Article pubs.acs.org/JPCB
Changes in Permittivity and Density of Molecular Liquids under High Pressure Vladimir D. Kiselev,*,† Dmitry A. Kornilov,† and Alexander I. Konovalov‡ †
Laboratory of High Pressure Chemistry, Butlerov Institute of Chemistry, Kazan Federal University, Kremlevskaya Str., 18, Kazan 420008, Russian Federation ‡ Arbuzov Institute of Organic and Physical Chemistry, Kazan Scientific Center of the Russian Academy of Sciences, Arbuzov Str., 8, Kazan 420088, Russian Federation ABSTRACT: We collected and analyzed the density and permittivity of 57 nonpolar and dipolar molecular liquids at different temperatures (143 sets) and pressures (555 sets). No equation was found that could accurately predict the change to polar liquid permittivity by the change of its density in the range of the pressures and temperatures tested. Consequently, the influence of high hydrostatic pressure and temperature on liquid permittivity may be a more complicated process compared to density changes. The pressure and temperature coefficients of permittivity can be drastically larger than the pressure and temperature coefficients of density, indicating that pressure and particularly temperature significantly affect the structure of molecular liquids. These changes have less influence on the density change but can strongly affect the permittivity change. The clear relationship between the tangent and secant moduli of the permittivity curvatures under pressure for various molecular liquids at different temperatures was obtained, from which one can calculate the Tait equation coefficients from the experimental values of the pressure influence on the permittivity at ambient pressure.
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INTRODUCTION Studying the influence of pressure on fluid properties illuminates not only special features of the internal structures
is studied. For polar and especially ionic reactions, the influence of increasing values of the relative permittivity under pressure (εP) is particularly important in the analysis of the rate, equilibrium, and volume parametric values.1−5 This allows us to distinguish between the following components in the reactionrate change under pressure: reaction-rate change due to the contributions caused by the work energy PΔV‡ and the change in interaction energy of initial and transition states with the solvent due to the increase in the relative permittivity under pressure, even in cases where the magnitude of ΔV‡ is zero.3,6 The compressibility of a liquid at an elevated external hydrostatic pressure up to several kbar corresponds to a decrease in intermolecular voids when the molecules approach each other under pressure.1−15 In all cases, the compressibility curves V−P in this pressure range are reliably described by the Tait relation (1)7,8 ΔV /V1 = (V1 − VP)/V1 = (dP − d1)/dP
Figure 1. Superposition of the energy curves of the intermolecular repulsion (1) and attraction (2, 3) with the formation of lower (4) and larger (5) intermolecular interaction energy, depending on the intermolecular equilibrium distance (R4, R5) between the solvent molecules.
= C1·ln[(B1 + P)/(B1)]
Hence, the compressibility coefficient (βT) can be calculated for any pressure as βT = ∂(ln dP)/∂P = (dP /d1) ·[C1/(B1 + P)]
of liquids but also the interaction of the solvent with the solute. These issues clearly emerge when the effect of pressure on the rate and equilibrium of chemical reactions is examined, when the volume of activation and the reaction volume are determined, and when the process of solvent electrostriction © 2014 American Chemical Society
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Received: February 7, 2014 Revised: March 12, 2014 Published: March 17, 2014 3702
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Table 1. Parameters of the Relation F(ε) = a + bdP between Functions (F(ε)) and density (dP) of n-Heptane, Tetrachloromethane, 1-Chlorobutane, 2,2,2-Trifluoroethanol, and Acetonitrile in a Given Pressure Range (R = Correlation Coefficient) 1 no.
a
2 function F(ε)
1 2 3 4 5 6
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
7 8 9 10 11 12
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
13 14 15 16 17 18
ε ε− ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
19 20 21 22 23 24
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
25 26 27 28 29 30
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
3 a
4 b
5
6
100a/F(ε),1
R
2
n-Heptane (20 °C; ε1 = 1.931; ΔP/bar = 1−3334)a 0.7558 1.7032 39.34% 0.9999 −0.2442 1.7032 −26.51 0.9999 −0.0220 0.5969 −5.70 1 0.0050 0.7472 0.97 1 0.0206 0.3138 8.77 0.9999 −0.0322 0.4242 −12.49 1 Tetrachloromethane (25 °C; ε1 = 2.2320; ΔP/bar = 1−1118)a 0.6076 1.0251 27.22 1 −0.3924 1.0251 −31.85 1 −0.0416 0.3380 −8.42 1 −0.0077 0.4154 −1.18 1 0.0275 0.1664 9.45 1 −3.0489 2.9755 −18.26 0.9996 1-Chlorobutane (30 °C; ε1 = 7.06; ΔP/bar = 1−4780)b −3.3939 11.913 −48.07 0.9921 −4.3939 11.913 −72.50 0.9921 −0.1833 2.1013 −11.06 0.9927 −0.0056 2.2704 −0.28 0.9927 0.3586 0.3559 53.70 0.9910 −0.8990 2.6680 −62.34 0.9921 2,2,2-Trifluoroethanol (20 °C; ε1 = 27.79 ΔP/bar = 1−508)c −8.3298 25.941 −29.97 0.9984 −9.3298 25.941 −34.83 0.9984 0.8989 2.4226 21.05 0.9983 1.2652 2.4841 26.78 0.9983 0.7841 0.0828 87.19 0.9980 1.9711 5.7683 32.53 0.9984 Acetonitrile (25 °C; ε1 = 35.95 ΔP/bar = 1−3000)d −2.1233 48.417 −5.91 0.9999 −3.1233 48.417 −8.94 0.9999 1.9595 3.8653 39.22 1 2.3708 3.9385 43.39 1 0.8539 0.0857 92.72 0.9984 −0.5888 10.763 −7.48 0.9999
7
8
9
F(ε),1/d1
F(ε),P/dP
[8]/[7]
2.81 1.3473 0.5647 0.7544 0.3436 0.3773
2.63 1.4051 0.5698 0.7531 0.3387 0.3849
0.9359 1.0430 1.0092 0.9983 0.9857 1.0201
1.4086 0.7775 0.3117 0.4105 0.1837 1.0538
1.3793 0.7965 0.3138 0.4109 0.1824 1.2009
0.9792 1.0244 1.0067 1.0009 0.9930 1.1395
8.0369 6.8984 1.8863 2.2566 0.7614 1.6416
8.6956 7.7229 1.9356 2.2782 0.7059 1.8122
1.0820 1.1195 1.0261 1.0096 0.9271 1.1039
19.9612 19.2429 3.0683 3.3928 0.6459 4.3531
20.2233 19.5383 3.0371 3.3495 0.6199 4.4154
1.0131 1.0153 0.9898 0.9872 0.9596 1.0143
45.74 44.47 6.36 6.95 0.92 7.87
46.14 45.06 5.99 6.51 0.93 9.34
1.0089 1.0133 0.9423 0.9362 1.0127 1.1863
From ref 20. Hereinafter, ε1 means the value of relative permittivity at ambient pressure (1 bar). bFrom ref 21. cFrom ref 22. dFrom ref 23.
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Here d1 and V1 as well as dP and VP are the solvent density and volume at atmospheric and at elevated pressures (P), respectively, and C1 and B1 are the constants for each of the liquids at a given temperature. Accumulated experimental data confirm that the constants C1 and B1 are independent of the pressure, and C1 practically does not depend on the temperature. B1 usually decreases with increasing temperature, which corresponds to an increase in the compressibility coefficient with an increase in temperature. In general, an increase in the energy of intermolecular interactions in the liquid is accompanied by a denser packing and lower compressibility.1,2,11 Imposing a steeper energy dependence for intermolecular repulsions (1) and the curves of intermolecular interactions (2 and 3) with a decrease in the intermolecular distance gives the potential energy curves of intermolecular stabilization (4 and 5) in a liquid (Figure 1). In general, as the intermolecular interaction energy (curve 2) weakens, the equilibrium intermolecular distance (R4, Figure 1) increases, which increases the void volume, leading to increasing compressibility.
