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Supporting Information

The viscosity of dilute water vapor revisited: New

reference values from experiment and theory for

temperatures between (250 and 2500) K

Robert Hellmann and Eckhard Vogel

Institut fur Chemie, Universitat Rostock, 18059 Rostock, Germany

E-mail: robert.hellmann@uni-rostock.de

To whom correspondence should be addressed

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http://localhost/var/www/apps/conversion/tmp/scratch_3/robert.hellmann@uni-rostock.dehttp://localhost/var/www/apps/conversion/tmp/scratch_3/robert.hellmann@uni-rostock.de

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The re-evaluated results of the fourteen measured isochores of Teske et al. S1 are sum-

marized in TableS1. The isochoric viscosity values had to be extrapolated to the limit of

zero density to enable a comparison with the theoretically computed 0 values and with

other viscosity correlations. In addition, the extrapolated0 values are needed to judge the

reliability of the experimental data measured particularly at the highest temperatures. The

procedure to obtain these values is presented below.

First, the re-evaluated data points, which were not recorded at exactly the same temper-

atures for the fourteen isochores, were converted into isothermal values using a first-order

Taylor series in temperature,

(Tiso) =(Texp) +

T

(Tiso Texp) + RN. (S1)

The temperature of the isotherms,Tiso, corresponds to the mean of the experimental temper-

aturesTexpdetected at the same approximate adjustment of the thermostat for the individual

measuring series. The temperature coefficient (/T) required in eqS1was derived from

the following relationship:

(T) =Sexp

A ln (TR) +

B

TR+

C

T2R+

D

T3R+ E

, (S2)

where TR = T /(298.15 K) and S = 10 Pa s. The coefficients A, B, C, D, and E were

determined in a fit to the experimental data of each isochoric series. The values of the two

re-measured experimental points of each isochore were not included in the fit. Thus, the

influence of any permanent alteration of the samples or of the suspension system of the

viscometer as a result of the measurements at high temperatures should have been avoided.

It was verified that the remainder RNin eqS1is insignificant compared with the experimental

uncertainty.

The quasi-experimental viscosity values of each isotherm were correlated as a function of

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Table S1: Re-Evaluated Viscosity Data for Water Vapor of Teske et al.S1 (Series 1 to 10)and of Four Additional Series (Series A to D)a

T/K /Pa s T/K /Pa s T/K /Pa s T/K /Pa s T/K /Pa s

Series A Series 1 Series B Series 2 Series 3= 0.018 kg m3 = 0.019 kg m3 = 0.023 kg m3 = 0.025 kg m3 = 0.037 kg m3

300.35 9.746 298.30 9.678 297.76 9.649 298.02 9.654 298.08 9.661311.52 10.132 310.80 10.107 311.12 10.108 311.77 10.129 310.87 10.104327.19 10.671 331.18 10.809 325.62 10.628 324.54 10.574 325.96 10.628339.05 11.090 338.83 11.081 338.47 11.067 338.47 11.060 338.71 11.072352.69 11.576 352.75 11.577 352.84 11.575 352.91 11.575 353.60 11.602366.97 12.079 367.84 12.130 367.44 12.101 366.37 12.060 368.22 12.138384.22 12.726 380.80 12.599 382.98 12.673 380.42 12.574 382.14 12.649394.87 13.127 394.39 13.106 395.16 13.118 394.72 13.100 396.61 13.194410.22 13.699 408.87 13.658 409.25 13.649 409.76 13.680 411.18 13.758423.87 14.231 423.26 14.212 423.35 14.191 422.75 14.184 423.80 14.254438.75 14.816 438.77 14.819 438.48 14.782 440.41 14.886 438.48 14.837468.21 16.009 466.81 15.960 468.74 15.983 469.77 16.069 467.91 16.026497.08 17.229 498.89 17.303 496.27 17.105 495.87 17.139 498.46 17.360

b 526.03 18.417 b b 528.41 18.672299.38 9.722 298.20 9.681 301.55 9.792 302.04 9.800 299.75 9.728380.86 12.599 384.49 12.744 381.03 12.605 380.92 12.596 384.51 12.747

