Recommended Values of the Viscosity in the Limit of Zero Density for R134a and Six Vapors of Aromatic Hydrocarbons as Well as of the Initial Density Dependence of Viscosity for R134a. Revisited from Experiment Between 297 K and 631 K

Previously published experimental viscosity data at low density, originally obtained using all-quartz oscillating-disk viscometers for R134a and six vapors of aromatic hydrocarbons in the temperature range between 297 K and 631 K at most, were re-evaluated after an improved re-calibration. The relative combined expanded (k=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document}) uncertainty of the re-evaluated data are 0.2 % near room temperature and increases to 0.3 % at higher temperatures. The re-evaluated data for R134a as well as for the vapors of mesitylene, durene, diphenyl, fluorobenzene, chlorobenzene, and p-dichlorobenzene were arranged in approximately isothermal groups and converted into quasi-isothermal viscosity data using a first-order Taylor series in temperature. Then, the data for R134a were evaluated by means of a series expansion truncated at first order to obtain the zero density and initial density viscosity coefficients, η(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta ^{(0)}$$\end{document} and η(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta ^{(1)}$$\end{document}. For the six aromatic vapors, the Rainwater–Friend theory for the initial density dependence of the viscosity was used to derive η(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta ^{(0)}$$\end{document} values. Finally, reliable η(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta ^{(0)}$$\end{document} and also η(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta ^{(1)}$$\end{document} values for R134a were selected as reference values in the measured temperature range to be applied when generating a new viscosity formulation.


Introduction
Numerous studies dealing with the viscosity of gases and particularly of organic vapors were accomplished applying all-quartz oscillating-disk viscometers at the University of Rostock over four decades. For these studies reported by Vogel and co-workers, a large number of measurement series were performed using continuously renewed instruments of this viscometer type. Unfortunately, the viscosity values following from these studies were seriously affected by a calibration carried out for the relative measurements. In general, the calibration was based on experimental reference values for the viscosity of noble gases and of nitrogen at low densities and at room temperature. With regard to the six vapors of the aromatic hydrocarbons under discussion, the applied reference values were still older than those, which were used for the subsequent measurements concerning the initial density dependence of the viscosity of some other gases and vapors (see Ref. [1]). The old reference values for the viscosity of argon and nitrogen of Kestin and Leidenfrost [2] are nowadays considered as obsolete. Since accurate theoretically calculated viscosity values for argon at zero density [3] as well as improved experimentally based [4] and theoretical [5] viscosity values for nitrogen at low density became available and the information about the measurements with the previous all-quartz oscillating-disk viscometer still exists, the formerly measured data of Vogel and co-workers should be re-evaluated.

Oscillating-Disk Viscometer, Calibration, and Uncertainty
The primary measurements were performed applying two marginally dissimilar all-quartz oscillating-disk viscometers with small gaps (see Figs. 1 and 2 as well as Fig. 1 of Ref. [1]). The oscillating disk of the viscometer in Fig. 1 of the present paper had a radius R = 17.48 mm and a thickness d = 1.55 mm . The moment of inertia amounted to I = 503.0 g ⋅ mm 2 , whereas the upper and lower gaps b 1 and b 2 between oscillating and fixed disks were 1.10 mm. The quartz-glass suspension strands of the viscometers had a length and a diameter of 170 mm and about (35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45) μ m, respectively, resulting in a period in vacuo 0 at 298.15 K of 24.61 s and 24.33 s. The design principle for the viscometer was specified in Ref. [19]. Quartz glass was employed as the building material on account of its small thermal expansion coefficient so that changes in the dimensions of the viscometer could be neglected and measurements in a large temperature range became possible. In addition, the logarithmic decrement in vacuo Δ 0 of a suspension strand made of fused quartz is negligibly small compared to that of a metal wire in the temperature range of the measurements [20]. Moreover, quartz glass is distinguished by a high chemical stability so that a possible decomposition of the organic vapors at higher temperature is only marginally influenced by a catalytic effect.
In the case of the aromatic hydrocarbon vapors, the logarithmic decrement Δ of the damped harmonic oscillation was derived from length measurements applying the silicon-coated mirror at the lower end of the long thin rod which was fused underside the oscillating disk. The mirror was employed together with a high-precision scale and a telescope at a distance of 2 m from the axis of the viscometer to register the turning points of the damped oscillation. The second parameter, the period of the harmonic oscillation, was deduced from time measurements with a stop watch. For this, the crossing of the zero position was observed with the telescope. To reduce the uncertainty, the time was taken for at least ten full oscillations. For the measurements on R134a, and Δ were obtained only from time measurements applying, in connection with the viscometer shown in Fig. 1 of Ref. [1], an optoelectronic system which consists of a 1 mW helium-neon laser and two photoreceivers, stationarily located at fixed positions on an optical bench at the distance of 2 m from the viscometer (see Ref. [21]). Both procedures are distinguished by the same uncertainties with regard to Δ and , because the movement of the disk is externally initiated by rotating a mechanical device through a small angle forward and reverse to the basic position. Five to 20 individual oscillations are taken after starting an oscillation run and passing some periods for the decay of perturbations. The relative uncertainty in Δ is 0.05 % , while that in is 0.005 %.
