Critical Behavior of the Electrical Conductivity of ... - Springer Link

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Journal of Solution Chemistry [josc]

QC: FJQ pp529-josl-375134

June 7, 2002

11:22

Style file version June 5th, 2002

C 2002) Journal of Solution Chemistry, Vol. 31, No. 5, May 2002 (°

Critical Behavior of the Electrical Conductivity of Concentrated Electrolytes: Ethylammonium Nitrate in n-Octanol Binary Mixture A. Oleinikova and M. Bonetti∗ Received November 5, 2001; revised March 1, 2002 The electrical conductivity κ of highly concentrated binary ionic mixtures of ethylammonium nitrate in n-octanol at a critical salt mole fraction x = 0.766 and at an off-critical one x = 0.908 was measured over an extended temperature range above the critical consolute point. Far from the critical temperature Tc , the conductivity is accurately described by the Vogel–Fulcher–Tammann (VFT) law. However, in a temperature range 1T = (T − Tc ) ≤ 3 K, the conductivity exhibits a monotonous deviation from the VFT behavior. This anomaly is finite at Tc and, for the critical mixture, its amplitude is ∼ = 0.23% of κ(Tc ). The asymptotic behavior of the conductivity anomaly is described by a power law τ (1−α) , with τ = (T − Tc )/Tc , the reduced temperature, and α, the critical exponent of the specific heat anomaly at constant pressure. This critical anomaly is similar to the one observed in other highly concentrated critical electrolytes. The degree of dissociation of the salt for the critical mixture, αdiss ∼ = 0.78 ± 0.04, is estimated from the value of the Walden product computed at Tc , and accounts for the effective free ion concentration in the reduced critical coordinates of the system. KEY WORDS: Highly concentrated binary ionic mixture; ionic conductivity; critical anomaly; Walden product; ion association.

1. INTRODUCTION During the last decade, the study of the critical properties of concentrated electrolytes has received particular attention because of the existence of organic salts, which are completely miscible at room temperature in organic solvents of low to moderate dielectric constant.(1,2) Light scattering(3,4) and small-angle neutron scattering,(5) coexistence curve determination(6–8) and shear viscosity measurements(9,10) performed near the critical point have shown that the binary ionic mixtures behave mostly as neutral systems. Indeed, a 3-D Isinglike Service de Physique de l’Etat Condensé, CEA-Saclay, F-91191 Gif sur Yvette Cedex, France; email: [email protected] 397 C 2002 Plenum Publishing Corporation 0095-9782/02/0500-0397/0 °

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Oleinikova and Bonetti

critical behavior(3–10) with possibly a crossover from mean field to Ising-like criticality(11,12) is generally observed. From the theoretical point of view, the location of the critical point and the shape of the coexistence curve depend on the interaction potential between ions and whether ion–dipole and dipole–dipole interactions are accounted for.(13–19) Experimentally, low values of the dielectric constant of the solvent (ε < 10) and of the ion dissociation constant (αdiss 100 kHz, with a minimum around 150 kHz. Our measurements were performed in a reduced frequency interval 100 Hz < f < 200 kHz with a 50 mV amplitude sine-wave excitation. Typically, 20–50 frequencies were scanned at each temperature and each datum was averaged over 50 measurements. A sixth-order polynomial in 1/ f fitted the conductivity data in the frequency range 1–200 kHz and extrapolation to infinite frequency produced a free-polarization impedance.(22,23,28) Cell calibration was carried out at the same frequencies and temperatures as in the experiment using a standard KCl reference solution of known conductivity. The cell constant was K = 0.810 ± 0.8 × 10−3 cm−1 . Thermal expansion of the glass reservoir and the platinum electrodes were negligible in that temperature range.(29) The absolute accuracy of the conductivity measurements is estimated to be 0.1% and the relative one to 0.001%.

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Fig. 1. Modulus of the impedance |Z | measured in the critical mixture ethylammonium nitrate (EAN) in n-octanol at T = 318.55 K, as a function of frequency f . The cell constant is K = 0.810 ± 0.8 × 10−3 cm−1 . The two arrows indicate the frequency range where a sixthorder polynomial in 1/ f is used to fit the data. The specific conductivity reported in Tables Ia, b, and c is calculated by extrapolating the polynomial to infinite frequency.