Series 4 Series 5 Series 6 Series C Series 7= 0.040 kg m3 = 0.067 kg m3 = 0.079 kg m3 = 0.092 kg m3 = 0.108 kg m3

297.39 9.644 297.88 9.633 299.06 9.688 298.16 9.654 298.38 9.665311.45 10.124 310.96 10.090 311.30 10.105 311.13 10.098 311.39 10.106325.00 10.593 324.45 10.568 331.47 10.813 325.16 10.587 325.42 10.593339.39 11.099 341.28 11.145 338.79 11.055 339.49 11.091 340.17 11.107352.77 11.577 352.17 11.536 352.79 11.563 352.03 11.536 353.16 11.571367.03 12.093 367.10 12.073 366.96 12.079 365.89 12.040 366.77 12.068381.09 12.605 380.59 12.563 381.09 12.600 380.14 12.565 381.33 12.601395.38 13.131 394.29 13.072 395.87 13.160 394.56 13.105 395.00 13.115409.46 13.663 408.32 13.606 409.39 13.674 408.37 13.630 409.37 13.660423.70 14.226 422.72 14.169 423.44 14.220 422.63 14.185 423.62 14.216438.99 14.825 438.04 14.775 438.71 14.826 437.65 14.786 438.05 14.801467.51 15.938 468.25 15.999 467.24 15.994 466.44 15.959 467.69 16.022497.04 17.168 495.48 17.141 498.62 17.347 496.94 17.227 497.08 17.272

b b 527.94 18.652 b b

301.33 9.800 299.66 9.709 297.66 9.649 297.33 9.630 299.88 9.717382.12 12.646 380.12 12.555 381.10 12.604 380.24 12.573 381.62 12.627

Series D Series 8 Series 9 Series 10= 0.129 kg m3 = 0.165 kg m3 = 0.197 kg m3 = 0.235 kg m3

298.69 9.676 298.17 9.672 298.57 9.663 300.07 9.704310.44 10.076 311.24 10.116 313.01 10.145 312.25 10.123326.81 10.640 324.54 10.573 325.15 10.570 324.74 10.557338.27 11.045 339.97 11.110 338.02 11.008 338.38 11.024352.79 11.558 352.05 11.512 352.03 11.502 352.37 11.518366.46 12.052 366.48 12.041 366.29 12.020 365.86 12.003380.61 12.568 380.60 12.553 380.43 12.542 381.70 12.592394.75 13.096 393.95 13.058 394.43 13.074 394.48 13.078408.69 13.654 408.30 13.621 408.76 13.634 408.34 13.617423.17 14.208 422.69 14.184 422.94 14.193 422.56 14.182437.93 14.791 437.71 14.795 437.84 14.805 437.27 14.793

466.96 15.983 465.99 15.958 466.15 15.974 466.87 16.031497.01 17.273 494.69 17.211 496.09 17.277 495.53 17.297

b b b b

298.99 9.697 299.96 9.714 299.72 9.697 297.72 9.647380.91 12.596 380.21 12.562 380.43 12.567 383.10 12.686

a Standard uncertainties u are u(T) = 0.05 K for T < 400 K, u(T) = 0.1 K for T > 400 K, and u() =

0.001 kg m3. The relative combined expanded (k= 2) uncertainty Uc,r is Uc,r() = 0.002 forT 450 K. The samples consisted of

doubly distilled and degassed water.b No data point at the highest temperature.

