Research: Science and Education
Temperature Dependence of Viscosities of Common Carrier Gases Trent S. Sommers and Tal M. Nahir* Department of Chemistry, California State University, Chico, CA 95929-0210; *
[email protected] A simplified derivation, based on a rigid-sphere model, suggests that the viscosity, η, of a gas increases with temperature at a rate proportional to T 1兾2, where T is the absolute temperature (1). However, real gases show an exponential dependence that is significantly different from 1兾2, and the calculation of the magnitude of their low-pressure viscosity should include a contribution from a temperature-dependent collision integral,
TM
η = 0.02669
(1)
σ2Ω
where η is the viscosity in µPa s, M is the molecular weight of the gas, σ is the molecular diameter in nm, and Ω is the collision integral. This equation is commonly associated with the Chapman–Enskog theory, which assumes binary elastic collisions, description of motion by classical mechanics, and intermolecular forces acting between centers of molecules (or atoms) (2). In this article, we describe two theoretical approaches for the calculations of the viscosities of real gases and provide experimental verification for these predictions. The relatively simple measurements are performed using a gas chromatograph and can be reproduced by students as part of an experiment for either analytical or physical chemistry laboratory.
A different way to calculate transport properties of gases invoked the law of corresponding states with σ and ε as scaling factors (6). The expression for the collision integral in this case shows a polynomial expansion about lnTr, which corresponds to the behavior at the low-temperature limit. Using this approach, the computation of viscosities of common carrier gases for gas chromatography was reported (7). Interestingly, later calculations using this theoretical model led to a suggestion of a relatively simple expression for the viscosity, (3) η = cT x where c is a constant and x is near 0.7 (8). Experimental Results Gas chromatography is a convenient method to probe physicochemical properties (9). Here, we assume that a steady-state flow of the mobile phase through the column is adequately described using Poiseuille’s law and the ideal gas law and show that the determination of the viscosity of a carrier gas can be accomplished by measuring the time a marker compound travels the full length of the column without interaction with the stationary phase, tm. This quantity, also known as the holdup time, is obtained by integrating the expression for the flow rate over the full length of the column,
Theoretical Considerations
tm =
A more realistic description of transport properties of gases models the existence of intermolecular forces by a potential energy function. An earlier report in this Journal has already demonstrated how to relate second virial coefficients to intermolecular potentials (3). In a similar way, Ω can be calculated by sequentially computing the angle of deflection, the cross section, and, finally, the collision integral (4). In an effort to avoid this time-consuming process, an empirical equation was suggested for Ω in terms of the Lennard–Jones 12-6 potential (5). To a good approximation, the collision integral for the calculation of real-gas viscosities can be expressed using only three of the original five terms,
Ω = 1.16145 Tr − 0.14874 + 0.52487 e−0.77320 Tr
(2)
−2.43787 Tr
+ 2.16178 e
where T r = k B T兾ε is the reduced temperature, k B is Boltzmann’s constant, and ε is the minimum of the pair-potential energy (2). Note that if the only contribution to Ω is from the first term on the right side of eq 2, then the dependence of the viscosity on temperature is approximately proportional to T 0.65. www.JCE.DivCHED.org
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32 L 2 3rc 2
η
pin3 − pout3
( pin2
− pout2
)
2
(4)
where L and rc are the length and the radius of the column, respectively, and pin and pout are the inlet and outlet pressures, respectively (10). Therefore, the temperature dependence of the viscosity is the same as that of the holdup time. Our measurements were conducted using a gas chromatograph and a mass-selective detector (GC–MS) with O2 as a detectable and unretained marker compound (11). An Agilent 7683 Series injector was used to accurately reproduce the introduction of 1.0-µL air samples at a 10:1 split ratio into a 1.5-mm i.d. liner (Agilent 18740-80200) and a Hewlett-Packard HP-5MS column (30-m × 0.250-mm × 0.25-µm nominal dimensions) in a Hewlett-Packard HP 6890 Plus Series GC system. Detection was accomplished by a Hewlett-Packard HP 5973 mass-selective detector operating at near vacuum, pout ≈ 0 (several mPa pressure measured by a Hewlett-Packard 59864B ionization gauge controller). Mass spectra were taken in the m兾z range from 31.5 to 32.5, and holdup times were recorded at the apex of peaks. Isothermal runs were carried out with an absolute inlet pressure of pin = 150 kPa (gauge pressure of 50 kPa vs 100 kPa ambient pressure). The relatively low pressure in the