Conductivity was measured at thermal equilibrium that corresponds to constant values in time of the electrical conductivity and the transmitted laser beam intensity. The sample homogeneity was probed by the measurement of turbidity at different vertical positions in the cell. Indeed, close to the critical point, gravityinduced concentration gradients might settle in the sample because of the divergence of the concentration fluctuations. As a matter of fact, in the temperature interval (T − Tc ) < 0.03 K, we observed concentration gradients in the sample. Therefore, we report here conductivity measurements that were performed with a density homogeneous sample. The 0.02% reproducibility of the measurements obtained between different temperature runs is lower than the one observed in critical mixtures of TBAP in 1-dodecanol and 1,4-butanediol solvents,(22,23) which comes presumably from some chemical decomposition of EAN when it is heated above 50◦ C.(6,30) More details of the experiment can be found in Oleinikova and Bonetti.(23) The critical temperature Tc was taken coincident to the temperature at which the intensity of the transmitted laser beam vanished, because of the critical opalescence. At this temperature, we also noticed a steep variation of the electrical conductivity, and a meniscus settled in the middle of the sample cell after few hours.

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Ethylammonium Nitrate in n-Octanol

401

2.3. Density and Equivalent Conductivity The density ρ of the mixture is computed assuming volume additivity: ρEAN ρoct ρ= xw ρoct + (1 − xw ) ρEAN

(1)

where ρEAN and ρoct are the densities of EAN and n-octanol, respectively, and xw is the salt weight fraction. Equation (1) does not account for the mixing volume and is accurate within 0.4%.(6,9) In the reduced temperature range 5 × 10−4 < τ = (T − Tc )/Tc , no (1 − α) anomaly of the thermal expansion of the mixture volume was observed(9) . The density ρi of the ith component is computed from the Lorentz–Lorenz (L–L) relation:(31) ¶ µ 2 n i − 1 3 Mi 1 ρi = (2) n 2i + 2 4π NA αi where n i is the refractive index measured at λ = 632.8 nm, Mi is the molar weight, αi is the molecular polarizability, and NA is Avogadro’s number. In the temperature interval between 303 and 321 K, the polarizability αi is assumed constant, and the temperature dependence of the refractive index of EAN and n-octanol is, respectively, given by: ηEAN = 1.375 96 + 21.727 73 T −1 and n oct = 1.297 27 + 37.889 16 T −1 , where T is the temperature in K.(6) From the density of ηEAN , ρEAN = 1.2106 g-cm−3 at T = 298.15 K, given by Hadded et al.,(32) in good agreement with the values of Perron et al.(33) and Allen et al.,(34) and, from the n-octanol density, ρoct = 0.827 g-cm−3 (T = 293.15 K) (Aldrich), one infers 3/4π Mi /NA 1/αi = 4.5037 and 3.2242 g-cm−3 for EAN and n-octanol, respectively. Within the temperature interval 298 and 323 K, the density of EAN computed from the L–L relation are within ±0.1% of the temperaturedependent density given by the linear regression ρEAN = 1.225 − 0.00063 T (◦ C) calculated from the experimental data given by Allen et al.(34) The density values of n-octanol calculated from Eq. (2) are within 0.25% of the reference density data.(35) The slight difference in density might come from the purity of n-octanol used in the present experiment. The specific conductivity κ (mS-cm−1 ) from EAN and from the two binary mixtures is shown as a function of temperature T in Fig. 2. The conductivity of the critical mixture EAN in n-octanol was measured from two distinct temperature runs. A first run (Run 1) spanned an extended temperature interval far from Tc . In this interval, the temperature was lowered by steps of −100 mK. A second run (Run 2) covered the critical region very near to Tc and the temperature steps were of −10 mK. Tables Ia, b, and c give, for the three samples, the values of the specific conductivity κ and the equivalent conductivity 3 = κ/c (mS-cm2 -mol−1 ), as a function of temperature T. c = Ns /V (mol-cm−3 ) is the salt molar concentration, where Ns is the number of mole of EAN, and V = 6i m i /ρ is the volume of the

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Fig. 2. Specific conductivity κ as a function of temperature T . Symbols, experimental conductivity from: (♦) ethylammonium nitrate (EAN); (1) off-critical mixture of EAN in n-octanol at a salt mole fraction x = 0.908; (h) and (°) critical mixture EAN in n-octanol at critical salt mole fraction xc = 0.766.

