Concentration Dependence of Molal Conductivity and Dielectric

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Concentration Dependence of Molal Conductivity and Dielectric Constant of 1-Alcohol Electrolytes using the Compensated Arrhenius Formalism Allison M. Fleshman, Matt Petrowsky, and Roger Frech J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp312243d • Publication Date (Web): 25 Mar 2013 Downloaded from http://pubs.acs.org on April 5, 2013

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The Journal of Physical Chemistry

Concentration Dependence of Molal Conductivity and Dielectric Constant of 1-Alcohol Electrolytes Using the Compensated Arrhenius Formalism Allison M. Fleshman, Matt Petrowsky, and Roger Frech∗ Department of Chemistry and Biochemistry, University of Oklahoma, Norman, OK E-mail: [email protected]

Phone: (1) 405 3254811. Fax: (1)405 3256111

∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment


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Abstract The molal conductivity of liquid electrolytes with low static dielectric constants (εs < 10) decreases to a minimum at low concentrations (region I) and increases to a maximum at higher concentrations (region II) when plotted against the square root of the concentration. This behavior is investigated by applying the compensated Arrhenius formalism (CAF) to the molal conductivity, Λ, of a family of 1-alcohol electrolytes over a broad concentration range. A scaling procedure is applied that results in an energy of activation (Ea ) and an exponential prefactor (Λ0 ) that are both concentration-dependent. It is shown that the increasing molal conductivity in region II results from the combined effect of (1) a decrease in the energy of activation calculated from the CAF, and (2) an inherent concentration dependence in the exponential prefactor that is partly due to the dielectric constant. Keywords: ionic mobility, ionic association, permittivity, hydrogen bonding, activation energy

Introduction A fundamental question concerning liquid electrolytes is: how does the ionic conductivity change with the addition of salt? In particular, for liquid and polymer electrolyte systems with a low dielectric constant, three distinct regions are observed when the molal conductivity is plotted versus the square root of the salt concentration, 1–5 depicted schematically in Figure 1, where Λ is defined as the ionic conductivity, σ , divided by the total salt concentration, c. An initial decrease to a

Figure 1: Schematic of molal conductivity vs. square root of the salt concentration depicting three different regions labelled I, II, & III for a low permittivity electrolyte.

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minimum is observed at low concentrations (region I), then an increase to a maximum (region II), followed by a decrease (region III). The three regions are labeled following the notation of Albinsson et al. 1 This behavior is seen in both protic and aprotic liquid electrolytes as well as polymer electrolytes. 1,3,6–9 The focus of this work is to explain the decrease in region I and the increase in region II for 1-alcohol based electrolytes. Previous studies have proposed several explanations for the minimum and maximum behavior. These interpretations are based primarily on changes in ionic association, where both the extent of association and nature of associating species varies with concentration. The initial decrease in region I of both polymer electrolytes and organic liquid electrolytes is ascribed to a decrease in the number of “free” ions by the formation of neutral ion pairs. 1,3,8 One explanation for the subsequent increase of Λ in region II is that a shift in the association equilibrium from neutral pairs back into “free” ions results in a “redissociation” effect. 6–8 Another explanation, however, claims that the low dielectric constant allows for the formation of triple ions introducing more charge carriers and thus an increase in Λ. 2,10 Ferry et al., however, determined that the increase in region II in polypropylene glycol LiCF3 SO3 systems is not governed by the variation of population of neutral ion pairs, and instead postulated that an increase of the ionic mobilities with concentration causes the increase in Λ. 9 Spectroscopic data were used to determine the percentage of “free,” contact-ion pair, and aggregate ions across the concentrations corresponding to the increase of Λ in region II. They concluded that the ionic mobilities must increase over the concentration range, because the percent of “free” ions and contact-ion pairs each remained constant. 9,11 We have observed regions I, II, and III in electrolyte systems that do not exhibit ionic association, 12 therefore we agree with the interpretation of Ferry et al.; the increase of Λ in region II is due to an increase in the ionic mobility with concentration. In this work, we show that the transition from region I to region II and the increase in region II can be explained in terms of the ionic mobilities using information gained from the compensated Arrhenius formalism (CAF) rather than arguments for ionic association. Recently, Yamaguchi et al. explored the equivalent conductivity in a computational study using Brownian dynamics. 13–15 Simulations at different concentrations show a minimum in the equivalent conductivity when the Coulombic

3



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interactions are strong, as is the case in solvents of low dielectric constant. The CAF takes an unconventional view of mass and charge transport that assumes transport to be a thermally activated process. This formalism has been successfully tested with ionic conductivity, 16–20 dielectric relaxation, 21 and self-diffusion 20,22 of organic liquid electrolytes and pure protic and aprotic solvents. The CAF assumes the temperature-dependent conductivity can be written in an Arrhenius form as �

� −Ea σ = σ0 (εs (T )) exp . RT

(1)

Here σ0 is the exponential prefactor, εs is the solution static dielectric constant, Ea is the energy of activation, R is the gas constant, and T is temperature. This formalism postulates that the temperature dependence of the exponential prefactor is due to the temperature dependence of the dielectric constant. A scaling procedure can be performed that removes the exponential prefactor allowing for the calculation of Ea , which governs the thermally activated process. 16,18,19 We have previously used the CAF to describe the differences between two concentrations of 1-alcohol electrolytes: 0.035 m (located at the dilute boundary of region II) and 0.35 m (located at the upper boundary of region II). 16 Here, we continue the previous study to include a broad range of concentrations that begins in region I and spans all of region II for 1-alcohol electrolytes. The relationship between the conductivity and dielectric constant afforded by the CAF can be extended to the ionic mobilities. The general expression for the conductivity of an electrolyte is σ = ∑ zi ci F µi

