Dimerization and Viscosity of Acetic Acid Vapor

Dimerization and Viscosity of Acetic Acid Vapor By Eckhard Vogel∗ and Eckard Bich Universität Rostock, Institut für Chemie, Albert-Einstein-Straße 3a, 18059 Rostock, Germany

Dedicated to Prof. Dr. Andreas Heintz on the occasion of his 65th birthday (Received July 16, 2012; accepted in revised form November 30, 2012) (Published online January 28, 2013)

Acetic Acid Vapor / Dimerization / Measurement / Viscosity Coefficient Results of new relative measurements on the vapor of acetic acid are reported. The measurements were based on a single calibration at room temperature with a theoretically calculated viscosity value of argon. Nineteen isochoric series, differing in density, were performed in an all-quartz oscillating-disk viscometer from 298 to 598 K and for densities between 0.5 and 61 mol m−3 . The uncertainty of the experimental data is ±0.5% at low densities and ±(0.3 − 0.4)% at high densities increasing with temperature. This is inferior compared with measurements on simple gases due to adsorption effects on quartz at low temperatures and to thermal decomposition at high temperatures. Isotherms recalculated from the original isochoric data show with decreasing density a strong curvature with negative slope, particularly at low temperatures. The further evaluation accounts for the fact that a strong dimerization depending on temperature occurs in acetic acid vapor. The isotherms, resulting from an association model which considers additional interactions between monomers and dimers, show a smaller curvature in terms of the mole fraction of monomers. Viscosity values for monomers and dimers were deduced using a simplified equation for the viscosity of a gas mixture. These values are restricted to 570 K due to thermal decomposition and low mole fractions of the dimers at high temperature. Furthermore, viscosity values for the saturated vapor could be determined at low temperatures.

1. Introduction Phenomena of strong intermolecular interactions, in particular hydrogen bonds in alkanols and carboxylic acids, have been investigated extensively using different experimental and theoretical methods. Although there exists a strong tendency to look into the molecular details of such an association using quantum mechanical ab initio calculations, the macroscopic thermodynamic behavior is of large importance for the test of the theoretical calculations. This holds especially for larger molecules with a complicated structure. Numerous papers have been dedicated to acetic acid vapor as a representative of a strong dimerization. Theoretical calculations have demonstrated that the potential energy surface of the acetic acid dimer is characterized by a global minimum corresponding to a cyclic structure with two very strong O–H· · · O hydrogen bonds as well as * Corresponding author. E-mail: [email protected]

This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder.

Z. Phys. Chem. 227 (2013) 315–331 / DOI 10.1524/zpch.2013.0319 © by Oldenbourg Wissenschaftsverlag, München

E. Vogel and E. Bich

by a number of further energy minima related to less stable cyclic and linear structures with O–H· · · O or C–H· · · O hydrogen bonds or both of these bonds simultaneously [1]. Evidence on the cyclic structure of the dimer has been supplied by infrared and Raman spectroscopic studies and other experiments (see [1,2] and references therein). The quantum cluster equilibrium model has been used at the B3LYP/6-31+G* level of theory to determine cluster populations for formic acid vapor [3]. Unfortunately, this model has not yet been applied to acetic acid vapor. Most of the experimental work on thermophysical properties has been dealing with the volumetric behavior of acetic acid vapor, but only a few papers have been directed to its transport properties. The evaluation of the experiments requires to consider the formation of dimers in the best possible way. A model based on an ideal mixture consisting of monomers and dimers should be used at least. A more sophisticated evaluation taking into account further intermolecular interactions between the components in the vapor could be applied if the uncertainty of the experimental data is low. In a previous paper [4] of our group the volumetric behavior of acetic acid vapor was studied based on own measurements between 410 K and 574 K as well as on experimental data from literature between room temperature and 467 K. The evaluation was performed with an equation of state for a reactive fluid so that the dimerization constant K 2 as well as second and higher interaction pressure virial coefficients for the mixture consisting of monomers and dimers could be derived from the experimental data base. The investigation has now been extended to viscosity measurements on acetic acid vapor in a very large temperature range (298 K to 598 K), since Timrot et al. [5] reported results only for a restricted one (307 K to 422 K). Influences due to the slip effect and to adsorption at low temperatures as well as due to thermal decomposition at high temperatures have to be considered, particularly at low densities of the substance. Furthermore, it has been tested, whether it is possible to perform viscosity measurements at somewhat higher densities, for which partial condensation occurs, and to deduce the viscosity of the saturated vapor.

