Oklahoma/ Arkansas MRSEC Grant No. DMR-0520550. Supporting Information Available: The temperature-dependent diffusion coefficient and static dielectric constant data used. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Bockris, J.; Reddy, A. Modern Electrochemistry, (sections 4.4.7 4.4.12), 2nd ed.; Plenum Press: New York, 1998; Vol. 1.

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