Determination of the Relative Permittivity, Îµr, and Conductance, G, of

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Determination of the Relative Permittivity, εr, and Conductance, G, of Methylbenzene at Temperatures between (303 and 393) K and Pressures below 60 MPa with a Concentric Cylinder Capacitor at a Frequency of 1 kHz Yijie Shen, Anthony R. H. Goodwin,* and Laurent Pirolli Schlumberger Technology Corporation, 150 Gillingham Lane, Sugar Land, Texas 77478, United States

Kenneth N. Marsh and Eric F. May Centre for Energy, School of Mechanical & Chemical Engineering, The University of Western Australia, Crawley WA 6009, Australia ABSTRACT: The real part of the complex relative electric permittivity, εr(T, p), of liquid methylbenzene has been determined with an estimated expanded relative uncertainty of ± 0.018 % from measurements of the complex capacitance of a concentric cylinder capacitor at temperatures between (293 and 393) K and pressures below 60.6 MPa. The measurements were corrected for the isothermal compressibility of the solid that formed the capacitor. The isothermal compressibility was determined by comparison of measurements of εr(He, 303 K and 393 K, p) with ab initio results from quantum mechanics. The εr(T, p) values obtained for methylbenzene were combined with the amount-of-substance density, ρ, which was obtained from the equation of state reported by Lemmon and Span (J. Chem. Eng. Data 2006, 51, 785−850), to calculate the molar polarizability, P, that was fit with an expanded (k = 2) relative uncertainty of ± 0.03 % by three parameters. The εr(T, p) reported by Kandil et al. (J. Chem. Eng. Data 2008, 53, 1056−1065) from measurements with a radio frequency re-entrant cavity, a technique with entirely different sources of systematic error, were converted to P·ρ and they differed from 1·10−3 at T = 290 K to −2.5·10−3 at T = 406 K (with ⟨P·ρ⟩ ≈ 0.31 giving a relative difference of between 0.3 % and −0.8 %, respectively) at temperatures that overlap ours. At T = 298 K the measurements reported by Mospik (J. Chem. Phys. 1969, 50, 2559−2569) at a pressure between (0.1 and 20) MPa when converted to the product P·ρ are within −0.8·10−3 (a relative difference of about −0.26 %) of our smoothing equation while at T = 273 K, about 20 K below our lowest temperature, the differences for P·ρ are −0.4·10−3 (a relative difference of about −0.13 %).

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INTRODUCTION

WORKING EQUATIONS In the low-frequency limit, the real part of the complex quantity εr ≡ ε′ − iε″, is the relative electric permittivity εr {Re(εr) = ε′ = εr} while the imaginary part, Im(εr) = ε″ = (ωε0ρR)−1 accounts for electrical dissipation within the fluid owing to the electrical resistivity, ρR, that is related to the conductance, G, through G = A/(ρRl) where A and l are the surface area and length, respectively, of concentric cylinders that form the capacitor; the electrical conductivity σ is related to ρR by σ = ρR−1. For methylbenzene, σ ≈ 8.6·10−12 S·m−1 (our measurements give G ≈ 3·10−12 S and within our apparatus l ≈ 8.3·10−4 m and ⟨A⟩ ≈ 2.91·10−4 m2) so that σ ≪ 1 S·m−1, and thus Im(εr) = 0. Furthermore, dispersion and absorption, which can give rise to frequency-dependence in εr, are negligible at frequencies below about 1011 Hz.26

In this article, the relative electric permittivity, εr, of liquid methylbenzene, an industrially important substance, is reported from measurements of the capacitance of a concentric cylinder at a frequency of 1 kHz with a relative expanded uncertainty of ± 0.018 % at temperatures between (303 and 393) K and pressures below 60 MPa. The pressure dependence of the capacitor was determined by comparing measurements of εr(He, T, p) with values obtained from ab inito calculations. The measurements of εr(C7H8, T, p) were intended to validate the results reported previously by Kandil et al.1 The εr(C7H8, T, p) reported in ref 1 were obtained at a frequency of about 340 MHz with a re-entrant cavity,2 which utilizes different principles, and has quite different sources of systematic error to that used herein. The εr(C7H8, T, p) values reported in this manuscript are also compared with literature values that differ from these results by up to about 4 %.1,3−23 © 2013 American Chemical Society