2. Experimental section 2.1 Experimental equipment An all-quartz oscillating-disk viscometer with small gaps was used for the measurements on acetic acid vapor. The construction and the characteristics of the viscometer were given in Fig. 1 and Table 1 of Ref. [6]. Principles of design and construction were reported previously [7]. Details of the manufacturing were improved in the course of time to facilitate the assembly of the viscometer. Fused quartz glass is characterized by a small thermal expansion coefficient and a small internal logarithmic decrement of the suspension strand and is accordingly best qualified for measurements in a large temperature range. But the dimensions of the viscometer cannot be determined with an accuracy needed for absolute measurements due to its construction and assembly using quartz glass. The decrement Δ and the period τ (of about 25 s) of the damped harmonic oscillation are obtained from time measurements by means of an opto-electronic system [8].


316

Fig. 1. The viscosity η of acetic acid vapor as a function of the molar density ρ – low isotherms. , 298.15 K; , 312.15 K; , 326.15 K; , 339.15 K; , 354.15 K; , 367.65 K; , 382.15 K; , 395.65 K; , 410.65 K; , 326.15 K (re-measurements); , 410.65 K (re-measurements); , saturated vapor of isotherms; ——–, saturated vapor line.

◦

•

The relative uncertainties in Δ and in τ are usually ±0.05% and ±0.005%, respectively. The temperature function of the period τ0 (T) in vacuo was measured separately, whereas results of Whitelaw [9] were employed for the logarithmic decrement Δ0 in vacuo. The temperature of the oscillating-disk viscometer is regulated by a specially designed air-bath thermostat consisting of three heating zones operated separately. The generated vertical temperature profile of the thermostat ensures a stable density stratification inside the viscometer and prevents convection. The temperature is determined with a 10 Ω platinum resistance thermometer together with a resistance measuring bridge, each calibrated. The uncertainty of the temperature measurements is ±50 mK at 300 K and ±150 mK at 600 K.

2.2 Experimental procedure As already indicated, only relative measurements can be performed with our viscometer. Their implementation is based on the measuring theory of Newell [10] which is designed for absolute measurements using an oscillating-disk viscometer with small gaps. For absolute measurements, the Newell constant CN has to be calculated solely from the dimensions of the specific viscometer. Then the independently determined density of the gas ρ and the measured quantities Δ and τ (including Δ0 and τ0 in vacuo) result in the viscosity η. The situation concerning our measurements with the all-quartz oscillating-disk viscometer is a special one, since the gas densities ρ are comparably


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Dimerization and Viscosity of Acetic Acid Vapor


Table 1. Viscosity of Acetic Acid Vapor. T/K

η/μPa s

T/K

η/μPa s

T/K

η/μPa s

Series 1a ρ = 0.491 mol m−3

Series 2a ρ = 0.492 mol m−3

Series 3 ρ = 0.964 mol m−3

300.03 312.13 326.39 339.64 353.45 367.81 382.27 396.39 410.85 424.81 440.63 454.18 469.08 483.10 498.13 512.65 528.20 548.82 569.42 598.97 326.65

297.48 312.88 324.89 339.68 352.70 366.55 380.92 395.10 413.43 423.43 439.91 452.13 466.39 482.06

7.640 8.281 8.832 9.517 10.083 10.654 11.191 11.683 12.275 12.569 13.076 13.432 13.874 14.315

b

b

297.75 314.33 325.05 339.81 353.22 366.73 381.83 394.81 409.90 423.90 438.58 452.45 467.24 481.68 496.25 510.84 527.61 547.35 568.71 598.02 324.71

7.814 8.254 8.837 9.426 10.037 10.656 11.230 11.744 12.239 12.679 13.170 13.573 14.022 14.433 14.875 15.289 15.766 16.395 17.008 17.920 8.904

7.364 7.997 8.457 9.137 9.762 10.371 10.989 11.477 11.990 12.443 12.898 13.304 13.740 14.161 14.587 15.003 15.489 16.078 16.709 17.614 8.503




299.73 312.98 326.93 340.63 354.68 368.89 383.19 397.03 411.88 425.92 440.48 454.57 469.27 483.75 499.12 513.18 527.49 549.63 569.93 602.21 326.14

297.35 311.38 324.96 338.21 352.65 366.57 381.98 394.60 412.24 423.50 438.39 452.15 467.39 481.00 496.14 510.27 525.37 547.36 568.26 596.93 324.93

299.60 313.58 327.45 341.73 354.37 368.51 383.43 397.62 411.59 425.81 440.30 454.20 469.23 483.53 498.26 513.43 528.48 548.99 569.33 599.27 331.99

7.281 7.700 8.228 8.811 9.449 10.103 10.739 11.310 11.874 12.359 12.840 13.277 13.717 14.143 14.588 14.995 15.415 16.058 16.679 17.683 8.244

7.218 7.635 8.120 8.655 9.292 9.931 10.628 11.158 11.853 12.246 12.745 13.180 13.646 14.045 14.489 14.894 15.327 15.971 16.589 17.443 8.138