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density by means of a series expansion truncated at first order,

(T, ) =0(T) + 1(T). (S3)

As discussed in section 2.1 of the main article, the reliability of the two 0 values at room

temperature resulting from the extrapolation to the limit of zero density is problematic due

to the reduced number of data points in the unsaturated vapor. Even small uncertainties of

the experimental data could have resulted in unreliable 0 values, since the available data

points correspond to nearly the same density, 0.020 kg m3. The comparatively large

values of the initial-density viscosity coefficient 1 for both isotherms at room temperature

(see Table 1 in the main article) indicate that this could be the case. Therefore, the 0

values at room temperature were also corrected to the limit of zero density applying the

Rainwater-Friend theoryS2,S3 for the initial density dependence of the transport properties.

The initial density dependence of the viscosity is characterized by the second viscosity

virial coefficient,

B(T) =M1(T)

0(T) , (S4)

whereMis the molar mass. Tables of the reduced second viscosity virial coefficient B as a

function of the reduced temperature T were reported by Bich and Vogel,S4,S5 while Vogel

et al.S6 proposed an improved correlation for B(T) in the range 0.3 T 100,

B(T) =

B(T)

NA3 =

6i=0

biTi/4 + b7T

5/2 + b8T11/2, (S5)

where NA

is Avogadros constant and T = kB

T /. The coefficients biare given in ref S6,

and the scaling factors for water were determined by Teske et al. S1 in a fit of eq S5 to

experimentalB values to be = 0.48873 nm and /kB= 459.85 K.

To calculate the viscosity in the limit of zero density from an experimental (T, ) value

at a moderately low density , the following relation, which results from eqs S3S5, was

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used:

0(T) = (T, )

1 + NA3B(T)/M

. (S6)

The 0 values of the isotherms at about 299 K in Table 1 of the main article were obtained

by averaging the results obtained for the three data points at densities < s.

Finally, viscosity values for the saturated vapor were deduced for the low-temperature

isotherms. The densities at saturation, s, were computed using the equation of state by

Wagner and Pru, S7 while the respective viscosityswas obtained by taking the mean of the

viscosity values for densities > sof the considered isotherm. The results for the saturated

vapor at low temperatures are given in Table S2.

Table S2: Viscosity s at the Saturated Vapor Density s Derived from the Re-EvaluatedQuasi-Experimental Isotherms of the Measurements of Teske et al. S1

T/K na s/kg m3 (s s)/Pa s

298.49 11 0.023520 9.668 0.003311.38 8 0.046856 10.104 0.003326.23 5 0.095843 10.617 0.004339.09 2 0.16793 11.047 0.002299.51b 11 0.024902 9.709 0.004

a Number of quasi-experimental points used for averaging the viscosity at densities > s.b Re-measurements at lower temperature after the highest temperature had been attained.

References

(S1) Teske, V.; Vogel, E.; Bich, E. Viscosity measurements on water vapor and their evalu-

ation. J. Chem. Eng. Data 2005, 50, 20822087.

(S2) Friend, D. G.; Rainwater, J. C. Transport properties of a moderately dense gas.Chem.

Phys. Lett. 1984, 107, 590594.

(S3) Rainwater, J. C.; Friend, D. G. Second viscosity and thermal-conductivity virial co-

efficients of gases: Extension to low reduced temperature. Phys. Rev. A 1987, 36,

40624066.

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(S4) Bich, E.; Vogel, E. The initial density dependence of transport properties: Noble gases.

Int. J. Thermophys. 1991, 12, 2742.

(S5) Bich, E.; Vogel, E. InTransport Properties of Fluids: Their Correlation, Prediction and

Estimation; Millat, J., Dymond, J., de Castro, C. A. N., Eds.; Cambridge University

Press: Cambridge, 1996; Chapter 5.2, pp 7282.

(S6) Vogel, E.; Kuchenmeister, C.; Bich, E.; Laesecke, A. Reference correlation of the vis-

cosity of propane. J. Phys. Chem. Ref. Data 1998, 27, 947970.

(S7) Wagner, W.; Pru, A. The IAPWS formulation 1995 for the thermodynamic properties

of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data2002, 31, 387535.

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