1 Introduction

The group of Professor Span at Ruhr University Bochum, Germany, has reported about viscosity measurements on gaseous mixtures including the respective pure gases. These measurements concerning the binary systems carbon dioxide–nitrogen [1] and ethane–methane [2] were performed up to 2 MPa in the temperature range from 253.15 K to 473.15 K using a rotating-cylinder viscometer and determining the decay rate of the cylinder [3].

The experimental viscosity data have already been compared by the authors of Refs. [1,2,3] with ab initio theoretically computed viscosity values. This concerns the theoretical values which were calculated by one of us (R. H.), alone or together with different colleagues, for the pure gases nitrogen [4], carbon dioxide [5], methane [6], and ethane [7] as well as for the binary mixtures carbon dioxide–nitrogen [8] and ethane–methane [9]. In addition, correlations for the viscosity values in the limit of zero density were proposed by Laesecke and Muzny for carbon dioxide [10] and methane [11, 12]. These correlations were generated by means of symbolic regression using the theoretically calculated viscosity values of Hellmann for these gases.

However, these mixture data were not employed to determine values for the binary diffusion coefficient. In the past, this was frequently done using experimental viscosity data of binary mixtures, although, as we know today, not in the right manner. The problem is that a ratio of two effective cross-sections for viscosity and diffusion, \(A_{12}^*\), is additionally needed in a twofold way. At first, \(A_{12}^*\) is required to deduce the interaction viscosity in the limit of zero density, \(\eta ^{(0)}_{12}\). For this step, it is not significant whether the ratio is computed using a spherical or a nonspherical potential for the interaction of the molecules. But for the second step, to compute the diffusion coefficient or, more precisely, the product of molar density and diffusion coefficient in the limit of zero density, \((\rho D_{12})^{(0)}\), it is highly significant what this ratio \(A_{12}^*\) is calculated for. In the case of a nonspherical potential, \(A_{12}^*\) and also \((\rho D_{12})^{(0)}\) are several percent higher. This is in agreement with theoretically computed values of the diffusion coefficients for dilute gas mixtures as can be seen in Ref. [13] for the carbon dioxide–ethane binary gaseous mixture applying the diffusion coefficients of Abe et al. [14] derived from viscosity measurements on this mixture. Since the nonspherical potentials for both intermolecular interactions are available by the work of Hellmann et al. [8, 9], it is reasonable to also prove this for the binary gaseous mixtures under discussion.

In a recent publication of one of us (E. V.) [15], the results of viscosity measurements on eight gaseous and vapor mixtures were re-evaluated and particularly analyzed to determine the interaction viscosity, \(\eta _{12}^{(0)}\), and subsequently to derive the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), both in the limit of zero density. Three procedures were used in Ref. [15] to deduce \(\eta _{12}^{(0)}\) applying the information on the measured viscosities for the eight gaseous and vapor mixtures. The first procedure could be used for the gaseous mixtures carbon dioxide–ethane [16] and propane–isobutane [17] as well as for the vapor mixture methanol–triethylamine [18]. In this procedure, density series for only one unique mole fraction are required for \(\eta _\textrm{mix}^{(0)}(T)\) and \(\eta _\textrm{mix}^{(1)}(T)\) to be derived. For the vapor mixture methanol–triethylamine, viscosity values at a unique mole fraction were especially created from five of six isochoric series of measurements at nearly the same mole fraction. The values of \(\eta _\textrm{mix}^{(0)}(T)\) were employed together with newly determined values \(\eta ^{(0)}_i\) (\(i=1,2\)) of the pure gases at the temperatures of quasi-isotherms for the mixtures. The root of a quadratic relationship resulting from the Chapman–Enskog theory of dilute gases in its first-order approximation [19] yielded the required \(\eta _{12}^{(0)}\) values. \(A_{12}^*\) appearing in this relationship was also needed. However, a nonspherical potential was only available for the gaseous mixture carbon dioxide–ethane, so that \(A_{12}^*\) had to be computed for the other two mixtures using the effective spherical potential of the interaction of the molecules. But the nonspherical potential for the interaction of the molecules is to be preferred.

In addition, a further scheme to determine \(\eta _{12}^{(0)}\) could be applied using a modification of the Enskog theory of hard spheres based on the initial density viscosity coefficients for the mixtures and for the pure gases, \(\eta _\textrm{mix}^{(1)}(T)\) and \(\eta _i^{(1)}(T)\). This procedure implies that the uncertainty of the initial density viscosity coefficients is very low. In this procedure, a quadratic relationship is also generated, whose root provides the interaction viscosity in the limit of zero density. However, in Ref. [15], the uncertainties of \(\eta ^{(1)}_i\) of the pure gases and vapors as well as of \(\eta _\textrm{mix}^{(1)}(T)\) for the particular mixtures were too large, so that reasonable results for \(\eta _{12}^{(0)}(T)\) could not be deduced. In this paper, we will try to derive \(\eta _{12}^{(0)}(T)\) by means of \(\eta _\textrm{mix}^{(1)}(T)\) and \(\eta _i^{(1)}(T)\) assuming that their uncertainties are lower than those of Ref. [15].

The second and third procedures used in Ref. [15] to determine experimentally based \(\eta _{12}^{(0)}(T)\) values for the six binary vapor mixtures methanol with triethylamine (all three procedures were applied to this system), benzene, and cyclohexane as well as benzene with toluene, p-xylene, and phenol shall not be discussed in detail here.

The results of the measurements at Ruhr University Bochum are suitable for applying at least the first procedure to deduce \(\eta _{12}^{(0)}(T)\) and \((\rho D_{12})^{(0)}\) values, since the viscosity data were determined as density series for the pure gases and at constant mole fractions y for the gaseous mixtures.

2 Results of the Viscosity Measurements at Ruhr University Bochum

The dynamic viscosity was measured in Bochum relative to helium, for which the most accurate viscosity values of the dilute gas (in the limit of zero density) were used. These values resulted from ab initio calculations with a relative standard uncertainty of 0.001 % (\(k = 1\)) [20]. Moreover, the residual damping, usually measured in vacuum, was calibrated in the complete temperature range from 253.15 K to 473.15 K by means of viscosity values ab initio calculated with relative standard uncertainties of 0.001 % for helium [20], of 0.1 % for neon [21], and of 0.1 % for argon [22]. Moreover, reference values of Berg and Moldover [23] were used at 298.15 K (with relative standard uncertainties of 0.001 % for helium [20], of 0.031 % for neon, and of 0.027 % for argon). The viscosity of pure carbon dioxide was reported in Ref. [3], whereas the results for the other pure gases were listed in Refs. [1, 2]. The relative expanded combined uncertainty (\(k = 2\)) in viscosity (including uncertainties of temperature and pressure) for all pure gases and the gaseous mixtures under discussion amounted to from 0.14 % to 0.49 %, depending on temperature, with the lower values corresponding to 298.15 K [1,2,3]. These values of the uncertainties indicate the measurements as very accurate ones.