RESULTS AND DISCUSSION
Most of the changes of fluid properties (e.g., refractive index, relative permittivity, viscosity) at elevated hydrostatic pressures up to a few kbar are mainly caused by changes in compressibility, i.e., from a reduction of intermolecular voids and by increasing the number of molecules per volume unit.1,2,4,9−15 The compressibility (eq 1) has been determined with high accuracy for a large number of molecular liquids.1,2,4,9−15 The search for correlations among liquid density (d), its refractive index (nD), and relative permittivity (ε) is of great interest. In this work the pressure influence on variation of the density and permittivity of 57 nonpolar and dipolar liquids was considered. As indicated in the original papers, all tested fluids were purified and the values of density, refractive index and relative permittivity at atmospheric pressure and chosen temperatures correspond to the known literature data. Several relationships (3)−(8) for the general type F(ε)/d = constant have been proposed16−19 as follows: 3703
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Table 1 is 1.05) at pressures up to 1 kbar generates an error of about 50−100% for the calculation of changes of the permittivity functions from the liquid density data (Table 1). This is because the density change itself (dP=1000/dP=1) at pressures of 1 kbar does not exceed the values of 1.05− 1.10.4−15 The ratio of the “constants” [8]/[7] (Table 1, column 9) is equal to the ratio of the limit values [F(ε),P/dP]/[F(ε),1/d1] and differs markedly from the expected unity. This can be considered as evidence that the change in the relative permittivity under pressure is not only determined by the increase of the number of molecules per volume unit, i.e., by the density change of the fluid. Numerous studies in this area have shown that the change of relative permittivity under pressure can be considered as a more complex process than the compressibility. It was suggested19,25 that the pressure effect on the relative permittivity of polar liquids, ∂ ln ε/∂P, is due to at least three factors: (1) the change in fluid density ∂ ln d/∂P, (2) the deformation of molecules induced by pressure ∂μ/∂P, and (3) the change in the internal liquid structure induced by external pressure ∂g/∂P, as shown in eq 9:
Figure 2. Influence of the fluid density changes, Δd/dP, in the range of pressures and temperatures on the permittivity change, Δε/(εP − 1), according to the data from Table 2. Blue points (◆): n-hexane (N = 62), n-heptane (N = 48), 2-methylbutane (N = 38). Red points (■): methanol (N = 10), acetonitrile (N = 18), ethyl acetate (N = 12), anisole (N = 10), o-xylene (N = 6), 3-methyl-1-butanol (N = 13), 1pentanol (N = 24), chlorobenzene (N = 9), cumene (N = 11), 2methyl-1-butanol (N = 16). Green points (▲): data from Table 2. Total number of points is 555.
(ε − 1)(2ε + 1)/9εd = P0 = (4πN /3)[α + (gμ2 /3kT )] (9)
ε /d = const
(3)
(ε − 1)/d = const
(4)
(ε 0.5 − 1)/d = const
(5)
(ε − 1)/(ε 0.5 + 0.4)d = const
(6)
(ε − 1)/(ε + 2)d = const
(7)
(ε − 1)(2ε + 1)/9εd = const
(8)
Here, N is the Avogadro number, α is the electronic polarizability, μ is the dipole moment, k is the Boltzmann constant, T is the absolute temperature, and g is a correction factor in the expanded Kirkwood equation (9), reflecting a possible change in dipole orientation of the molecules relative to each other under pressure.19 Using an appropriate value of g in magnitude and sign, everyone can calculate the changes in the magnitude of P0 in eq 9 in the range of pressure and temperature. Additionally, high pressure may increase the proportion of more compact conformers. For example, for 1,2dichloroethane, an increase in pressure may lead to the accumulation of the more compact and more polar gauche form.26,27 The more compact and polar cis-1,2-dichloroethene (d20 = 1.2818; ε20 = 9.20, μ = 1.89 D) has also been found to increase compared to trans-1,2-dichloroethene (d20 = 1.2546; ε20 = 2.14, μ = 0 D) in a mixture of geometric isomers at an elevated temperature in the presence of traces of iodine at a pressure up to 3 kbar.28 Let us assume that at a pressure increasing up to 1 kbar, 1% of the less stable but space-saving cis-form will be generated in addition to the trans-1,2-dichloroethene. At 1 kbar the expected density growth is 6%,13 plus 0.2% density growth due to the occurrence of 1% of the cis-form with a slightly higher density. Similarly, the permittivity [Δε/(εP − 1)] increases by 6% due to the compression plus almost 7% due to the occurrence of 1% of the dipolar cis-form. Very large differences of pressure and temperature influences on the density (Δd/d) and permittivity [Δε/(ε − 1)] values were observed for a series of isomeric alcohols with varying shielding of OH groups.29−32 This means that the pressure and particularly the temperature of a fluid can significantly affect its molecular structure. These changes have only a small influence on the density but they strongly disturb the permittivity. For nonpolar compounds such complications are not expected, so the permittivity changes should be caused only by the density changes. It is clear that the change of density (Δd/dP) needs to be compared with that of the dielectric susceptibility [Δε/(εP −1)] because the permittivity of a vacuum is equal to unity by definition. From eq 4 for the different pressures follows:
All of the collected data clearly confirm the presence of a linear relationship (R > 0.99 for eqs 3 to 8) between the changes of any function of relative permittivity, F(ε), and the liquid density (d) in the investigated pressure range (ΔP). Note that the total variation in the density in the studied pressure and temperature range rarely exceeds 10−15%. This may be the cause for the observed linear dependence on the general smooth curve of F(ε) vs d. With an extrapolation of the density to the limit (dP → 0), the value of F(ε) in eqs 4−8 is also required to be zero, or in eq 3 to be unity, because the value of ε in a vacuum is equal to unity by definition. The experimental correlation F(ε) = a + bdP, exemplified for five molecular liquids in Table 1, shows that the limit segment (a) is in all cases not equal to zero and its value can be either positive or negative. This means that in the whole range of density changes (from 0 to dP), the relationships between F(ε) and dP in eqs 3−8 are not strictly linear but consist of convex (for a > 0) or concave (for a < 0) curves. From column 5 (Table 1) it follows that the ratio a/F(ε),1 can be significant, which prevents the constancy of the ratio F(ε),P/dP for any of the eqs 3−8. For nonpolar nheptane (ε = 1.921), tetrachloromethane (ε = 2.232), and lowpolar 1-chlorobutane (ε = 7.06), the Eykman empirical eq 6 is significantly more accurate than the others, as has been often noted for a wide range of nonpolar liquids.16,24 For polar 2,2,2trifluoroethanol (ε = 27.79) and acetonitrile (ε = 35.95), none of these equations performs reliably, meaning that reliable options for the relationship F(ε)/d = constant are absent and, perhaps, do not exist at all. It is important to note that changing the “constant” values in eqs 3−8 by only 5% (ratio of [8]/[7] in 3704
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Table 2. Values of the Coefficients C and B (bar−1) of the Tait Equation (14) for Calculation of the Tangent, ∂P/∂ ln(εP − 1), and Secant, (εP − 1)/Δε, Moduli of the Permittivity (ε) Change in the Given Range of the Pressure (1 − Pmax, P/bar) and Temperature (T/°C)a liquid n-heptane20
2-methylbutane34
tetrachloromethane34
toluene34
1-pentanol32
2-pentanol32
3-pentanol32
2-methyl-1-butanol32
3-methyl-1-butanol32
2-methyl-2-butanol32
diethyl ether21 1-clorobutane21 o-xylene21 di-n-propyl ether21 chlorobenzene21 cumene21 anisole21 ethyl acetate35 diisopropyl ether35 tetrahydrofuran35 dichloromethane35 2,2,2-trifluoroethanol22
T 0 10 20 30 49 60 80 100 25 0 −25 −50 50 25 0 25 0 −25 20 30 40 50 20 30 40 50 20 30 40 50 20 30 40 50 20 30 40 50 20 30 40 50 30 30 30 30 30 50 30 30 30 50 30 50 30 50 30 50 20
ε1 1.9494 1.9353 1.921 1.9065 1.8792 1.8638 1.8338 1.8028 1.8275 1.8692 1.9103 1.9512 2.1789 2.2286 2.2786 2.3807 2.4401 2.504 15.366 14.149 12.932 11.715 14.314 12.836 11.358 9.88 14.092 12.433 10.774 9.115 16.112 14.758 13.404 12.05 15.636 14.454 13.272 12.09 6.283 5.966 5.649 5.332 4.17 7.06 2.514 3.255 5.65 5.216 2.322 4.28 5.984 5.422 3.805 3.519 7.261 6.272 8.649 7.757 27.79