solution, with m i , the mass of the ith component and ρ, the density of the solution given by Eq. (1). The absolute accuracy of the equivalent conductivity is estimated to be 0.4%. 3. RESULTS AND DISCUSSION 3.1. Background Conductivity and Critical Anomaly The specific conductivity of EAN, and the background conductivity of the two binary mixtures measured far from Tc are fitted by the empirical Vogel–Fulcher– Tammann (VFT) equation:(36–38) µ ¶ B κVFT = κ0,VFT exp − (3) (T − T0 ) In Eq. (3), T0 is the temperature at which the conductivity vanishes and the parameter B is related to a temperature-dependent activation energy.(39) The amplitude κ0,VFT is assumed to be constant in the present analysis. Table II gives the values of the VFT parameters and the temperature range, where the fit by Eq. (3) was performed. The values of the parameter B and the temperature T0 are shown as

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Ethylammonium Nitrate in n-Octanol Table Ia.

403

Specific Electrical Conductivity and Equivalent Conductivity of Ethylammonium Nitrate (EAN)a

T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

312.834 312.935 313.035 313.135 313.236 313.335 313.435 313.535 313.635 313.735 313.835 313.935 314.035 314.134 314.233 314.332 314.432 314.532 314.630 314.729 314.828 314.927

34.8582 34.9469 35.0407 35.1288 35.2143 35.3018 35.3882 35.4765 35.5622 35.6467 35.7348 35.8217 35.9074 35.9952 36.0802 36.1672 36.2529 36.3408 36.4274 36.5145 36.6019 36.6881

3.138 3.146 3.155 3.163 3.171 3.179 3.187 3.195 3.203 3.211 3.219 3.226 3.234 3.242 3.250 3.258 3.266 3.274 3.282 3.290 3.298 3.306

315.026 315.125 315.222 315.321 315.420 315.518 315.618 315.716 315.813 315.911 316.010 316.108 317.231 318.195 319.154 320.105 321.047 321.987 322.921 323.850 324.787 325.722

36.7764 36.8668 36.9509 37.0387 37.1270 37.2140 37.3014 37.3875 37.4740 37.5626 37.6499 37.7376 38.7504 39.6282 40.5122 41.4089 42.2888 43.1747 44.0648 44.9600 45.8949 46.7857

3.314 3.322 3.330 3.338 3.346 3.354 3.362 3.370 3.378 3.386 3.394 3.402 3.495 3.576 3.657 3.740 3.820 3.902 3.984 4.066 4.153 4.235

a The

conductivity measurements were obtained from two distinct temperature runs (see text). The conductivity is measured at thermal equilibrium that corresponds to a constant value of the sample turbidity. The relative accuracy of the specific conductivity is 0.001%. The absolute accuracy of the equivalent conductivity is estimated to be 0.4%.

a function of the salt mole fraction x in Fig. 3. The temperature T0 decreases for increasing values of x as it is observed in other concentrated electrolytes.(23,40) The parameter B increases with x and has the largest value for the EAN sample. This suggests that EAN is more structured than the binary mixtures because of stronger hydrogen bonding.(41) The VFT law describes well the conductivity of EAN in the whole investigated temperature range, as it is shown by the ±0.05% deviations in Fig. 4a. The electrical conductivity of the off-critical and critical mixtures shows, however, a pronounced and monotonous deviation from the background VFT law (Fig. 4b and 4c). This deviation becomes larger as the temperature of the mixture approaches the demixing temperature and might be interpreted as an anomaly of the electrical conductivity.(22,23) For the critical mixture, the amplitude of the critical anomaly is defined as 1κc = κ(Tc ) − κVFT (Tc ), where we assume that the additive background conductivity described by the VFT law holds down to the critical temperature Tc .

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Table Ib. Specific Electrical Conductivity and Equivalent Conductivity of Binary Mixture of EAN in n-Octanol with Off-Critical Salt Mole Fraction x = 0.908a T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

313.204 313.305 313.405 313.504 313.605 313.704 313.804 313.904 314.005 314.104 314.203 314.302 314.402 314.501 314.600 314.699 314.798 314.898 314.997 315.097 315.194 315.293 315.391 315.490 315.589 315.687 315.785

26.5356 26.6176 26.6886 26.7654 26.8461 26.9139 26.9883 27.0586 27.1384 27.2049 27.2709 27.3409 27.4102 27.4865 27.5529 27.6224 27.6961 27.7635 27.8351 27.9079 27.9793 28.0481 28.1172 28.1847 28.2575 28.3256 28.3932