(2)

i=1

where zi is the valence of the ith ion, ci is the concentration of species i, F is Faraday’s constant, and µi is the ionic mobility of species i. For a simple monovalent electrolyte that is completely dissociated in solution, zi =1 and c+ = c− = cm , where cm is the total concentration of the electrolyte

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in units of molality (moles of solute per kg solvent). Eq. (2) then becomes σ = c+ F µ+ + c− F µ− = cm F (µ+ + µ− )

(3)

Dividing Eq. (3) by the total salt concentration, cm , results in the familiar expression for the molal conductivity, Λ of our simple electrolyte Λ=

σ = F (µ+ + µ− ). cm

(4)

If discrete, ionically associated species do form, the relationship in Eq. (4) becomes more complicated. To isolate the effect of the mobility on Λ such that Eq. (4) is valid, we select tetrabutylammonium (Tba) trifluoromethanesulfonate (CF3 SO3 , labelled Tf) as the salt in our system. The bulky butyl groups protect the charge on the nitrogen and hinder association. We have previously shown that TbaTf exists only as “free” ions in DMSO, 12 and we will show similar evidence in 1-alcohol systems at the highest concentration measured here. Here we combine the relationship of Eq. (4) with the results of the CAF to determine the origin of the concentration-dependent behavior of Λ for simple monovalent, low permittivity 1-alcohol electrolytes that do not exhibit ionic association. We show that the transition from region I to region II and the increase in Λ in region II result from two distinct concentration-dependent factors: (1) the concentration dependence contained in the exponential prefactor and (2) the concentration dependence of the energy of activation.

Experimental All chemicals were acquired from either Sigma Aldrich or Alfa Aesar, stored in a nitrogen atmosphere glovebox (≤1 ppm H2 O), and used as received. Solutions were prepared in the glovebox at an ambient temperature of 27◦ C and allowed to stir for at least 24 hours before measurements were taken. All solution compositions are molal concentrations (moles solute per kg of solvent, 5



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abbreviated m). We use the molality scale because it is independent of thermal expansion by the liquids, whereas the molarity scale is temperature-dependent. Both conductivities and dielectric constants were measured using an HP 4192A impedance analyzer with an Agilent 16452A liquid test fixture. The electrodes are nickel-plated circular disks made of 54 % Fe, 17 % Co, and 29 % Ni and have a diameter of 38 mm; the electrodes are spaced 2 mm apart. The capacitance (C), conductance (G), and phase angle (θ ) were measured with a logarithmic sweep over a frequency range 1 kHz to 13 MHz. The instrument was set to parallel circuit and averaging (slow) mode. A short compensation was performed at 10 MHz. A Huber ministat 125 bath was used to regulate the temperature to ±0.1◦ C from 5–85◦ C, in 10◦ C increments. The conductivity, σ , is calculated from the measured conductance, G, through the equation σ = L × G × A−1 , where L is the electrode gap and A is the electrode area. The static dielectric constant, εs , is calculated from the measured capacitance C using εs = α × C × C0−1 , where α is a variable to account for stray capacitance, and C0 is the atmospheric capacitance. 23 Measuring the static dielectric constant of an ionically

conducting solution is complicated because the capacitance in the limit of low frequency is artificially high due to electrode polarization effects. 24–28 A detailed description of the technique used to determine the solution dielectric constant and conductivity, as well as the error associated with the measurements has been given previously. 16 Infrared data were collected with a Bruker IFS66V Fourier Transform Infrared (FTIR) spectrometer with a potassium bromide beamsplitter. Data were recorded with a spectral resolution of 1 cm−1 over the range 500 – 4000 cm−1 . The data were averaged over 64 scans under N2 purge. Samples were placed between two sodium chloride (2 mm thickness and 25 mm diameter) windows and secured in a Harrick temperature-controlled demountable liquid cell (model TFC-M25-3). The temperature was controlled using both a Neslab coolflow CFT-33 refrigerated regulator and an Omega CN9000A digital temperature controller.

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Results and discussion

R ! 104 (S kg cm-1 mol-1)

Concentration dependence of molal conductivity, Λ

R ! 104 (S kg cm-1 mol-1)

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80

1-hexanol

15"C 45"C 85"C

60 40 20 0 12 1-decanol

10 8 6 4 2 0 0.0

0.2

0.4 0.6 c1/2 (mol/kg)1/2

0.8

Figure 2: Molal conductivity vs. square root of concentration for TbaTf 1-hexanol (top) and TbaTf 1-decanol (bottom) for 15◦ C (blue squares), 45◦ C (red circles), and 85 ◦ C (gray crosses). Figure 2 shows the molal conductivity versus square root of the concentration for solutions of TbaTf 1-hexanol (top) and 1-decanol (bottom) at three temperatures: 15, 45, and 85◦ C. The behavior of Λ in the vicinity of the minimum is best observed when the data are plotted as a function of c1/2 . The 1-hexanol data at 15 and 45 ◦ C do not show region II behavior with increasing concentration, but the 85◦ C data show a distinct increase in Λ at concentrations greater than about 0.14 m1/2 . For all temperatures shown here, Λ of 1-decanol electrolytes demonstrates a distinct transition from region I (decrease with concentration) to region II (increase at about 0.07 m1/2 ). For both solvents, the increase of Λ in region II becomes more apparent at higher temperatures. The molal conductivity decreases as the chain length increases, similar to the behavior of the dielectric constant. 16,18,19 This same behavior has been observed for solutions of LiTf in 1-alcohols 7



at 25◦ C. 12 Increasing the temperature and extending the chain length of the alcohol enhances the distinct behavior characterizing regions I and II, which is in part due to the magnitude and concentration dependence of the dielectric constant. As mentioned in the Introduction, the increase in Λ in region II has been attributed to changes in ionic association. We choose TbaTf as the salt because it is known to not form discrete ionically associating species. 12,29–32 Figure 3 shows IR spectra for 0.6 m TbaTf 1-decanol. A single

1.0 (A.U.)