Pmax
C
B
1000(εP − 1)/Δε
∂P/∂ ln(εP − 1)
3334 3334 3334 3334 3334 3334 3334 3334 2028 2028 2028 2028 1977 1217 508 2028 2028 2028 3500 3500 3500 3500 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4090 4780 2310 4850 4850 5029 4840 4750 5029 5060 5016 5046 5057 5102 4883 5121 508
0.1096 0.11 0.1097 0.1088 0.1088 0.108 0.108 0.107 0.1097 0.109 0.109 0.09 0.119 0.116 0.136 0.11 0.11 0.114 0.1201 0.122 0.125 0.128 0.119 0.133 0.148 0.167 0.126 0.147 0.1717 0.201 0.0933 0.108 0.124 0.143 0.121 0.128 0.135 0.142 0.223 0.2415 0.2723 0.334 0.1598 0.108 0.1064 0.161 0.1027 0.1127 0.1125 0.1097 0.131 0.139 0.162 0.154 0.163 0.2337 0.162 0.147 0.1144
768 717 656 594 504 451 371 302 400 509 653 888 700 832 1220 965 1192 1429 1141 1062 980 895 524 566 605 638 397 463 530 589 684 757 823 879 1196 1144 1086 1004 233 322 489 876 478 606 1018 656 1000 888 915 1177 930 874 30 354 1290 2079 1115 852 719
10947 10442 9340 9835 8423 7870 7144 6361 7221 8498 9792 14730 9590 10908 12282 12783 14923 16535 13279 12338 11388 10426 7852 7408 6916 6384 6288 5914 5504 5000 11893 11025 10123 9195 13558 12438 11311 10215 2688 2926 3294 3916 5543 9504 13735 6708 14032 11766 12036 14823 10456 9432 5137 4840 10690 10896 9642 8761 10029
7007 6518 5460 5980 4632 4176 3435 2822 3646 4670 5991 9867 5882 7172 8971 8773 10836 12535 9500 8705 7840 6992 4403 4256 4088 3820 3151 3150 3087 2930 7334 7009 6637 6147 9884 8938 8044 7070 1046 1333 1796 2623 2991 5611 9568 4075 9737 7882 8133 10729 7099 6288 2654 2299 7914 8896 6883 5796 6285
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Table 2. continued liquid
2,2,3,3-tetrafluoro-1-propanol22 2,2,3,3,3-pentafluoro-1-propanol22 n-hexane33
benzene33
tetrachloromethane33
chlorobenzene33
methanol33
benzyl alcohol33
acetone33
nitrobenzene33
tricloromethane33 fluorobenzene33 bromobenzene33 iodobenzene33 2-chlorotoluene33 3-chlorotoluene33 4-chlorotoluene33 1,2-dichlorobenzene33 1,3-dichlorobenzene33 1,1,1-trichloroethane33 1,1-dichloroethane33 1,2-dichloroethane33 1-chlorobutane33 chlorocyclohexane33 bromocyclohexane33
T 25 30 40 50 25 50 25 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30
ε1
Pmax
C
B
1000(εP − 1)/Δε
∂P/∂ ln(εP − 1)
26.8 25.82 23.96 22.2 21.18 17.9 18.56 14.58 1.8887 1.8658 1.841 2.2832 2.2533 2.2229 2.2375 2.208 2.1773 5.6895 5.437 5.199 33.6 30.83 28.24 13.74 12,31 11.04 21.24 19.75 18.34 35.72 33.05 30.6 4.6362 4.3183 5.2693 4.9279 5.2876 5.0196 4.5494 4.3818 4.638 4.3978 5.6061 5.3101 6.1413 5.7969 9.8425 9.0534 4.9165 4.6681 6.9886 6.4101 9.9008 8.9859 10.152 9.1144 6.694 6.4228 7.9505 7.3905 8.0026
503 506 509 512 509 506 515 501 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000
0.124 0.121 0.1147 0.119 0.162 0.133 0.1298 0.138 0.0927 0.092 0.092 0.0915 0.0922 0.0992 0.0964 0.1032 0.0978 0.1103 0.1007 0.1006 0.108 0.0957 0.0975 0.14 0.1148 0.088 0.1048 0.1096 0.102 0.156 0.1146 0.1043 0.11 0.097 0.0816 0.098 0.0797 0.0685 0.0655 0.0612 0.0612 0.0892 0.07254 0.06685 0.09633 0.07259 0.1141 0.1004 0.08188 0.0715 0.1027 0.0978 0.0993 0.09346 0.186 0.1497 0.09844 0.09583 0.1154 0.1052 0.08577
753 688 560 520 1174 725 573 423 434 365 313 713 650 633 671 658 537 1158 927 824 866 651 584 1896 1309 801 639 581 433 1972 1265 1021 676 438 525 535 928 648 1151 862 862 816 750 546 1060 605 1062 775 845 602 567 426 546 385 919 589 561 415 1036 795 825
9544 9208 8510 7834 10018 8674 7629 5973 9026 8241 7581 12467 11625 10630 11354 10472 9708 14564 13559 12511 12058 11229 10284 16855 15349 13982 10127 9120 8185 15623 14974 14033 10000 8632 11495 9654 17151 15637 24432 21218 21218 14010 16271 14367 15617 14120 13200 12020 15631 14290 9580 8460 9671 8351 7290 6727 9918 8503 12816 11668 14678
6073 5686 4882 4370 7247 5451 4414 3065 4682 3967 3402 7792 7050 6383 6962 6376 5491 10499 9206 8191 8019 6803 5990 13543 11402 9102 6097 5301 4245 12641 11038 9789 6145 4515 6437 5459 11644 9460 17573 14085 14085 9148 10339 8168 11004 8334 9308 7719 10320 8420 5521 4356 5498 4119 4941 3935 5699 4331 8977 7557 9619
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Table 2. continued liquid cyclohexanone33 2-butanone33 propionitrile33 benzonitrile33 nitromethane33 dichloromethane33 toluene33 propylene carbonate36
γ-butyrolactone36
acetonitrile23
T 50 30 50 30 50 30 50 30 50 30 50 30 30 15 25 35 45 15 25 35 10 25 40
ε1 7.4988 15.421 14.262 17.675 16.038 27.901 25.716 24.899 23.122 35.78 32.671 8.6381 2.3614 67.42 64.93 62.73 60.543 43.14 41.77 40.55 38.38 35.95 33.73
Pmax
C
B
1000(εP − 1)/Δε
∂P/∂ ln(εP − 1)
2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 3000 3000 3000
0.0889 0.0655 0.0697 0.1193 0.0895 0.0747 0.07126 0.06797 0.06056 0.1294 0.12098 0.1014 0.0818 0.0632 0.0619 0.0546 0.05517 0.0506 0.04534 0.0505 0.1124 0.1106 0.1104
770 707 628 732 392 630 464 1038 708 1405 1130 600 636 1208 1089 840 789 1144 896 946 1101 939 820
13514 17316 15062 9735 8814 14068 12206 21802 18753 14378 13037 10050 12924 25988 24520 22976 21746 31140 28965 27066 13845 12497 11358
8661 10794 9010 6136 4380 8434 6511 15271 11691 10858 9340 5917 7775 19114 17593 15385 14293 22609 19762 18733 9795 8490 7426
a All values of the coefficients C and B were obtained in this work by recalculation of the experimental data of referenced liquids in accordance with the Tait eq 14.
9.93 for 2,2,2-trifluoroethanol, and 12.65 for acetonitrile (Table 1). Experimental values of these angular coefficients are 1.20, R = 0.9998 for n-heptane; 1.28, R = 0.9999 for tetrachloromethane; 1.54, R = 0.9965 for 1-chlorobutane; 1.33, R = 0.9991 for 2,2,2-trifluoroethanol; and 1.07, R = 1 for acetonitrile. This means that the Clausius−Mossotti eq 12 is not suitable for the description of polar liquids. It is interesting that for the nonpolar liquids a linear dependence of Δε/(εP − 1) vs Δd/dP, caused by the pressure and temperature changes of permittivity and density, is commonly found (Figure 2). Some of the polar compounds (e.g., methanol, ethyl acetate, acetonitrile, primary alcohols, Table 2) also lie on this correlation line. Secondary and tertiary alcohols, ethers, esters, and nitro derivatives have enhanced values of angular coefficients, which are very sensitive to changes in temperature (Figure 2) . As follows from the “noncrossing rule” of V−P curves, the initial compressibility of a liquid at atmospheric pressure will define the compressibility curve at high pressure, and Δd/dP vs P curves never cross each other for different liquids.13 A linear correlation has been observed between the tangent (1/β0) and the secant bulk modulus value at 1000 bar (V0/ΔV1kbar) for different liquids in a wide temperature range, according to eq 13:
Figure 3. Relation between the tangent bulk modulus values at atmospheric pressure (∂P/∂ ln(εP − 1)) and secant bulk modulus values at 1000 bar ((εP=1000 − 1)/Δε) of permittivity of all liquids given in Table 2.
(1/d1 − 1/dP)/C2 = (1/(ε1 − 1) − 1/(εP − 1)
(10)
(εP − ε1)/(εP − 1) = [(ε1 − 1)/C2d1] × (dP − d1)/dP = 1 × (dP − d1)/dP
(11)
1/β1 = ( −4559 ± 23) + (0.9865 ± 0.0010)(1000V1/ΔV )
For liquids with a high relative permittivity, eq 8 can be written as (ε −1)/d = 9Po/2,19 which also leads to relation 11. From eq 7 a similar calculation leads to eq 12: (εP − ε1)/(εP − 1) = [(ε1 + 2)/3] × (dP − d1)/dP
R = 0.9999; N = 272
(13)
The total correlation contains all known data about the compressibility of fluids of different classes and at different temperatures (cyclic, linear and branched alkanes, alkenes, alcohols, aldehydes, ketones, carboxylic acids, ethers, esters, nitriles, halo- and nitro-derivatives, amines, amides, heterocycles, difunctional compounds, and even mercury). This indicates that in the studied range of pressure there are no
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From 11 it follows that the slope of the dependence Δε/(εP − 1) vs Δd/dP in all cases should be equal to unity. The predicted angular coefficients [(ε1 + 2)/3] from eq 12 are 1.31 for nheptane, 1.41 for tetrachloromethane, 3.02 for 1-chlorobutane, 3707
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different temperatures has revealed the means to calculate the coefficients of the Tait equation (14) from the experimental values of the pressure influence on the permittivity at ambient pressure.