2.819 2.828 2.836 2.844 2.853 2.860 2.868 2.876 2.884 2.892 2.899 2.906 2.914 2.922 2.929 2.937 2.945 2.952 2.960 2.968 2.975 2.983 2.990 2.998 3.006 3.013 3.020

315.883 315.981 316.080 316.178 316.275 316.374 316.471 316.570 316.667 316.763 316.860 316.958 317.231 318.196 319.154 319.250 319.346 319.441 319.536 319.632 320.106 321.05 321.988 322.923 323.855 324.788 325.723

28.4620 28.5332 28.6017 28.6705 28.7387 28.8082 28.8770 28.9430 29.0135 29.0823 29.1493 29.2187 29.4086 30.0959 30.7809 30.8501 30.9184 30.9858 31.0559 31.1239 31.4649 32.1531 32.8441 33.5376 34.2364 34.9379 35.6498

3.028 3.035 3.043 3.050 3.058 3.065 3.073 3.080 3.088 3.095 3.102 3.110 3.130 3.205 3.280 3.287 3.294 3.302 3.309 3.317 3.354 3.429 3.504 3.580 3.656 3.732 3.810

a See

legend, Table Ia.

Along a path of constant critical concentration, the temperature dependence of the specific electrical conductivity reads:(22,23) κ = κVFT (τ ) + κcrit (τ ) where

µ κcrit = κ0 τ

1−α

Bcrit + τ κ0

(4)

¶ + 1κc

(5)

is the critical contribution.(24) The critical exponent α = 0.109 is that of the specific heat anomaly at constant pressure.(25) κ0 /(1 − α) and Bcrit are, respectively, the critical amplitude and the critical fluctuation-induced additive constant.(42)

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Table Ic. Specific Electrical Conductivity and Electrical Conductivity of Critical Binary Mixture of EAN in n-Octanol with a Critical Salt Mole Fraction x c = 0.766a T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

325.394 325.190 324.987 324.784 324.582 324.379 324.177 323.974 323.773 323.570 323.368 323.165 322.963 322.760 322.559 322.357 322.155 321.952 321.750

22.8966 22.7931 22.6886 22.5844 22.4811 22.3764 22.2713 22.1666 22.0635 21.9596 21.8566 21.7534 21.6500 21.5470 21.4444 21.3422 21.2399 21.1369 21.0348

3.203 3.188 3.173 3.159 3.144 3.129 3.114 3.099 3.084 3.069 3.054 3.040 3.025 3.010 2.996 2.981 2.966 2.952 2.937

318.316 318.296 318.276 318.256 318.235 318.216 318.195 318.186 318.175 318.165 318.155 318.145 318.134 318.124 318.115 318.104 318.094 318.085 318.074 318.064 318.054 318.044 318.034

19.2675 19.2568 19.2462 19.2356 19.2254 19.2153 19.2052 19.2001 19.1950 19.1904 19.1854 19.1799 19.1744 19.1698 19.1648 19.1597 19.1547 19.1492 19.1442 19.1391 19.1337 19.1286 19.1236

2.685 2.684 2.682 2.681 2.679 2.678 2.676 2.676 2.675 2.674 2.674 2.673 2.672 2.671 2.671 2.670 2.669 2.668 2.668 2.667 2.666 2.666 2.665

T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

321.549 321.347 321.144 320.942 320.739 320.538 320.337 320.135 319.932 319.731 319.529 319.326 319.125 318.925 318.724 318.524 318.325 318.126

20.9359 20.8342 20.7318 20.6305 20.5285 20.4276 20.3271 20.2255 20.1240 20.0230 19.9220 19.8205 19.7196 19.6187 19.5170 19.4154 19.3125 19.2111

2.923 2.908 2.894 2.879 2.865 2.850 2.836 2.822 2.807 2.793 2.778 2.764 2.750 2.735 2.721 2.706 2.692 2.677

318.023 318.014 318.003 317.993 317.984 317.974 317.963 317.953 317.943 317.933 317.923 317.914 317.903 317.893 317.883 317.873 317.863 317.852 317.843 317.832 317.822 317.813 317.802

19.1181 19.1131 19.1077 19.1031 19.0976 19.0922 19.0876 19.0822 19.0767 19.0722 19.0668 19.0618 19.0568 19.0514 19.0459 19.0409 19.0360 19.0310 19.0260 19.0206 19.0157 19.0102 19.0053