Unfortunately, an absolute measurement procedure is not feasible in the case of an all-quartz oscillating-disk viscometer. For absolute measurements, Newell [22] developed an appropriate theory, in which a so-called Newell constant C N has to be computed from the dimensions of the viscometer, which must be known with very high accuracy, Here N(u) is a constant value for a considered viscometer [23]. Since all connections between the single parts of the all-quartz oscillating-disk viscometer have to be connected by fusing, the dimensions cannot be ascertained with an accuracy necessary for absolute measurements. For relative measurements, the working equation according to Newell is given as This equation enables to calculate the Newell constant C N using one reliable reference viscosity value at an experimentally known low density of the calibration gas, especially at room temperature, including the measurement quantities Δ and (as well as Δ 0 and 0 in vacuo). The experimental density of the examined vapors was also comparatively small so that the contributions of the terms with f and h in (1) (2) Eq. 2 are entirely insignificant and even the term with a amounts to less than 0.1 % of the main contribution (see Ref. [24] [11], of methane and hydrogen sulfide [25], of nitrogen and carbon monoxide [13], and of acetic acid vapor [26]     Two possibilities exist for the choice of a suitable (0) 298.15 value for the re-calibration when the previous viscometer was calibrated with argon. Both a theoretically computed viscosity value and an experimentally based viscosity datum could be used. We preferred to apply a theoretical (0) 298.15 value. Vogel et al. [28] used an ab initio potential energy curve for the argon atom pair (see Refs. [29,30]) and the kinetic theory of dilute monatomic gases to calculate theoretical viscosity values in the limit of zero density. The relative uncertainty of the (0) 298. 15 value was supposed to be < 0.1 % . Another theoretically computed viscosity value for argon at 298.15 K obtained by Mehl was compared with that by Vogel et al. in Figure 7.2b of Ref. [31]. This figure shows that the relative uncertainty of both theoretical values could actually be lowered to 0.07 %. The theoretical viscosity values for argon were originally listed by Vogel et al. with only five significant digits. Hellmann [3] repeated the calculations using one digit more. The obsolete experimentally based (0) 298.15 value of Kestin and Leidenfrost [2] deviates from the new theoretical value of Hellmann at 298.15 K by 0.313 % . The coefficient of the temperature dependence ( ∕ T) in Table 1 was derived using Ref. [3].
The new experimentally based viscosity datum (0) 298.15 for argon recommended by Berg and Moldover [4] is distinguished by a very low value of the relative uncertainty: u r ( ) = 0.00 027 . To obtain this value, Berg and Moldover critically assessed the results of viscosity measurements relative to helium using 18 instruments for 11 gases near 298.15 K, which were extrapolated to zero density. Then, they used a fitting process considering altogether 235 viscosity ratios, the values of which were anchored to the highly accurate, theoretically computed (0) 298.15 value for helium reported by Cencek et al. [32]. In this way, similar to the improvement of a signalto-noise ratio, the relative uncertainty of the (0) 298.15 values for all 11 gases could be reduced. For an individual gas, the situation is somewhat inferior. Thus, Xiao et al. [33] re-evaluated previous measurements of May et al. [34] for argon carried out with a two-capillary viscometer to obtain viscosity ratios relative to helium between 200 K and 400 K. Applying again the (0) 298.15 value for helium by Cencek et al., the relative uncertainty of the argon viscosity data increased to u r ( ) = 0.00 038 . But, employing another measurement technique instead of the two-capillary viscometer (relative to helium), only an increased relative uncertainty can be achieved, probably twice as high. The difference between the theoretical viscosity value of Hellmann [3] and the experimentally based viscosity datum (0) 298.15 of Berg and Moldover amounts to 0.058 % and is within the relative uncertainty of the theoretical value.
For nitrogen, Berg and Moldover [4] recommended the (0) 298.15 datum of Table 1, which was determined by the same procedure already explained for argon. Note that the experimentally based (0) 298.15 value of Kestin and Leidenfrost [2] deviates from the recommended value of Berg and Moldover at 298.15 K by 0.215 % . In addition, the value of Berg and Moldover for nitrogen is distinguished by a deviation of 0.03 % from a theoretically based value (0) 298.15 = 17.701 μPa ⋅ s , which was proposed by Hellmann [5]. Hellmann applied for his calculations an ab initio potential energy surface for the nitrogen molecule pair and the kinetic theory of molecular gases. Since the computed theoretical values are too low by about 0.3 % between 300 K and 700 K compared with the best experimental viscosity data, Hellmann suggested to increase his theoretical viscosity values by a factor 1.003. This was based on the finding that the temperature dependence of the theoretical values is in excellent agreement with that of the best experimental data. Therefore, the temperature derivative ( ∕ T) for nitrogen was derived from the theoretical values of Hellmann.
An uncertainty analysis applying the new (0) 298. 15 value for argon including the temperature and density derivatives ( ∕ T) and ( ∕ ) T , all given in Table 1, was performed in Ref. [1] and should not be repeated here. As a summary, the relative combined expanded ( k = 2 ) uncertainty of the experimental viscosity data for any gas or organic vapor results as U c,r ( ) = 0.002 at room temperature and, assuming a slight increase with temperature, as U c,r ( ) = 0.003 at higher temperatures.
The series of measurements on a certain gas or organic vapor were often supplemented by additional measurement series on the calibration gas. As long as no change of the Newell constant was detected, it was concluded that the suspension system of the viscometer did not undergo a change as a result of an alteration due to the measurements at high temperatures.