7.259 7.640 8.075 8.607 9.157 9.790 10.463 11.068 11.666 12.198 12.705 13.166 13.633 14.062 14.501 14.951 15.395 16.004 16.625 17.555 8.282

T/K

η/μPa s

Series 4 ρ = 1.619 mol m−3 297.56 311.70 325.36 338.90 352.73 366.44 381.23 394.89 409.53 423.45 438.80 452.24 467.89 481.29 495.99 510.82 525.22 547.77 568.27 597.24 324.84

7.269 7.750 8.288 8.877 9.512 10.140 10.788 11.336 11.867 12.338 12.823 13.232 13.697 14.091 14.519 14.934 15.359 16.023 16.626 17.485 8.295

Series 8 ρ = 6.051 mol m−3 298.03 310.89 324.54 338.92 354.66 366.83 381.72 394.94 410.41 424.96 438.76 452.72 467.41 481.42 496.12 510.66 525.77 548.12 570.82 598.39 325.12

7.221 7.559 7.931 8.408 9.029 9.572 10.223 10.821 11.450 12.012 12.537 13.014 13.488 13.918 14.367 14.793 15.230 15.890 16.540 17.386 8.022

a Series of measurement was not included in further evaluation due to adsorption at low temperatures and thermal decomposition at high temperatures. b Measurement was stopped due to thermal decomposition.


318

Table 1. (continued) T/K

η/μPa s

T/K

η/μPa s

T/K

η/μPa s




297.32 311.25 325.12 338.56 353.12 366.87 382.12 395.06 410.09 424.20 438.79 452.62 467.55 481.62 496.65 511.06 529.88 548.73 574.26 598.48 324.91

325.32 340.10 354.60 367.87 385.66 396.45 410.31 424.81 440.02 453.13 467.74 482.13 496.78 512.17 526.35 549.04 569.03 597.52 324.77

327.13 338.34 353.87 367.09 381.30 395.37 409.99 424.42 438.63 452.80 467.38 481.67 496.87 511.64 526.71 548.06 570.70 599.27 325.09

7.196 7.552 7.926 8.310 8.842 9.384 10.051 10.617 11.278 11.830 12.410 12.907 13.400 13.859 14.325 14.767 15.302 15.877 16.598 17.356 7.994

7.952 8.344 8.781 9.288 9.980 10.494 11.090 11.700 12.290 12.795 13.321 13.807 14.269 14.754 15.175 15.863 16.455 17.321 7.996

7.975 8.269 8.717 9.151 9.714 10.313 10.940 11.550 12.141 12.679 13.211 13.708 14.209 14.686 15.158 15.810 16.499 17.356 7.972




355.26 367.58 381.48 395.75 412.46 424.26 438.69 453.59 467.63 482.16 497.35 512.26 527.55 550.01 568.83 597.79 409.99

353.38 367.55 381.28 396.20 410.95 425.35 439.12 453.16 467.17 481.93 497.33 511.41 526.06 549.27 568.96 597.52 411.34

367.82 382.45 395.77 410.69 424.61 439.77 453.56 467.61 482.13 496.71 511.65 527.63 551.39 570.96 598.70 409.84

8.734 9.109 9.630 10.222 10.907 11.436 12.049 12.631 13.139 13.675 14.200 14.681 15.170 15.878 16.449 17.318 10.860

8.675 9.089 9.616 10.205 10.834 11.436 12.028 12.586 13.120 13.635 14.168 14.635 15.081 15.858 16.433 17.298 10.872

9.080 9.555 10.077 10.710 11.294 11.919 12.483 13.043 13.584 14.097 14.610 15.121 15.879 16.474 17.343 10.704

T/K

η/μPa s


339.00 353.07 367.70 382.25 396.42 409.97 425.13 439.01 452.99 467.22 481.99 497.10 511.80 526.69 548.20 569.45 599.35 410.07

8.282 8.672 9.136 9.691 10.296 10.894 11.520 12.102 12.655 13.182 13.712 14.213 14.683 15.150 15.819 16.488 17.387 10.922

Series 16 ρ = 42.88 mol m−3 367.52 381.95 395.32 410.85 424.41 438.51 452.74 468.84 481.71 497.84 512.55 527.99 548.73 573.04 598.37 411.12

9.078 9.483 9.979 10.605 11.183 11.784 12.355 12.992 13.478 14.034 14.567 15.094 15.789 16.546 17.320 10.623

small so that the contributions of higher terms to CN are negligible. Therefore, the approximately known characteristics of the viscometer are used together with the working equation of the Newell theory to derive CN . For that purpose the viscosity and the dens-


319



Table 1. (continued) T/K

η/μPa s

T/K

η/μPa s



367.96 381.96 396.04 410.38 424.41 439.49 453.16 467.46 483.01 497.15 511.99 529.63 547.60 573.56 598.15 410.16