2.664 2.663 2.663 2.662 2.661 2.660 2.660 2.659 2.658 2.657 2.657 2.656 2.655 2.655 2.654 2.653 2.652 2.652 2.651 2.650 2.649 2.649 2.648 (Continued)

Run 1

Run 2

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Oleinikova and Bonetti Table Ic. (Continued)

T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

T (K)

κ (mS-cm−1 )

3 (S-cm2 -mol−1 )

317.792 317.782 317.772 317.762 317.751 317.742 317.731 317.721 317.712 317.701 317.691 317.680 317.671 317.661 317.650 317.641 317.631 317.620 317.610 317.600 317.590 317.579 317.570 317.559 317.549 317.540 317.529 317.519 317.509 317.499 317.489 317.479 317.469 317.459 317.448 317.438

19.0003 18.9949 18.9899 18.9846 18.9796 18.9738 18.9693 18.9639 18.9586 18.9541 18.9482 18.9433 18.9379 18.9330 18.9272 18.9223 18.9169 18.9120 18.9071 18.9018 18.8964 18.8915 18.8862 18.8813 18.8755 18.8702 18.8649 18.8599 18.8547 18.8493 18.8440 18.8387 18.8338 18.8285 18.8232 18.8179

2.647 2.647 2.646 2.645 2.644 2.644 2.643 2.642 2.641 2.641 2.640 2.639 2.638 2.638 2.637 2.636 2.635 2.635 2.634 2.633 2.633 2.632 2.631 2.630 2.630 2.629 2.628 2.627 2.627 2.626 2.625 2.624 2.624 2.623 2.622 2.621

317.429 317.418 317.408 317.398 317.388 317.378 317.369 317.358 317.348 317.338 317.329 317.318 317.308 317.299 317.289 317.278 317.268 317.258 317.248 317.237 317.228 317.218 317.207 317.197 317.187 317.177 317.166 317.156 317.146 317.136 317.126 317.116 317.106 317.095 317.086

18.8131 18.8078 18.8020 18.7972 18.7919 18.7866 18.7813 18.7761 18.7708 18.7659 18.7602 18.7550 18.7497 18.7440 18.7392 18.7339 18.7283 18.7230 18.7178 18.7121 18.7068 18.7016 18.6959 18.6903 18.6846 18.6794 18.6737 18.6685 18.6629 18.6572 18.6516 18.6464 18.6403 18.6338 18.6290

2.621 2.620 2.619 2.618 2.618 2.617 2.616 2.615 2.615 2.614 2.613 2.612 2.612 2.611 2.610 2.609 2.609 2.608 2.607 2.606 2.606 2.605 2.604 2.603 2.602 2.602 2.601 2.600 2.599 2.599 2.598 2.597 2.596 2.595 2.595

a The

conductivity measurements were obtained from two distinct temperature runs (see text).

The electrical conductivity of the critical mixture was measured from two separate runs in the temperature range 7 × 10−5 < τ < 4 × 10−3 . A slight shift between the background conductivities was, however, noticed and, to make them coincide, the amplitudes κ0,VFT of the VFT law were adjusted by a 0.2% factor. Obviously, this modifies value of the term 1κc in Eq. (5) without altering the

c (mol-cm−1 ) κ0,VFT T = 317 K (mS-cm−1 ) B (K)

T0 (K)

T range for VFT fitting (K)

407 0.766 0.731

7.2 × 10−3 416.2 ± 30 372.4 ± 17 197.1 ± 3 320.3 < T < 325.4

1 1 11.0 × 10−3 1691.7 ± 75 609.8 ± 15 155.8 ± 2 312.9 < T < 325.7 0.908 0.891 9.4 × 10−3 962.0 ± 30 507.3 ± 10 171.8 ± 2 316.6 < T < 325.7

xw

≈0.23

— —

0.407 ± 0.02

0.283 ± 0.03 (0.283)

|1κc |/ κ(Tc ) (%) η0,VFT (mPa-s)

T0,η (K)

470.2 ± 7

(197.1)

672.1 ± 20 158.2 ± 3 (672.1) (158.2)

Bη (K)

ax

pp529-josl-375134

and xw are the mole and weight fraction of EAN, respectively. c is the salt molar concentration. κ0,VFT , B, and T0 are the parameters of the VFT-like conductivity, and η0,VFT , Bη and T0,η are those of the VFT-like shear viscosity. The VFT parameters of the shear viscosity for the off-critical mixture (x = 0.908) are set equal to those of EAN and are given in parentheses. T0,η for the critical sample has almost the same value as T0 , and is set to this value in the VFT fit of the shear viscosity. |1κc |/κc (Tc ) is the relative value of the conductivity anomaly at Tc = 317.065 ± 0.01 K, where |1κc | = 0.0615 ± 2 × 10−4 mS-cm−1 is the critical contribution.