Absorbance

15C 35C 55C 0.5

Absorbance (A.U.)

1.5

0.6 m TbaTf 1-decanol

71C

pure 1-decanol 35C

0.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1010

1020

1030 1040 1050 Wavenumbers (cm-1)

1060

1070

Figure 3: IR spectra of 0.6 m TbaTf 1-decanol at 15, 35, 55, and 71◦ C and pure 1-decanol at 35◦ C. peak is seen at 1032 cm−1 in the SO3 symmetric stretching region of the Tf anion, which has been assigned to the "free" ion. 31 This demonstrates that there is no spectroscopically detectable indication of ionic association of the Tf anion in 1-decanol at the highest concentration over several temperatures. A spectrum of the TbaTf crystal (available in the supporting information) also shows a single band at 1032 cm−1 . Therefore, the behavior of Λ in Figure 2 for TbaTf 1-alcohol electrolytes cannot be due solely to changes in ionic association with concentration, i.e., the variation of contact ion-pair populations. There could, however, be changes in the number of solvent separated ion pairs, which are difficult to detect with vibrational spectroscopy. 33

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Compensated Arrhenius formalism The CAF provides an energy of activation and exponential prefactor for 1-alcohol electrolytes that each have a concentration dependence contributing to the behavior of Λ in Figure 2. The scaling procedure to determine Ea has been given in detail elsewhere, 16,19 but is briefly summarized here. Reference conductivities, σr , are determined from a plot of isothermal conductivity versus dielectric constant for TbaTf-alcohol electrolytes at the same concentration. This plot is defined as the reference curve at the reference temperature, Tr . An example of the reference curve is discussed later in the text and given in Figure 6. The temperature dependent conductivity, σ (T ) in Eq. (1), is divided by the reference conductivity corresponding to the same value of the dielectric constant. The natural logarithm is taken of σ /σr and the resulting expression is called the compensated Arrhenius equation (CAE), shown as Eq. (5). A plot of the natural log of the scaled conductivity versus reciprocal temperature is called a compensated Arrhenius plot, and Ea is calculated from either the slope or the intercept. �

σ (T, εs ) ln σr (Tr , εs )

�

=−

Ea 1 Ea + R T R Tr

(5)

The CAF is applied to the temperature dependent conductivity of TbaTf 1-alcohol solutions over a concentration range of 0.00042 – 0.60 m. The simple Arrhenius equation (SAE): σ = σ0 exp[−Ea /R T ], is also used for comparison where σ0 is assumed to be temperature independent. Figures 4 and 5 show simple Arrhenius plots (left axis, filled circles, SAE) and compensated Arrhenius plots (right axis, open diamonds, CAE) for the highest and lowest concentrations of TbaTf of the two end members of the alcohol family studied here: 1-hexanol and 1-decanol over the temperature range 5 – 85◦ C (15 – 85◦ C for 1-decanol). Conductivity and dielectric constant data for all concentrations measured are available in the Supporting Information. Table 1 gives Ea values that are determined from the CAE plots and those SAE plots that demonstrate linearity. The CAE Ea value for each family member is calculated by averaging the Ea values obtained from the slope and the intercept of the compensated Arrhenius

9



TbaTf 1-hexanol

TbaTf 1-decanol 4

-6.0

0.6 mol kg-1

-6.5

2

0

-7.5 -8.0 -8.5

2

CAE: R =0.999 SAE: R2=0.999

-2

-1 ln m(T)

-7.0

-8

-2 -3

-9 2

-10

CAE: R =0.999 SAE: R2=0.998

-4 -5

-12.0 0.00042 mol kg

0

-1

0.00042 mol kg

-15.4

2

£"m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

ln m(T)

0.6 mol kg-1

-7

£"m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

-1

0

-13.0

-13.5

-2

2

CAE: R =0.998 SAE: R2=0.961

-15.6 ln m(T)

0

3.0 3.2 3.4 T-1! 103 (K-1)

-15.8 -4 -16.0

2

CAE: R =0.999 SAE: R2=0.683

-6

-16.2

-4 2.8

-2

3.6

2.8

Figure 4: Simple Arrhenius plots (left axis, filled circles) and compensated Arrhenius plots (right axis, open diamonds) for 0.6 m (top) and 0.00042 m (bottom) TbaTf 1hexanol.

3.0 3.2 T-1! 103 (K-1)

£"m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

ln m(T)

-12.5

£"m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

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3.4

Figure 5: Simple Arrhenius plots (left axis, filled circles) and compensated Arrhenius plots (right axis, open diamonds) for 0.6 m (top) and 0.00042 m (bottom) TbaTf 1decanol.

plot. These Ea values for each member are then averaged together at each concentration. The non-scaled conductivity data (filled circles in Figures 4 and 5) for the lowest concentrations show the greatest deviation from Arrhenius-like behavior. As the concentration increases, the nonscaled conductivity data become more linear with T −1 , which allows for the calculation of an Ea based on the SAE. The transition from non-Arrhenius to Arrhenius behavior occurs between 0.035 and 0.1 m for both 1-hexanol and 1-decanol (data not shown). The linearity of the non-scaled conductivity at higher concentrations has been discussed previously. 16 Applying the CAF to the dilute electrolytes corrects the non-linearity and yields Arrhenius-like behavior (open diamonds). Table 1 shows a distinct decreasing trend in Ea values from the CAF with increasing concentration. This decrease is most likely due to a reduction of the intermolecular interactions of the solvent with the presence of TbaTf, which could be acting as a structure-breaker reducing the extended hydrogen bonding network. It is important to note that the same Ea values given in Table 1