382.67 395.20 410.92 424.14 439.11 452.74 467.42 481.45 496.72 511.38 526.93 548.42 569.50 598.80 409.89

9.090 9.473 9.968 10.548 11.140 11.762 12.344 12.912 13.462 14.003 14.507 15.115 15.702 16.520 17.294 10.578

9.500 9.905 10.523 11.100 11.709 12.281 12.879 13.415 13.961 14.478 15.018 15.718 16.405 17.325 10.503

T/K

η/μPa s

Series 19 ρ = 61.16 mol m−3 381.66 396.17 412.44 424.56 439.48 453.19 468.03 482.29 496.76 511.19 527.71 547.66 571.21 600.37 410.96

9.474 9.899 10.505 11.026 11.639 12.212 12.802 13.352 13.903 14.414 14.971 15.654 16.421 17.442 10.475

ity of the gas used for the calibration are needed independently. Only one reference viscosity value at room temperature and at low density is enough to calibrate the apparatus, because the obtained Newell constant will not change with temperature due to the properties of quartz glass mentioned above. The viscometer was calibrated repeatedly with argon in the course of the prolonged work, in which only a very small change in the viscometer constant occurred. For that purpose, a theoretically calculated viscosity value in the limit of zero density, η(0) 298.15 , together with the temperature and density derivatives of the viscosity, (∂η/∂T)ρ and (∂η/∂ρ)T , was used for the calibration according to Ref. [6]: η(T, ρ) = η(0) 298.15 + (T − 298.15)(∂η/∂T)ρ + ρ(∂η/∂ρ)T , with

(1)

η(0) 298.15 = 22.552 ± 0.02 μPa s , (∂η/∂T)ρ = 0.0635 μPa s K−1 , (∂η/∂ρ)T = 11.099 nPa s m3 kg−1 . According to the sound error analysis in Ref. [6,11,12], the relative uncertainty of the measurements on simple gases is conservatively estimated to be ±0.15% at room temperature and up to ±0.20% at higher temperatures, whereas the reproducibility does not exceed ±0.1% in the measured temperature range. The results of the high-precision measurements on argon [6] and on nitrogen [11] have proven that the performance of the all-quartz oscillating-disk viscometer is well established. But it remains to be tested whether such low uncertainties can also be achieved in the case of substances like acetic acid, which are more difficult to handle.


320

321

2.3 Measurements and results Acetic acid of quality Suprapur was supplied by Merck. In a special glass apparatus the substance was purified by fractional crystallization and the final sample was characterized by a melting point of 16.52 ± 0.02 ◦ C showing the purity to be 99.9% at least. In the same apparatus the substance was dried by molecular sieves 4A, degassed by repeated freezing and evacuation and filled into small glass ampules with long necks. The ampules were closed by fusing the necks with a flame simultaneously using a vacuum degassing equipment. The mass of substance was obtained from weighing the empty and filled ampules. Then the viscometer was filled by sublimation of the samples from the broken up ampules using a further glass apparatus. Finally, the viscometer was closed by fusing its filling pipe. The density of the series of measurements was obtained from the mass of the sample and the volume of the viscometer. Due to the different manipulations during the filling process, the uncertainty of the density determination is ±2%. Nevertheless, the allocation error of the density determination does practically not increase the total uncertainty in the viscosity (see Ref. [6]). Nineteen isochoric series of measurements were carried out on acetic acid vapor in a maximum temperature range from 298 to 598 K for densities between 0.5 and 61 mol m−3 . In the case of the series with the lowest densities, a special procedure was applied, since the amount of substance needed for a direct filling of the viscometer was too small to be handled safely in the process of closing the ampules by fusing their necks. Hence the viscometer was connected with auxiliary volumes, into which a certain percentage of the sublimated sample could be evaporated to decrease the density. The density used for these series of measurements resulted from the ratio of the viscometer and auxiliary volumes. For a number of points at low temperatures a certain amount of substance existed as liquid so that the measurements were performed in the saturated vapor assuming that the condensation did occur on the walls of the viscometer vessel and not on the oscillating-disk system. Furthermore, the measurements of some of the series concerning higher densities were started at a higher temperature so that most of the substance existed as vapor. A check-measurement at a lower temperature following that at the highest one was performed in each series in order to test for thermal decomposition. The first seven series were carried out at such low densities that the results should be influenced by the slip effect, which was investigated on noble gases and nitrogen for this viscometer type by Vogel et al. [13]. The correction for the slip effect increases the results for the two series at the lowest density by at most 0.45% and that for the next series by only 0.2% or less. The experimental results of all nineteen series of measurements including the correction for slip are summarized in Table 1. The original isochoric data were adjusted to isothermal values η(Tnom ) by means of a first-order Taylor expansion in terms of temperature: ∂η η(Tnom ) = η(Texp ) + ΔT + RN , (2) ∂T ρ (3) ΔT = Tnom − Texp . The nominal temperature Tnom was obtained by averaging the corresponding temperatures Texp of all series of measurements and choosing the next half a degree (in centi-




◦

Fig. 2. The viscosity η of acetic acid vapor as a function of the molar density ρ – high isotherms. , 424.65 K; , 439.15 K; , 453.15 K; , 467.65 K; , 482.15 K; , 497.15 K; , 511.65 K; , 527.15 K; , 548.65 K; , 570.15 K; ⊕, 598.65 K.