The isothermal measurements at Ruhr University Bochum were taken up to 2 MPa, meaning that the density series include, at least, a linear density contribution or that even a quadratic member in the density series could appear. The density series for the pure gases and for the gaseous mixtures at a constant mole fraction enabled to determine apart from the viscosities in the limit of zero density, \(\eta _i^{(0)}\) and \(\eta _\textrm{mix}^{(0)}\), respectively, in any case the viscosity coefficients of the initial density dependence, \(\eta _i^{(1)}\) and \(\eta _\textrm{mix}^{(1)}\), respectively. The authors of Refs. [1, 2] themselves evaluated the isothermal viscosity measurements dependent on the density for the three pure gases nitrogen, ethane, and methane as well as for the gaseous mixtures of carbon dioxide–nitrogen and ethane–methane with density series restricted to the first (\(\textrm{N}_2\) and \(\textrm{CO}_2{-}\textrm{N}_2\)) or second (\(\textrm{C}_2\textrm{H}_6\), \(\textrm{CH}_4\), and \(\textrm{C}_2\textrm{H}_6{-}\textrm{CH}_4\)) power,

$$\begin{aligned} \eta _i(T,\rho )= & {} \eta _i^{(0)}(T) + \eta _i^{(1)}(T) \rho + \eta _i^{(2)}(T) \rho ^2~,\hspace{1cm} i=1,2~, \end{aligned}$$
(1)
$$\begin{aligned} \eta _\textrm{mix}(T,\rho )= & {} \eta _\textrm{mix}^{(0)}(T) + \eta _\textrm{mix}^{(1)}(T) \rho + \eta _\textrm{mix}^{(2)}(T) \rho ^2~. \end{aligned}$$
(2)

They reported in Refs. [1, 2] the obtained values for \(\eta _{i}^{(0)}\), \(\eta _{i}^{(1)}\) [or for the second viscosity virial coefficient \(B_{\eta _i} = \eta _{i}^{(1)} / \eta _{i}^{(0)}\)], \(\eta _\textrm{mix}^{(0)}\), and \(\eta _\textrm{mix}^{(1)}\), which we employed in this work. However, in the case of carbon dioxide, the \(\eta _i^{(0)}\) and \(\eta _i^{(1)}\) values were not listed in Ref. [3]. Therefore, we derived these coefficients from the experimental data of that report. For this, Eq. 1 was limited to the first order in density.

For our purpose, to determine the interaction viscosity, \(\eta _{12}^{(0)}(T)\), and the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), each in the limit of zero density, isotherms should be regarded or, at least, quasi-isothermal data should be generated for the viscosities \(\eta _{i}^{(0)}(T)\) of the pure gases and of the respective values \(\eta _\textrm{mix}^{(0)}(T)\) of the gaseous mixtures as well as for the initial density viscosity coefficients \(\eta _{i}^{(1)}(T)\) of the pure gases and \(\eta _\textrm{mix}^{(1)}(T)\) of the respective gaseous mixtures. However, the experimental data of the pure gases and of the mixtures for a specified mole fraction were not determined at exactly the same temperatures. To create quasi-isothermal viscosity values, the experimental data of the series of measurements on the pure gases and on the mixtures of carbon dioxide–nitrogen and ethane–methane are most suitable, since they are distinguished by only small temperature differences in adjacent measurement points. They were arranged to isotherms, meaning that quasi-isotherms were created for the three mole fractions of the mixtures and for the pure gases in a body. In the following, the different viscosities are all labeled as \(\eta (T)\). They were converted into quasi-isothermal values using a first-order Taylor series in temperature,

$$\begin{aligned} \eta (T_{\textrm{iso}}) = \eta (T_\textrm{exp}) + \left( \frac{\partial \eta }{\partial T}\right) _\rho (T_{\textrm{iso}} - T_{\textrm{exp}}) + R_\textrm{N}. \end{aligned}$$
(3)

The temperature of the isotherms \(T_{\textrm{iso}}\) conforms to the mean of the experimental temperatures \(T_{\textrm{exp}}\) of the available data points for the pure gases and for the gaseous mixtures measured at the specified mole fraction. It was checked that the remainder \(R_\textrm{N}\) is negligible compared to the experimental uncertainty. The temperature coefficient \((\partial \eta /\partial T)_\rho\) needed in Eq. 3 was deduced from the following relationship,

$$\begin{aligned} \eta (T) = S \exp \left[ A \ln \left( T_{\textrm{R}}\right) +\frac{B}{T_{\textrm{R}}}+\frac{C}{T_{\textrm{R}}^2}+\frac{D}{T_{\textrm{R}}^3}+E\right] , \end{aligned}$$
(4)

where \(T_{\textrm{R}} = T/(298.15\, \textrm{K})\) and \(S=10\, \upmu \textrm{Pa}\cdot \textrm{s}\) or \(S=10\, \upmu \textrm{Pa}\cdot \textrm{s}\cdot \textrm{L}\cdot \textrm{mol}^{-1}\). The coefficients A, B, C, D, and E were derived from a fit to the experimental data of each series of measurements for the pure gases and for the mixtures at a specified mole fraction. The determined experimental data of the check measurements at approximately 298.15 K of the pure gases each and of the gaseous mixtures at the particular mole fraction were incorporated in the fit. Note that the reason for using a Taylor series instead of employing any fitting function and computing afterwards isothermal viscosity values is the assumption that the experimental errors are kept in this way.

The resulting \(\eta _\textrm{mix}^{(0)}\) and \(\eta _\textrm{mix}^{(1)}\) values are specified for the three mole fractions \(y_\mathrm {N_2}\) of the gaseous mixture carbon dioxide–nitrogen in Table 1 and for the three mole fractions \(y_\mathrm {CH_4}\) of the gaseous mixture ethane–methane in Table 2. \(\eta _i^{(0)}\) and \(\eta _i^{(1)}\) values of the pure components at the quasi-isothermal temperatures are also included in the tables.

Table 1 Values for the viscosity in the limit of zero density, \(\eta _\textrm{mix}^{(0)}\), and for the initial density viscosity coefficients, \(\eta _\textrm{mix}^{(1)}\), resulting from the quasi-experimental isotherms of the measurements of Humberg et al. [1] on the gaseous mixture carbon dioxide–nitrogen including \(\eta _i^{(0)}\) and \(\eta _i^{(1)}\) of the pure components (data for carbon dioxide from Ref. [3])
Full size table
Table 2 Values for the viscosity in the limit of zero density, \(\eta _\textrm{mix}^{(0)}\), and for the initial density viscosity coefficients, \(\eta _\textrm{mix}^{(1)}\), resulting from the quasi-experimental isotherms of the measurements of Humberg et al. [2] on the gaseous mixture ethane–methane including \(\eta _i^{(0)}\) and \(\eta _i^{(1)}\) of the pure components
Full size table

In order to estimate more precisely the uncertainty of the measurements at Ruhr University Bochum, we calculated for this investigation the viscosity in the limit of zero density for the pure gases and for the three mole fractions at the required temperatures in the third-order approximation of the sophisticated kinetic theory of polyatomic gases [24, 25]. In addition, Crusius et al. in Ref. [8] and Hellmann in Ref. [9] recommended for the gaseous mixture carbon dioxide–nitrogen and for the gaseous mixture ethane–methane, respectively, scaled correlations to consider the deviations of their theoretical results for the viscosity of pure carbon dioxide gas, of pure nitrogen gas, and of pure methane gas from the best experimental viscosity data. The viscosity of ethane did not require any scaling. The recommended scaling of the pure gases has been applied to other gaseous mixtures in Refs. [13, 26,27,28,29,30] and is represented for the mixtures under discussion by a temperature-independent factor which depends linearly on the mole fraction:

$$\begin{aligned} \eta _\textrm{rec}^{(0)}= & {} \eta _\textrm{cal}^{(0)} (1.0055 \;y_\mathrm {CO_2} + 1.0024 \;y_\mathrm {N_2}) ~, \end{aligned}$$
(5)
$$\begin{aligned} \eta _\textrm{rec}^{(0)}= & {} \eta _\textrm{cal}^{(0)} (0.9955 \;y_\mathrm {CH_4} + \;y_\mathrm {C_2H_6}) ~. \end{aligned}$$
(6)

Here \(\eta _\textrm{rec}^{(0)}\) and \(\eta _\textrm{cal}^{(0)}\) are the recommended and the according to the kinetic theory calculated viscosity values, respectively.