sharp changes in the processes determining the compressibility and no phase transformations or chemical processes take place. The main goal of the current research is to determine the presence or absence of a similar “noncrossing rule” for Δε/(εP − 1)−P curves. It can be expected that a variety of reasons leading to a change in the permittivity under pressure (volume change, molecule deformation, change of the dipole−dipole interaction, change of the conformational equilibrium, change of the molecular structure of OH complexes) will gradually vary with pressure because all studied polar and nonpolar liquids are reliably described by eq 14, recalculated in this work for all used data (Table 2) with correlation coefficients of R > 0.999. On the basis of these considerations, it should be expected that the linear relationship between the tangent [(εP→1 − 1)∂P/∂ε] and secant [(ε P=1000 − 1)/Δε P=1000 ] moduli for curves is implemented similarly to that found previously for the process of compressibility.13 All values of the coefficients C and B, ∂P/ ∂ln(ε − 1) = B/C, and (εP − 1)/Δε (Table 2) were obtained in this work by recalculation of the experimental data in accordance with the Tait equation (14). Δε /(εP − 1) = C ln[(B + P)/B]
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Corresponding Author
*V. D. Kiselev: e-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (Project No 12-03-00029), the U.S. Civilian Research and Development Foundation (CRDF), and the Ministry of Science and Education of the Russian Federation (Joint Program “Fundamental Research and High Education”, Grant REC 007).
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REFERENCES
(1) Bridgman, P. The Physics of High Pressure; Bell and Sons: London, 1958. (2) Isaacs, N. S. Liquid Phase High Pressure Chemistry; WileyInterscience: Chichester, New York-Brisbane-Toronto, 1981. (3) Kiselev, V. D.; Kashaeva, E. A.; Konovalov, A. I. Pressure Effect on the Rate and Equilibrium Constants of the Diels-Alder Reaction 9Chloroanthracene with Tetracyanoethylene. Tetrahedron 1999, 55, 1153−1162. (4) Marcus, Y.; Hefter, G. T. The Compressibility of Liquids at Ambient Temperature and Pressure. J. Mol. Liq. 1997, 73, 61−74. (5) Marcus, Y.; Hefter, G. T. On the Pressure and Electric Field Dependencies of the Relative Permittivity of Liquids. J. Solution Chem. 1999, 28, 575−592. (6) Kiselev, V. D.; Konovalov, A. I. Effect of Solvent Polarity on the Activation Volume of Polar Reactions. Dokl. Chem., Engl. Transl. 1998, 363, 426−428. (7) Tait, P. G. Physics and Chemistry of the Voyage of H.M.S. Challenger; H.M.S.O.: London, 1888. (8) Dymond, J. H.; Malhotra, R. The Tait Equation: 100 years on. Int. J. Thermophys. 1988, 9, 941−951. (9) Hayward, A. T. J. Compressibility Equations for Liquids: a Comparative Study. J. Appl. Phys. 1967, 18, 965−977. (10) Hayward, A. T. J. How to Measure Isothermal Compressibility of Liquids Accurately. J. Phys. D: Appl. Phys. 1971, 4, 938−950. (11) Whalley, E. The Compression of Liquids in Experimental Thermodynamics; Butterworths: London, 1975. (12) Gibson, R. E.; Loeffler, O. H. Pressure-Volume-Temperature Relations in Solutions. II. The Energy-Volume Coefficients of Aniline, Nitrobenzene, Bromobenzene and Chlorobenzene. J. Am. Chem. Soc. 1939, 61, 2515−2522. (13) Kiselev, V. D.; Bolotov, A. V.; Satonin, A. P.; Shakirova, I. I.; Kashaeva, E. A.; Konovalov, A. I. Compressibility of Liquids. Rule of Noncrossing V−P Curvatures. J. Phys. Chem. B 2008, 112, 6674−6682. (14) Hayward, A. T. J. Precise Determination of the Isothermal Compressibility of Mercury at 20 °C and 192 bar. J. Phys. D: Appl. Phys. 1971, 4, 951−955. (15) le Noble, W. J. Organic High Pressure Chemistry; Elsevier: Amsterdam, The Netherlands, 1988. (16) Gibson, R. E.; Kincaid, J. F. The Influence of Temperature and Pressure on the Volume and Refractive Index of Benzene. J. Am. Chem. Soc. 1938, 60, 511−518. (17) Owen, B. B.; Brinkley, S. R. The Effect of Pressure upon the Dielectric Constants of Liquids. Phys. Rev. 1943, 64, 32−36. (18) Rosen, J. S. Refractive Indices and Dielectric Constants of Liquids and Gases under Pressure. J. Chem. Phys. 1949, 17, 1192− 1197.
The resulting relation (Figure 3) is described by eq 15: (εP → 1 − 1)∂P /∂ε = −1308 + 757.9[(εP = 1000 − 1)/Δε] R = 0.984; N = 143
AUTHOR INFORMATION
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It should be noted that Figure 3 includes all the liquids (Table 2) that were presented in Figure 2. The average uncertainty in the slope is ±1.6%, which can be attributed to experimental errors, especially for the value (εP − 1)∂P/∂ε. For example, the pressure influence at 30 °C on the permittivity of 1-chlorobutane according to the data from various sources21,33 differs by nearly 15%. Equation 15 allows the calculation of the initial change (εP − 1)∂P/∂ε at atmospheric pressure when the value of the permittivity change under 1 kbar is known. Conversely, with the known initial changes (εP − 1)∂P/∂ε at atmospheric pressure, the value of (ε1000 − 1)/Δε at 1 kbar can be calculated from eq 15. Then, with the known value of (ε1 − 1)∂P/∂ε equal to B/C and that calculated from the eq 15 value of Δε/ (εP=1000 − 1) = C ln[(B + 1000)/B], we can determine the constants B and C to describe the whole curve by eq 14.
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CONCLUSIONS From this analysis it follows that the influence of a high external hydrostatic pressure on the liquid permittivity is more complicated than the effect on the liquid compressibility. The pressure and temperature coefficients of the permittivity can be drastically larger than the changes in pressure and temperature coefficients of density. This means that the pressure and particularly the temperature significantly affect the structure of a liquid. For this reason, the changes in permittivity under pressure and temperature bear much additional information on the intermolecular interactions of a fluid. Especially valuable information can be provided from the sharp difference in the change of the values Δε/(εP − 1) and Δd/dP (Figure 2). This allows specifying the pressure and temperature influence on the change of the rate and equilibrium of a chemical reaction. Here we assume that a variety of reasons leading to a change in the permittivity in the studied range of pressure will gradually vary without saturation. The observed clear relation (15) between the tangent and secant bulk moduli for various liquids at 3708
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(19) Kirkwood, J. G. The Dielectric Polarization of Polar Liquids. J. Chem. Phys. 1939, 7, 911−919. (20) Scaife, W. Relative Permittivity of Organic Liquids as a Function of Pressure and Temperature. J. Phys. A: Math. Gen. 1971, 4, 413−425. (21) Skinner, J. F.; Cussler, E. L.; Fuoss, R. M. Pressure Dependence of Dielectric Constant and Density of Liquids. J. Phys. Chem. 1968, 72, 1057−1064. (22) Tanaka, Y.; Xiao, Y. F.; Matsuo, S. Relative Permittivity of Fluoroalcohols at Temperatures from 293 to 323 K and Pressures up to 50 MPa. Fluid Phase Equilib. 2000, 170, 139−149. (23) Srinivasan, K. R.; Kay, R. L. The Pressure Dependence of the Dielectric Constant and Density of Acetonitrile at Three Temperatures. J. Solution Chem. 1977, 6, 357−367. (24) Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents, 4th ed.; Wiley-Interscience: New York, 1986. (25) Chen, T.; Dannhauser, W.; Johary, G. P. Study of the Pressure Dependence of Dielectric Polarization. J. Chem. Phys. 1969, 50, 2046− 2052. (26) Sabharwal, R. J.; Huang, Y.; Song, Y. High-Pressure Induced Conformational and Phase Transformations of 1,2-Dichloroethane Probed by Raman Spectroscopy. J. Phys. Chem. B 2007, 111, 7267− 7273. (27) Murugan, N. A.; Hugosson, H. W.; Agren, H. Solvent Dependence on Conformational Transition, Dipole Moment, and Molecular Geometry of 1,2-Dichloroethane: Insight from Car− Parrinello Molecular Dynamics Calculations. J. Phys. Chem. B 2008, 112, 14673−14677. (28) Ewald, A. H.; Hamann, S. D.; Stutchebury, J. E. The Effect of Pressure on Some Free-radical Reactions of 1,2-Dichlorethylene. Trans. Faraday Soc. 1957, 53, 991−996. (29) Jacobs, I. S.; Lawson, A. W. An Analysis of the Pressure Dependence of the Dielectric Constant of Polar Liquids. J. Chem. Phys. 1952, 20, 1161−1164. (30) Gilchrist, A.; Earley, J. E.; Cole, R. E. Effect of Pressure on Dielectric Properties and Volume of 1-Propanol and Glycerol. J. Chem. Phys. 1957, 26, 196−200. (31) Johary, G. P.; Dannhauser, W. Dielectric Study of the Pressure Dependence of Intermolecular Association in Isomeric Octyl Alcohols. J. Chem. Phys. 1968, 48, 5114−5122. (32) Bennett, R. G.; Hall, G. H.; Calderwood, J. H. The Pressure Dependence of the Static Permittivity of Pentanol Isomers. J. Phys. D: Appl. Phys. 1973, 6, 781−789. (33) Hartmann, H.; Neumann, A.; Schmidt, A. P. Zur Druckabhängigkeit der statischen Dielektrizitätskonstante von Flüssigkeiten. Ber. Bunsen-Ges. Phys. Chem. 1968, 72, 877−880. (34) Mopsik, F. I. Dielectric Properties of Slightly Polar Organic Liquids as a Function of Pressure, Volume, and Temperature. J. Chem. Phys. 1969, 50, 2559−2569. (35) Shornack, L. G.; Eckert, C. A. Effect of Pressure on the Density and Dielectric Constant of Polar Solvents. J. Phys. Chem. 1970, 74, 3014−3020. (36) Cote, J.-F.; Brouillette, D.; Desnoyers, J. E.; Rouleau, J.-F.; StArnaud, J.-M.; Perron, G. Dielectric Constants of Acetonitrile, γButyrolactone, Propylene Carbonate, and 1,2-Dimethoxyethane as a Function of Pressure and Temperature. J. Solution Chem. 1996, 25, 1163−1173.