•

grade). The remainder RN in Eq. (2) was proven to be negligible in comparison with the experimental uncertainty. The temperature derivative of viscosity (∂η/∂T)ρ was calculated using Eq. (4) with the respective coefficients which were derived from a fit to the original isochoric data: 5 Bi T + C , TR = η(T) = S exp A ln TR + , S = 10.0 μPa s . (4) i TR 298.15 K i=1 The isotherms for acetic acid vapor are illustrated as a function of the filled molar density in Fig. 1 for lower temperatures and in Fig. 2 for higher temperatures. Both figures make evident that thermal decomposition occurs at high temperatures. Figure 1 clarifies that the viscosity values of the check-measurements at 326.15 K concerning the isochoric series at lower densities are increased by about 0.6%, whereas the values remeasured at 410.65 K for the series at higher densities are increased by approximately 0.3%. Figure 2 shows that the effect of thermal decomposition at high temperatures is distinctly enhanced at low densities. The results of series 1 and 2 are clearly increased. This applies primarily for series 2, for which the measurements were stopped at 490 K due to an excessive increase of the viscosity (see also Table 1). The onset of the thermal decomposition remains undetected from the viscosity measurements. But the earlier pρT measurements of our group [4] on acetic acid vapor from 410 K to 574 K at densities between 20 and 51 mol m−3 had not indicated thermal decomposition so that only the highest isotherm may be influenced at higher densities. As already mentioned, some of the measured points were performed in the saturated vapor at low temperatures for densities larger than the density of the saturated vapor.


322

Table 2. Viscosities of the saturated vapor and of the monomers and dimers.

a

Temperature

Saturated density

T/K

ρ/mol m−3

298.15 312.15 326.15 339.15 354.15 367.65 382.15 395.65 410.65 424.65 439.15 453.15 467.15 482.15 497.15 511.65 527.15 548.65 570.15 598.65

1.569 3.199 5.788 9.812 16.50 25.77 40.22 57.67

Viscosity saturated vapor monomer measured from fit η/μPa s η/μPa s η1 /μPa s 7.239 7.588 7.966 8.305 8.708 9.083 9.485 9.881

7.620 7.972 8.277 8.712 9.075 9.441 9.822

dimer η2 /μPa s

9.64 9.94 10.25 10.49 10.85 11.27 11.64 12.10 12.34 12.80 13.20 13.60 14.02 14.44 14.86 15.30 15.92 16.53

7.22 7.46 7.67 7.98 8.22 8.45 8.68 8.84 9.17 9.35 9.58 9.82 9.99 10.20 10.40 10.50 10.87 11.71

a

a

Values are falsified due to thermal decomposition and low x2 range.

The saturated densities at these temperatures were calculated by means of the information given by Bich et al. [4] (see below). The viscosity values at densities larger than the saturated one are nearly constant and correspond virtually to the saturated vapor. Hence these viscosity values were averaged and assigned to the saturated vapor density. The averaged viscosity data are given in column 3 of Table 2 as measured values for the saturated vapor together with the corresponding saturated densities in column 2. The saturated vapor data are also illustrated in Fig. 1. In contrast to simple gases, there is no indication concerning the value which is approximated in the so-called limit of zero density. The viscosity of a moderately dense, simple gas or vapor can be analyzed with a density series for the viscosity in which only a linear contribution is considered: η(T, ρ) = η(0) (T) + η(1) (T)ρ = η(0) (T)[1 + Bη (T)ρ] .

(5)

Here, η(0) is the viscosity coefficient in the limit of zero density, η(1) is the initial density viscosity coefficient, and Bη is the second viscosity virial coefficient. In practice, measurements at very low densities or even in the limit of zero density are not feasible so that η(0) has to be determined by extrapolating isotherms with a sufficient number of experimental points at moderately low densities to the limit of zero density. The slope of the isotherms corresponds to the initial density viscosity coefficient η(1) . But Fig. 1


323



makes evident that the isotherms at low temperatures are not characterized by a linear density dependence. The reason for the pronounced curvature of the isotherms is the strong tendency to dimerization of acetic acid vapor. In addition, Fig. 1 may reveal that the viscosity values at very low densities and at the lowest temperatures could be influenced by adsorption. In this case the actual vapor density is decreased compared with the filled one, so that the viscosity is additionally raised. Nevertheless, a linear negative slope cannot be discovered so that the viscosity coefficient in the limit of zero density is not accessible this way. The measured results are characterized by a somewhat larger reproducibility of ±0.2% compared with that for simple gases. This becomes obvious when considering the results of the two series at about 25.8 mol m−3 . Consequently, the uncertainty of the measurements has also to be increased. It is estimated to be ±0.5% for the series at low densities due to adsorption effects at low temperatures and to thermal decomposition at high temperatures. For the series at higher densities the uncertainty is assumed to be ±0.3% at lower temperatures and ±0.4% at higher ones.

