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Viscosity of Gases
Theory
The kinetic theory of gases provides a relatively simple model
description of gaseous molecules. This theory allows one to find
relations between transport properties of gases and the simple theory,
which treats molecules as hard spheres. The gaseous transport property
that is most easily and accurately measured is the viscosity of gases.
The theory allows one to estimate the hard sphere collision diameter
of molecules from the measurement of the gas viscosity. Later in the
course, we will find that collision diameters or collision cross
sections are useful in the estimation of limits on the rate constants
for reactions involving gaseous molecules. In this experiment, you
will measure the viscosity of various gases, determine the temperature
dependence of the viscosity, and calculate the collision diameter and
cross section.
In studying gas phase reaction dynamics, it is important to know the
effective sizes, e.g., diameters, d, of the molecules involved in the
elementary steps of the reaction. One of the fundamental assumptions
this collision theory is that molecules do not react unless they
collide. The figure below illustrates what we mean by the collision
diameter, σ, for two spheres approaching each other.
It can be seen that the collision diameter is given by
σ = 1/2 (dA + dB)
(1)
The most common method for estimating the hard sphere diameter is
through measurements of gas viscosity. From the application of gas
kinetic theory to "hard sphere" molecules, the square of the collision
diameter, σ, may be expressed as follows:
 (2)
where M is the molecular weight, η is the viscosity, NA is Avogadro's
number, and the other symbols have the usual meanings.
Equation (2) can be rearranged to obtain an expression for the gas
viscosity in terms of the collision diameter
 (3)
The cgs unit of viscosity is called the poise, P, which is equal to 1
g/(cm s) (1µP = 10-7 Pa s). This equation suggests that η should be
independent of gas pressure and should vary with temperature as T1/2.
The former prediction is likely to seem counterintuitive, but it has
been verified over a relatively wide range of pressures. The reasons
that η becomes pressure-dependent at low and high pressures stem from
departures from laminar flow; some of these complications will be
discussed below. The T1/2 dependence has been confirmed experimentally
and is unusual because gases demonstrate the opposite type of
temperature dependence from liquids, that is, liquid viscosities
decrease with increasing temperature
Practice
The availability of simple equations relating viscosity, μ, to
temperature, t, and pressure, p, would be of convenience to the
engineer or other worker in determining μ. In this article, modified
simple equations developed to enable one to conveniently make precise,
accurate calculations of μ for dry air, nitrogen, carbon dioxide,
helium, argon and oxygen, using readily available hand-held
calculators are presented.
The equations were fitted to experimental data selected from the
literature on the basis of claimed accuracy and precision, and of
internal consistency. The sets of data published by Kestin and his
collaborators, references 2-11, met these criteria and were used to
develop the equations.
Equations
Equation 1
For oxygen, the equation is:
μ = 0.0190395 + 6.50043 x 10-5 t - 8.97542 x 10-8 t2 + 8.97542 x 10-7 p +6.13118 x 10-10 p2
Using Equation 1, t = 20oC and p = 14.69595 PSI, μ = 0.020317 cP.
Equation 2
For nitrogen, the equation is:
μ = 0.0167214 + 3.92728 x 10-5 t + 1.22474 x 10-7 t2 + 8.56087 x 10-7 p + 4.69295 x 10-10 p2
Using Equation 2, t = 20oC and p = 14.69575 PSI, μ = 0.017569 cP.
Equation 3
For carbon dioxide, the equation is:
μ = 0.0137339 + 4.41133 x 10-5 t + 1.12987 x 10-7 t2 + 4.33063 x 10-8 p + 2.67625 x 10-9 p2
Using Equation 3, t = 20oC and p = 14.69595 PSI, μ = 0.014663 cP.
Equation 4
For helium, the equation is:
μ = 0.0185975 + 5.30773 x 10-5 t - 1.04983 x 10-7 t2 - 4.82504 x 10-8 p
Using Equation 4, t = 20oC and p = 14.69595 PSI, μ = 0.019616 cP.
Equation 5
For argon, the equation is:
μ = 0.0208762 + 7.02190 x 10-5 t - 3.30712 x 10-8 t2 + 1.19960 x 10-6 p + 7.24294 x 10-10 p2
Using Equation 5, t = 20oC and p = 14.69595 PSI, μ = 0.022285 cP.
Uncertainties
The estimate of residual standard deviation, RSD, is the estimate of
the standard deviation of (μcalc. - μmeas.), where μcalc. is the
calculated value of μ using the equations above and μmeas. is the
corresponding value measured in references 2-12. The estimate of the
relative residual standard deviation, RRSD, is the ratio of the RSD to
the mean μmeas. The RSD is used in this article as an estimate of
precision.
Values of RRSD are compared with estimates of precision in the
references. The precision estimated in the literature was 0.05 percent.
For air, the RSD was 0.000010 cP and the RRSD was 0.05 percent. For
nitrogen, the RSD was 0.000005 cP and the RRSD was 0.03 percent. For
carbon dioxide, the RSD was 0.000004 cP and the RRSD was 0.02 percent.
For helium, the RSD was 0.000004 cP and the RRSD was 0.02 percent. For
argon, the RSD was 0.000007 cP and the RRSD was 0.03 percent. For
oxygen, the RSD was 0.000006 cP and the RRSD was 0.03 percent.
The following conversion factors can be used to convert from atm to
other units of pressure:
1 atm = 101,325 Pa
1 atm = 0.101325 MPa
1 atm = 14.69595 PSI
1 atm = 760 millimeters of mercury
Conclusions
Equations, fitted to experimental data, for the calculation of μ for
dry air, nitrogen, carbon dioxide, helium, argon and oxygen have been
developed. The estimates of residual standard deviation for the fits
are in agreement with the estimates of precision for the experimental
measurements |
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