The correlation for the viscosity values in the limit of zero density derived by Laesecke and Muzny for carbon dioxide [10] employing symbolic regression is based on the theoretically calculated viscosity values of Hellmann [5] and includes already the scaling factor 1.0055. Therefore, it is not surprising that the correlated viscosity values of Laesecke and Muzny for pure carbon dioxide are characterized by a tendency to be only smaller by 0.02 % up to 0.04 % than the scaled recommended values of Crusius et al. [8] according to Eq. 5.

In the case of methane, Laesecke and Muzny [11, 12] used guided symbolic regression, too, and the theoretically calculated viscosity values by Hellmann et al. [6]. Then in a second step, Laesecke and Muzny adjusted their formulation to the recalibrated experimental data of May et al. [31]. In doing so, they determined a factor that leads to the following alternative to Eq. 6:

$$\begin{aligned} \eta _\textrm{rec}^{(0)} = \eta _\textrm{cal}^{(0)} (0.993388 y_\mathrm {CH_4} + y_\mathrm {C_2H_6}) ~. \end{aligned}$$
(7)

The experimentally based viscosity data \(\eta _\textrm{mix}^{(0)}(T)\) have already been compared for the gaseous mixtures carbon dioxide–nitrogen and ethane–methane in some figures of Refs. [1, 2, 8] with the theoretically calculated viscosity values \(\eta _\textrm{cal}^{(0)}\) and with the recommended ones \(\eta _\textrm{rec}^{(0)}(T)\) according to Eqs. 5 and 6, respectively, so that there is no need to repeat it here.

3 Values for the Interaction Viscosity and for the Product of Molar Density and Diffusion Coefficient in the Limit of Zero Density

This report aims at providing reliable values for the limit of zero density of the interaction viscosity, \(\eta _{12}^{(0)}(T)\), and of the product of molar density and binary diffusion coefficient, \((\rho D_\textrm{12})^{(0)}(T)\). For that purpose, two different procedures dependent on the available information could be used to determine \(\eta _{12}^{(0)}(T)\).

We would like to start with the formalism of the second procedure, since the first procedure employs the respective result in the limit of zero density of this second procedure. The measurements on the binary gaseous mixtures carbon dioxide–nitrogen and ethane–methane could be evaluated by means of the linear-in-density expansions of Eq. 2, so that the second procedure could apply the resulting \(\eta _\textrm{mix}^{(1)}(T)\) values including the initial density viscosity coefficients \(\eta _i^{(1)}(T)\) of the pure gases. This procedure is based on a modification of the Enskog theory of hard-spheres mixtures. Unfortunately, the Rainwater–Friend theory [32,33,34] describing \(\eta _i^{(1)}(T)\) of a pure gas has not been extended to mixtures. Hence, modifications of the Enskog theory for hard-spheres mixtures have to be adopted when representing the initial-density viscosity coefficient of a real dense gaseous mixture. These were described by Di Pippo et al. [35], Kestin et al. [36], as well as Vesovic and Wakeham [37, 38]. Hendl and Vogel [39,40,41] developed a variant of this theory in which they proposed to apply the zero-density viscosity values of the pure components and of one mixture at a particular mole fraction to correlate the mixture viscosities at other mole fractions and different moderate densities. This methodology was extended by Humberg et al. [1, 2] to correlate the results of their viscosity measurements at temperatures from 253 K to 473 K up to 2 MPa on the gaseous mixtures carbon dioxide–nitrogen and ethane–methane.

In the Enskog theory, the viscosity of a binary hard-spheres mixture can be represented in the measured range of moderate densities considering terms linear in density by

$$\begin{aligned} \eta _\textrm{mix} = - \left| \begin{array}{ccc} H_{11} &{} H_{12} &{} Y_1\\ H_{21} &{} H_{22} &{} Y_2\\ Y_1 &{} Y_2 &{} 0 \end{array}\right| \Bigg / \left| \begin{array}{cc} H_{11} &{} H_{12} \\ H_{21} &{} H_{22} \end{array}\right| , \end{aligned}$$
(8)

which, taking into account that \(H_{12}\equiv H_{21}\), leads to

$$\begin{aligned} \eta _\textrm{mix} = \frac{Y_1^2 H_{22} + Y_2^2 H_{11} - 2\,Y_1 Y_2 H_{12}}{H_{11}H_{22} - H_{12}^2}. \end{aligned}$$
(9)

The quantities \(H_{ij}\), \(Y_i\), and a contact value of the radial distribution function \(\chi _{ij}\) are given by density series:

$$\begin{aligned} H_{ij}= & {} H_{ij}^{(0)} + H_{ij}^{(1)}\rho ~, \end{aligned}$$
(10)
$$\begin{aligned} Y_i= & {} y_i + y_i^{(1)}\rho ~, \end{aligned}$$
(11)
$$\begin{aligned} \chi _{ij}= & {} 1 + \chi _{ij}^{(1)}\rho ~. \end{aligned}$$
(12)

After inserting Eqs. 10 and 11 into Eq. 9, expanding into a density series, neglecting all density terms beyond linear ones, and comparing with the coefficients of the density series of Eq. 2, one obtains

$$\begin{aligned} \eta _\textrm{mix}^{(0)}= & {} \left[ y_1^2 H_{22}^{(0)} + y_2^2 H_{11}^{(0)} - 2\,y_1 y_2 H_{12}^{(0)}\right] \left[ H_{11}^{(0)}H_{22}^{(0)} - {H_{12}^{(0)}}^2\right] ^{-1}~, \end{aligned}$$
(13)
$$\begin{aligned} \eta _\textrm{mix}^{(1)}= & {} \left[ y_1^2 H_{22}^{(1)} + y_2^2 H_{11}^{(1)} - 2\,y_1 y_2 H_{12}^{(1)} + 2\,y_1 y_1^{(1)} H_{22}^{(0)} \right. \nonumber \\{} & {} \left. + 2\,y_2 y_2^{(1)} H_{11}^{(0)} - 2\,y_1 y_2^{(1)} H_{12}^{(0)} - 2\,y_2 y_1^{(1)} H_{12}^{(0)}\right. \nonumber \\{} & {} \left. - \eta _\textrm{mix}^{(0)} \left( H_{11}^{(0)}H_{22}^{(1)} + H_{11}^{(1)}H_{22}^{(0)} - 2\,H_{12}^{(0)}H_{12}^{(1)}\right) \right] \left[ H_{11}^{(0)}H_{22}^{(0)} - {H_{12}^{(0)}}^2\right] ^{-1} ~. \end{aligned}$$
(14)