Received: January 29, 2013 Accepted: March 30, 2013 Published: April 12, 2013 1340

dx.doi.org/10.1021/je400102p | J. Chem. Eng. Data 2013, 58, 1340−1348

Journal of Chemical & Engineering Data

Article

Figure 1. Cross sections A−A′ and B−B′ through the concentric cylinder capacitor mounted within a pressure vessel. The capacitance was formed from between tubes C1 and C2 supported from two plates D and electrically isolated from D by 12 sapphire spheres E of nominal outer diameters of 2.0 mm. The capacitor was held together by a 316 stainless steel spring F. Fluid entered the pressure vessel through two ports machined to accept Autoclave Engineers coned-and-threaded fittings G. The supporting disks also contained holes H to permit fluid flow through between cylinders C1 and C2. For the sake of clarity, only one hole is shown per disk H. Electrical connnections between either C1 or C2 and the external electronics were effected with Inconcel 718 wire I, which was looped to resemble a spring, and Greene Tweed electrical feedthroughs J.

pressure assuming the solid is isotropic so that all linear dimensions decrease by κT p/3; (∂V/∂p)T arises because the volume of an elastic solid decreases with increasing hydrostatic pressure by the product of the isothermal compressibility, κT, and pressure p.

The capacitance formed between two concentric cylinders is given by C(T , p) = 2πε0εr(T , p)l /{ln(b/a)} −12

(1)

−1

where ε0 = 8.854 187 817·10 F·m is the electric constant, εr(T, p) is the relative electric permittivity of the fluid contained between the cylinders that form the capacitor at temperature T and pressure p, b is the internal radius of the larger diameter cyclinder, and a is the external radius of the inner cylinder each of length l. In this work εr(T, p) was determined with an expression reported by Buckley et al.24 and Schmidt and Moldover:25 εr(T , p) =

C(T , p) (1 + κT p /3) C(T , p = 0)

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EXPERIMENTAL SECTION Apparatus, Experimental Procedures, and Calibration. Techniques for the measurement of relative electric permittivity, including refractive index, for both electrically conducting fluids and insulators, have been discussed by Moldover et al.26 On the basis of the information in ref 26, for measurements with methylbenzene a guarded parallel plate capacitor would suffice while a guarded concentric cylinder capacitor would be preferred. The concentric cylinder capacitor design, shown in Figure 1, was adapted from that of Goodwin and Hill27 and is similar in concept to both that used by Fernandez et al.28 for relative

(2)

In eq 2 the term (1 + κT p/3), where κT is the isothermal compressibility {κT = −(∂V/∂p)T/V}, accounts for the decrease in measured capacitance that arises from increasing hydrostatic 1341



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The temperature of the stirred thermostat fluid, which at thermal equilibrium is assumed equal to the temperature of the fluid within the capacitor, was determined on the International Temperature Scale of 1990 (ITS-90) using a nominal 100 Ω platinum resistance thermometer. The resistance and thus temperature were determined with an ASL F100 bridge with an expanded uncertainty of about ± 0.01 K. The center of the sensing element was located at a height in the same horizontal plane as equivalent to l/2. The δT = ± 0.01 K results in a negligible uncertainty in εr of δ|εr| < 0.00002 (a relative uncertainty of about ± 0.001 %) because over the range of temperatures and pressures |(∂εr/∂T)p| < 0.002 K−1. Pressures were measured with a resonant quartz transducer (Quartzdyne model QHB009-16-200 serial number 157 972 with a maximum operating pressure of 110 MPa and maximum operating temperature of 473 K) with a relative uncertainty cited by the manufacturer of about ± 0.02 % of full scale. When the pressure transducer was calibrated against an oil-lubricated dead-weight gauge it was found to have an uncertainty of δp/ MPa = {0.0001·(p/MPa) + 0.022}; the pressure of 0.022 MPa is comparable with the cited relative uncertainty of the transducer of about ± 0.02 % for the full-scale pressure of 110 MPa. In the temperature and pressure range investigated the δp was less than ± 0.029 MPa and, when combined, in the worst case, with (∂εr/∂p)T ≈ 0.002 MPa−1, corresponds to a potential uncertainty in εr of 0.000058 (or relative uncertainty of about 0.003 %). The required derivatives with respect to pressure and temperature were determined from literature values described in the results and discussion section below. Pressures were generated in the system with an ISCO model 100 DX positive displacement pump with an upper operating pressure of about 68 MPa. Prior to measurements, the apparatus was evacuated, with a turbo-molecular pump, to a pressure (as indicated by an ionization gauge located near the pump) of less than 10−2 Pa for a time of at least 24 h. Before filling the apparatus with methylbenzene the capacitance was measured. Materials. Methylbenzene from Riedel de Haën with stated mole fraction purity greater than 0.997, containing a mole fraction of less than 0.00001 of water and an unquantified amount of benzene was used as listed in Table 2. Chemical

electric permittivity measurements on liquid water and that reported by Younglove and Straty.29 The concentric cylinder capacitor of Figure 1 was machined from UNS S17400 type 630, formally known as 17-4 PH,30 and the chemical composition is given in Table 1, as two cylinders C1 (of Table 1. Chemical Constituents i and Mass Fraction, w, Composition of UNS S1740033 i