3. Theoretically based analysis 3.1 Theoretical background and approximations The first two coefficients of the viscosity density expansion (Eq. 5) have a sound theoretical basis and can usually be evaluated separately in appropriate manner. The transport properties of molecular gases in the limit of zero density, here η(0) , are related to generalized cross sections by means of the formal kinetic theory [14– 16]. These cross sections are determined by the dynamics of the binary collisions in the gas and are connected with the intermolecular potential energy hypersurface that describes the molecular interactions. The intermolecular potential is typically derived assuming the zero-point vibrationally averaged configuration so that the collision dynamics requires only to treat the molecules as rigid rotors. Whereas the thermal conductivity demands special corrections to take the effects of the vibrational degrees of freedom [17] into consideration, the findings for the viscosity using the rigid-rotor assumption are consistent with the experimental data. This has been demonstrated for linear molecules like nitrogen [18] and carbon dioxide [19], but also for spherical-top and asymmetric-top molecules, such as methane [20], water [21], and hydrogen sulfide [22]. In the case of such a large polyatomic molecule like acetic acid, the described procedure can hardly be applied until now so that the zero-density viscosity coefficient would have to be analyzed using the kinetic theory of dilute monatomic gases of Chapman and Enskog [23,24]. The second viscosity virial coefficient Bη and consequently the initial density viscosity coefficient η(1) of Eq. (5) can be treated in the case of a pure, simple gas with the Rainwater–Friend theory [25–28]. This theory is based on the Lennard-Jones (12–6) potential for the interactions in the moderately dense gas. As already demonstrated in Sect. 2.3, our measurements, at least at low temperatures, are characterized by a pronounced dimerization of the acetic acid molecules. That means the measured substance does not correspond to a pure fluid, but represents a gaseous mixture. Hence a theory for the density dependence of the viscosity of a moderately dense gaseous mixture


324

325

would be needed for the evaluation. Unfortunately, the Rainwater–Friend theory has not been extended to mixtures. Therefore, modifications of the Enskog theory for hardsphere mixtures would have to be used for the desription of the initial-density viscosity coefficient of the real dense vapor mixture under consideration [29–32]. In summary, Fig. 1 illustrates that the isotherms at low temperatures cannot be evaluated based on a linear density dependence of the viscosity of a pure gas. In fact, the experimental results refer to the viscosity of a vapor mixture at low densities. But it is a problem to decide whether the density range of the measurements is actually affected by the initial density dependence. In a first approximation, the gaseous density region of the mixture under consideration is assumed to be not large enough to adequately derive zero-density and initial density viscosity coefficients for monomers and dimers of acetic acid vapor. With regard to the further evaluation, the volumetric behavior for a fluid characterized by a dimerization reaction had to be taken into account. For that purpose the equation of state of Bich et al. [4] for a reactive fluid, here acetic acid vapor, was used. In this way viscosity values as a function of the mole fraction of monomers or dimers could be obtained. The resulting isotherms should finally be evaluated using an equation proposed by Wilke [33] for the viscosity of a gas mixture in which the viscosity for the interaction between monomers and dimers is not explicitly considered: 2 xi η(0) i (T) , η (T, x) = 2 j=1 x j φij i=1 (0)

φij =

(0) 1/2 [1 + (η(0) (M j /Mi )1/4 ]2 i /η j ) . [8(1 + Mi /M j )]1/2

(6) (7)

(0) Here η(0) , η(0) 1 , and η2 are the viscosities in the limit of zero density for the mixture and for monomers and dimers. x1 and x2 are the mole fractions of monomers and dimers, M1 and M2 are their molar masses. The approximate zero-density viscosity coefficients of monomers and of dimers should be determined this way.

3.2 Volumetric behavior of acetic acid vapor The volumetric behavior of a moderately dense gas consisting of molecules with weak interactions may be illustrated by the usual virial equation of state: p = ρ + B(T)ρ 2 + C(T)ρ 3 + D(T)ρ 4 · · · . RT

(8)

Here B, C, and D are the second, third, and fourth pressure virial coefficients. In the case of strong intermolecular interactions, this equation, considering monomers as the only component, becomes alternately divergent. The formation of dimers in acetic acid vapor may be described as a reaction equilibrium: 2A A2 .