The elements of the density series of Eqs. 10–12 are defined as:

$$\begin{aligned} H_{ii}^{(0)} (i=1,2)= & {} \frac{y_i^2}{\eta _i^{(0)}} + \sum _{j=1, j\ne i}^2 \frac{2\,y_i y_j}{\eta _{ij}^{(0)}} \frac{m_i m_j}{(m_i + m_j)^2} \left( \frac{5}{3\,A_{ij}^*} + \frac{m_j}{m_i}\right) ~, \end{aligned}$$
(15)
$$\begin{aligned} H_{ii}^{(1)} (i=1,2)= & {} \frac{y_i^2 \chi _{ii}^{(1)}}{\eta _i^{(0)}} + \sum _{j=1, j\ne i}^2 \frac{2\,y_i y_j \chi _{ij}^{(1)}}{\eta _{ij}^{(0)}} \frac{m_i m_j}{(m_i + m_j)^2} \left( \frac{5}{3\,A_{ij}^*} + \frac{m_j}{m_i}\right) ~, \end{aligned}$$
(16)
$$\begin{aligned} H_{ij}^{(0)} (j\ne i)= & {} - \frac{2\,y_i y_j}{\eta _{ij}^{(0)}} \frac{m_i m_j}{(m_i + m_j)^2} \left( \frac{5}{3\,A_{ij}^*} - 1\right) ~,\end{aligned}$$
(17)
$$\begin{aligned} H_{ij}^{(1)} (j\ne i)= & {} - \frac{2\,y_i y_j \chi _{ij}^{(1)}}{\eta _{ij}^{(0)}} \frac{m_i m_j}{(m_i + m_j)^2} \left( \frac{5}{3\,A_{ij}^*} - 1\right) ~,\end{aligned}$$
(18)
$$\begin{aligned} \chi _{ij}^{(1)}= & {} \zeta _3 + \frac{3\,\sigma _{ii}\sigma _{jj}}{2\,\sigma _{ij}}\zeta _2~,\end{aligned}$$
(19)
$$\begin{aligned} \zeta _k= & {} \frac{\pi }{6}N_\textrm{A}\sum _{i=1}^2 y_i \sigma _{ii}^k~,\end{aligned}$$
(20)
$$\begin{aligned} y_i^{(1)} (i=1,2)= & {} \frac{4}{5}y_i \sum _{j=1}^2 \frac{m_j}{m_i + m_j} y_j b_{ij}~,\end{aligned}$$
(21)
$$\begin{aligned} b_{12}= & {} \frac{2}{3}\pi N_\textrm{A}\sigma _{12}^3~, \end{aligned}$$
(22)

where \(b_{12}\) is the cross second virial coefficient.

Equations 13, 15, and 17 correspond to the well-established kinetic theory of dilute monatomic gases and gaseous mixtures by Chapman and Enskog [19] in its first-order approximation for representing the viscosity of a two-component gaseous mixture in the limit of zero density and form the basis of the first procedure to derive \(\eta _{12}^{(0)}\). Here, \(y_2=1-y_1\) is the mole fraction of the second component, \(m_1\) and \(m_2\) are the molecular masses of components 1 and 2, whereas \(\eta _{1}^{(0)}\) and \(\eta _{2}^{(0)}\) are the zero-density viscosities of the pure components. In this procedure, the interaction viscosity \(\eta _{12}^{(0)}\) corresponds to the viscosity of a hypothetical one-component gas with the molecular mass \(m_{12}\) and the interaction potential of the unlike particles and is given as

$$\begin{aligned} \eta _{12}^{(0)}(T)= & {} \frac{\left( \pi m_{12} k_{\mathrm B}T \right) ^{1/2}}{4 \pi \sigma _{12}^2 \mathfrak {S}_\eta ^*(T_{12}^*)}~,\end{aligned}$$
(23)
$$\begin{aligned} m_{12}= & {} \frac{2m_1 m_2}{m_1+m_2}~,\end{aligned}$$
(24)
$$\begin{aligned} T_{12}^*= & {} \frac{k_{\mathrm B}T}{\varepsilon _{12}}~. \end{aligned}$$
(25)

\(T_{12}^*\) is the reduced temperature, while \(\mathfrak {S}_\eta ^*\) represents a reduced effective cross-section, which considers all the dynamic and statistical information about the binary collisions and, if the first-order kinetic theory framework is applied to real experimental data, incorporates in an effective manner contributions of higher-order kinetic theory approximations. In principle, the scaling parameters \(\varepsilon _{12}\) and \(\sigma _{12}\) can be derived from those of the pure components by means of the usual approximate mixing rules,

$$\begin{aligned} \sigma _{12}= & {} \frac{\sigma _1 + \sigma _2}{2}~, \end{aligned}$$
(26)
$$\begin{aligned} \varepsilon _{12}= & {} \left( \varepsilon _1 \varepsilon _2\right) ^{1/2}~. \end{aligned}$$
(27)

The scaling parameters for the pure components can be obtained via the viscosity of a pure gas i in the limit of zero density according to the kinetic theory of dilute monatomic gases by Chapman and Enskog [19],

$$\begin{aligned} \eta _i^{(0)}(T) = \frac{\left( \pi m_i k_{\mathrm B}T \right) ^{1/2}}{4 \pi \sigma _i^2 \mathfrak {S}_\eta ^*(T_i^*)}~. \end{aligned}$$
(28)

Then, the analysis of the experimental \(\eta ^{(0)}_i\) values is based on the extended corresponding states principle within the kinetic theory of monatomic gases, in which the temperature dependence of the reduced effective cross-section \(\mathfrak {S}_\eta ^*(T_i^*)\) is represented by the following functional form,

$$\begin{aligned} \ln \mathfrak {S}^*_\eta (T_i^*) = \sum _{k=0}^4 a_k\big (\ln T_i^*\big )^k~. \end{aligned}$$
(29)

In a first step, a universal correlation using the coefficients \(a_k\) of the functional \(\mathfrak {S}_\eta ^*\) for the rare gases published by Bich et al. [42] is applied to derive the scaling factors \(\sigma _i\) and \(\varepsilon _i/k_\textrm{B}\). After that, a second step follows typically, in which an individual correlation is performed and new coefficients \(a_k\) are deduced by fitting Eqs. 28 and 29 to the experimental \(\eta ^{(0)}_i\) values. This second step is not necessary if, as in our case, the scaling factors \(\varepsilon _i\) and \(\sigma _i\) are used in Eqs. 26 and 27 to compute the scaling parameters \(\sigma _{12}\) and \(\varepsilon _{12}\). The reason is that the reduced effective cross-section \(\mathfrak {S}_\eta ^*(T_{12}^*)\) in Eq. 23 is calculated by using the coefficients \(a_k\) for a universal correlation.

\(A^*_{12}\) is a temperature dependent ratio of two effective cross-sections for the viscosity (enumerator) and diffusion (denominator),

$$\begin{aligned} A^*_{12} = \frac{5}{6} \frac{\mathfrak {S}(2000)_{12}}{\mathfrak {S}^\prime (1000)_{12}}~. \end{aligned}$$
(30)

It is often approximated by a constant value of 1.10. In the framework of the corresponding states principle, \(A_{12}^*\) can be determined more accurately as a function of temperature for a spherical potential of the intermolecular interaction using universal correlations for the needed two cross-sections [42]. However, for molecules it is more appropriate to compute this ratio applying the nonspherical potential of the intermolecular interaction.