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NOTE ADDED AFTER ASAP PUBLICATION This paper was published ASAP on March 25, 2014. The Results and Discussion section was updated. The revised paper was reposted on April 3, 2014.
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Changes in Permittivity and Density of Molecular Liquids under High Pressure Vladimir D. Kiselev,*,† Dmitry A. Kornilov,† and Alexander I. Konovalov‡ †
Laboratory of High Pressure Chemistry, Butlerov Institute of Chemistry, Kazan Federal University, Kremlevskaya Str., 18, Kazan 420008, Russian Federation ‡ Arbuzov Institute of Organic and Physical Chemistry, Kazan Scientific Center of the Russian Academy of Sciences, Arbuzov Str., 8, Kazan 420088, Russian Federation ABSTRACT: We collected and analyzed the density and permittivity of 57 nonpolar and dipolar molecular liquids at different temperatures (143 sets) and pressures (555 sets). No equation was found that could accurately predict the change to polar liquid permittivity by the change of its density in the range of the pressures and temperatures tested. Consequently, the influence of high hydrostatic pressure and temperature on liquid permittivity may be a more complicated process compared to density changes. The pressure and temperature coefficients of permittivity can be drastically larger than the pressure and temperature coefficients of density, indicating that pressure and particularly temperature significantly affect the structure of molecular liquids. These changes have less influence on the density change but can strongly affect the permittivity change. The clear relationship between the tangent and secant moduli of the permittivity curvatures under pressure for various molecular liquids at different temperatures was obtained, from which one can calculate the Tait equation coefficients from the experimental values of the pressure influence on the permittivity at ambient pressure.
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INTRODUCTION Studying the influence of pressure on fluid properties illuminates not only special features of the internal structures
is studied. For polar and especially ionic reactions, the influence of increasing values of the relative permittivity under pressure (εP) is particularly important in the analysis of the rate, equilibrium, and volume parametric values.1−5 This allows us to distinguish between the following components in the reactionrate change under pressure: reaction-rate change due to the contributions caused by the work energy PΔV‡ and the change in interaction energy of initial and transition states with the solvent due to the increase in the relative permittivity under pressure, even in cases where the magnitude of ΔV‡ is zero.3,6 The compressibility of a liquid at an elevated external hydrostatic pressure up to several kbar corresponds to a decrease in intermolecular voids when the molecules approach each other under pressure.1−15 In all cases, the compressibility curves V−P in this pressure range are reliably described by the Tait relation (1)7,8 ΔV /V1 = (V1 − VP)/V1 = (dP − d1)/dP
Figure 1. Superposition of the energy curves of the intermolecular repulsion (1) and attraction (2, 3) with the formation of lower (4) and larger (5) intermolecular interaction energy, depending on the intermolecular equilibrium distance (R4, R5) between the solvent molecules.
= C1·ln[(B1 + P)/(B1)]
Hence, the compressibility coefficient (βT) can be calculated for any pressure as βT = ∂(ln dP)/∂P = (dP /d1) ·[C1/(B1 + P)]
of liquids but also the interaction of the solvent with the solute. These issues clearly emerge when the effect of pressure on the rate and equilibrium of chemical reactions is examined, when the volume of activation and the reaction volume are determined, and when the process of solvent electrostriction © 2014 American Chemical Society
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Received: February 7, 2014 Revised: March 12, 2014 Published: March 17, 2014 3702
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Table 1. Parameters of the Relation F(ε) = a + bdP between Functions (F(ε)) and density (dP) of n-Heptane, Tetrachloromethane, 1-Chlorobutane, 2,2,2-Trifluoroethanol, and Acetonitrile in a Given Pressure Range (R = Correlation Coefficient) 1 no.
a
2 function F(ε)
1 2 3 4 5 6
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
7 8 9 10 11 12
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
13 14 15 16 17 18
ε ε− ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
19 20 21 22 23 24
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
25 26 27 28 29 30
ε ε−1 ε0.5 − 1 (ε − 1)/(ε0.5 + 0.4) (ε − 1)/(ε + 2) (ε − 1)(2ε + 1)/9ε
3 a
4 b
5
6
100a/F(ε),1
R
2
n-Heptane (20 °C; ε1 = 1.931; ΔP/bar = 1−3334)a 0.7558 1.7032 39.34% 0.9999 −0.2442 1.7032 −26.51 0.9999 −0.0220 0.5969 −5.70 1 0.0050 0.7472 0.97 1 0.0206 0.3138 8.77 0.9999 −0.0322 0.4242 −12.49 1 Tetrachloromethane (25 °C; ε1 = 2.2320; ΔP/bar = 1−1118)a 0.6076 1.0251 27.22 1 −0.3924 1.0251 −31.85 1 −0.0416 0.3380 −8.42 1 −0.0077 0.4154 −1.18 1 0.0275 0.1664 9.45 1 −3.0489 2.9755 −18.26 0.9996 1-Chlorobutane (30 °C; ε1 = 7.06; ΔP/bar = 1−4780)b −3.3939 11.913 −48.07 0.9921 −4.3939 11.913 −72.50 0.9921 −0.1833 2.1013 −11.06 0.9927 −0.0056 2.2704 −0.28 0.9927 0.3586 0.3559 53.70 0.9910 −0.8990 2.6680 −62.34 0.9921 2,2,2-Trifluoroethanol (20 °C; ε1 = 27.79 ΔP/bar = 1−508)c −8.3298 25.941 −29.97 0.9984 −9.3298 25.941 −34.83 0.9984 0.8989 2.4226 21.05 0.9983 1.2652 2.4841 26.78 0.9983 0.7841 0.0828 87.19 0.9980 1.9711 5.7683 32.53 0.9984 Acetonitrile (25 °C; ε1 = 35.95 ΔP/bar = 1−3000)d −2.1233 48.417 −5.91 0.9999 −3.1233 48.417 −8.94 0.9999 1.9595 3.8653 39.22 1 2.3708 3.9385 43.39 1 0.8539 0.0857 92.72 0.9984 −0.5888 10.763 −7.48 0.9999
7
8
9
F(ε),1/d1
F(ε),P/dP
[8]/[7]
2.81 1.3473 0.5647 0.7544 0.3436 0.3773
2.63 1.4051 0.5698 0.7531 0.3387 0.3849
0.9359 1.0430 1.0092 0.9983 0.9857 1.0201
1.4086 0.7775 0.3117 0.4105 0.1837 1.0538
1.3793 0.7965 0.3138 0.4109 0.1824 1.2009
0.9792 1.0244 1.0067 1.0009 0.9930 1.1395
8.0369 6.8984 1.8863 2.2566 0.7614 1.6416
8.6956 7.7229 1.9356 2.2782 0.7059 1.8122
1.0820 1.1195 1.0261 1.0096 0.9271 1.1039
19.9612 19.2429 3.0683 3.3928 0.6459 4.3531
20.2233 19.5383 3.0371 3.3495 0.6199 4.4154
1.0131 1.0153 0.9898 0.9872 0.9596 1.0143
45.74 44.47 6.36 6.95 0.92 7.87
46.14 45.06 5.99 6.51 0.93 9.34
1.0089 1.0133 0.9423 0.9362 1.0127 1.1863
From ref 20. Hereinafter, ε1 means the value of relative permittivity at ambient pressure (1 bar). bFrom ref 21. cFrom ref 22. dFrom ref 23.