EAN EAN + n-octanol off-critical EAN + n-octanol critical

x

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Parameters of the VFT Law, Eq. (3) Describing the Background Electrical Conductivity and Shear Viscosity of EAN, and Two Binary Mixtures EAN in n-Octanola


Table II.

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Fig. 3. Parameter B(°) and Vogel temperature T0 (•) as a function of the salt mole fraction x. The values of the parameters B and T0 are obtained from a fit of the background conductivity with the VFT law, Eq. (3), in a temperature interval given in Table II.

critical behavior of the conductivity anomaly. From a fit with Eqs. (4) and (5), the values of the critical parameters can be inferred: κ0 = 26.84 ± 0.43 mS-cm−1 , Bcrit /κ0 = −1.43 ± 0.01, and |1κc | = 0.0615 ± 2. × 10−4 mS-cm−1 . The corresponding parameter values of the VFT-like background conductivity are given in Table II. Any change of the amplitude κ0,VFT within 0.2% modifies the values of the critical amplitude κ0 and the fluctuation-induced constant Bcrit /κ0 within their uncertainty. In the temperature range studied, additional correction-to-scaling terms(22,23,43) in Eq. (5) were not required to improve the quality of the fit. Figure 5 shows the percentage deviation between the specific conductivity and the fitted Eqs. (4) and (5). The relative value of the critical contribution is |1κc |/κ(Tc ) ≈ 0.23% (Table II). This value is smaller than the one obtained by Oleinikova and Bonetti(22,23) in the highly concentrated ionic mixtures of tetra-n-butylammonium picrate (TBAP) in 1-dodecanol and in 1,4-butanediol for which |1κc |/κ(Tc ) ≈ 1.6 and ≈0.45%, respectively. The value of the ratio Bcrit /κ0 ∼ = −1.43 ± 0.01 obtained by the fit is smaller by ∼ = −1.8 ± 0.3 obtained from = 26% from the value Bcrit /κ0 ∼ the binary mixtures TBAP in 1-dodecanol and 1,4-butanediol.(23) 3.2. Walden Product and Free Ion Concentration At working frequencies f > 100 kHz, the critical divergence of the shear viscosity η is not observed.(44) Therefore, a coupling of the anomaly of the electrical conductivity with that of the shear viscosity measured under quasistatic conditions

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Fig. 4. Anomaly of the specific conductivity (κ − κVFT )/κ, as a function of temperature T . Symbols, experimental data from: (♦) ethylammonium nitrate (EAN); (1) off-critical mixture of EAN in n-octanol at a salt mole fraction x = 0.908; (h) and (°) critical mixture of EAN in n-octanol at a critical salt mole fraction x c = 0.766.

might be excluded.(23,45) The variation of the Walden product (3 ηVFT ) as a function of the salt concentration accounts for any change of the free ion concentration in the mixture and allows the determination of the free ion concentration at the critical point.(22,23) ηVFT is the background shear viscosity, that was measured in the critical mixture EAN in n-octanol by Oleinikova and Bonetti.(9) The temperature dependence of the background shear viscosity is fitted by a VFT-like equation, ηVFT = η0,VFT exp[Bη /(T − T0,η )], whose parameter values are given in Table II. Figure 6 shows the variation of the Walden product (3 ηVFT ) as a function of temperature for EAN and the two binary mixtures. The Walden product of EAN extrapolated at T = 298 K agrees with the data of Perron et al.(33) From Fig. 6, it can be noticed that the value of the Walden product diminishes for increasing values of n-octanol concentration, indicating that the degree of dissociation of EAN is reduced.

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Fig. 5. Residuals between the specific conductivity of the critical mixture EAN in n-octanol (critical salt mole fraction xc = 0.766) and the fitted Eqs. (4) and (5), as a function of the reduced temperature τ .