The first procedure was applied for both the carbon dioxide–nitrogen and ethane–methane mixtures. In doing so, the values of the zero-density viscosity values \(\eta _\textrm{mix}^{(0)}(T)\) of the mixtures for a specified mole fraction \(y_2\) as well as \(\eta ^{(0)}_i\) (\(i=1,2\)) of the pure gases at the temperatures of the quasi-isotherms of Tables 1 and 2 were used. A rearrangement of Eqs. 13, 15, and 17 leads to a quadratic relationship, the root of which yields the required interaction viscosity. In a first step of the procedure, \(A_{12}^*\) values as a function of temperature following from the corresponding states principle by means of universal correlations for the needed two cross-sections [42] were used. These cross-sections are based on an effective spherical potential. The results of \(\eta ^{(0)}_{12}\) for the three mole fractions each of the carbon dioxide–nitrogen and of the ethane–methane mixtures including the average together with the applied \(A_{12}^*\) values are given in Tables 3 and 4. For a second step, we computed improved \(A_{12}^*\) values for the nonspherical carbon dioxide–nitrogen and ethane–methane potentials. This second step is motivated by findings described in Ref. [43]. In Fig. 7.9a of this reference, it is demonstrated that \(A_{12}^*\) values of molecules are distinctly increased by several percent if their nonspherical potentials are used instead of an effective spherical potential. The results for the improved \(A_{12}^*\) values computed applying the nonspherical potentials of the intermolecular interaction of carbon dioxide and nitrogen [8] and of ethane and methane [9] are listed in Tables 5 and 6. In addition, values of the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), are also provided in these tables. The interaction viscosity \(\eta ^{(0)}_{12}\) is related to that product according to the first-order approximation of the kinetic theory of dilute gases as

$$\begin{aligned} (\rho D_{12})^{(0)}= \frac{3}{5} \frac{m_1+m_2}{m_1\;m_2} \frac{1}{N_\textrm{A}} A_{12}^*\;\eta ^{(0)}_{12}~, \end{aligned}$$
(31)

where \(N_\textrm{A}\) is Avogadro’s constant. To gain experimentally based values of \((\rho D_{12,\textrm{exp}})^{(0)}\) by means of Eq. 31, the values for \(A_{12}^*\) were those chosen in Tables 3 and 4, respectively, as well as selected in Tables 5 and 6, respectively, i.e., those derived for the respective effective spherical potential and for the particular realistic nonspherical potentials each.

Table 3 Values for the limit of zero density of the interaction viscosity, \(\eta _{12}^{(0)}\), and of the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), resulting from the \(\eta _\textrm{mix}^{(0)}\) and \(\eta _i^{(0)}\) (\(i=1,2\)) values of Table 1 for the measurements of Humberg et al. [1] and using \(A_{12}^*\) values from the effective spherical potential for the gaseous mixture carbon dioxide–nitrogen
Full size table
Table 4 Values for the limit of zero density of the interaction viscosity, \(\eta _{12}^{(0)}\), and of the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), resulting from the \(\eta _\textrm{mix}^{(0)}\) and \(\eta _i^{(0)}\) (\(i=1,2\)) values of Table 2 for the measurements of Humberg et al. [2] and using \(A_{12}^*\) values from the effective spherical potential for the gaseous mixture ethane–methane
Full size table
Table 5 Values for the limit of zero density of the interaction viscosity, \(\eta _{12}^{(0)}\), and of the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), resulting from the \(\eta _\textrm{mix}^{(0)}\) and \(\eta _i^{(0)}\) (\(i=1,2\)) values of Table 1 for the measurements of Humberg et al. [1] and using \(A_{12}^*\) values from the nonspherical potential for the gaseous mixture carbon dioxide–nitrogen
Full size table
Table 6 Values for the limit of zero density of the interaction viscosity, \(\eta _{12}^{(0)}\), and of the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), resulting from the \(\eta _\textrm{mix}^{(0)}\) and \(\eta _i^{(0)}\) (\(i=1,2\)) values of Table 2 for the measurements of Humberg et al. [2] and using \(A_{12}^*\) values from the nonspherical potential for the gaseous mixture ethane–methane
Full size table

Equations 9–22 are usually applied as correlation and extrapolation schemes for the composition and temperature dependencies of the viscosity of moderately dense gaseous and vapor mixtures. These schemes usually include an additional modification of the Enskog theory (see below). In principle, Eq. 14 together with Eqs. 13 and 15–22 can be used to generate a quadratic relationship, whose root provides the interaction viscosity in the limit of zero density.

This second procedure uses the idea of the interaction viscosity for hard spheres \(\eta _{12}^{(0)}\) according to

$$\begin{aligned} \eta _{12}^{(0)} = \frac{5}{16} \frac{\left( \pi m_{12} k_\textrm{B} T \right) ^{1/2}}{\pi \sigma ^2_{12}}~, \end{aligned}$$
(32)

where \(m_{12}\) and \(\sigma _{12}\) correspond to those of Eqs. 24 and 26, respectively. \(\sigma _i\) are the diameters of the hard spheres. In addition, one needs the formalism for a pure hard-sphere gas,

$$\begin{aligned} \eta _i^{(0)}= & {} \frac{5}{16} \frac{\left( \pi m_i k_\textrm{B} T \right) ^{1/2}}{\pi \sigma ^2_i}~, \end{aligned}$$
(33)
$$\begin{aligned} \eta _i^{(1)}= & {} \eta _i^{(0)} \left( \frac{4}{5} b_i - \chi _i^{(1)} \right) ~,\end{aligned}$$
(34)
$$\begin{aligned} b_i= & {} \frac{2}{3} \pi N_\textrm{A} \sigma _i^3~,\end{aligned}$$
(35)
$$\begin{aligned} \chi _i= & {} 1 + \chi _i^{(1)} \rho ~,\end{aligned}$$
(36)
$$\begin{aligned} \chi _i^{(1)}= & {} \frac{5}{12} \pi N_\textrm{A} \sigma _i^3~. \end{aligned}$$
(37)

Here, \(b_i\) is the second virial coefficient and \(\chi _i\) is the equilibrium radial distribution function at contact, each for hard spheres.

In the adaptation of the Enskog theory of hard spheres to real molecules, it is assumed that the transport coefficients for real molecules have the same form as in the case of a hard-spheres gas. Then, the hard-spheres quantities can be replaced by real-gas quantities. This leads to the distinction among three real-gas diameters: \(\sigma _i^\eta\) for \(\eta _i^{(0)}\), \(\sigma _i^b\) for \(b_i\), and \(\sigma _i^\chi\) for \(\chi _i^{(1)}\). The main modification consists in the pressure p in the equation of state for hard spheres being replaced by the thermal pressure \(T (\partial p/ \partial T )_\rho\) of the real gas, which means

$$\begin{aligned} b_i = B_i + T \frac{dB_i}{dT}~,\hspace{1cm}b_{12} = B_{12} + T \frac{dB_{12}}{dT}~, \end{aligned}$$
(38)

where \(B_i\) is the second virial coefficient of the real gas and \(B_{12}\) is the cross second virial coefficient of the real gaseous mixture. The \(B_i\) and \(B_{12}\) values for the considered gases as well as for the respective gaseous mixtures were taken from Ref. [44]. Then, \(\sigma _i^b\) and \(\sigma _{12}^b\) can be determined directly from \(B_i(T)\) as well as \(B_{12}(T)\) data by means of

$$\begin{aligned} \sigma _i^b = \left( \frac{3}{2 \pi N_\textrm{A}} b_i\right) ^{1/3}~,\hspace{1cm} \sigma _{12}^b = \left( \frac{3}{2 \pi N_\textrm{A}} b_{12}\right) ^{1/3}~. \end{aligned}$$
(39)

Finally, \(\sigma _i^\chi\) follows from Eqs. 34 and 37 with \(b_i\), whereas \(\sigma _i^\eta\) results from \(\eta _i^{(0)}\). The scaling factors \(\sigma _i\) for the pure components as well as \(\sigma _{12}\) for the mixtures are needed in connection with the hard-spheres modification.