w

C Mn Ph S Si Cr Ni Ta Cu Fe

0.0004 0.004 0.0002 0.00005 0.005 0.155 0.045 0.003 0.035 0.75235

internal radius b) and C2 (of external radius a), each about 100 mm in length l, with a surface finish of 16 μm r.m.s. The cylinders were supported between two circular plates D, which were also machined from UNS S17400, to form a radial separation of about 0.83 mm and a nominal vacuum capacitance C(293 K, p = 0) from eq 1 of about 64 pF. These components were electrically isolated from each other and the supports with 12 sapphire spheres E of nominal diameter of 2 mm that were located in recesses. The supporting disks, cylinders and sapphire spheres were compressed with a tensioned spring F formed from type 316 stainless steel. The supporting disks also contained holes H to permit fluid to flow between cylinders C1 and C2; for the sake of clarity, in Figure 1 only one hole is shown per disk H. Electrical connnections between either C1 or C2 and the external electronics were effected with Inconcel 718 wire I, which was looped to resemble a spring, and electrical feedthroughs J manufactured by Greene Tweed, Houston, TX, USA. The pressure vessel, also shown in Figure 1, was designed to operate at a maximum pressure of 70 MPa at a temperature of 473 K; the cap was fabricated from BeCu with the remainder from UNS S17400. Fluid flowed into and out of the apparatus through two ports G machined to accept Autoclave Engineers coned-and-threaded connectors, shown in Figure 1, located at either end of the vessel. The capacitance and loss were measured with a ratio transformer bridge (Andeen-Hagerling model AH2700A opt C) driven at 7 V at frequencies between 50 Hz and 20 kHz. This instrument can resolve fractional changes in capacitance δC/C < 10−6 when the capacitance C is on the order of 1 pF. The relative standard uncertainty of the capacitance so obtained was ± 6·10−6 as determined by an algorithm provided by Andeen Hagerling for the capacitance, conductance, and frequencies used in this work. In this work, the AH2700 was operated solely at a frequency of 1 kHz. The concentric cylinder capacitor was connected to the bridge with a pair of coaxial cables (AH DCOAX-5-BNC) and the ground of both were connected to the pressure vessel. The thermostat used, which contained a polydimethylsiloxane polymer (CH3)3SiO[SiO(CH3)2]nSi(CH3)3 with 10 ≤ n ≤ 12 (supplied Dow Corning as fluid 200−20 with CAS# 63 148−62−9), has been described by Goodwin et al.31 previously.

Table 2. Chemical Substance, B, Supplier, and Stated Mole Fraction Purity, xi, and Final Mole Fraction Purity, xf. No Purification Was Performed B

supplier

xi

helium methylbenzene

Air Liquide Riedel de Haën

0.999998 0.998

xf

analysis

0.998

none Karl Fischer

compositional analyses were performed neither before nor after the measurements and we have assumed that there were no variations in chemical composition from those cited by the supplier during the time required to acquire the data. The samples were degassed by vacuum sublimation and dried over a 0.4 nm molecular sieve for a time of about 24 h. Prior to use, the molecular sieve was heated for at least 48 h to a temperature greater than 500 K while maintaining the pressure at less than 1 mPa with a turbo-molecular pumping station. A Karl Fischer titration of the methylbenzene gave a water mass fraction of less than 1·10−6. Although no other measurements were performed to identify the impurities in the samples it is plausible that there were hydrocarbons of similar normal 1342



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boiling temperature {T(lg, p = 0.1 MPa)} to those of the major constituent. The εr of these substances are between 2 and 2.5 thus a mole fraction x ≈ 0.01 of those chemicals, which is a factor of 10 greater than stated by the supplier and thus implausible, would introduce an uncertainty of 0.005 in the relative electric permittivity that is about 30 times greater than the uncertainty of the measurements reported here. Vacuum Capacitance and Calibration for Material Compressibility. Measurements of C(T, p = 0) are reported because these values are required to both determine εr(T, p) and characterize the stability of the capacitor as a function of time. The C(T, p = 0) determined at five temperatures between