10


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Table 1: Average energies of activation for TbaTf 1-alcohols calculated from the CAF (Eq. (5)) and from the simple Arrhenius expression (σ = σ0 exp(−Ea /RT )) (SAE). Ea values could not be determined from non-linear SAE plots and are left blank. CAE SAE c Ea Ea 1/2 −1 (m) (m ) (kJ mol ) (kJ mol−1 ) 0.00042 0.02 52.9 ± 0.3 0.0012 0.03 51.8 ± 0.5 0.0050 0.07 49.9 ± 0.3 0.012 0.11 48.2 ± 0.5 0.020 0.14 47.7 ± 0.4 0.035 0.19 47.9 ± 0.2 0.10 0.32 44.5 ± 0.2 20.3 ± 0.2 0.20 0.45 42.7 ± 0.3 22.9 ± 0.2 0.35 0.59 41.5 ± 0.2 25.4 ± 0.2 0.48 0.69 39.9 ± 0.2 25.2 ± 0.2 0.60 0.77 39.1 ± 0.2 26.7 ± 0.2 c1/2

result when the CAF is applied to Λ(T ) or σ (T ), as expected. The same is true when the SAE is applied to Λ(T ). The Ea values from the simple Arrhenius equation were calculated from 0.1 to 0.6 m and increase with increasing concentration. This concentration range corresponds to the range where Λ increases with concentration, particularly for 1-decanol and 1-hexanol at 85◦ C in Figure 2. An increasing Ea necessarily results in a decreasing conductivity predicted by the SAE. Hence the SAE provides an especially poor description of the temperature-dependent conductivity in these (and other) systems. The decrease in Ea observed for the CAE values is consistent with the increase in Λ observed in Figure 2 and illustrates one factor determining the concentration dependence of Λ. The other concentration-dependent component in Λ, and therefore in σ originates in the exponential prefactor partly through its dependence on the dielectric constant. Compensated Arrhenius formalism: exponential prefactor The CAF shows that the temperature dependence of the conductivity is due in part to the temperature dependence of the dielectric constant. Figure 6 shows conductivity data for 0.6 m (top), 0.1 m (middle), and 0.00042 m (bottom) TbaTf 1-alcohol solutions plotted as a function of dielectric

11



85"C 75"C 65"C (7) 55"C 45"C (8) 35"C 25"C (9) 15"C (10) 5"C

150 100

(6)

0.6 mol kg-1

m0 ! 10-3 (S cm-1)

m ! 105 (S cm-1)

200

50

(6)

0.1 mol kg-1

m0 ! 10-3 (S cm-1)

m ! 105 (S cm-1)

3 2

85"C 75"C 65"C 55"C 45"C 35"C 25"C 15"C 5"C

Ea = 39.1 kJ mol-1

1 0

25 20 (7)

15 (8)

10

(9) (10)

5

10 Ea = 44.5 kJ mol-1

8 6 4 2

0.1 mol kg-1

0

0 (6)

10

0.00042 mol kg-1

m0 ! 10-3 (S cm-1)

0.3 0.2 (7)

0.1

4

0.6 mol kg-1

0

m ! 105 (S cm-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(8) (9) (10)

8

Ea = 52.9 kJ mol-1

6 4 2 0.00042 mol kg-1

0

0.0 4

6

8

10 12 ¡s

14 16

18

4

6

8

10

12

14

16

18

¡s

Figure 6: Conductivity vs. dielectric constant for (top) 0.6 m (middle) 0.1 m and (bottom) 0.00042 m TbaTf 1-alcohol solutions. The numbers correspond to the curve for each family member (6) 1-hexanol, (7) 1-heptanol, (8) 1-octanol, (9) 1-nonanol, (10) 1-decanol.

Figure 7: Exponential prefactors vs. dielectric constant for (top) 0.6 m (middle) 0.1 m and (bottom) 0.00042 m TbaTf 1-alcohol solutions. Ea values correspond to average values calculated from the CAE and are given in Table 1.

constant. The data separate into five distinct curves, one for each solvent family member, and are labelled according to the number of carbons in the alkyl chain. Each isothermal curve defined by a particular symbol is a reference curve at that reference temperature, Tr , for the given concentration.

12


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10000

m0 (S cm-1)

Page 13 of 37

1000

100 85!C 45!C 5!C

10 0.0

0.1

0.2 0.3 0.4 c (mol kg-1)

0.5

0.6

Figure 8: The exponential prefactor versus concentration of TbaTf in 1-octanol at 5, 45, and 85◦ C. As an example, in the scaling procedure for 0.6 m TbaTf 1-octanol (the curve labelled 8), Tr was chosen as 45◦ C (corresponding to the red circles). The conductivity increases by three orders of magnitude over the concentration range presented here. At higher concentrations, the conductivity data show different behavior with dielectric constant than the lower concentrations because of a reduced temperature dependence of εs (demonstrated and discussed with Figure 10). The exponential prefactor is determined by dividing σ (T ) by the quantity exp[−Ea /R T ] using the Ea calculated from the CAF and averaged from the slope and the intercept. When plotted as σ0 versus εs , the data of Figure 6 collapse to single master curves shown in Figure 7. This supports the primary assumption of the CAF: the temperature dependence of σ0 is due to the temperature dependence of the dielectric constant. The master curves all show a similar, exponential-like dependence on εs , but the magnitudes vary with concentration. The concentration dependence of σ0 is better illustrated in Figure 8. There exists a trivial concentration dependence in σ0 due to the increased number of ions present. A plot of σ0 versus c should therefore result in a linearly increasing relationship of σ0 with c. Due to an additional concentration dependence contained in the mobility, this linear relationship is not observed, as shown in Figure 8. The concentration dependence of σ0 is different in two concentration regimes: the dilute region in which the value of σ0 is increasing sharply with concentration and the moderately concentrated region where σ0 either decreases slightly or levels off. This trend becomes more distinct at higher temperatures. The concentration dependence of σ0 is partly due to the concentration 13



dependence of εs , which will be discussed in greater detail in the next section. The concentration dependence of both Ea and σ0 contribute to the concentration dependence of Λ, however, a more direct comparison is afforded by dividing out the trivial concentration dependence of σ0 , i.e., defining Λ0 as σ0 /c. The behavior of Λ in region I and II results from both the concentration