The results averaged for the individual mole fractions of this second procedure using \(\eta _i^{(1)}(T)\) of the pure components as well as \(\eta _{\textrm{mix}}^{(1)}(T)\) of the gaseous mixtures are presented in Table 7 applying the \(A_{12}^*\) values calculated by us for the nonspherical potentials. The outcome using the \(A_{12}^*\) values computed for the spherical potentials is not given here, since the uncertainties of the initial density viscosity coefficients \(\eta ^{(1)}_i\) of the pure gases as well as of \(\eta _\textrm{mix}^{(1)}(T)\) for the particular mixtures are comparatively large, so that the results for the individual mole fractions illustrated as relative differences from the averaged interaction viscosity values \(\eta _{12,\mathrm {2nd\, proc,aver}}^{(0)}\) in Fig. 1 show substantial deviations.

Table 7 Averaged values for the limit of zero density of the interaction viscosity, \(\eta _{12}^{(0)}\), and of the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\), resulting from the \(\eta _\textrm{mix}^{(1)}\) and \(\eta _i^{(1)}\) values of Table 1 for the measurements of Humberg et al. [1] as well as for the gaseous mixture ethane–methane resulting from the \(\eta _\textrm{mix}^{(1)}\) and \(\eta _i^{(1)}\) values of Table 2 for the measurements of Humberg et al. [2], in each case using \(A_{12}^*\) values from the nonspherical potential
Full size table
Fig. 1
figure 1

Relative differences of the interaction viscosity values \(\eta _{12,\mathrm {2nd\;proc,mol}}^{(0)}(T)\) of the second procedure for the individual mole fractions of the gaseous mixtures carbon dioxide–nitrogen and ethane–methane from the averaged interaction viscosity values \(\eta _{12,\mathrm {2nd\;proc,aver}}^{(0)}(T)\), as a function of temperature T. Deviations: \(\Delta = 100\cdot (\eta _{12,\mathrm {2nd\;proc,mol}}^{(0)} - \eta _{12,\mathrm {2nd\;proc,aver}}^{(0)})/\eta _{12,\mathrm {2nd\;proc,aver}}^{(0)}\). Carbon dioxide–nitrogen: \(\circ\), \(y_\mathrm {N_2}=0.24788\); \(\vartriangle\), \(y_\mathrm {N_2}=0.49962\); \(\triangledown\), \(y_\mathrm {N_2}=0.74981\). Ethane–methane: •, \(y_\mathrm {CH_4}=0.2498\); \(\blacktriangle\), \(y_\mathrm {CH_4}=0.4942\); \(\blacktriangledown\), \(y_\mathrm {CH_4}=0.7512\)

Full size image

4 Discussion

The discussion of the results for the interaction viscosity \(\eta _{12}^{(0)}\) is started with the second procedure, which applies the initial density viscosity coefficients \(\eta _{i}^{(1)}\) and \(\eta _\textrm{mix}^{(1)}\). At the end of the previous section, it has already been stated that the uncertainties of the initial density viscosity coefficients are serious.

Without anticipating the results of the first procedure, it is significant that \(\eta _{12}^{(0)}\) following from \(\eta _{i}^{(0)}\) and \(\eta _\textrm{mix}^{(0)}\) amount to lower values. This is illustrated in Fig. 2, in which \(\eta _{12,\mathrm {2nd\;proc,aver}}^{(0)}(T)\), the averaged interaction viscosity values of the second procedure, are compared with those of the first procedure \(\eta _{12,\mathrm {1st\;proc,aver}}^{(0)}(T)\). The values of the second procedure are for the gaseous system carbon dioxide–nitrogen higher by (14–40) %, whereas those for the mixture ethane–methane are increased by up to 20 %.

Moreover, a recalculation of the viscosity values \(\eta _\textrm{mix}\) was performed and in Table 8 compared with the experimental values \(\eta _\mathrm {mix,\;exp}\) measured by Humberg et al. [1, 2] for the gaseous systems under discussion. This comparison is restricted to only one density series for both systems, at the middle mole fraction and at ambient temperature each. Ambient temperature was chosen because the measurements have the lowest uncertainty under this condition. The results of \(\eta _{12}^{(0)}\) for the first and second procedures at the given mole fraction and at the respective temperature are also listed. Three different ways for the recalculation were tested. The third column in the table corresponds to Eq. 2 applying the results of \(\eta _{12}^{(0)}\) of the first procedure in the coefficient \(\eta _\textrm{mix}^{(0)}\) (Eq. 13) and of \(\eta _{12}^{(0)}\) of the second procedure in the coefficient \(\eta _\textrm{mix}^{(1)}\) (Eq. 14) of the series expansion. The findings show that the experimental data \(\eta _\mathrm {mix,\;exp}\) are consistently described. In the fourth and fifth columns, the findings are reported which followed for the second and third ways if the ratio of determinants (Eq. 8) was computed using either \(\eta _{12}^{(0)}\) of the first or of the second procedures. The experimental data \(\eta _\mathrm {mix,\;exp}\) are underestimated using \(\eta _{12}^{(0)}\) of the first procedure, whereas they are overestimated applying \(\eta _{12}^{(0)}\) of the second procedure. Table 8 illustrates that the approximations included in the Enskog theory for binary hard-spheres mixtures at moderate densities are unsuitable to represent the viscosity of a real binary gas mixture. The conclusion is that the initial density viscosity coefficients of the pure gases \(\eta _{i}^{(1)}\) as well as of the gaseous mixture \(\eta _\textrm{mix}^{(1)}\) cannot be used to derive reasonable values of \(\eta _{12}^{(0)}\).

Fig. 2
figure 2

Relative differences of the averaged interaction viscosity values \(\eta _{12,\mathrm {2nd\;proc,aver}}^{(0)}(T)\) of the second procedure for the gaseous mixtures carbon dioxide–nitrogen and ethane–methane from the averaged interaction viscosity values \(\eta _{12,\mathrm {1st\;proc,aver}}^{(0)}(T)\) of the first procedure, as a function of temperature T. Deviations: \(\Delta = 100\cdot (\eta _{12,\mathrm {2nd\;proc,aver}}^{(0)} - \eta _{12,\mathrm {1st\;proc,aver}}^{(0)})/\eta _{12,\mathrm {1st\;proc,aver}}^{(0)}\). \(\square\), carbon dioxide–nitrogen; \(\blacksquare\), ethane–methane

Full size image
Table 8 Recalculated viscosity values, \(\eta _\textrm{mix}\), using the zero-density interaction viscosity values \(\eta _{12}^{(0)}\) resulting from the first procedure applying the zero-density viscosities \(\eta _{i}^{(0)}\) and \(\eta _\textrm{mix}^{(0)}\) and from the second procedure applying the initial density viscosity coefficients \(\eta _{i}^{(1)}\) and \(\eta _\textrm{mix}^{(1)}\), respectively, for the gaseous systems carbon dioxide–nitrogen of the measurements of Humberg et al. [1] and ethane–methane of the measurements of Humberg et al. [2]
Full size table

The results of the interaction viscosity \(\eta _{12}^{(0)}\) for the first procedure are summarized for the gaseous mixtures carbon dioxide–nitrogen and ethane–methane in Tables 3, 4, 5, and 6 and should now be investigated. For this, not only the cross-section ratio \(A_{12}^*\) but also the interaction viscosity in the limit of zero density, \(\eta _{12,\textrm{cal}}^{(0)}(T)\), were computed for the nonspherical potentials of the intermolecular interaction of the systems carbon dioxide–nitrogen and ethane–methane. This could only be done in the first-order approximation of the sophisticated kinetic theory for polyatomic molecules [24, 25], since these quantities are only defined for this approximation.