The linear thermal expansion coefficient for UNS S17400 obtained from the literature33 is L−1 (dL/dT) = 11.2·10−6 K−1 and compares favorably with C(T, p = 0)−1{dC(T, p = 0)/dT} = 11.0·10−6 K−1 assuming the variation of C(T, p = 0) with temperature arises solely from the dimensions of the parts formed from UNS S17400. The variation of dC(T, p = 0)/dt, where t is time, was less than the relative expanded uncertainty over a time of 0.5 a. Equation 2 was used to account for the variation of capacitance as a function of hydrostatic pressure that arises solely from the compressibility of the solid used to form the cylinders. This calculation requires the isothermal compressibility for UNS S17400 and it is to address the source of this value that we now turn. The isothermal compressibility is the 32,33 reciprocal of the bulk modulus K (= κ−1 T ). For UNS S17400 −12 −1 K = 196 GPa so that κT = 5.1·10 Pa and for Al2O3 (sapphire) K = 240 GPa and κT = 4.2·10−12 Pa−1.34 Because each metallic part that formed the capacitor was cut from a rod of UNS S167400, we anticipated, on the basis of the work of Buckely et al.24 and Schmidt and Moldover25 albeit both of uncertainty a factor of 10 less than reported here, the pressure dependence of C can be attributed solely to the mechanical properties of UNS S17400. We now digress to discuss the procedures, also recommended by Moldover et al.26 and Schmidt and Moldover,25 to confirm this conjecture and determine the compressibility of the solid material. To determine κT we measured εr(expt, He, T, p) at two temperatures of (303 and 391) K and pressures below 60 MPa and compared these results with εr(calc, He, T, p) determined from

Table 3. The Capacitance of the Capacitor Measured under Vacuum at p ≈ 0, C(T, p = 0), as a Function of Temperature T. The Expanded (k = 2) Uncertainties of the Measurements Are Provided T/K 296.285 303.244 333.633 363.009 393.082

± ± ± ± ±

C(T, p = 0)/pF 0.010 0.010 0.010 0.010 0.010

63.037841 63.037624 63.051159 63.065563 63.081424

± ± ± ± ±

0.000012 0.000012 0.000012 0.000012 0.000012

(296 and 393) K are provided in Table 3 and can be represented by C(T , p = 0)/pF = 63.029410 + 2.827597·10−4(T /K − 273.15) + 1.273164·10−6(T /K − 273.15)2

⎛ ε − 1⎞1 P(ρ , T ) = ⎜ r ⎟ ⎝ εr + 2 ⎠ ρ

(3)

−5

with a relative standard deviation of 1.39·10 , which is about 2 times the expanded uncertainty of the individual measurements of capacitance. The results are shown as deviations from eq 3 in Figure 2. The results are within 2 times the estimated

= A εr (T ){1 + bεr(T )ρ(p , T ) + cεr(T )ρ2 (p , T )} (4)

where P is the total polarization, Aεr is the molar polarizability arising from individual molecules in the absence of intermolecular interations, bεr and cεr are the second and third electric permittivity virial coefficients, and ρ is the amount-ofsubstance density; eq 4 is a truncated form of the infinite series. Each of these quantities was obtained from ab inito calculations for 4He. The Aεr = 0.517 254 19·10−6 m3·mol−1 reported by Lach et al.35 was used and the values of bεr(T) used at temperatures of (303 and 301) K were estimated from a quadratic function fit to the values calculated by Cencek, et al.36 at temperatures of (77, 243, 298, 303 and 323) K. The cεr was assumed to be independent of temperature and was obtained by fitting the P(He, 298.15 K, p) measured by Lallemand and Vidal37 to eq 4 constraining both Aεr and bεr to their corresponding ab initio values. Doing so returned cεr = −(1.60 ± 0.19) cm6·mol−2; this value is about 60 % larger in magnitude than that reported by Lallemand and Vidal37 who permitted Aεr and bεr to float in their regression. The uncertainty bound in cεr reported here arises from the estimated experimental uncertainty of the density measurements reported in ref 37. Over the range of temperatures and pressures measured in this work, the effect of varying cεr over this uncertainty bound on the calculated relative electric permittivity was less than 0.01 % of (εr − 1), which is an order of

Figure 2. Deviations ΔC = C(expt, T, p = 0) − C(calc, T, p = 0) of the measured capacitance C(expt, T, p = 0) from the calculated values C(calc) of eq 3 as a function of temperature T. The ordinate error bars represent the expanded (k = 2) uncertainty. ⧫, this work.

uncertainty but are not a smooth function of temperature. A plausible source of systematic errors, particularly at the lower temperatures, could arise from variations in the ability to evacuate the apparatus. However, repeated measurements lie within the estimated uncertainty and could neither confirm nor deny this postulate. 1343



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Table 4. The Relative Electric Permittivity, εr, and Conductance, G, for Methylbenzene Determined at a Frequency of 1 kHz at the Mean Temperature ⟨T⟩ and Pressure pa ⟨T⟩/K 303.150 ± 0.010

333.150 ± 0.010

363.150 ± 0.010

393.150 ± 0.010

εr

p/MPa 0.312 1.015 5.058 10.116 20.221 30.317 40.408 50.497 60.603 0.402 0.995 5.053 10.114 20.221 30.317 40.419 50.510 60.615 0.314 1.010 5.053 10.111 10.108 10.109 20.232 20.229 20.231 30.325 30.317 40.408 50.510 50.502 60.624 60.614 1.622 5.063 5.070 10.114 10.123 20.234 20.223 30.331 40.414