The investigation on R134a reported in 1996 was part of a round-robin project, in which a single source of supply should be applied. Thus, nine separate cylinders were filled and supplied by ICI Chemical and Polymers Ltd., UK, using special procedures for the cleanliness of the cylinders. The measurements extended over a larger density range up to somewhat more than 9 kg ⋅ m −3 .
For the original measurements on the respective aromatic vapors, the evacuated oscillating-disk viscometer was filled by sublimation of the corresponding substance from ampules with the weighed samples. These small glass ampules were filled in turn by sublimation with the final purified samples after drying with molecular sieves as well as degassing by repeated freezing and evacuating. If applicable, the melting point of the substance was determined using the specifically designed filling apparatus. After freezing the substance at the bottom of the viscometer employing liquid nitrogen, the filling line was fused for the quasi-isochoric measurements. Note that the measurement densities of the aromatic hydrocarbon vapors were at least fifteen times smaller than the highest density for R134a.

Results of the Re-evaluated Measurements
Firstly, the temperatures of the original measurements were, apart from those for R134a, converted to the ITS-90 temperature scale. Secondly, the associated argon and nitrogen measurements were used to re-calibrate the employed oscillating-disk viscometer applying the revalued viscosity values of Table 1. Thirdly, the values of the improved Newell constants C N for a respective measurement series were applied for calculating the re-evaluated viscosity data using the directly obtained Δ , , 0 , and values.
Because this investigation is aimed at the identification of the viscosity in the limit of zero density, (0) , and additionally for R134a of its initial density dependence, (1) , isotherms should be considered. Since quasi-isothermal re-evaluated data are required for this analysis, the experimental re-evaluated points of the three (for five of the aromatic vapors), four (for p-dichlorobenzene), or six (for R134a) isochoric series were grouped such that approximated isotherms resulted. To have at least three isothermal values for the evaluation, we were forced to include actually data points at temperatures differing by ±12 K . In addition, re-measurements were carried out at lower temperatures, after the highest temperature had been attained, to check for any thermal alteration or decomposition of the chemical substance. But it must be considered that the thermal alteration could be a reversible process. Then, a few experimental points at high temperatures or re-measured points had to be excluded owing to significantly increased viscosity values compared with other values of the quasi-isotherms. Hence, the isothermal groups could include a reduced number of data compared with the total number of series for a considered substance. For that reason, only one or two experimental data points could belong to an isotherm so that none or no trustworthy (0) and (1) (for R134a) values could be determined using the quasi-isothermal data. It should be restated that the density range of the isochores is rather restricted for the six aromatic vapors so that, in spite of the high reproducibility, small differences could particularly lead to unexpected values of the initial density dependence, (1) .
The re-evaluated grouped results of the measured isochores are listed for R134a and the six aromatic vapors in Tables 2, 3, 4, 5, 6, 7, and 8. Each row of the different series consisting of filled and empty spaces represents a grouped isotherm. The table captions include the reference in which the measurements were originally described. In a footnote of the tables, the calibration gas and the purity of the measured samples are provided. Moreover, detailed information is indicated in further footnotes concerning individual experimental points which were included or excluded compared to the original publication.
Note that one experimental datum of Series 3 for chlorobenzene at the lowest temperature (see Table 7) should not directly be used, because this point was apparently measured at a density higher than that of the saturated vapor sat at the measurement temperature.

Validation of the Viscosity in the Limit of Zero Density and of Its Initial Density Dependence
As already discussed, the primary data points of the respective isochores were not measured at exactly identical temperatures for a certain substance. The same is true for the re-evaluated counterparts, which could only be compressed to approximate isotherms. Then, the re-evaluated data were converted into quasi-isothermal values applying a first-order Taylor series in temperature, The temperature of the isotherms T iso conforms to the mean of the experimental temperatures T exp of the data for the distinct isochoric series measured at the respective approximate adjustment of the thermostat. The temperature derivative ( ∕ T) needed in Eq. 4 was deduced by means of the following equation, Table 2 Re-evaluated viscosity data for 1,1,1,2-tetrafluoroethane (R134a) 1 of Wilhelm and Vogel [6] 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.002 near room temperature (297 K) and U c,r ( ) = 0.003 up to 438 K.
The product was supplied in a cylinder by ICI Chemical and Polymers Ltd., UK. After evacuation of the viscometer, a certain amount of substance was sublimated in a special part of its filling system and frozen with liquid nitrogen to avoid that any solid particles from the cylinder could enter the viscometer. Then, the substance was introduced into the viscometer. The density of the used samples was determined by p, V, T measurements. The purity of the batch was > 99.9 % . The main impurity was R134 with a concentration of 850 ppm verified by means of gas chromatography. Water was in the sample at a concentration of 6 ppm determined by Karl-Fischer test 2 Experimental point, also not listed in Ref. [6], has to be left out as an outlier where T R = T∕(298.15 K) and S = 10 μPa ⋅ s . The coefficients A, B, C, D, and E were derived in a fit of Eq. 5 to the experimental data of each isochoric series, in which the values of the re-measured experimental point or points of each isochore were also included, meaning that the influence of any reversible thermal alteration of the samples remained hidden. It was proven that the remainder R N in Eq. 4 is insignificant compared to the experimental uncertainty. The only experimental point situated in the saturated vapor and reported for chlorobenzene in Table 7 does not correspond to the actual density at which the measurements of the isochore were carried out, but to the density of the saturated vapor sat also given in that table. The small excess amount of the substance was adsorbed at the inner wall of the viscometer. The value of sat was derived using a virial equation limited to the second virial coefficient B(T) according to Table 3 Re-evaluated viscosity data for 1,3,5-trimethylbenzene (mesitylene) vapor 1 of Vogel [7] 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.003 up to 631 K.