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Here d1 and V1 as well as dP and VP are the solvent density and volume at atmospheric and at elevated pressures (P), respectively, and C1 and B1 are the constants for each of the liquids at a given temperature. Accumulated experimental data confirm that the constants C1 and B1 are independent of the pressure, and C1 practically does not depend on the temperature. B1 usually decreases with increasing temperature, which corresponds to an increase in the compressibility coefficient with an increase in temperature. In general, an increase in the energy of intermolecular interactions in the liquid is accompanied by a denser packing and lower compressibility.1,2,11 Imposing a steeper energy dependence for intermolecular repulsions (1) and the curves of intermolecular interactions (2 and 3) with a decrease in the intermolecular distance gives the potential energy curves of intermolecular stabilization (4 and 5) in a liquid (Figure 1). In general, as the intermolecular interaction energy (curve 2) weakens, the equilibrium intermolecular distance (R4, Figure 1) increases, which increases the void volume, leading to increasing compressibility.
RESULTS AND DISCUSSION
Most of the changes of fluid properties (e.g., refractive index, relative permittivity, viscosity) at elevated hydrostatic pressures up to a few kbar are mainly caused by changes in compressibility, i.e., from a reduction of intermolecular voids and by increasing the number of molecules per volume unit.1,2,4,9−15 The compressibility (eq 1) has been determined with high accuracy for a large number of molecular liquids.1,2,4,9−15 The search for correlations among liquid density (d), its refractive index (nD), and relative permittivity (ε) is of great interest. In this work the pressure influence on variation of the density and permittivity of 57 nonpolar and dipolar liquids was considered. As indicated in the original papers, all tested fluids were purified and the values of density, refractive index and relative permittivity at atmospheric pressure and chosen temperatures correspond to the known literature data. Several relationships (3)−(8) for the general type F(ε)/d = constant have been proposed16−19 as follows: 3703
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Table 1 is 1.05) at pressures up to 1 kbar generates an error of about 50−100% for the calculation of changes of the permittivity functions from the liquid density data (Table 1). This is because the density change itself (dP=1000/dP=1) at pressures of 1 kbar does not exceed the values of 1.05− 1.10.4−15 The ratio of the “constants” [8]/[7] (Table 1, column 9) is equal to the ratio of the limit values [F(ε),P/dP]/[F(ε),1/d1] and differs markedly from the expected unity. This can be considered as evidence that the change in the relative permittivity under pressure is not only determined by the increase of the number of molecules per volume unit, i.e., by the density change of the fluid. Numerous studies in this area have shown that the change of relative permittivity under pressure can be considered as a more complex process than the compressibility. It was suggested19,25 that the pressure effect on the relative permittivity of polar liquids, ∂ ln ε/∂P, is due to at least three factors: (1) the change in fluid density ∂ ln d/∂P, (2) the deformation of molecules induced by pressure ∂μ/∂P, and (3) the change in the internal liquid structure induced by external pressure ∂g/∂P, as shown in eq 9:
Figure 2. Influence of the fluid density changes, Δd/dP, in the range of pressures and temperatures on the permittivity change, Δε/(εP − 1), according to the data from Table 2. Blue points (◆): n-hexane (N = 62), n-heptane (N = 48), 2-methylbutane (N = 38). Red points (■): methanol (N = 10), acetonitrile (N = 18), ethyl acetate (N = 12), anisole (N = 10), o-xylene (N = 6), 3-methyl-1-butanol (N = 13), 1pentanol (N = 24), chlorobenzene (N = 9), cumene (N = 11), 2methyl-1-butanol (N = 16). Green points (▲): data from Table 2. Total number of points is 555.
(ε − 1)(2ε + 1)/9εd = P0 = (4πN /3)[α + (gμ2 /3kT )] (9)
ε /d = const
(3)
(ε − 1)/d = const
(4)
(ε 0.5 − 1)/d = const
(5)
(ε − 1)/(ε 0.5 + 0.4)d = const
(6)
(ε − 1)/(ε + 2)d = const
(7)
(ε − 1)(2ε + 1)/9εd = const
(8)
Here, N is the Avogadro number, α is the electronic polarizability, μ is the dipole moment, k is the Boltzmann constant, T is the absolute temperature, and g is a correction factor in the expanded Kirkwood equation (9), reflecting a possible change in dipole orientation of the molecules relative to each other under pressure.19 Using an appropriate value of g in magnitude and sign, everyone can calculate the changes in the magnitude of P0 in eq 9 in the range of pressure and temperature. Additionally, high pressure may increase the proportion of more compact conformers. For example, for 1,2dichloroethane, an increase in pressure may lead to the accumulation of the more compact and more polar gauche form.26,27 The more compact and polar cis-1,2-dichloroethene (d20 = 1.2818; ε20 = 9.20, μ = 1.89 D) has also been found to increase compared to trans-1,2-dichloroethene (d20 = 1.2546; ε20 = 2.14, μ = 0 D) in a mixture of geometric isomers at an elevated temperature in the presence of traces of iodine at a pressure up to 3 kbar.28 Let us assume that at a pressure increasing up to 1 kbar, 1% of the less stable but space-saving cis-form will be generated in addition to the trans-1,2-dichloroethene. At 1 kbar the expected density growth is 6%,13 plus 0.2% density growth due to the occurrence of 1% of the cis-form with a slightly higher density. Similarly, the permittivity [Δε/(εP − 1)] increases by 6% due to the compression plus almost 7% due to the occurrence of 1% of the dipolar cis-form. Very large differences of pressure and temperature influences on the density (Δd/d) and permittivity [Δε/(ε − 1)] values were observed for a series of isomeric alcohols with varying shielding of OH groups.29−32 This means that the pressure and particularly the temperature of a fluid can significantly affect its molecular structure. These changes have only a small influence on the density but they strongly disturb the permittivity. For nonpolar compounds such complications are not expected, so the permittivity changes should be caused only by the density changes. It is clear that the change of density (Δd/dP) needs to be compared with that of the dielectric susceptibility [Δε/(εP −1)] because the permittivity of a vacuum is equal to unity by definition. From eq 4 for the different pressures follows:
All of the collected data clearly confirm the presence of a linear relationship (R > 0.99 for eqs 3 to 8) between the changes of any function of relative permittivity, F(ε), and the liquid density (d) in the investigated pressure range (ΔP). Note that the total variation in the density in the studied pressure and temperature range rarely exceeds 10−15%. This may be the cause for the observed linear dependence on the general smooth curve of F(ε) vs d. With an extrapolation of the density to the limit (dP → 0), the value of F(ε) in eqs 4−8 is also required to be zero, or in eq 3 to be unity, because the value of ε in a vacuum is equal to unity by definition. The experimental correlation F(ε) = a + bdP, exemplified for five molecular liquids in Table 1, shows that the limit segment (a) is in all cases not equal to zero and its value can be either positive or negative. This means that in the whole range of density changes (from 0 to dP), the relationships between F(ε) and dP in eqs 3−8 are not strictly linear but consist of convex (for a > 0) or concave (for a < 0) curves. From column 5 (Table 1) it follows that the ratio a/F(ε),1 can be significant, which prevents the constancy of the ratio F(ε),P/dP for any of the eqs 3−8. For nonpolar nheptane (ε = 1.921), tetrachloromethane (ε = 2.232), and lowpolar 1-chlorobutane (ε = 7.06), the Eykman empirical eq 6 is significantly more accurate than the others, as has been often noted for a wide range of nonpolar liquids.16,24 For polar 2,2,2trifluoroethanol (ε = 27.79) and acetonitrile (ε = 35.95), none of these equations performs reliably, meaning that reliable options for the relationship F(ε)/d = constant are absent and, perhaps, do not exist at all. It is important to note that changing the “constant” values in eqs 3−8 by only 5% (ratio of [8]/[7] in 3704