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.020 0.020 0.021 0.021 0.022 0.023 0.024 0.025 0.026 0.020 0.020 0.021 0.021 0.022 0.023 0.024 0.025 0.026 0.020 0.020 0.021 0.021 0.021 0.021 0.022 0.022 0.022 0.023 0.023 0.024 0.025 0.025 0.026 0.026 0.020 0.021 0.021 0.021 0.021 0.022 0.022 0.023 0.024

2.37527 2.37634 2.38253 2.38999 2.40447 2.41824 2.43058 2.44313 2.45469 2.31204 2.31340 2.32081 2.32958 2.34637 2.36168 2.37560 2.38883 2.40129 2.24824 2.24980 2.25867 2.26913 2.26912 2.26918 2.28831 2.28839 2.28835 2.30571 2.30557 2.32135 2.33603 2.33608 2.34978 2.34990 2.18753 2.19687 2.19705 2.20937 2.20955 2.23259 2.23281 2.25270 2.27067

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

G/nS 0.00042 0.00042 0.00042 0.00042 0.00042 0.00043 0.00043 0.00043 0.00044 0.00041 0.00041 0.00041 0.00041 0.00041 0.00042 0.00042 0.00042 0.00043 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00041 0.00041 0.00041 0.00042 0.00042 0.00042 0.00042 0.00038 0.00039 0.00039 0.00039 0.00039 0.00039 0.00039 0.00040 0.00040

2.2845 2.2811 2.2981 2.3522 2.5892 3.2982 3.6500 4.1401 5.5860 2.6063 2.5850 2.5659 2.5364 2.5107 2.4894 2.4361 2.3882 2.3334 4.9045 4.9535 4.9985 5.0780 5.0786 5.0899 5.1809 5.1993 5.2036 5.2420 5.2689 5.2612 5.2233 5.2215 5.1734 5.1611 2.9964 2.9586 2.8933 2.9430 2.9327 3.2525 3.2332 3.3658 3.3726

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.0064 0.0064 0.0065 0.0066 0.0073 0.0068 0.0069 0.0072 0.0076 0.0063 0.0062 0.0072 0.0072 0.0071 0.0070 0.0069 0.0067 0.0066 0.0070 0.0071 0.0072 0.0071 0.0071 0.0071 0.0072 0.0073 0.0073 0.0073 0.0074 0.0073 0.0073 0.0073 0.0072 0.0072 0.0062 0.0061 0.0060 0.0061 0.0060 0.0062 0.0062 0.0064 0.0064

The εr were adjusted to the stated temperature for each isotherm, and the corrections used were less than the expanded uncertainty (k = 2) that are listed.

a

the result κT = 3·10−12 Pa−1 with a relative standard deviation of 8.6·10−5 that is within with the estimated combined relative expanded uncertainty for εr of ± 1.8·10−4 by a factor of about 2; the same value of κT was used for both temperatures. Schmidt and Moldover25 demonstrated the importance of determining κT for the measurement of Aεr (He) where the correction for elastic deformation was stated to be 0.54 % of Aεr. Fortunately, for methylbenzene at liquid densities, the correction for compressibility at p < 60 MPa of eq 2 contributes relatively less than 0.006 % to εr and, in the worst case, the 41 % relative difference between the literature value of κT = 5.1·10−12 Pa−1 for UNS S17400 and that obtained in this work results in an additional relative uncertainty of less than 0.004 %, which is less

magnitude larger than the effect of varying bεr by its uncertainty.36 The densities of He were obtained from a virial equation of state containing terms up to and including the sixth virial coefficient. All of the virial coefficients were determined from ab initio calculations reported by Cencek et al.,38 Garberoglio et al.,39 Shaul et al.,40 and Schultz and Kofke41 as described by Moldover et al.42 The densities calculated with this ab initio virial equation of state had an estimated relative uncertainty of less than 10−4; further details of these calculations will be given in a future publication by Moldover et al.42 The value of κT in eq 2 was then adjusted to minimize the difference between εr(expt, He, T, p) and εr(calc, He, T, p) with 1344



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Table 5. The Relative Electric Permittivity εr(T, pr = 0.1 MPa) for Methylbenzene as a Function of Temperature T at a Frequency of 1 kHza

than the estimated relative expanded uncertainty by a factor of 4.5. The average absolute differences between εr(expt, He, T, p) and εr(expt, He, T, p) obtained from the determination of κT are taken as a measure of the anticipated uncertainty in the measurements and were combined, as described below, to obtain the expanded relative uncertainty in the relative permittivity of δεr/εr ≈ 1.8·10−4.