The initial product was of "pure" quality and supplied by Sojuzchimexport, Moscow, USSR. After a standard pretreatment and fractional distillation, the middle fractions were characterized by a boiling point T b = (437.1 ± 0.1) K . In a special glass apparatus, the substance was additionally purified by repeated freezing and evacuating, degassed, and dried by molecular sieve 4A. The final samples showed a content of 99.7 % mesitylene according to gas chromatography and a refractive index n 25 D = 1.4963 2 Experimental point listed in Ref. [7] has to be left out as an outlier  Table 4 Re-evaluated viscosity data for 1,2,4,5-tetramethylbenzene (durene) vapor 1 of Vogel [7] 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.003 up to 631 K.
Re-calibration with argon (series 1) and nitrogen (series 2 and 3). The applied product was supplied by Ferak, Berlin, Germany, in a quality "purified by zone melting". The melting point of the substance was determined to be T fus = (352.33 ± 0.02) K . In a special glass apparatus, the substance was degassed and dried by molecular sieve 4A and thereafter used for the measurements 2 Experimental point listed in Ref. [7] has to be left out since it suffered from thermal alteration at higher temperature  Table 5 Re-evaluated viscosity data for biphenyl (diphenyl) vapor 1 of Vogel [8] 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.003 up to 624 K.
Re-calibration with nitrogen (series 1 and 2) as well as with argon and nitrogen (series 3).
The initial product of "extra pure" quality was supplied by Berlin-Chemie, Berlin, Germany. After vacuum distillation, it was purified by zone melting. In a special glass apparatus, the substance was dried by molecular sieve 4A, degassed, and its melting point was determined to be T fus = (342.10 ± 0.02) K 2 Experimental point listed in Ref. [8] has to be left out due to an increase resulting from thermal alteration 3 Experimental point listed in Ref. [8] has to be left out as an outlier Here, p sat is the saturation pressure, which was calculated for chlorobenzene with an Antoine equation recommended by NIST [35]. The second virial coefficient B(T) in turn was computed using a polynomial in 1/T given in Ref. [10]. The values of the zero density viscosity (0) and of the initial density dependence (1) were derived from a fit of the quasi-isothermal viscosity values at a certain temperature T iso as a function of density applying a series expansion truncated at first order, It is clear that the density range for the aromatic vapors is badly restricted. Moreover, a fit using only three values, even though only two coefficients must be deduced, represents an invidious task. Already small uncertainties of the experimental data could have led to unreliable (0) and (1) values. In the case that only two experimental points were available for a grouped isotherm, (0) and (1) were directly calculated using Eq. 7. With one experimental point, it was not possible to derive (0) by means of Eq. 7. For R134a, it was distinctly easier to obtain appropriate (0) and (1) values (7) (T, ) = (0) (T) + (1) (T) . Table 6 Re-evaluated viscosity data for fluorobenzene vapor 1 of Kaussmann et al. [9] 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.002 near room temperature (304 K) and U c,r ( ) = 0.003 up to 605 K.
The initial product of "extra pure" quality was supplied by Ferak, Berlin, Germany. After fractional distillation, the middle fractions were characterized by a boiling point T boil = (357.6 ± 0.1) K . In a special glass apparatus, the substance was additionally purified by repeated freezing and evacuating, degassed, and dried by molecular sieve 4A. The final samples showed a content of 99.9 % fluorobenzene according to gas chromatography 2 Experimental point was not listed in Ref. [9] since it had to be left out due to an increase resulting from electric charging of the quartz glass oscillating-disk viscometer applying Eq. 7, since the number of quasi-isothermal viscosity values was five or six and the density range was at least ten times larger compared to that of the aromatic vapors. Therefore and particularly for the six aromatic vapors, the re-evaluated values of a quasi-isotherm were also rectified to the limit of zero density applying the Rainwater-Friend theory [36,37], which describes the initial density dependence of the transport properties. For the viscosity, the second viscosity virial coefficient is defined as where M is the molar mass. Bich and Vogel [38,39] published tables of the reduced second viscosity virial coefficient B * as a function of the reduced temperature T * . After that, Vogel et al. [40] proposed a reasonable correlation of B * (T * ) , valid in the range 0.3 ≤ T * ≤ 100, Table 7 Re-evaluated viscosity data for chlorobenzene vapor 1 of Ahlmeyer et al. [10] 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.002 near room temperature (319 K) and U c,r ( ) = 0.003 up to 630 K.
The initial product was of "for synthesis" quality and supplied by Laborchemie, Apolda, Germany. After a standard pretreatment and fractional distillation, the middle fractions were characterized by a boiling point T boil = (404.9 ± 0.1) K . In a special glass apparatus, the substance was additionally purified by repeated freezing and evacuating, degassed, and dried by molecular sieve 4A. The final samples showed a content of 99.94 % chlorobenzene according to gas chromatography 2 Experimental point listed in Ref. [10] was situated in the saturated vapor ( sat = 0.207 kg ⋅ m −3 ) 3 Experimental point listed in Ref. [10] has to be left out due to an increase resulting from thermal alteration Re-calibration with argon (series 1−4).