R0 (S kg cm-1 mol-1)

R " 104 (S kg cm-1 mol-1)

dependence of Ea and the concentration dependence of Λ0 . TbaTf 1-octanol

20 15 10 5 0 107 106 105 104 103

10-5 exp[-Ea/R T]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

10-6 10-7 10-8 85!C 45!C 5!C

10-9 10-10 0.0

0.2

0.4 0.6 c1/2 (mol kg-1)1/2

0.8

Figure 9: Concentration dependence (plotted as c1/2 ) of (top) molal conductivity, (middle) molal exponential prefactor, and (bottom) Boltzmann factor at 5, 45, and 85◦ C for TbaTf 1-octanol solutions. The top of Figure 9 illustrates the concentration dependence of the molal conductivity, Λ, for TbaTf 1-octanol at 5, 45, and 85◦ C. The transition of Λ from region I to region II is more apparent at the higher temperatures, similar to the data in Figure 2. The lower plots of Figure 9 separately show the two contributing factors of Λ: Λ0 and exp[−Ea /RT ]. At low concentrations, Λ0 dominates the behavior of Λ. At higher concentrations, after the transition from region I to region II, the 14


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Boltzmann factor dominates the behavior of Λ. The concentration range encompassing the transition is clearly seen in both Λ0 and exp[−Ea /RT ]. The concentration dependence of the Boltzmann term is contained in the concentration dependence of Ea , given in Table 1. The concentration dependence of Λ0 (and therefore the additional, non-trivial concentration dependence in σ0 ) is due to the concentration dependence of the dielectric constant. As the concentration increases, the values of Λ0 for the different temperatures begin to converge as a result of the reduced temperature dependence of the dielectric constant, which also reduces the concentration dependence of σ0 . Figure 10 plots εs versus temperature for three concentrations of TbaTf 1-octanol. The dielectric constants of the 0.00042 m and 0.035 m electrolytes show a similar temperature dependence but a slight offset in the magnitude of εs . As the concentration increases, εs changes less with temperature, shown by data of the 0.6 m electrolyte solution. This reduction in the temperature dependence of εs directly affects the decrease in Λ0 at higher concentrations. TbaTf 1-octanol 0.6 m 0.035 m 0.00042 m

12

¡s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 8 6 0

20

40 T (!C)

60

80

Figure 10: Dielectric constant vs. temperature of TbaTf 1-octanol at 0.6 m (grey crosses), 0.035 m (red circles) and 0.00042 m (blue triangles) from 5 – 85◦ C. The lines represent linear regression fits. The concentration dependence in εs is given in Figure 11, which shows the concentrationdependent behavior of εs at three temperatures. At low concentrations, εs increases with concentration and then levels off or slightly decreases at higher concentrations. Although the behavior of εs with concentration appears to be similar to that of σ0 , the intrinsic concentration dependence 15



14 12 ¡s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

10 8 85!C 45!C 5!C

6 0.0

0.1

0.2 0.3 0.4 c (mol kg-1)

0.5

0.6

Figure 11: Dielectric constant vs. concentration at 5, 45, and 85◦ C for TbaTf 1-octanol solutions. contained in σ0 prohibits a direct comparison between εs and σ0 . We conclude that one contribution of the concentration dependence in σ0 is due to the concentration dependence in εs , with an additional cm dependence that is independent of εs i.e., σ0 (εs (cm ), cm ). The origin of the increase of εs with increasing salt concentration is not well understood. Gestblom et al. 34,35 and others 5,36 attribute the increase in εs with salt concentration to an increase in the total number of dipoles by the addition of dipoles formed by ion pairs; the IR data presented in Figure 3 do not support this claim since TbaTf is unable to form discrete, ionically associated species in solution. Previous studies involving aqueous monovalent electrolytes have shown that the dielectric constant decreases with increasing concentration. 28,37 The variation of the dielectric constant in alcohol-based electrolytes is not consistent, however, and depends on the nature of the salt. Gestblom et al. measured εs in solutions of LiCl and CaCl2 ·2H2 O in 1-propanol 34 and 1hexanol 38 and found it to decrease with concentration for LiCl but increase for CaCl2 ·2H2 O. They also reported that εs remains constant with increasing concentration of CaCl2 ·2H2 O in ethanol

while it decreases with LiCl and increases with Ca(NO3 )2 ·4H2 O. 38 The use of hydrated salts affects the dielectric constant behavior with concentration differently depending on the system, and therefore complicates the analysis. Sigvartsen et al. reported an increase in εs with increasing concentrations of TbaClO4 in several different non-aqueous solvents ranging in dielectric constant from 3 – 20. 5 It is unclear why εs increases with concentration to a maximum and then decreases, but the inherent concentration dependence of εs directly affects Λ0 , and therefore σ0 . 16


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Comparison to previous work: Ea of TbaTf 1-alcohol solutions We have previously shown that the average Ea calculated from the CAF for TbaTf 1-alcohol solutions increases with concentration from approximately 32.6 to 39.5 kJ mol−1 over the concentration range 5.10 × 10−5 to 0.0066 m for short to moderate chain length alcohols, 18 which does not agree with the Ea values presented here (52.9 – 49.9 kJ mol−1 ). Short chain alcohols have higher dielectric constants (εs > 10) than the moderate to long chain alcohols and therefore do not show region II behavior. Consequently, we expect that these members may exhibit somewhat different values of Ea in addition to their different concentration behavior. We have shown that selecting short alkyl chain family members for the scaling procedure in aprotic solvents (e.g., methyl–butyl) results in a lower average Ea value. 20,39 The 1-alcohol members used in the previous study included ethanol, 1-propanol, 1-butanol, and 1-hexanol, which resulted in Ea values approximately 10 kJ mol−1 less than the long chain alcohols studied here. The Ea for 0.005 m was recalculated using the same 1-alcohol members from the previous study, and was found to be 40 ± 1 kJ mol−1 , which is consistent with the values reported. 18