T/K 303.150 333.150 363.150 393.150

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RESULTS AND DISCUSSION The relative electric permittivity εr for methylbenzene obtained from eq 2 with measurements of C(T, p) combined with C(T, p = 0) from eq 3 for the concentric cylinder capacitor are listed in Table 4 along with the measured conductance G. The values reported were obtained at a frequency of 1 kHz at temperatures between 303 K and 393 K and pressures below 60.6 MPa. At any temperature and pressure in this range, the observed variation of C(f) at frequencies f between 50 Hz and 20 kHz was found to be less than the expanded uncertainty of an individual measurement albeit with an exponential decay for which C( f) decreased with increasing f. In addition, small corrections were applied to the εr(T, p) of Table 4 to reduce all values to the stated temperature for each isotherm; these corrections are less than the cited uncertainty. The uncertainties, listed in Table 4, are at a confidence interval of 0.995 (k = 2), and were obtained by combining in quadrature uncertainties arising from the uncertainty of each capacitance measurement and the calibration with He to determine the isothermal compressibility with δT(dεr/dT) and δp(dεr/dp). The major source of the expanded uncertainty (by at least factor of 5) arises from the uncertainty of the isothermal compressibility for which the contribution is δεr ≈ ± 0.0004 (which is a relative uncertainty of about 0.016 %). The next most significant and quantifiable contribution to the uncertainties arises from dεr/dp that was estimated from a combination of our results and the δp listed in Table 4 for the pressure gauge to be δεr ≈ ± 0.00007 (which is a relative uncertainty of about 0.003 %). The contribution to the uncertainty arising from the capacitance bridge of δC/C contributes δεr ≈ ± 0.00006 (which is a relative uncertainty of about 0.0025 %) while the uncertainty arising from dεr/dT was estimated from a combination of our results and the δT listed in Table 4 to be δεr ≈ ± 0.00002 (which is a relative uncertainty about 0.001 %). The required derivatives were determined from an analysis of the results. In the absence of a chemical analysis for these fluids, the contribution to the uncertainty arising from the uncertainty in composition was assumed to be zero. The εr(T, pr = 0.1 MPa) values were obtained by extrapolating the εr(T, p) of Table 4 for each isotherm with an equation quadratic in pressure to pr = 0.1 MPa. The values of εr(T, pr) so determined are listed in Table 5 and represented by

∑ Ai{(T /K) − 273.15}i i=0

0.010 0.010 0.010 0.010

0.1 0.1 0.1 0.1

2.37499 2.31178 2.24826 2.18371

± ± ± ±

0.00042 0.00042 0.00042 0.00042

The εr(T, pr = 0.1 MPa) values were obtained by extrapolating the εr(T, p) from Table 4 for each isotherm with a second order polynominal in pressure to pr = 0.1 MPa; the εr(T, p) values of Table 4 were adjusted to the stated temperature for each isotherm and these corrections were less than the expanded uncertainty. The expanded (k = 2) uncertainties are cited. a

Figure 3. Deviations Δεr = εr(extra, T, p = 0.1 MPa) − εr(calc, T, p = 0.1 MPa) of the εr(extra, T, p = 0.1 MPa) obtained at each T by extrapolation of the measurements listed in Table 4 with a quadratic in pressure values εr(calc) of eq 5 as a function of temperature T. The ordinate axis is about the expanded (k = 2) uncertainty; ⧫, this work.

⎛ ε − 1⎞1 P(ρ , T ) = ⎜ r ⎟ = A εr + Aμ /T + Bεr ρ ⎝ εr + 2 ⎠ ρ

(5)

with a standard deviation of the mean σ(⟨ε′⟩) that gives 100·σ(⟨εr⟩)/εr = ± 1.9·10−2 when the parameters were as follows: A0 = 2.439029 and A1 = −0.0021246. The εr(T, pr) of Table 5 are shown as deviations from eq 5 in Figure 3 for which the ordinate axis is equal the expanded uncertainty. The measurements of εr(T, p) of Table 4 were represented by an expanded form of eq 4 for the total polarization P of 1345