The applied product was supplied by Ferak, Berlin, Germany, in a quality "purified by zone melting". The melting point of the substance was determined to be T fus = (326.29 ± 0.02) K . In a special glass apparatus, the substance was degassed and dried by molecular sieve 4A and thereafter used for the measurements 2 Experimental point listed in Ref. [8] has to be left out due to an increase resulting from thermal alteration In Eqs. 9 and 10, N A and k B are Avogadro's and Boltzmann's constants, respectively. The coefficients b i are listed in Table 9. The scaling factors ∕k B and are usually the parameters for the Lennard-Jones 12-6 potential. Of course, none of the considered substances complies with the Lennard-Jones 12-6 potential so that the theoretical guidance for the representation of the initial density dependence of the viscosity is inadequate for them. But Eqs. 9 and 10 enable a secure extrapolation both to low and to high temperatures if appropriate values for the energy scaling factor ∕k B and the length scaling factor can be derived from the measurements. At low temperatures T or better at low reduced temperatures T * , negative values for (1) (T) and B (T) , respectively, are observed, which become less negative with increasing temperature, followed by a transition to positive values. After passing through a maximum, the values of B (T) decrease monotonically and attain negative values at very high reduced temperatures, only proven experimentally for helium. In principle, it is reasonable to derive optimized scaling factors ∕k B and by fitting the theoretical results for the Rainwater-Friend theory, presented by Eqs. 9 and 10, to experimental B (T) values of a selected substance resulting from Eq. 8. Typically, the (0) and (1) values following from the series expansion according to Eq. 7 should be employed to calculate the required experimental B values. In Ref. [1], this was recently demonstrated for measurements on twelve gases and vapors, but those experiments extended over a distinctly larger density range and included more isochoric measurement series. When using a fit to three and four quasi-isothermal re-evaluated data points or a simple calculation from two re-evaluated data points, Eq. 7 may yield B (T) values which are implausible and do not correspond to the pattern of the explained general behavior of second viscosity virial coefficients. Consequently, the fit of Eq. 9 to such values will not be satisfactory. In principle, only reliable B (T) values should be included to determine the optimized scaling factors ∕k B and . For the aromatic vapors, less than half of the quasi-isotherms provided appropriate B (T) values so that the derivation of the scaling factors was a complicated task and, aside from that, pretty subjective and arbitrary. The lastly taken selection of reasonably sensitive B (T) values yielded the scaling factors for the aromatic vapors listed in Table 10 together with the information which isotherms could be or were not considered in the fit. Since the scaling factors for the aromatic vapors are rather insecure, the correction to the limit of zero density by means of the Rainwater-Friend theory was additionally performed using the scaling factors of similar substances. The rationale for this approach is that the density range over which the extrapolation has to be done is relatively small and the scaling factors for very similar substances are available. As an example, the molecular sizes and the electrostatic surface potentials of benzene and fluorobenzene, shown in Fig. 1b of Ref. [41], are not so different. In addition, the parameters of the 11-6-8 potential deduced from the measured viscosity coefficients (without extrapolation to the limit of zero density) and reported in 1982 (Ref. [42]) do not differ strongly (benzene: = 2.317 , = 0.4993 nm , ∕k B = 582.9 K ; fluorobenzene: = 2.447 , = 0.5102 nm , ∕k B = 579.0 K ). Hence, the scaling factors for benzene, toluene, p-xylene, and phenol, derived in Ref. [1] by employing B values following from reevaluated viscosity data measured in density ranges similar to R134a, are also given in Table 10. These scaling factors were partly applied in the next step as substitute for the scaling factors of the aromatic vapors considered in this paper. The scaling factors for R134a are also given in Table 10; they proved to be much more reliable. 2 Isotherms at 352.63 K, 608.87 K, and 629.47 K as well as re-measurements at 372.08 K excluded from fit using Eq. 9 3 Only isotherms at 377.03 K, 447.07 K, and 586.42 K included in fit using Eq. 9 4 Only isotherms at 409.98 K, 440.87 K, 474.04 K, and 599.25 K included in fit using Eq. 9 5 Only isotherms at 348.17 K, 365.49 K, and 397.67 K as well as re-measurements at 532.43 K and 427.11 K included in fit using Eq. 9 6 Only isotherms at 320.10 K and 342.24 K as well as a fictitious value of (1) = −6.00 μPa ⋅ s ⋅ L ⋅ mol −1 at 408.75 K included in fit using Eq. 9 7 Only isotherms at 378.96 K, 410.39 K, 443.55 K, and 471.20 K included in fit using Eq. 9 8 Isotherms at densities similar to those for R134a included in fit using Eq. 9

Gas
Reference Finally, the scaling factors were used to rectify the re-evaluated (T, ) values of each grouped isotherm to the limit of zero density. In doing so, the next relation following from Eqs. 7−10 was applied, Since data points from several isochoric series typically belonged to a quasi-isotherm, the (0) values resulted by averaging the corresponding values for each reevaluated point of the quasi-isotherm. Furthermore, the B values for all grouped isotherms computed via the fit were employed to deduce (1) values using the (0) values corrected with Eq. 11.