Fig. 6. Walden product (3 ηVFT ) as a function of temperature T. 3 is the equivalent conductivity computed from the measured specific conductivity (see Tables Ia, b, and c), assuming volume additivity (see text). ηVFT is the shear viscosity fitted with a VFT-like equation whose parameters are given in Table II. Symbols, experimental data from: (♦) ethylammonium nitrate (EAN); (1) off-critical mixture of EAN in n-octanol at a salt mole fraction x = 0.908; (h) and (°) critical mixture of EAN in n-octanol at critical salt mole fraction xc = 0.766.

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A detailed study of the Walden product for EAN in acetonitrile(33) shows a linear concentration dependence at large values of the salt concentration. A similar trend is observed for TBAP in solvents with low dielectric constant.(23) The linear dependence of the Walden product with the salt concentration permits an estimation of the degree of dissociation αdiss according to following relation:(23,46) αdiss =

(3 ηVFT ) (3 ηVFT )∗

(6)

where (3 ηVFT )∗ is the limiting Walden product with the fully dissociated salt. EAN and infinitely diluted EAN solutions in low dielectric solvents, such as EAN in acetonitrile,(33) are completely dissociated systems. These are characterized, respectively, by Walden products with comparable values, (3 ηVFT )∗ = 67.8 mScm2 -Pa-s-mol−1 (T = 317 K) and 64.8 mS-cm2 -Pa-s-mol−1 .(33,47) Therefore, from these limiting values, one can infer from Eq. (6) the degree of dissociation of EAN at Tc in the critical mixture, αdiss ∼ = (0.78 ± 0.04). 3.3. Critical Coordinates According to the Restricted Primitive Model (RPM), for a 1-1 electrolyte, the reduced critical temperature Tc∗ and free ion number density cc∗ are defined by scaling the critical temperature Tc by the Coulomb energy e2 /4π ε ε0 a, and the critical free ion number density ccα = 2 αdiss NA cc , with cc , with critical salt molar concentration, by the ion volume a 3 . Tc∗ =

4π ε ε0 a kB Tc e2

cc∗ = ccα a 3

(7) (8)

In Eq. (7), e is the elementary charge, ε0 is the vacuum permeability, kB is Boltzmann’s constant, and ε is the dielectric constant of the solvent. Theory and numerical simulations of the RPM model locate the critical temperature Tc∗ between 0.049 and 0.063, and the critical concentration cc∗ between 0.005 and 0.080.(48) In absence of any reliable theory for estimating the effective dielectric constant of the medium surrounding the ions in highly concentrated electrolytes, the value ε = 8.5 of n-octanol has been used in Eq. (7). This assumption is motivated by the large degree of dissociation of the salt and by the extremely small value of the Debye screening length (see below) that indicates that the interaction between neighboring ions is short range and suggests that the mixture might be considered as ˚ for the ionic radii,(30) the infinitely diluted. Indeed, taking an average value a ∼ = 4A ∗ ∼ reduced critical temperature of the critical mixture Tc = 0.063 is in fair agreement with the RPM predictions and the estimation Tc∗ = 0.060 given by Weingärtner et al.(30) However, the corresponding reduced critical concentration cc∗ ∼ = 0.43 is significantly larger than the value predicted by the RPM model.

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The Debye screening length in reduced coordinates is:(23) µ ∗ 2 ¶1/2 Tc a λD,c = 4π cc∗

(9)

˚ is found for the critical mixture EAN in n-octanol, a value λD,c = (0.43 ± 0.04) A ˚ obtained in the highly concentrated electrolytes smaller than λD,c = (2.1 ± 0.2) A of TBAP in 1,4-butanediol and 1-dodecanol solvents.(23) In such mixtures, the dielectric constant of the solvent varies between 4.6 to 25.9, and λD,c is almost constant and close to the RPM predictions. In summary, a critical anomaly of the electrical conductivity in the highly concentrated critical solution EAN in n-octanol has been detected. The electrical conductivity remains finite at the critical consolute point, and the temperature dependence of the critical part is consistent with a (1 − α) power law predicted by the theory of short-range fluctuations.(24) The determination of the Walden product at the critical point allows an estimation of the degree of dissociation of the salt and shows that almost 78% of EAN is dissociated. This large free ion concentration leads to a small value of the Debye screening length λD,c . The small λD,c value—and, hence, short-range interactions—might explain the almost pure Ising-like behavior observed for the critical mixture over a rather extended temperature range.(4) REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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