−1

(6)

with the results Aεr = 37.12 cm ·mol , Aμ = 541.5 K·cm ·mol−1, and Bεr = −552.1 cm6·mol−2 that represented the data with a relative standard deviation of 100·σ(⟨P⟩)/P = ± 0.03. This functional form is similar to that used by Harvey and Lemmon43 although the parameters of eq 6 were assumed independent of temperature. In eq 6, ρ is the amount-ofsubstance density that was obtained from REFPROP44 using the equation of state published by Lemmon and Span.45 The quantity Aμ determined in this work can be related to the dipole moment μ of methylbenzene through Aμ = Lμ2(T)/9ε0kT where L is Avogardo’s number and k is Boltzmann’s constant. However, at the average temperature of the data measured in this work, the value Aμ = 541.5 K·cm3·mol−1 corresponds to an effective dipole moment of μ = 9.87·10−28 C·m (in units of debye symbol D the value is μ = 296 D),46 which is 3 orders of magnitude larger than μ(g, C7H8, T = 298.15 K, p = 0.1 MPa) = 1.2·10−30 C·m (equivalent to 0.36 D).44 This difference indicates that Aμ obtained in this work can only be interpreted as an adjustable parameter used to correlate the results listed in Table 4. Figure 4 shows the deviations of the measured εr(T, p) of Table 4 converted to {εr(T, p) − 1}/{εr(T, p) + 2} from the dimensionless quantities P·ρ calculated from a combination of eq 6 and ref 45 as implemented within REFPROP.44 The {εr(T, p) − 1}/{εr(T, p) + 2} values of Table 4 show a systematic undulation from eq 6, albeit within the assigned estimated 3

1

εr(T , pr ) =

± ± ± ±

εr

pr/MPa

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Article

Figure 4. Deviations ΔP·ρ = P(expt)·ρ − P(calc)·ρ of the total polarizability P(expt) determined by combination of the measured relative electric permittivity εr(expt) of methylbenzene listed in Table 4 with the amount-of-substance density ρ of ref 44 (that was implemented within ref 45) as deviations from the calculated values P(calc) obtained from eq 6 at pressure p: ○, T = 303.15 K; □, T = 333.15 K; Δ, T = 363.15 K; ◊, T = 393.15 K. The ordinate error bars represent the relative expanded (k = 2) uncertainty of P·ρ (expt, T, p) listed in Table 4.

Figure 5. Deviations Δεr = εr(expt) − εr(calc) of the measured relative electric permittivity εr(expt) of methylbenzene as deviations from the calculated values εr(calc) of eq 5 at p = 0.1 MPa: ⧫, this work; ●, ref 1; gray filled triangle with black outline, ref 3; gray filled and outline triangle, ref 4; gray cross, ref 5; gray diamond, ref 6; △, ref 7; ○, ref 8; +, ref 9; gray filled square, ref 10; light gray filled square with black outline, ref 11; □, ref 12; gray asterisk, ref 13; gray plus, ref 14; ∗, refs 15 and 17;·, ref 16; dark gray filled square with black outline, ref 18; gray filled diamond with black outline, ref 19; ◇, ref 20; gray filled circle with black outline, ref 21; ■, ref 22; gray filled circle (appears atop the results reported in ■, ref 22), ref 23. The recommendations of Maryott and Smith51 are not shown in Figure 1 because they are coincident with the values reported by Tangl.4 The dashed lines at 0 indicate an extrapolation of eq 5 while the solid line indicates the temperature range of the fit to eq 5.

expanded uncertainty, which implies the functional form is not entirely suitable; inclusion of higher order terms were unjustified because they did not significantly reduce the standard deviation of the fit. The εr(T, p) of Table 4 were not fit to the empirical expression reported by Owen and Brinkley47 that is analogous to the Tait equation48 for density and used previously by Kandil et al.1 Comparison with Literature Data. The εr(p = 0.1 MPa) reported by other workers1,3−23 are shown as deviations from eq 5 in Figure 5. The εr(p = 0.1 MPa) values of ref 1 that were obtained with the re-entrant cavity show a systematic relative deviation from eq 5 with differences that increase from about 0.1 % to −0.6 % with increasing temperature and exceed the relative expanded uncertainty of 0.01 % cited in ref 1. In the overlapping temperature range, the values reported by Mospik12 and Lewis and Smyth8 agree with those of Table 5 within relative differences of −0.2 % and ± 1 %, respectively, while those of refs 3, 15, 17, and 19 are all systematically below those of Table 5 by a relative difference of 2 % which is about 7 times the combined uncertainties. At temperatures below 293 K, comparisons of the measurements reported by both Mospik12 and Lewis and Smyth8 with values obtained by extrapolation of eq 5 show, perhaps not unsurprisingly, an increasing deviation with decreasing temperature to give a relative difference of about 1 % at 223 K and 1.5 % at 213 K. The values of εr(T, p = 0.1 MPa) reported in ref 1 from measurements with a MicroElectricalMechanical System (MEMS) interdigitated comb capacitor are not shown in Figure 5 because of the systematic errors reported in ref 1. The {εr(T, p) − 1}/{εr(T, p) + 2} values obtained from εr(T, p) of Table 4 are shown in Figure 6 as deviations from the P·ρ calculated with eq 6 and ref 45 along with the measurements reported in refs 1, 10, 11, and 12 also converted to {εr(T, p) − 1}/{εr(T, p) + 2} albeit limited to pressures up to p = 203 MPa; an extrapolation of eq 6 in pressure by a factor of 3.4. The {εr(T, p) − 1}/{εr(T, p) + 2} calculated from the εr(T, p) reported by Kandil et al.,1 which were obtained with a radio frequency re-entrant cavity, differ, as Figure 6 shows, by between 1·10−3 at T = 290 K and −2.5·10−3 at T = 406 K from