All of the resulting (0) and (1) values for R134a and for the respective aromatic vapors are summarized in Tables 11,12,13,14,15,16, and 17. The number of quasi-isothermal experimental points considered for a grouped isotherm is specified in Column 2 of these tables. The results of the fit of Eq. 7 to the re-evaluated quasiisothermal data are indicated in Columns 3−5: the (0) and (1) values are given together with their individual standard deviations (0) and (1) and with the standard deviation for each isotherm. Footnotes concerning Column 1 clarify that re-measurements were performed or an experimental point in the saturated vapor was taken into account, whereas footnotes dealing with Column 2 state that only one or two data points had been considered. Finally, footnotes of Column 4 are related to the fact that an (1) value was implausible. In Columns 6 and 7 of Table 11 for R134a as well as in the respective columns of the first row of each quasi-isotherm of Tables 12, 13 Table 10 for the considered substance, are listed. In the second row and in some cases in the third row of Columns 6 and 7 for the respective quasiisotherms of Tables 12, 13 The evaluation of the quasi-isothermal experimental data with the series expansion (Eq. 7) is discussed using the data for the second viscosity virial coefficient B (T) computed from the (0) (T) and (1) (T) data with Eq. 8. In Figs. 3, 4, and 5, exemplarily for R134a, diphenyl, and chlorobenzene, the experimentally based B (T) data are compared with the B (T) values calculated applying the scaling factors ∕k B and , which were derived for the Rainwater-Friend theory by means of Eqs. 9 and 10. Figure 3 for R134a shows that the B (T) data of the experiments of Wilhelm and Vogel [6] nearly perfectly agree with the Rainwater-Friend theory and, in addition, that the B (T) data for the re-measured isotherms at 317.54 K, 338.35 K, and 352.84 K only marginally deviate from those for the isotherms with increasing temperature. The reason for this agreement consists in the comparatively large experimentally probed density range. Figure 4 demonstrates that, in the case of diphenyl, only four of eight experimentally based B data are in some degree suitable to derive acceptable scaling factors. Apart from chlorobenzene, the situation is similar for the other aromatic vapors meaning that about half of the experimental B data points conforms to the Rainwater-Friend theory. Furthermore, the subsequent calculations of B values with scaling factors of the substituted and/or of the aromatic hydrocarbon vapors discussed in this paper prove that the choice of the aromatic hydrocarbon vapor can be somewhat arbitrary as long as a certain similarity is met. In Fig. 4 for diphenyl, the scaling factors of mesitylene are less appropriate, whereas the scaling factors of durene yield a B (T) curve whose trend is roughly like that resulting from the scaling factors of diphenyl itself. Whereas the differences between the corresponding curves (here obtained with the scaling factors of durene and diphenyl itself) are comparatively large at low temperatures, they decrease with Table 11 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Wilhelm and Vogel [6] on R134a 1 Here and in Tables 12,13,14,15,16,and 17, n is the number of quasi-experimental points included in the fit of Eq. 7 2 Here and in Tables 12,13,14,15,16, and 17, the value of (0) RF results from Eq. 11 by means of the Rainwater-Friend theory using the parameters of Table 10 received from a fit of B (T) values, which were computed applying (0) and (1) from Columns 3 and 4 3 Here and in Tables 12,13,14,15,16, and 17, the value of (1) RF follows from a fit of B (T) values, which were calculated applying (0) and (1) from Columns 3 and 4, and then employing the (0) RF value of Column 6 4 Re-measurements at lower temperature after the highest temperature had been attained 5 Only two data points so that (0) and (1) were directly calculated 6 Value of (1) is implausible increasing temperature T. Analogous statements can be made for mesitylene, durene, fluorobenzene, and p-dichlorobenzene, respectively. Figure 5 illustrates that only two experimental B data points are appropriate for chlorobenzene, whereas most of the B data completely differ with the temperature function B (T) of the Rainwater-Friend theory. Therefore, a fictitious B (T) datum at 408.75 K that corresponds to (1) = −6.00 μPa ⋅ s ⋅ L ⋅ mol −1 was selected. This choice seems to be quite reasonable when the computed B curve, resulting with the deduced scaling factors for chlorobenzene, is compared with the B curves following from the scaling factors for toluene and phenol. Table 12 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Vogel [7] on 1,3,5-trimethylbenzene (mesitylene) vapor. Values of (0) and (1) in the second row of each quasi-isotherm were derived with scaling factors of p-xylene given in Table 10 1 Only two data points so that (0) and (1) were directly calculated 2 Value of (1) is implausible 3 Re-measurements at lower temperature after the highest temperature had been attained

Recommended Values for the Viscosity in the Limit of Zero Density and of Its Initial Density Dependence
As already explained, this report aims at providing reliable values for the viscosity in the limit of zero density, (0) (T) , and, if possible, for its initial density dependence, (1) (T) . In principle, the (0) data of Columns 3 of Tables 11, 12, 13, 14, 15, 16, and 17, resulting from the fit of Eq. 7 to the quasi-isothermal viscosity data, should be preferred and the (0) RF values of Columns 6 at the same temperatures should not be considered. This is certainly valid for R134a and holds in general when (0) (T) and (1) (T) data can reliably be derived with Eq. 7 from quasi-isotherms, whose corresponding measurements extend over a comparatively large density range. This was the case for the substances considered in Ref. [1]. However, for the aromatic vapors considered in this paper, the density range of the experiments is so small that a reasonable evaluation of Table 13 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Vogel [7] on 1,2,4,5-tetramethylbenzene (durene) vapor. Values of (0) and (1) in the second row of each quasi-isotherm were derived with scaling factors of mesitylene given in Table 10 1 Value of (1) is implausible 2 Only one data point so that (0) and (1) could neither be obtained from a fit of Eq. 7 nor directly be calculated 3 Only two data points so that (0) and (1) were directly calculated 4 Re-measurements at lower temperature