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Table 2. Values of the Coefficients C and B (bar−1) of the Tait Equation (14) for Calculation of the Tangent, ∂P/∂ ln(εP − 1), and Secant, (εP − 1)/Δε, Moduli of the Permittivity (ε) Change in the Given Range of the Pressure (1 − Pmax, P/bar) and Temperature (T/°C)a liquid n-heptane20
2-methylbutane34
tetrachloromethane34
toluene34
1-pentanol32
2-pentanol32
3-pentanol32
2-methyl-1-butanol32
3-methyl-1-butanol32
2-methyl-2-butanol32
diethyl ether21 1-clorobutane21 o-xylene21 di-n-propyl ether21 chlorobenzene21 cumene21 anisole21 ethyl acetate35 diisopropyl ether35 tetrahydrofuran35 dichloromethane35 2,2,2-trifluoroethanol22
T 0 10 20 30 49 60 80 100 25 0 −25 −50 50 25 0 25 0 −25 20 30 40 50 20 30 40 50 20 30 40 50 20 30 40 50 20 30 40 50 20 30 40 50 30 30 30 30 30 50 30 30 30 50 30 50 30 50 30 50 20
ε1 1.9494 1.9353 1.921 1.9065 1.8792 1.8638 1.8338 1.8028 1.8275 1.8692 1.9103 1.9512 2.1789 2.2286 2.2786 2.3807 2.4401 2.504 15.366 14.149 12.932 11.715 14.314 12.836 11.358 9.88 14.092 12.433 10.774 9.115 16.112 14.758 13.404 12.05 15.636 14.454 13.272 12.09 6.283 5.966 5.649 5.332 4.17 7.06 2.514 3.255 5.65 5.216 2.322 4.28 5.984 5.422 3.805 3.519 7.261 6.272 8.649 7.757 27.79
Pmax
C
B
1000(εP − 1)/Δε
∂P/∂ ln(εP − 1)
3334 3334 3334 3334 3334 3334 3334 3334 2028 2028 2028 2028 1977 1217 508 2028 2028 2028 3500 3500 3500 3500 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4090 4780 2310 4850 4850 5029 4840 4750 5029 5060 5016 5046 5057 5102 4883 5121 508
0.1096 0.11 0.1097 0.1088 0.1088 0.108 0.108 0.107 0.1097 0.109 0.109 0.09 0.119 0.116 0.136 0.11 0.11 0.114 0.1201 0.122 0.125 0.128 0.119 0.133 0.148 0.167 0.126 0.147 0.1717 0.201 0.0933 0.108 0.124 0.143 0.121 0.128 0.135 0.142 0.223 0.2415 0.2723 0.334 0.1598 0.108 0.1064 0.161 0.1027 0.1127 0.1125 0.1097 0.131 0.139 0.162 0.154 0.163 0.2337 0.162 0.147 0.1144
768 717 656 594 504 451 371 302 400 509 653 888 700 832 1220 965 1192 1429 1141 1062 980 895 524 566 605 638 397 463 530 589 684 757 823 879 1196 1144 1086 1004 233 322 489 876 478 606 1018 656 1000 888 915 1177 930 874 30 354 1290 2079 1115 852 719
10947 10442 9340 9835 8423 7870 7144 6361 7221 8498 9792 14730 9590 10908 12282 12783 14923 16535 13279 12338 11388 10426 7852 7408 6916 6384 6288 5914 5504 5000 11893 11025 10123 9195 13558 12438 11311 10215 2688 2926 3294 3916 5543 9504 13735 6708 14032 11766 12036 14823 10456 9432 5137 4840 10690 10896 9642 8761 10029
7007 6518 5460 5980 4632 4176 3435 2822 3646 4670 5991 9867 5882 7172 8971 8773 10836 12535 9500 8705 7840 6992 4403 4256 4088 3820 3151 3150 3087 2930 7334 7009 6637 6147 9884 8938 8044 7070 1046 1333 1796 2623 2991 5611 9568 4075 9737 7882 8133 10729 7099 6288 2654 2299 7914 8896 6883 5796 6285
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Table 2. continued liquid
2,2,3,3-tetrafluoro-1-propanol22 2,2,3,3,3-pentafluoro-1-propanol22 n-hexane33
benzene33
tetrachloromethane33
chlorobenzene33
methanol33
benzyl alcohol33
acetone33
nitrobenzene33
tricloromethane33 fluorobenzene33 bromobenzene33 iodobenzene33 2-chlorotoluene33 3-chlorotoluene33 4-chlorotoluene33 1,2-dichlorobenzene33 1,3-dichlorobenzene33 1,1,1-trichloroethane33 1,1-dichloroethane33 1,2-dichloroethane33 1-chlorobutane33 chlorocyclohexane33 bromocyclohexane33
T 25 30 40 50 25 50 25 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 20 35 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30 50 30
ε1
Pmax
C
B
1000(εP − 1)/Δε
∂P/∂ ln(εP − 1)
26.8 25.82 23.96 22.2 21.18 17.9 18.56 14.58 1.8887 1.8658 1.841 2.2832 2.2533 2.2229 2.2375 2.208 2.1773 5.6895 5.437 5.199 33.6 30.83 28.24 13.74 12,31 11.04 21.24 19.75 18.34 35.72 33.05 30.6 4.6362 4.3183 5.2693 4.9279 5.2876 5.0196 4.5494 4.3818 4.638 4.3978 5.6061 5.3101 6.1413 5.7969 9.8425 9.0534 4.9165 4.6681 6.9886 6.4101 9.9008 8.9859 10.152 9.1144 6.694 6.4228 7.9505 7.3905 8.0026
503 506 509 512 509 506 515 501 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000
0.124 0.121 0.1147 0.119 0.162 0.133 0.1298 0.138 0.0927 0.092 0.092 0.0915 0.0922 0.0992 0.0964 0.1032 0.0978 0.1103 0.1007 0.1006 0.108 0.0957 0.0975 0.14 0.1148 0.088 0.1048 0.1096 0.102 0.156 0.1146 0.1043 0.11 0.097 0.0816 0.098 0.0797 0.0685 0.0655 0.0612 0.0612 0.0892 0.07254 0.06685 0.09633 0.07259 0.1141 0.1004 0.08188 0.0715 0.1027 0.0978 0.0993 0.09346 0.186 0.1497 0.09844 0.09583 0.1154 0.1052 0.08577
753 688 560 520 1174 725 573 423 434 365 313 713 650 633 671 658 537 1158 927 824 866 651 584 1896 1309 801 639 581 433 1972 1265 1021 676 438 525 535 928 648 1151 862 862 816 750 546 1060 605 1062 775 845 602 567 426 546 385 919 589 561 415 1036 795 825
9544 9208 8510 7834 10018 8674 7629 5973 9026 8241 7581 12467 11625 10630 11354 10472 9708 14564 13559 12511 12058 11229 10284 16855 15349 13982 10127 9120 8185 15623 14974 14033 10000 8632 11495 9654 17151 15637 24432 21218 21218 14010 16271 14367 15617 14120 13200 12020 15631 14290 9580 8460 9671 8351 7290 6727 9918 8503 12816 11668 14678
6073 5686 4882 4370 7247 5451 4414 3065 4682 3967 3402 7792 7050 6383 6962 6376 5491 10499 9206 8191 8019 6803 5990 13543 11402 9102 6097 5301 4245 12641 11038 9789 6145 4515 6437 5459 11644 9460 17573 14085 14085 9148 10339 8168 11004 8334 9308 7719 10320 8420 5521 4356 5498 4119 4941 3935 5699 4331 8977 7557 9619
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Table 2. continued liquid cyclohexanone33 2-butanone33 propionitrile33 benzonitrile33 nitromethane33 dichloromethane33 toluene33 propylene carbonate36
γ-butyrolactone36
acetonitrile23
T 50 30 50 30 50 30 50 30 50 30 50 30 30 15 25 35 45 15 25 35 10 25 40
ε1 7.4988 15.421 14.262 17.675 16.038 27.901 25.716 24.899 23.122 35.78 32.671 8.6381 2.3614 67.42 64.93 62.73 60.543 43.14 41.77 40.55 38.38 35.95 33.73
Pmax
C
B
1000(εP − 1)/Δε
∂P/∂ ln(εP − 1)
2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 3000 3000 3000
0.0889 0.0655 0.0697 0.1193 0.0895 0.0747 0.07126 0.06797 0.06056 0.1294 0.12098 0.1014 0.0818 0.0632 0.0619 0.0546 0.05517 0.0506 0.04534 0.0505 0.1124 0.1106 0.1104
770 707 628 732 392 630 464 1038 708 1405 1130 600 636 1208 1089 840 789 1144 896 946 1101 939 820
13514 17316 15062 9735 8814 14068 12206 21802 18753 14378 13037 10050 12924 25988 24520 22976 21746 31140 28965 27066 13845 12497 11358
8661 10794 9010 6136 4380 8434 6511 15271 11691 10858 9340 5917 7775 19114 17593 15385 14293 22609 19762 18733 9795 8490 7426
a All values of the coefficients C and B were obtained in this work by recalculation of the experimental data of referenced liquids in accordance with the Tait eq 14.
9.93 for 2,2,2-trifluoroethanol, and 12.65 for acetonitrile (Table 1). Experimental values of these angular coefficients are 1.20, R = 0.9998 for n-heptane; 1.28, R = 0.9999 for tetrachloromethane; 1.54, R = 0.9965 for 1-chlorobutane; 1.33, R = 0.9991 for 2,2,2-trifluoroethanol; and 1.07, R = 1 for acetonitrile. This means that the Clausius−Mossotti eq 12 is not suitable for the description of polar liquids. It is interesting that for the nonpolar liquids a linear dependence of Δε/(εP − 1) vs Δd/dP, caused by the pressure and temperature changes of permittivity and density, is commonly found (Figure 2). Some of the polar compounds (e.g., methanol, ethyl acetate, acetonitrile, primary alcohols, Table 2) also lie on this correlation line. Secondary and tertiary alcohols, ethers, esters, and nitro derivatives have enhanced values of angular coefficients, which are very sensitive to changes in temperature (Figure 2) . As follows from the “noncrossing rule” of V−P curves, the initial compressibility of a liquid at atmospheric pressure will define the compressibility curve at high pressure, and Δd/dP vs P curves never cross each other for different liquids.13 A linear correlation has been observed between the tangent (1/β0) and the secant bulk modulus value at 1000 bar (V0/ΔV1kbar) for different liquids in a wide temperature range, according to eq 13:
Figure 3. Relation between the tangent bulk modulus values at atmospheric pressure (∂P/∂ ln(εP − 1)) and secant bulk modulus values at 1000 bar ((εP=1000 − 1)/Δε) of permittivity of all liquids given in Table 2.