Figure 6. Deviations ΔP·ρ = P(expt)·ρ − P(calc)·ρ of the total polarizability P(expt) determined by a combination of the measured relative electric permittivity εr(expt) of methylbenzene listed in Table 4 with the amount-of-substance density ρ of ref 44 (that was implemented within ref 45) as deviations from the calculated values P(calc) obtained from eq 6 at pressure p: ○, T = 303.15 K; □, T = 333.15 K; △, T = 363.15 K; ◇, T = 393.15 K; ●, T = 290 K ref 1; ■, T = 310 K ref 1; ▲, T = 320 K ref 1; ⧫, T = 340 K ref 1; +, T = 357 K ref 1; ∗, T = 390 K ref 1; gray filled circle, T = 405 K ref 1; gray filled diamond with black outline, T = 223 K ref 12; gray filled square with black outline, T = 273 K ref 12; gray filled circle with black outline, T = 298 K ref 12; gray filled triangle, T = 303.15 K ref 11; gray filled diamond, T = 303 K ref 10.

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the calculated value of P·ρ obtained from eq 6 combined with ρ of ref 45 (these are relative difference of between −0.8 % and 0.3 %, respectively). The determination of the relative electric permittivity with a re-entrant cavity and a concentric cylinder capacitor is very different.26 Nevertheless, the agreement between the results of these different techniques is considered satisfactory and indicates that P can be obtained to better than 1 % with a re-entrant cavity and that the purity of the substance might be the limiting factor. The values of εr(T, p) reported in ref 1 from measurements with a MicroElectricalMechanical System (MEMS) interdigitated comb capacitor are neither shown in Figure 5 nor Figure 6 because of the systematic errors reported in ref 1. The measurements of εr(T, p) reported by Mospik12 and pressures between (0.1 and 20) MPa were converted to the quantity {εr(T, p) − 1}/{εr(T, p) + 2} for comparison with the quantity P·ρ calculated from a combination of eq 6 and ρ obtained with the equation of state reported in ref 45 as implemented in ref 44. At T = 298 K the data from ref 12 lie within −0.8·10−3 (a relative difference of about −0.26 %) of eq 6 that is within the combined expanded uncertainty while at T = 273 K, about 20 K below our lowest temperature, the differences are about −0.4·10−3 (a relative difference of about −0.13 %). At T = 248 K the differences are about 0.07·10−3 while at T = 223 K they are less than 0.4·10−3. These differences suggest that eq 6 can be extrapolated to temperatures and pressures outside of the range fit. The P·ρ obtained from the measurements of εr(T = 303 K, p = 10 MPa) reported in ref 11 combined with ρ from ref 45 differ from eq 6 by 1.9·10−3 (a relative difference of about −0.6 %). The {εr(T, p) − 1}/{εr(T, p) + 2} obtained from the εr(T = 303 K, p < 406 MPa) of ref 10 differ from eq 6 and ref 45 by 1.9·10−3 (a relative difference of about −0.6 %) at p = 101 MPa and by 3.1·10−3 at p = 202 MPa (a relative difference of about 1 %) from an extrapolation, by a factor of 3.4 in pressure, of eq 6 combined with ref 45. To the best of our knowledge the only measured conductivity, σ(l, C7H8, T = 298.15 K, p = 0.1 MPa), was reported by Forster,49 which has been cited by Chow et al.,50 for oxygen-free methylbenzene as (8 ± 5)·10−13 S·m−1 that is at a temperature 5 K below the lowest temperature reported in Table 4. The result reported in ref 49 is an order of magnitude lower than those listed in Table 4 albeit at a temperature at least 5 K higher than ref 49; our measurements give G ≈ 3·10−12 S as determined within an apparatus for which l ≈ 8.3·10−4 m and ⟨A⟩ ≈ 2.91·10−4 m2 so that σ ≈ 8.6·10−12 S·m−1.

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Article

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +1 281 285 8071. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to thank David Kempher, Kalim Ullah, Carlos De Bourg, Benoit LeDenmat and Sami Trevino, all of Schlumberger Technology Corporation, for providing the design, engineering, quality, and procurement of the components, repectively, used to form the capacitor and the pressure vessel. 1347



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Determination of the Relative Permittivity, Îµr, and Conductance, G, of

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