after the highest temperature had been attained 7.588 ± 0.001 −6.783 Table 14 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Vogel [8] on biphenyl (diphenyl) vapor. Values of (0) and (1) in the second and third rows of each quasi-isotherm were derived with scaling factors of mesitylene and durene, respectively, given in Table 10 1 Value of (1) is implausible 2 Only two data points so that (0) and (1) were directly calculated 3 Re-measurements at lower temperature after the highest temperature had been attained 4 Only one data point so that (0) and (1) could neither be obtained from a fit of Eq. 7 nor directly be calculated the quasi-isotherms is generally hindered. In addition, when the (0) data of Columns 3 arose from only two experimental points and the (1) (T) values and accordingly the B (T) values are implausible, the (0) data had to be substituted by the (0) RF values of Columns 6 at the corresponding temperatures. Furthermore, if only one quasi-experimental viscosity datum was available and only (0) RF values could be deduced by means Table 15 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Kaussmann et al. [9] on fluorobenzene vapor. Values of (0) and (1) in the second row of each quasi-isotherm were derived with scaling factors of benzene given in Table 10 1 Only one data point so that (0) and (1) could neither be obtained from a fit of Eq. 7 nor directly be calculated 2 Only two data points so that (0) and (1) were directly calculated 3 Value of (1) is implausible 4 Re-measurements at lower temperature after the highest temperature had been attained 11.397 ± 0.001 −0.567 Table 16 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Ahlmeyer et al. [10] on chlorobenzene vapor. Values of (0) and (1) in the second and third rows of each quasi-isotherm were derived with scaling factors of toluene and phenol, respectively, given in Table 10 1 An experimental point of Series 3 situated in the saturated vapor ( sat = 0.207 kg ⋅ m −3 ) was used 2 Value of (1) is implausible 3 Only two data points so that (0) and (1) were directly calculated 4 Only one data point so that (0) and (1) could neither be obtained from a fit of Eq. 7 nor directly be calculated 5 Re-measurements at lower temperature after the highest temperature had been attained of the Rainwater-Friend theory, the respective values of the Columns 6 had to be applied in any case.
In conclusion, the experimentally based (0) (T) and (1) (T) data deduced from the series expansion of Eq. 7 are recommended for R134a, but not for the aromatic hydrocarbon vapors considered in this paper. For these substances, the (0) RF (T) values of Column 6, given in the first row of each quasi-isotherm and following from the Rainwater-Friend theory with the determined scaling factors for the respective substance, should be preferred. This is further based on the fact that the use of Eq. 11 is only connected with a change in the viscosity of −(0.35 to 0.56) % at the lowest temperatures up Table 17 Values for the viscosity in the limit of zero density, (0) , and for its initial density dependence, (1) , resulting from the re-evaluated quasi-experimental isotherms of the measurements of Vogel [8] on 1,4-dichlorobenzene (p-dichlorobenzene) vapor. Values of (0) and (1) in the second row of each quasiisotherm were derived with scaling factors of p-xylene given in Table 10 1 Value of (1) is implausible 2 Only one data point so that (0) and (1) could neither be obtained from a fit of Eq. 7 nor directly be calculated 3 Only two data points so that (0) and (1) were directly calculated 4 Re-measurements at lower temperature after the highest temperature had been attained to (−0.05 to + 0.12) % at the highest temperatures, when the highest measured density is corrected to zero density. Then, the (0) data for R134a and the respective (0) RF values (T)  Table 10: -----, according to the range of the quasi-isotherms; ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , extrapolated down to 298 K and up to 1073 K. Values calculated using the scaling factors of durene given in Table 10: ---, according to the range of the quasi-isotherms; ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , extrapolated down to 298 K and up to 1073 K were correlated applying the following equations which appropriately extrapolate both to low and to high temperatures: In Eq. 12, (0) is in μPa ⋅ s and T in K. The functional form of (T) was chosen by using symbolic regression [43], implemented in the Eureqa software package, along the lines of Laesecke and Muzny [44,45], correlating the dilute gas viscosities for carbon dioxide and methane, as well as of Hellmann [46], who correlated the dilute gas viscosity for ethane. The coefficients f i ( i = 1, ..., 4 ) obtained by fitting Eqs. 12 and 13 to the (0) data for R134a and to the (0) RF values of the aromatic hydrocarbon vapors are listed in Table 18. The functional form of Eq. 13 has no physical relevance and, consequently, the parameters f 1 to f 4 have no physical significance for the different chemical species. The extrapolation behavior of these correlations is illustrated in Figs. 6 and 7. Figure 6 shows that the correlations for R134a and for fluorobenzene appropriately extrapolate down to 1 K. But the correlations for the other hydrocarbon vapors do not adequately extrapolate to such a low temperature. Thus, the correlations for chlorobenzene and p-dichlorobenzene yield reasonable ▪ , isotherms included in the fit of Eq. 9; ⊞ , fictitious value at 408.75 K included in the fit of Eq. 9; ◻ , isotherms excluded from the fit of Eq. 9. Values calculated with the scaling factors = 1.13 855 nm and ∕k B = 365.35 K using the theoretical B * function of the Rainwater-Friend theory corresponding to Eq (9): ---. Values calculated using the scaling factors of toluene given in Table 10: -----, according to the range of the quasi-isotherms; ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , extrapolated down to 298 K and up to 1073 K. Values calculated using the scaling factors of phenol given in Table 10: ---, according to the range of the quasi-isotherms; ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , extrapolated down to 298 K and up to 1073 K (0) viscosity values down to 150 K, whereas the correlation for mesitylene provides meaningful values down to 70 K and those for durene and diphenyl down to 200 K. This is also demonstrated in this figure and certainly sufficient for all practical purposes. Figure 7 shows that the correlations for all seven substances reliably extrapolate up to 1500 K.