The average values of the experimentally based interaction viscosity \(\eta ^{(0)}_{12,\textrm{exp}}\) given in Table 3 for the mixture carbon dioxide–nitrogen are compared in Fig. 3 with the theoretical values \(\eta ^{(0)}_{12,\textrm{cal}}\) computed at the temperatures of this paper. The figure shows that the experimentally based data, applying \(A_{12}^*\) values derived from the corresponding states principle using universal correlations for a spherical potential [42], are roughly between \(0.5\,\%\) and \(0.85\,\%\) too high. Some scatter should be caused by experimental uncertainties. Furthermore, the figure illustrates a comparison of the \(\eta ^{(0)}_{12,\textrm{cal}}\) values with the experimentally based \(\eta ^{(0)}_{12,\textrm{exp}}\) data of Table 5, applying the \(A_{12}^*\) values for the nonspherical potential. The figure shows that the agreement is only slightly improved. But one has to take into account that the quantity \(\eta ^{(0)}_{12,\textrm{cal}}\) could only be computed in the first-order approximation. Hence, the existing deviations are partly due to the comparison with an effective experimental quantity that only has a counterpart in the kinetic theory of gases at the lowest-order approximation.

The average values of the interaction viscosity \(\eta ^{(0)}_{12,\textrm{exp}}\) listed in Table 4 for the mixture ethane–methane are checked in Fig. 4 against the theoretical values \(\eta ^{(0)}_{12,\textrm{cal}}\) calculated at the temperatures of this investigation. The figure illustrates in comparison with Fig. 3 that the experimentally based data are better represented for this gaseous mixture, but the relative differences are negative. This concerns both the deviations applying the \(A_{12}^*\) values derived from the corresponding states principle using universal correlations for a spherical potential [42], and those resulting with the experimentally based \(\eta ^{(0)}_{12,\textrm{exp}}\) data of Table 6 applying the \(A_{12}^*\) values for the nonspherical potential. Surprisingly, the \(\eta ^{(0)}_{12,\textrm{exp}}\) data derived with the \(A_{12}^*\) values for the nonspherical potential are less optimally represented by the theoretically computed values \(\eta ^{(0)}_{12,\textrm{cal}}\).

Fig. 3
figure 3

Comparison of the theoretically calculated values of the interaction viscosity \(\eta _{12,\textrm{cal}}^{(0)}(T)\) for the gaseous mixture carbon dioxide–nitrogen with the average of the experimentally based zero-density viscosity data \(\eta _{12,\textrm{exp}}^{(0)}(T)\) determined by means of Eqs. 13, 15, and 17 using \(A_{12}^*\) values derived either from the corresponding states principle for a spherical potential given in Table 3 or from the kinetic theory of polyatomic gases for a nonspherical potential listed in Table 5, as a function of temperature T. Deviations: \(\Delta = 100\cdot (\eta _{12,\textrm{exp}}^{(0)} - \eta _{12,\textrm{cal}}^{(0)})/\eta _{12,\textrm{cal}}^{(0)}\). \(\square\), using \(A_{12}^*\) values derived from the corresponding states principle for a spherical potential [42]; \(\blacksquare\), using \(A_{12}^*\) deduced from the kinetic theory of polyatomic gases for the nonspherical potential [8]

Full size image
Fig. 4
figure 4

Comparison of the theoretically calculated values of the interaction viscosity \(\eta _{12,\textrm{cal}}^{(0)}(T)\) for the gaseous mixture ethane–methane with the average of the experimentally based zero-density viscosity data \(\eta _{12,\textrm{exp}}^{(0)}(T)\) determined by means of Eqs. 13, 15, and 17 using \(A_{12}^*\) values derived either from the corresponding states principle for a spherical potential given in Table 4 or from the kinetic theory of polyatomic gases for a nonspherical potential listed in Table 6, as a function of temperature T. Deviations: \(\Delta = 100\cdot (\eta _{12,\textrm{exp}}^{(0)} - \eta _{12,\textrm{cal}}^{(0)})/\eta _{12,\textrm{cal}}^{(0)}\). \(\square\), using \(A_{12}^*\) values derived from the corresponding states principle for a spherical potential [42]; \(\blacksquare\), using \(A_{12}^*\) deduced from the kinetic theory of polyatomic gases for the nonspherical potential [9]

Full size image

The experimentally based data of the product \((\rho D_{12,\textrm{exp}})^{(0)}\) of Table 3 for the carbon dioxide–nitrogen mixture, applying \(A_{12}^*\) values derived from the corresponding states principle using universal correlations for a spherical potential [42], are compared in Fig. 5 with theoretical values \((\rho D_{12,\textrm{cal}})^{(0)}\), which were again computed for the nonspherical potential of the interaction between the molecules by means of the kinetic theory for polyatomic gases in its third-order approximation [8]. Figure 5 shows that deviations between \(-4.0\,\%\) and \(-5.4\,\%\) occur. Figure 7 of Ref. [8] illustrates deviations of experimental data \((\rho D_{12,\textrm{exp}})^{(0)}\) of different authors from theoretical values \((\rho D_{12,\textrm{cal}})^{(0)}\) computed for the nonspherical potential. Crusius et al. [8] found that the data of Robjohns and Dunlop [45] given in the temperature range between 277 K and 323 K agree with their calculations within \(\pm 0.2\,\%\), whereas the data of most other authors are characterized by deviations \(>\pm 1\,\%\). Moreover, Fig. 5 of this paper demonstrates that for the \(A_{12}^*\) values computed for the nonspherical potential and applied for Table 5 the experimentally based \((\rho D_{12,\textrm{exp}})^{(0)}\) data nearly perfectly agree with the computed \((\rho D_{12,\textrm{cal}})^{(0)}\) values of this paper. The relative differences amount to \(+0.33\) % up to \(+0.53\) %.

Fig. 5
figure 5

Comparison of the theoretically calculated values of the product of molar density and binary diffusion coefficient in the limit of zero density \((\rho D_{12,\textrm{cal}})^{(0)}(T)\) for the gaseous mixture carbon dioxide–nitrogen with experimentally based data \((\rho D_{12,\textrm{exp}})^{(0)}(T)\), determined by means of Eq. 31 applying \(\eta _{12,\textrm{exp}}^{(0)}(T)\) and \(A_{12}^*\) values derived either from the corresponding states principle for a spherical potential given in Table 3 or from the kinetic theory of polyatomic gases for a nonspherical potential listed in Table 5, as a function of temperature T. Deviations: \(\Delta = 100\cdot [(\rho D_{12,\textrm{exp}})^{(0)} - (\rho D_{12,\textrm{cal}})^{(0)}]/(\rho D_{12,\textrm{cal}})^{(0)}\). \(\circ\), using \(A_{12}^*\) values derived from the corresponding states principle for a spherical potential [42]; •, using \(A_{12}^*\) deduced from the kinetic theory of polyatomic gases for a nonspherical potential [8]

Full size image

The experimentally based data of the product \((\rho D_{12,\textrm{exp}})^{(0)}\) of Table 4 for the ethane–methane mixture, applying \(A_{12}^*\) values derived from the corresponding states principle using universal correlations for a spherical potential [42], are compared in Fig. 6 with theoretical values \((\rho D_{12,\textrm{cal}})^{(0)}\), which were again computed for the nonspherical potential of the interaction between the molecules by means of the kinetic theory for polyatomic gases in its third-order approximation [9]. Figure 6 shows that deviations between \(-2.5\,\%\) and \(-3.7\,\%\) appear. Figure 11 of Ref. [9] illustrates deviations of experimental data \((\rho D_{12,\textrm{exp}})^{(0)}\) of a few authors from theoretical values \((\rho D_{12,\textrm{cal}})^{(0)}\) computed for the nonspherical potential. Hellmann [9] found that the data of Arora et al. [46] given in the temperature range from 275 K to 323 K differ from his calculations only by about \(-0.5\,\%\), whereas the data of the other authors show deviations \(>\pm 1\,\%\). Moreover, Fig. 6 of this paper demonstrates that for the \(A_{12}^*\) values computed for the nonspherical potential and applied for Table 6 the experimentally based \((\rho D_{12,\textrm{exp}})^{(0)}\) data are lower by 0.05 % up to 0.30 % than the computed \((\rho D_{12,\textrm{cal}})^{(0)}\) values of this paper.