(9)




Consequently, Eq. (8) can be replaced by the convergent virial equation of a binary mixture composed of monomers and dimers: 2 2 2 2 p ρi + Bij ρi ρ j + Cijk ρi ρ j ρk + Dijkl ρi ρ j ρk ρl + · · · . = RT i=1 i, j=1 i, j,k=1 i, j,k,l=1

(10)

Bij , Cijk , and Dijkl represent the second, third, and fourth pressure virial coefficients for the mixtures of monomers and dimers, whereas ρi are the molar densities of the components. The dimerization constant K 2 can be expressed by the activities z i of monomers and dimers: z2 (11) K2 = 2 , z1 2 2 3 2 x j Bij + ρmix x j xk Cijk z i = ρi exp 2ρmix 2 j=1 j,k=1 2 4 3 + ρmix x j xk xl Dijkl + · · · , ρmix = ρ1 + ρ2 , ρi = xi ρmix . (12) 3 j,k,l=1 The balance between the experimentally given molar density ρ and the molar densities of monomers and dimers in the mixture is: ρ = ρ1 + 2 ρ2 .

(13)

The preceding relationships (Eq. 10 to Eq. 13) represent the volumetric behavior for molecules with strong interactions, for which not only the dimerization constant K 2 but also the interaction pressure virial coefficients have to be taken into account. Bich et al. [4] derived values for the dimerization constant and for some relevant virial coefficients by a fit of these equations to isotherms experimentally determined by themselves or by other groups. In the present paper the coefficients of polynomials in 1/T were deduced by fitting to the values of the virial coefficients B12 , B22 , C222 , and D2222 as well as of the dimerization constant K 2 reported in Tables 4 and 5 of the paper by Bich et al. These polynomials were used, with the correlation for the dimerization constant extrapolated up to 570 K, to determine the molar densities and the mole fractions of monomers and dimers for the filled experimental molar densities at the nominal temperatures of the viscosity isotherms.

3.3 Viscosities of monomers and dimers in acetic acid vapor The viscosity values for acetic acid vapor are shown as a function of the mole fraction of monomers x1 in Fig. 3 for lower temperatures and in Fig. 4 for higher temperatures. It is to note that the highest values of x1 correspond to the lowest molar densities ρ. In Fig. 3, viscosity values at densities larger than the saturated ones are omitted. Figure 3 makes evident that the viscosity is much less dependent on the mole fraction than on the experimental molar density (see Fig. 1). Hence the viscosity is adequately represented as that of a vapor mixture of monomers and dimers. In addition, both figures


326

Fig. 3. The viscosity η of acetic acid vapor as a function of the mole fraction of monomers x1 – low isotherms. , 298.15 K; , 312.15 K; , 326.15 K; , 339.15 K; , 354.15 K; , 367.65 K; , 382.15 K; , 395.65 K; , 410.65 K; , 326.15 K (re-measurements); , 410.65 K (re-measurements); , saturated vapor of isotherms; ——–, saturated vapor line.

•

◦

Fig. 4. The viscosity η of acetic acid vapor as a function of the mole fraction of monomers x1 – high isotherms. , 424.65 K; , 439.15 K; , 453.15 K; , 467.65 K; , 482.15 K; , 497.15 K; , 511.65 K;

, 527.15 K; , 548.65 K; , 570.15 K.

◦

•

enable secured statements concerning adsorption effects at low temperatures and thermal decomposition at high temperatures. Although the viscosity values of the series of measurements 1 and 2 at the highest mole fractions x1 for the three isotherms between


327



Fig. 5. The viscosity η of monomers and of dimers in acetic acid vapor as a function of temperature. , Monomers, present paper; , dimers, present paper; ⊕, monomers, data scanned from Fig. 2 of Timrot et al. [5]; , monomers, determined from experimental data of Timrot et al. [5]; , dimers, data scanned from Fig. 2 of Timrot et al. [5]; , dimers, determined from experimental data of Timrot et al. [5].

◦

354 K and 382 K appear to be in compliance with the other series of measurements at lower x1 values, Fig. 3 reveals that at lower temperatures the viscosity values of the series 1 and 2 are excessively increased, in particular at 298 K. The reason for that is the adsorption of acetic acid molecules on quartz-glass, which decreases the molar density in the vapor compared with the filled density and increases simultaneously the mole fraction of the monomers and the viscosity (see also Sect. 2.3). Furthermore, the viscosity values of series 1 and 2 at 410.65 K seem to be again increased compared with the values of the other series. This tendency is continuing heightened for the isotherms at higher temperatures in Fig. 4 in consequence of the thermal decomposition, which is proportionately increased at low molar density and at large values of x1 . In conclusion, the viscosity values of the complete series of measurements 1 and 2 and those of the series 3 to 7 (ρ < 5 mol m−3 ) above 420 K were excluded from the further evaluation of the isotherms. Moreover, the viscosity data of the isotherm 598 K are definitely influenced by thermal decomposition. Finally, the viscosities of the monomers η1 and of the dimers η2 are derived using Eq. (6). The results are given in column 5 and column 6 of Table 2 and in Fig. 5. It is to note that after exclusion of series 1 and 2 the isotherm 298 K could not yield any result. Figure 5 makes evident that the uncertainty of the viscosity values for the dimers is increased at higher temperatures, since the mole fraction of the dimers x2 becomes small (x2 < 0.1 above 500 K) and less suitable for a reasonable fit of Eq. (6) to the values of the isotherms. In addition, viscosity values at the saturated density were determined for the lowtemperature isotherms by extrapolation of the fitted curves. They are given in column 4