(1/d1 − 1/dP)/C2 = (1/(ε1 − 1) − 1/(εP − 1)
(10)
(εP − ε1)/(εP − 1) = [(ε1 − 1)/C2d1] × (dP − d1)/dP = 1 × (dP − d1)/dP
(11)
1/β1 = ( −4559 ± 23) + (0.9865 ± 0.0010)(1000V1/ΔV )
For liquids with a high relative permittivity, eq 8 can be written as (ε −1)/d = 9Po/2,19 which also leads to relation 11. From eq 7 a similar calculation leads to eq 12: (εP − ε1)/(εP − 1) = [(ε1 + 2)/3] × (dP − d1)/dP
R = 0.9999; N = 272
(13)
The total correlation contains all known data about the compressibility of fluids of different classes and at different temperatures (cyclic, linear and branched alkanes, alkenes, alcohols, aldehydes, ketones, carboxylic acids, ethers, esters, nitriles, halo- and nitro-derivatives, amines, amides, heterocycles, difunctional compounds, and even mercury). This indicates that in the studied range of pressure there are no
(12)
From 11 it follows that the slope of the dependence Δε/(εP − 1) vs Δd/dP in all cases should be equal to unity. The predicted angular coefficients [(ε1 + 2)/3] from eq 12 are 1.31 for nheptane, 1.41 for tetrachloromethane, 3.02 for 1-chlorobutane, 3707
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different temperatures has revealed the means to calculate the coefficients of the Tait equation (14) from the experimental values of the pressure influence on the permittivity at ambient pressure.
sharp changes in the processes determining the compressibility and no phase transformations or chemical processes take place. The main goal of the current research is to determine the presence or absence of a similar “noncrossing rule” for Δε/(εP − 1)−P curves. It can be expected that a variety of reasons leading to a change in the permittivity under pressure (volume change, molecule deformation, change of the dipole−dipole interaction, change of the conformational equilibrium, change of the molecular structure of OH complexes) will gradually vary with pressure because all studied polar and nonpolar liquids are reliably described by eq 14, recalculated in this work for all used data (Table 2) with correlation coefficients of R > 0.999. On the basis of these considerations, it should be expected that the linear relationship between the tangent [(εP→1 − 1)∂P/∂ε] and secant [(ε P=1000 − 1)/Δε P=1000 ] moduli for curves is implemented similarly to that found previously for the process of compressibility.13 All values of the coefficients C and B, ∂P/ ∂ln(ε − 1) = B/C, and (εP − 1)/Δε (Table 2) were obtained in this work by recalculation of the experimental data in accordance with the Tait equation (14). Δε /(εP − 1) = C ln[(B + P)/B]
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Corresponding Author
*V. D. Kiselev: e-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (Project No 12-03-00029), the U.S. Civilian Research and Development Foundation (CRDF), and the Ministry of Science and Education of the Russian Federation (Joint Program “Fundamental Research and High Education”, Grant REC 007).
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(14)
REFERENCES
(1) Bridgman, P. The Physics of High Pressure; Bell and Sons: London, 1958. (2) Isaacs, N. S. Liquid Phase High Pressure Chemistry; WileyInterscience: Chichester, New York-Brisbane-Toronto, 1981. (3) Kiselev, V. D.; Kashaeva, E. A.; Konovalov, A. I. Pressure Effect on the Rate and Equilibrium Constants of the Diels-Alder Reaction 9Chloroanthracene with Tetracyanoethylene. Tetrahedron 1999, 55, 1153−1162. (4) Marcus, Y.; Hefter, G. T. The Compressibility of Liquids at Ambient Temperature and Pressure. J. Mol. Liq. 1997, 73, 61−74. (5) Marcus, Y.; Hefter, G. T. On the Pressure and Electric Field Dependencies of the Relative Permittivity of Liquids. J. Solution Chem. 1999, 28, 575−592. (6) Kiselev, V. D.; Konovalov, A. I. Effect of Solvent Polarity on the Activation Volume of Polar Reactions. Dokl. Chem., Engl. Transl. 1998, 363, 426−428. (7) Tait, P. G. Physics and Chemistry of the Voyage of H.M.S. Challenger; H.M.S.O.: London, 1888. (8) Dymond, J. H.; Malhotra, R. The Tait Equation: 100 years on. Int. J. Thermophys. 1988, 9, 941−951. (9) Hayward, A. T. J. Compressibility Equations for Liquids: a Comparative Study. J. Appl. Phys. 1967, 18, 965−977. (10) Hayward, A. T. J. How to Measure Isothermal Compressibility of Liquids Accurately. J. Phys. D: Appl. Phys. 1971, 4, 938−950. (11) Whalley, E. The Compression of Liquids in Experimental Thermodynamics; Butterworths: London, 1975. (12) Gibson, R. E.; Loeffler, O. H. Pressure-Volume-Temperature Relations in Solutions. II. The Energy-Volume Coefficients of Aniline, Nitrobenzene, Bromobenzene and Chlorobenzene. J. Am. Chem. Soc. 1939, 61, 2515−2522. (13) Kiselev, V. D.; Bolotov, A. V.; Satonin, A. P.; Shakirova, I. I.; Kashaeva, E. A.; Konovalov, A. I. Compressibility of Liquids. Rule of Noncrossing V−P Curvatures. J. Phys. Chem. B 2008, 112, 6674−6682. (14) Hayward, A. T. J. Precise Determination of the Isothermal Compressibility of Mercury at 20 °C and 192 bar. J. Phys. D: Appl. Phys. 1971, 4, 951−955. (15) le Noble, W. J. Organic High Pressure Chemistry; Elsevier: Amsterdam, The Netherlands, 1988. (16) Gibson, R. E.; Kincaid, J. F. The Influence of Temperature and Pressure on the Volume and Refractive Index of Benzene. J. Am. Chem. Soc. 1938, 60, 511−518. (17) Owen, B. B.; Brinkley, S. R. The Effect of Pressure upon the Dielectric Constants of Liquids. Phys. Rev. 1943, 64, 32−36. (18) Rosen, J. S. Refractive Indices and Dielectric Constants of Liquids and Gases under Pressure. J. Chem. Phys. 1949, 17, 1192− 1197.
The resulting relation (Figure 3) is described by eq 15: (εP → 1 − 1)∂P /∂ε = −1308 + 757.9[(εP = 1000 − 1)/Δε] R = 0.984; N = 143
AUTHOR INFORMATION
(15)
It should be noted that Figure 3 includes all the liquids (Table 2) that were presented in Figure 2. The average uncertainty in the slope is ±1.6%, which can be attributed to experimental errors, especially for the value (εP − 1)∂P/∂ε. For example, the pressure influence at 30 °C on the permittivity of 1-chlorobutane according to the data from various sources21,33 differs by nearly 15%. Equation 15 allows the calculation of the initial change (εP − 1)∂P/∂ε at atmospheric pressure when the value of the permittivity change under 1 kbar is known. Conversely, with the known initial changes (εP − 1)∂P/∂ε at atmospheric pressure, the value of (ε1000 − 1)/Δε at 1 kbar can be calculated from eq 15. Then, with the known value of (ε1 − 1)∂P/∂ε equal to B/C and that calculated from the eq 15 value of Δε/ (εP=1000 − 1) = C ln[(B + 1000)/B], we can determine the constants B and C to describe the whole curve by eq 14.
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CONCLUSIONS From this analysis it follows that the influence of a high external hydrostatic pressure on the liquid permittivity is more complicated than the effect on the liquid compressibility. The pressure and temperature coefficients of the permittivity can be drastically larger than the changes in pressure and temperature coefficients of density. This means that the pressure and particularly the temperature significantly affect the structure of a liquid. For this reason, the changes in permittivity under pressure and temperature bear much additional information on the intermolecular interactions of a fluid. Especially valuable information can be provided from the sharp difference in the change of the values Δε/(εP − 1) and Δd/dP (Figure 2). This allows specifying the pressure and temperature influence on the change of the rate and equilibrium of a chemical reaction. Here we assume that a variety of reasons leading to a change in the permittivity in the studied range of pressure will gradually vary without saturation. The observed clear relation (15) between the tangent and secant bulk moduli for various liquids at 3708
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NOTE ADDED AFTER ASAP PUBLICATION This paper was published ASAP on March 25, 2014. The Results and Discussion section was updated. The revised paper was reposted on April 3, 2014.
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