Fig. 6
figure 6

Comparison of the theoretically calculated values of the product of molar density and binary diffusion coefficient in the limit of zero density \((\rho D_{12,\textrm{cal}})^{(0)}(T)\) for the gaseous mixture ethane–methane with experimentally based data \((\rho D_{12,\textrm{exp}})^{(0)}(T)\), determined by means of Eq. 31 applying \(\eta _{12,\textrm{exp}}^{(0)}(T)\) and \(A_{12}^*\) values derived either from the corresponding states principle for a spherical potential given in Table 4 or from the kinetic theory of polyatomic gases for a nonspherical potential listed in Table 6, as a function of temperature T. Deviations: \(\Delta = 100\cdot [(\rho D_{12,\textrm{exp}})^{(0)} - (\rho D_{12,\textrm{cal}})^{(0)}]/(\rho D_{12,\textrm{cal}})^{(0)}\). \(\circ\), using \(A_{12}^*\) values derived from the corresponding states principle for a spherical potential [42]; •, using \(A_{12}^*\) deduced from the kinetic theory of polyatomic gases for the nonspherical potential [9]

Full size image

The conclusion of Ref. [43] (page 243) that inelastic collisions have a fractionally larger effect on the transfer of mass than on the transfer of momentum is justified for the gaseous systems under discussion. This is obvious if the influence of the \(A_{12}^*\) values, deduced applying the corresponding states principle with universal correlations for the spherical potentials [42] or calculated by Crusius et al. [8] and by Hellmann [9] for the nonspherical potentials of the interaction between the molecules by means of the kinetic theory for polyatomic gases, is examined for the interaction viscosity \(\eta _{12}^{(0)}\) and for the product \((\rho D_{12})^{(0)}\) in Figs. 3 and 5 as well as in Figs. 4 and 6, respectively. Furthermore, the argumentation of the authors of Ref. [43] that the determination of self-diffusion and binary diffusion coefficients by means of their experimental viscosities in the limit of zero density, \(\eta _i^{(0)}\) and \(\eta _{12}^{(0)}\), respectively, requires the multiplication by sensible \(A_i^*\) and \(A_{12}^*\) values based on nonspherical potentials for the interaction between the molecules is verified.

5 Summary and Conclusions

Humberg et al. [1, 2] reported on viscosity measurements on the two gaseous mixtures carbon dioxide–nitrogen and ethane–methane including the pure gases applying a rotating-cylinder viscometer between 0.1 MPa and 2.0 MPa in the temperature range of 253.15 K to 473.15 K. The isothermal viscosity data were evaluated by these authors with density series restricted to at most the second order to derive the zero-density viscosities and initial density viscosity coefficients. In the present paper, these data were used to determine the zero-density limits of the interaction viscosity, \(\eta _{12}^{(0)}\), and the product of molar density and diffusion coefficient, \((\rho D_{12})^{(0)}\). Because the data of Humberg et al. for a specific mole fraction were not determined at exactly the same temperatures, in a first step, quasi-isotherms were created for the three mole fractions of the mixtures and for the pure gases.

Two procedures were used to deduce \(\eta _{12}^{(0)}\). In the first procedure, \(\eta _{12}^{(0)}\) was obtained from the values of the zero-density viscosities of the mixtures \(\eta _\textrm{mix}^{(0)}(T)\) and the respective values of the pure gases, \(\eta ^{(0)}_1\) and \(\eta ^{(0)}_2\), using the Chapman–Enskog theory of dilute monatomic gases [19] in its first-order approximation. The quantity \(A_{12}^*\) appearing in this relationship represents the ratio of two effective cross-sections for viscosity and diffusion and can be computed for the effective spherical potential of the interaction of the molecules or, if one applies the more appropriate kinetic theory of polyatomic gases [24, 25] instead of the Chapman–Enskog theory, for a nonspherical potential [8, 9]. However, it was found that \(\eta _{12}^{(0)}\) is fairly insensitive to \(A_{12}^*\), so that the differences between \(\eta _{12}^{(0)}\) values obtained with \(A_{12}^*\) values for the spherical and nonspherical potentials are almost negligible. The obtained \(\eta _{12}^{(0)}\) values agree very well with those calculated theoretically, despite the fact that \(\eta _{12}^{(0)}\) and \(A_{12}^*\) are only defined in the first-order kinetic theory approximation and the experimentally based \(\eta _{12}^{(0)}\) values thus represent effective quantities that implicitly account for higher-order kinetic theory contributions. The experimentally based \(\eta _{12}^{(0)}\) values are better represented by the theoretically calculated ones for the mixture ethane–methane (within 0.3 %) than for the mixture carbon dioxide–nitrogen (within 0.9 %), with the relative differences being negative for the former and positive for the latter.

The second procedure to determine \(\eta _{12}^{(0)}\) uses a modification of the Enskog theory of hard spheres based on the initial density viscosity coefficients for the mixtures and the pure gases. In Ref. [15], the uncertainties of the initial density quantities were found to be too large for obtaining reasonable results for the interaction viscosity, and, unfortunately, in the present paper this problem was encountered as well, even though \(A_{12}^*\) values for the nonspherical potentials were used. The \(\eta _{12}^{(0)}\) values are higher by up to 40 % for the carbon dioxide–nitrogen system and by up to 20 % for the ethane–methane system than the respective \(\eta _{12}^{(0)}\) values obtained with the first procedure. It is therefore concluded that the approximations included in the Enskog theory for binary hard-spheres mixtures at moderate densities are unsuitable to represent the viscosity of a real binary gas mixture.

The experimentally based data of the product \((\rho D_{12,\textrm{exp}})^{(0)}\), applying the \(\eta _{12}^{(0)}\) values from the first procedure and \(A_{12}^*\) values derived from the corresponding states principle using universal correlations for a spherical potential [42] as well as \(A_{12}^*\) values computed for the nonspherical potential, are compared with theoretical values \((\rho D_{12,\textrm{cal}})^{(0)}\) computed for the nonspherical potential by means of the kinetic theory for polyatomic gases in its third-order approximation [8, 9]. While the relative deviations of \((\rho D_{12,\textrm{exp}})^{(0)}\) from \((\rho D_{12,\textrm{cal}})^{(0)}\) for the spherical potentials reach up to \(-5.4\,\%\) for the system carbon dioxide–nitrogen and up to \(-3.7\,\%\) for the system ethane–methane, the deviations for the nonspherical potentials are considerably reduced, being within 0.33 % to 0.53 % for the system carbon dioxide–nitrogen and within \(-0.05\,\%\) to \(-0.30\,\%\) for the system ethane–methane. Therefore, the traditional way of estimating self-diffusion and binary diffusion coefficients from experimental viscosity data in the limit of zero density should use suitable values based on nonspherical potentials for \(A_i^*\) and \(A_{12}^*\), respectively, whenever possible.