Summary and Conclusion It is clear that models based on ionic association do not adequately describe the behavior of the molal conductivity with concentration for low dielectric constant electrolytes (shown in Figure 1) because the same behavior is seen using TbaTf as the salt (Figure 2), which exists as spectroscopically “free” ions. The decrease of Λ in region I and the increase in region II can, however, be described by the combined effect of the concentration dependence of both Ea and the molal exponential prefactor, Λ0 , the latter having a dependence on the dielectric constant. It is therefore necessary to use the CAF to compensate for this dielectric constant dependence to calculate an appropriate Ea . The CAF postulates that all of the temperature dependence of σ0 is due to the temperature dependence of εs , or σ0 (εs (T )). Extending this assumption to the definition for Λ given in Eq. (4) 17



55

Ea (kJ mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 37

50

45

40 0.0

0.2

0.4 0.6 c1/2 (mol kg-1)1/2

0.8

Figure 12: Energy of activation vs. square root of the concentration for TbaTf 1-alcohol solutions. The inset is molal conductivity versus square root of the concentration at 5, 45, and 85◦ C for TbaTf 1-octanol solutions. yields: Λ = Λ0 (εs (T, cm ), cm )e

−Ea (cm ) RT

= F (µ+ + µ− ).

(6)

The sum of the ionic mobilities is the only quantity on the right hand side of the equation that can have a concentration dependence. The addition of TbaTf alters the Ea as well as the magnitude of the dielectric constant and its temperature dependence, which in turn, affects the sum of the ionic mobilities, (µ+ + µ− ). It appears that (µ+ + µ− ) decreases in region I, becomes approximately concentration independent through the minimum, and increases with concentration in region II. We suggest that (µ+ + µ− ) has both a temperature and concentration dependence that is governed by changes in the intermolecular interactions. The calculated CAF Ea values (Figure 12) show a decreasing trend with concentration that is divided into two different behaviors. A sharp decrease at low concentrations followed by a gradual decrease. The reduction of Ea is likely due to a reduced hydrogen bonding network caused by the addition of TbaTf. Further studies are needed to determine the extent that a reduction in hydrogen bonding will have on the average Ea of the electrolyte system given that the relationship between dielectric constant and the extent of hydrogen bonding is not straightforward. If the dielectric constant changes with concentration, the effect will also be observed in the concentration dependence of Λ0 . It is therefore essential to include the dielectric constant when describing the concentration dependence of charge transport. A molecular-level 18


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understanding of how the ions affect the dielectric constant of the solution is still unavailable. It is clear, however, that the source of the concentration dependence of Λ originates from the concentration dependence of Ea and Λ0 , the latter being due in part to the concentration dependence of εs .

Acknowledgement We would like to thank the Army Research Office for support of this work through Grant No. W911NF-10-1-0437. We thank Matthew Johnson in the OU Physics Department and his research group: Jeremy D. Jernigen, R. S. P. Bokalawela, and Chris Crowe for support in automating our impedance analyzer system. We thank John Moore Furneaux for implementing temperature control for the Bruker IR system and Whitney Booher for help in data collection.

Supporting Information Available All temperature dependent conductivity and dielectric constant data for each concentration of TbaTf used for the CAF analysis are available in the Supporting Information. A vibrational spectrum of the TbaTf crystal is also given. This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Albinsson, I.; Mellander, B.-E.; Stevens, J. R. Ionic Conductivity in Poly(propylene Glycol) Complexed With Lithium and Sodium Triflate. J. Chem. Phys. 1992, 96, 681–690. (2) Bruce, P. G.; Vincent, C. A. Polymer Electrolytes. J. Chem. Soc. Faraday Trans. 1993, 89, 3187–3203. (3) Gray, F. M. An Interpretation of Raman Spectral Data For Polymer Electrolytes in the Light of New Evidence For Ion association in Dilute Solution. J. Polym. Sci. Pol. Phys. 1991, 29, 1441–1445. 19



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(4) Kraus, C. A.; Fuoss, R. M. Properties of Electrolytic Solutions. I. Conductance as Influenced by the Dielectric Constant of the Solvent Medium. J. Am. Chem. Soc. 1933, 55, 21–36. (5) Sigvartsen, T.; Gestblom, B.; Noreland, E.; Songstad, J. Conductometric and Dielectric Behavior of Solutions of Tetrabutylammonium Perchlorate in Solvents of Low and Medium Permittivity. Acta Chem. Scand. 1989, 43, 103–115. (6) Cameron, G. G.; Ingram, M. D.; Sorrie, G. A. The Mechanism of Conductivity of Liquid Polymer Electrolytes. J. Chem. Soc. Faraday T. 1 1987, 83, 3345–3353. (7) Cavell, E. A. S.; Knight, P. C. Effect of Concentration Changes on Permittivity of Electrolyte Solutions. Z. Phys. Chem. 1968, 57, 331–4. (8) Stygar, J.; Biernat, A.; Kwiatkowska, A.; Lewandowski, P.; Rusiecka, A.; Zalewska, A.; Wieczorek, W. Effect of Cation and Salt Concentration on Conductivity and Microstructure Characteristics of Polyether Electrolytes Doped With Alkali Metal Perchlorates. J. Phys. Chem. B 2004, 108, 4263–4267. (9) Ferry, A.; Jacobsson, P.; Torell, L. M. The Molar Conductivity Behavior in Polymer Electrolytes At Low Salt Concentrations; a Raman Study of Poly(propylene Glycol) Complexed With LiCF3 SO3 . Electrochim. Acta 1995, 40, 2369–2373. (10) Fuoss, R. M.; Kraus, C. A. Properties of Electrolytic Solutions. IV. The Conductance Minimum and the Formation of Triple Ions Due to the Action of Coulomb Forces. J. Am. Chem. Soc. 1933, 55, 2387–2399. (11) Pehlivan, ˙I. B.; Georén, P.; Marsal, R.; Granqvist, C. G.; Niklasson, G. A. Ion Conduction of Branched Polyethyleneimine–lithium Bis(trifluoromethylsulfonyl) Imide Electrolytes. Electrochim. Acta 2011, 57, 201–206. (12) Petrowsky, M.; Frech, R. Concentration Dependence of Ionic Transport in Dilute Organic Electrolyte Solutions. J. Phys. Chem. B 2008, 112, 8285–8290. 20


