# Statistical Thermodynamics and Rate Theories/Viscosity

## What is Viscosity?

The viscosity of a fluid (gas or liquid) is a measurement of the resistance said fluid has to its deformation by stress. For a liquid, this is a relatively easy concept to understand. Water, for example, has a lower viscosity than maple syrup. The concept is slightly less tangible when considering a gas. The system we will analyze in order to better understand the viscosity of a gas is two parallel plates where the gas is contained between them.

In order to establish this relationship, we must consider the bottom plate stationary. That is,

${\displaystyle u_{bottom}=0}$

The top plate is considered to be moving with constant velocity. That is,

${\displaystyle u_{top}=U}$

where ${\displaystyle U}$  is a constant velocity.

When a gas is filling the gap between the two plates, the gas molecules exert a force of drag on the motion of the upper plate. This Shear Stress is the previously mentioned force of drag per unit area between the two parallel planes. This relationship is shown by the following formula:

${\displaystyle \tau ={\frac {F(drag)}{A}}}$

where ${\displaystyle \tau }$  is the shear stress of the gas on the motion of the upper plate. The shear stress can also be represented by the relationship:

${\displaystyle \tau =-\eta {\frac {U}{a}}}$

where a is the distance between the plates and ${\displaystyle \eta }$  is the viscosity coefficient of the specific molecule between the plates.

## Viscosity Coefficient

As previously mentioned, the viscosity coefficient is a specific constant for the gas being examined. The Kinetic Theory of Gases must be considered in order to derive the equation for the viscosity coefficient. It is assumed from this theory that:

• The particles are hard spheres which implies that elastic collisions occur when particles are within the radius of another molecule.
• The distributions of these velocities follow the Maxwell Distribution of molecular speeds.

From these requirements determined by the Kinetic Theory of Gases, the simple version of the viscosity coefficient can be derived.

${\displaystyle \eta _{simple}={\frac {1}{3}}\left({\frac {2}{\pi }}\right)^{3/2}{\frac {(mk_{B}T)^{1/2}}{\sigma ^{2}}}}$  [1]

This value of ${\displaystyle \eta _{simple}}$  is derived by a simple method for the derivation of transport properties of hard spheres. A more rigorous analysis shows that the true values for the viscosity coefficients of gasses follows this formula more closely:

${\displaystyle \eta _{collision}={\frac {5}{16\pi ^{1/2}}}{\frac {(mk_{B}T)^{1/2}}{\sigma ^{2}}}}$  [2]

The only difference between these two determinations of the viscosity coefficient is the numerical factor. The ratio of these two equations is determined here:

${\displaystyle {\frac {\eta _{simple}}{\eta _{collision}}}={\frac {{\frac {1}{3}}\left({\frac {2}{\pi }}\right)^{3/2}}{\frac {5}{16\pi ^{1/2}}}}=0.96}$  [3]

This ratio shows that the simple derivation is essentially correct and is close enough where the viscosity coefficient of gasses is considered.

When looking at the formula for the viscosity coefficient, it is clear that the viscosity if a gas is not dependent on very many factors. The viscosity coefficient only depends on the mass of the molecule,${\displaystyle m}$ , the collision cross section, ${\displaystyle \sigma }$ , and the Temperature of the gas. Since the viscosity coefficient of a gas is proportional to the viscosity of the gas, the impacts of these values are outlined below:

• The viscosity of the gas increases as the mass of the molecule increases.
• The viscosity of the gas decreases as the collisional cross section of the molecule increases.
• The viscosity of the gas increases as the temperature of the molecule increases.

These general trends are the most important take away of the viscosity coefficient.

## Viscosity Comparisons

We will now look at some examples of the viscosity coefficients of some molecules and determine how the trends outlined above apply. The first example we will look at is the viscosity coefficients of N2 and CO2 at 273.15 K. The value of ${\displaystyle \eta }$  for these molecules respectively are 16.6 and 13.7 respectively. Even though the mass of COTemplate:Sub is much higher than that of N2, the viscosity coefficient of CO2 is slightly lower than that of NTemplate:Sub. This is due to the fact that since carbon dioxide is much larger in its collision cross section, this outweighs the fact that it is heavier than nitrogen. This is most because the collision cross section is squared while the mass of the molecule is square-rooted. This concludes that the viscosity coefficient of the molecule depends more on its collision cross section than its mass.

Another example of the phenomenon outlined above is the viscosity coefficients of helium and argon. The viscosity coefficients of these molecules are 29.7 and 21.0 ${\displaystyle \times 10^{-5}}$  Pa s. Helium is much lighter than argon but it has a much larger viscosity coefficient due to its much smaller collision cross section.

One example which shows a greater dependence on the mass than the collision cross section is shown when examining the viscosity coefficients of argon and krypton. The viscosity coefficients of these molecules at 273.15 K are 21.0 and 23.3 ${\displaystyle \times 10^{-5}}$  Pa s, respectively. This is because the collision cross sections of the two atoms are relatively close so the masses begin to have a larger impact. The mass of krypton is almost double of that of argon so this drastic difference in mass combined with the relatively close collision cross sections makes it so that the viscosity of krypton is larger than the viscosity of argon.

## References

1. McQuarrie, D. A., Statistical Mechanics, University Science Books, 2000, QC 174.8 M3
2. McQuarrie, D. A., Statistical Mechanics, University Science Books, 2000, QC 174.8 M3
3. McQuarrie, D. A., Statistical Mechanics, University Science Books, 2000, QC 174.8 M3