328

329

of Table 2 and can be compared with the values, which were obtained by averaging the results measured in the saturated vapor and listed in column 3.

3.4 Comparison with data from the literature Timrot et al. [5] investigated several carboxylic acid vapors with an all-quartz oscillating-disk viscometer using a simplified measuring theory for the evaluation of the relative viscosity measurements. They estimated the uncertainty of their results to be ±(0.5 − 0.7)%. Acetic acid vapor was measured in the temperature range 307 K to 422 K at pressures between 0.017 atm and at most 1.23 atm. The resulting eight isotherms showed a trend similar to the curves in Fig. 1 of the present paper. In addition, Timrot et al. established isotherms as a function of the mole fraction of monomers, in which they calculated x1 according to an ideal association model by means of older values for the dimerization constant of Pimentel and McClellan [34]. Finally, they derived viscosity values for monomers and dimers of acetic acid but communicated them only at seven temperatures, partly different from those of their isotherms, as an insert of Fig. 2 of their paper. After scanning the insert, the η1 and η2 values were determined. They are illustrated in Fig. 5 in comparison with the η1 and η2 values of the present paper. The figure makes evident that, apart from the η2 values at higher temperatures, the agreement in the overlapping temperature range is quite reasonable, although Timrot et al. used in their evaluation only an ideal model for the dimerization in acetic acid vapor. Hence, we repeated the evaluation of the viscosity isotherms of Timrot et al. (Table 2 of their paper) according to the procedure of the present paper, in which not only the dimerization but also additional interactions between monomers and dimers are considered (see Sect. 3.2). The results are also demonstrated in Fig. 5. The agreement becomes inferior for the monomers, in particular at lower temperatures, and somewhat better for the dimers at higher teperatures. The reason for that remains unclear. It could be possible that a compensation of the experimental uncertainty and of the simpler dimerization model occurs.

4. Conclusions An all-quartz oscillating-disk viscometer was applied for relative measurements on acetic acid vapor. Nineteen isochoric series were performed from 298 to 598 K and for densities between 0.5 and 61 mol m−3 . Isotherms recalculated from the original isochoric data show with decreasing density a strong curvature with negative slope, particularly at low temperatures. The uncertainty of the measurements is estimated to be ±0.5% for the series at low densities due to adsorption effects at low temperatures and to thermal decomposition at high temperatures. For the series at higher densities the uncertainty is assumed to be ±0.3% at lower temperatures and ±0.4% at higher ones. The onset of the thermal decomposition could not be detected. The isotherms, in particular those at low temperatures, cannot be evaluated based on a linear density dependence of the viscosity of a pure gas. They correspond to the viscosity of a vapor mixture at low densities. The further evaluation considered the strongly temperature dependent dimerization in acetic acid vapor. For that purpose the




volumetric behavior using the equation of state by Bich et al. [4], which includes additional interactions between monomers and dimers, was taken into account. In this way viscosity values as a function of the mole fraction of monomers or dimers were obtained. These isotherms, showing a smaller curvature in terms of the mole fraction of the monomers, were evaluated using an equation proposed by Wilke [33] for the viscosity of a gas mixture. Viscosity values for monomers and dimers, restricted to 570 K due to thermal decomposition and low mole fractions of the dimers at high temperatures, were finally deduced. They were compared with viscosity values of Timrot et al. [5] reported for temperatures up to 422 K only. The agreement is reasonable in view of the different evaluation models used. Viscosity data were measured at densities larger than the saturated one at low temperatures. These values, nearly constant and corresponding virtually to the saturated vapor, were averaged and assigned to the saturated vapor density. Viscosity values for the saturated vapor were also derived by extrapolation of the fitted isotherms and compared with the values measured in the saturated vapor. The values from both approaches are consistent within ±0.3% on average.

Acknowledgement We greatly thank the glassblower Mr. Matthias Auer for his help with the filling of the viscometer and the laboratory assistants Mrs. Edeltraud Hoffmann and Mrs. Anika Rose for their help with the measurements.

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

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