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(13) Yamaguchi, T.; Shimoda, Y.; Koda, S. Effects of Hydrodynamic Interaction on the Equivalent Conductivity Minimum of Electrolyte Solutions in Solvents of Low Dielectric Constant. J. Chem. Phys. 2013, 138, 24503–24510. (14) Yamaguchi, T.; Akatsuka, T.; Koda, S. Brownian Dynamics Simulation of a Model Simple Electrolyte in Solvents of Low Dielectric Constant. J. Chem. Phys. 2011, 134, 244506– 244514. (15) Yamaguchi, T.; Matsuoka, T.; Koda, S. Dynamic Mechanism of Equivalent Conductivity Minimum of Electrolyte Solution. J. Chem. Phys. 2011, 135, 164511–164521. (16) Fleshman, A. M.; Petrowsky, M.; Jernigen, J. D.; Bokalawela, R. S. P.; Johnson, M. B.; Frech, R. Extending the Compensated Arrhenius Formalism to Concentrated Alcohol Electrolytes: Arrhenius vs. Non-Arrhenius Behavior. Electrochim. Acta 2011, 57, 147–152. (17) Petrowsky, M. Ion Transport in Liquid Electrolytes. Ph.D. Dissertation, University of Oklahoma, 2008. (18) Petrowsky, M.; Frech, R. Salt Concentration Dependence of the Compensated Arrhenius Equation For Alcohol-based Electrolytes. Electrochim. Acta 2010, 55, 1285–1288. (19) Petrowsky, M.; Frech, R. Temperature Dependence of Ion Transport: the Compensated Arrhenius Equation. J. Phys. Chem. B 2009, 113, 5996–6000. (20) Bopege, D. N.; Petrowsky, M.; Fleshman, A. M.; Frech, R.; Johnson, M. B. Temperature Dependence of Ion Transport in Dilute Tetrabutylammonium Triflate-Acetate Solutions and Self-Diffusion in Pure Acetate Liquids. J. Phys. Chem. B 2012, 116, 71–76. (21) Petrowsky, M.; Frech, R. Application of the Compensated Arrhenius Formalism to Dielectric Relaxation. J. Phys. Chem. B 2009, 113, 16118–16123.

21



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(22) Petrowsky, M.; Frech, R. Application of the Compensated Arrhenius Formalism to SelfDiffusion: Implications For Ionic Conductivity and Dielectric Relaxation. J. Phys. Chem. B 2010, 114, 8600–8605. (23) Agilent 16452A Liquid Test Fixture Operation and Service Manual. 2000. (24) Chang, H.-C.; Jaffe, G. Polarization in Electrolytic Solutions. I. Theory. J. Chem. Phys. 1952, 20, 1071–1077. (25) Johnson, J. F.; Cole, R. H. Dielectric Polarization of Liquid and Solid Formic Acid. J. Am. Chem. Soc. 1951, 73, 4536–4540. (26) Maruska, H.; Stevens, J. Technique For Extracting Dielectric Permittivity From Data Obscured by Electrode Polarization. Transactions on Electrical Insulation 1988, 23, 197–200. (27) Scheider, W. Theory of the Frequency Dispersion of Electrode Polarization. Topology of Networks With Fractional Power Frequency Dependence. J. Phys. Chem. 1975, 79, 127–136. (28) Van Beek, W. M.; Mandel, M. Static Relative Permittivity of Some Electrolyte Solutions in Water and Methanol. J. Chem. Soc. Faraday Trans. 1978, 74, 2339–2351. (29) Bacelon, P.; Corset, J.; Loze, C. Anion Solvation III. Infrared Spectroscopic Determination of Solvent Acceptor Numbers and their Use in Understanding Anion Solvation. J. Solution Chem. 1983, 12, 23–31. (30) Bacelon, P.; Corset, J.; Loze, C. Anion Solvation II. Solvation of Thiocyanate and Halide Anions in Mixtures of Protic and Aprotic Solvents. J. Solution Chem. 1983, 12, 13–22. (31) Frech, R.; Huang, W.; Dissanayake, M. Ionic Association of Lithium Triflate in Glymes, Model Solvents, and High Molecular Weight Poly(ethylene Oxide). Mater. Res. Soc. Symp. Proc. 1995, 369, 523–534. (32) Frech, R.; Huang, W. Anion-solvent and Anion-cation Interactions in Lithium and Tetrabutylammonium Trifluoromethanesulfonate Solutions. J. Solution Chem. 1994, 23, 469–481. 22