The (0) cal values calculated with Eqs. 12 and 13 are exemplarily compared for diphenyl in Fig. 8 and for chlorobenzene in Fig. 9 with corresponding (0) data and (0) values of Tables 14 and 16, that means with the experimentally based (0) data of Columns 3, deduced by means of a fit with Eq. 7, and with (0) RF values of Columns 6, derived using Eq. 11 and the scaling factors of different aromatic vapors. The relative deviations Δ in these figures are marked in the majority of cases with error bars calculated from the standard deviations given in Columns 3 and 6. Note that in the case of only one experimentally based datum, no values for were derived and  no error bars could be plotted. Figure 8 shows that the four (0) data points of those quasi-isotherms, to which Eq. 9 was fitted to determine the scaling factors of diphenyl, deviate from the (0) cal values, which were calculated applying Eqs. 12 and 13 with the respective coefficients of Table 18, by < ±0.03 % , whereas the excluded four (0) data points differ by up to +0.50 % . The (0) RF values which were obtained with the scaling factors of diphenyl itself and were employed to derive the coefficients of Table 18, are represented by Eqs. 12 and 13 within ±0.04 % . In contrast, the (0) RF values which are based on the scaling factors of mesitylene deviate by −0.26 % to +0.02 % , while those obtained with the scaling factors of durene differed by −0.15 % to −0.01 % , each systematically increasing with temperature. Figure 9 illustrates that the two (0) data points at 320 K and 342 K, to which Eq. 9 including a third fictitious datum was fitted to determine the scaling factors of chlorobenzene, differ from the (0) cal values, which were calculated applying Eqs. 12 and 13 with the respective coefficients of Table 18, by < ±0.11 % , whereas the excluded eight (0) data points deviate by up to +0.90 % . The (0) RF values which were obtained with the scaling factors of chlorobenzene and were applied to determine the coefficients of Table 18 are represented by Eqs. 12 and 13 within ±0.09 % . In contrast, the (0) RF values which resulted with the scaling factors of toluene differ by −0.12 % to +0.10 % , while those obtained with the scaling factors of phenol deviate by −0.02 % to +0.17 % , each to a large extent systematically increasing with temperature. However, the differences ∆ (five of the aromatic hydrocarbon vapors), of four (p-dichlorobenzene), and of six (R134a) isochoric series were grouped such that approximated isotherms followed. Then, the re-evaluated data of the quasi-isothermal groups were converted into isothermal values using a first-order Taylor series in temperature.
A series expansion in density truncated at first order was applied to derive (0) and (1) data from the isothermal viscosity data. This procedure could appropriately be employed to deduce (0) and (1) data for R134a. Since the density range of the measurements on the six aromatic hydrocarbon vapors was insufficiently large, the resulting (0) and primarily the obtained (1) data had such large uncertainties that half of them had to be refused. In particular, if only two experimental data points belonged to an isotherm, the reliability of the (0) and (1) data was distinctly reduced. Hence, the Rainwater-Friend theory for the initial density dependence of the viscosity was applied to determine (0) and (1) values. For this, optimized scaling factors of length and energy were deduced by means of a fit of theoretical second viscosity virial coefficients, B (T) , to the comparatively reliable experimentally based B (T) data, which were obtained from the (0) and (1) data deduced with the series expansion in density. Furthermore, (0) and (1) values were deduced applying scaling factors for similar aromatic hydrocarbon vapors dealt with in this paper and for aromatic hydrocarbon vapors re-evaluated in Ref. [1]. All resulting (0) and (1) values are summarized in tables for the respective substance.
In principle, the (0) and (1) data resulting from the series expansion in density should be preferred. This holds for R134a, for which, apart from the re-measurements, all (0) and (1) data of Columns 3 and 4 of Table 11 are recommended. For the six aromatic hydrocarbon vapors, only the (0) values derived with the Rainwater-Friend theory applying the optimized scaling factors of energy and length of the respective vapor can be recommended, while the (1) values are somewhat doubtful. The selected (0) values are once more summarized in Table 19. The relative combined expanded ( k = 2 ) uncertainty is estimated to be U c,r ( ) = 0.003 at all temperatures. The reason for this judgement consists in that the re-evaluated experimental data are characterized by U c,r ( ) = 0.002 near room temperature and by an increase to U c,r ( ) = 0.003 at higher measurement temperatures, while the shift to the limit of zero density amounts to at most −0.5 % at the lowest temperatures decreasing to ±0.1 % at the highest ones.
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Table 19
Recommended viscosity values in the limit of zero density for six aromatic hydrocarbons 1 1 The relative combined expanded ( k = 2 ) uncertainty is U c,r ( ) = 0.004 at the given temperatures of the quasi-isotherms