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Journal of Chemical Education
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Research: Science and Education
column, which was below the critical pressures for the three gases investigated here, is consistent with the assumption that the viscosity is independent of pressure in these measurements (12). Figure 1 shows that the slopes for three common carrier gases are different from 1兾2. Table 1 summarizes the results and shows similar trends for temperature dependence in both experimental results and theoretical predictions (calculated using eqs 1 and 2 and values for σ and ε兾kB from ref 2). The
somewhat low experimental values for He and H2 might initially suggest the need for quantum-mechanical methods (13); however, the calculated magnitudes of quantum corrections for hydrogen (14) and helium (15) are relatively small under the current experimental conditions. Instead, this discrepancy could be related to the permeation of these gases through the walls of the column, which would result in experimental holdup times gradually shorter than predicted by theory as the temperature increases (16). Summary
2.0
He N2
1.9
Log (tm / s)
1.8
Literature Cited 1.7
1/2 H2
1.6
1.5 2.48
2.50
2.52
2.54
2.56
2.58
2.60
Log(T / K) Figure 1. Dependence of experimental holdup times (in seconds) on absolute temperature for three common carrier gases. The dashed line with a slope of 1/2 was added for comparison.
Table 1. Comparison between Experimental Results and Theoretical Predictions for the Temperature Dependence of Carrier-Gas Viscosities Gas
a
xexpa
xthe
He
0.629 (0.001)
b
0.649 (0.000)
H2
0.659 (0.003)
0.678 (0.001)
N2
0.699 (0.001)
0.697 (0.001)
Defined in eq 3.
b
Values in parentheses are standard deviations for the slopes based on eight data points shown in Figure 1.
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This article describes both theoretical and experimental evidence that the dependence of viscosities of real gases on temperature is greater than that predicted by the kinetic theory for ideal gases. In particular, the experimental results were obtained using common modern instrumentation and could be reproduced by students in analytical or physical chemistry laboratory experiments. While the measurements shown here have focused on the practical examples of carrier gases in gas chromatography, the overall methodology could be extended to similar measurements with other gases, such as argon, as long as they are compatible with the instrument.
Journal of Chemical Education
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1. Halpern, A. M. J. Chem. Educ. 2002, 79, 214–216. 2. Poling, B. E.; Prausnitz J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: San Francisco, 2000. 3. Reid, B. P. J. Chem. Educ. 1996, 73, 612–615. 4. Storck, U. Z. Angew. Math. Mech. 1998, 8, 555–563. 5. Neufeld, P. D.; Janzen, A. R.; Aziz, R. A. J. Chem. Phys. 1972, 57, 1100–1102. 6. Kestin, J.; Ro, S. T.; Wakeman, W. Physica 1972, 58, 165– 211. 7. Hawkes, S. J. Chromatographia 1993, 37, 399–401. 8. Hinshaw, J. V.; Ettre, L. S. J. High Resol. Chromatogr. 1997, 20, 471–481. 9. Laub, R. J.; Pescok, R. L. Physicochemical Applications of Gas Chromatography; Wiley-Interscience: New York, 1978. 10. Nahir, T. M.; Morales, K. M. Anal. Chem. 2000, 72, 4667– 4670. 11. Quintanilla-López, J. E.; Lebrón-Aguilar, R.; GarcíaDomínguez, J. A. J. Chromatogr. A 1997, 767, 127–136. 12. Atkins, P. W. Physical Chemistry, 6th ed.; W. H. Freeman: New York, 1998. 13. Mason, E. A.; Spurling, T. H. The Virial Equation of State; Pergamon: Oxford, 1969. 14. Niblett, P. D.; Takayanagi, K. Proc. Roy. Soc. 1959, A250, 222– 247. 15. Hurly, J. J.; Moldover, M. R. J. Res. Natl. Inst. Stand. Technol. 2000, 105, 667–688. 16. Cahill, J. E.; Tracy, D. H. J. High Resol. Chromatogr. 1998, 21, 531–539.
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