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(33) Hefter, G. When Spectroscopy Fails: The Measurement of Ion Pairing. Pure Appl. Chem. 2006, 78, 1571–1586. (34) Gestblom, B.; Mehrotra, S.; Sjöblom, J. Dielectric Properties of Electrolytes in 1-propanol. J. Solution Chem. 1986, 15, 55–68. (35) Gestblom, B.; Svorstoel, I.; Songstad, J. Dielectric Properties of Solutions of Some Onium Salts in Dichloromethane. J. Phys. Chem. 1986, 90, 4684–4686. (36) Cachet, H.; Cyrot, A.; Fekir, M.; Lestrade, J. C. Dielectric Relaxation of Lithium Perchlorate and Tetrabutylammonium Bromide Solutions. A Model of Ion Pairs. J. Phys. Chem. 1979, 83, 2419–2429. (37) Gulich, R.; Köhler, M.; Lunkenheimer, P.; Loidl, A. Dielectric Spectroscopy on Aqueous Electrolytic Solutions. Radiat. Environ. Bioph. 2009, 48, 107–114. (38) Gestblom, B.; Sjöblom, J. Dielectric Properties of Alcohol Electrolyte Solutions. J. Solution Chem. 1986, 15, 259–268. (39) Bopege, D. N.; Petrowsky, M.; Frech, R.; Johnson, M. B. J. Solution Chem. 2012, accepted for publication.

23


R0 exp[-Ea/RT]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

R ! 104


c1/2

!


Page 24 of 37

Page 25 of 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43



R ! 104 (S kg cm-1 mol-1)

The Journal of Physical Chemistry Page 26 of 37

R ! 104 (S kg cm-1 mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

80

1-hexanol

15"C 45"C 85"C

60 40 20 0 12 1-decanol

10 8 6 4 2 0 0.0

0.2

0.4 (mol/kg)

0.6

ACS Paragon 1/2Plus Environment 1/2

c

0.8

Page 27 of 37

1.0 (A.U.)

Absorbance

15C 35C 55C 0.5

Absorbance (A.U.)

1.5

0.6 m TbaTf 1-decanol

71C

pure 1-decanol 35C

0.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39


1010

1020

1030 1040 1050 Wavenumbers (cm-1)


1060

1070

TbaTf 1-hexanol

The Journal of Physical Chemistry Page 28 of 37

4

-6.0

0.6 mol kg

2

-7.0 0

-7.5 -8.0 -8.5

CAE: R2=0.999 SAE: R2=0.999

-2

-12.0 0.00042 mol kg-1

2

-12.5 0 -13.0

-13.5

-2

CAE: R2=0.998 SAE: R2=0.961 ACS Paragon Plus Environment

2.8

3.0 3.2 3.4 T-1× 103 (K-1)

-4 3.6

£ m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

ln m(T)

ln m(T)

-6.5

£ m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

-1

TbaTf 1-decanol

Page 29 of 37 The Journal of Physical Chemistry

0.6 mol kg-1

-7

0

-8

-2 -3

-9 -10

CAE: R2=0.999 SAE: R2=0.998

-4 -5

0.00042 mol kg-1

-15.4 -15.6

0

-2

-15.8 -4 -16.0 -16.2

CAE: R2=0.999 SAE: R2=0.683

-6


2.8

3.0 3.2 T-1× 103 (K-1)

3.4

£ m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

ln m(T)

ln m(T)

-1

£ m(T,¡S ) ¥ ln¤m r (Tr ,¡S )¦

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


200 m × 105 (S cm-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

85°C 75°C 65°C (7) 55°C 45°C (8) 35°C 25°C (9) 15°C (10) 5°C

150 100

(6)

Page 30 of 37

0.6 mol kg-1

50 0 (6)

(7)

15 (8)

10

(9) (10)

5 0

(6)

0.3

0.00042 mol kg-1

-1

m × 10 (S cm )

0.1 mol kg-1

20

5

-1

m × 10 (S cm )

25

5

0.2 (7)

0.1

(8) (9) (10)

0.0 4

6

8

10 12 14 16 ACS Paragon Plus Environment ¡s

18

m0 × 10-3 (S cm-1)

m0 × 10-3 (S cm-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

m0 × 10-3 (S cm-1)

Page 31 of 37

4 3 2


85°C 75°C 65°C 55°C 45°C 35°C 25°C 15°C 5°C

Ea = 39.1 kJ mol-1

1 0.6 mol kg-1

0 10 Ea = 44.5 kJ mol-1

8 6 4 2

0.1 mol kg-1

0 10 8

Ea = 52.9 kJ mol-1

6 4 2 0.00042 mol kg-1

0 4

6

8

10

12 ¡s

14


16

18

10000

m0 (S cm-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


Page 32 of 37

1000

100 85°C 45°C 5°C

10 ACS Paragon Plus Environment 0.0 0.1 0.2 0.3 0.4 c (mol kg-1)

0.5

0.6

R × 104 (S kg cm-1 mol-1)

Page 33 of 37

exp[-Ea/R T]

R0 (S kg cm-1 mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

20

The Journal of Physical Chemistry TbaTf 1-octanol

15 10 5 0 107 106 105 104 103

10-5 10-6 10-7 10-8 85°C 45°C 5°C

10-9 10-10 0.0

0.4Environment 0.6 ACS0.2 Paragon Plus c

1/2

-1 1/2

(mol kg )

0.8

TbaTf 1-octanol


¡s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Page 34 of 37

0.6 m 0.035 m 0.00042 m

12 10 8 6 0


20

40 T (°C)

60

80

Page 35 of 37

¡s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


14 12 10 8 85°C 45°C 5°C

6 0.0


0.1

0.2 0.3 0.4 c (mol kg-1)

0.5

0.6

55

Ea (kJ mol-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21


Page 36 of 37

50

45

40 0.0


0.2

1/2

c

0.4 0.6 -1 1/2 (mol kg )

0.8

R0 exp[-Ea/RT]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


R × 104

Page 37 of 37

c1/2


Concentration Dependence of Molal Conductivity and Dielectric

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