Gas Kinetic Theory

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Lecture date: Wednesday, 2014.09.10 (lecture recording)

Play With a Maxwell-Boltzmann Distribution[edit]

Brandon whipped up a Maxwell-Boltzmann distribution graphing script in Sage Math. Sage is an open-source computer algebra system that can be embedded right inside a web page; check out Brandon's other engineering math demos in Sage to get an idea of what it can do.

The Utility of Molecular Statistical Distributions[edit]

Why are we so concerned with the statistical distributions of molecular properties like speed and kinetic energy? With knowledge of these distributions, it becomes possible to estimate all sorts of macroscopic properties: rate coefficients, diffusivities, &etc.

Consider that many properties are in the "Arrhenius form" and demonstrate an exponential dependence on temperature. This occurs because the relative speed of colliding molecules decides how they react and diffuse. Reactions often involve only the most energetic molecules in a system; while small changes in the scaling factor (e.g. temperature and molecular mass) cause only small changes in the mean molecular velocity, they cause very large changes in the occupancy of the highest energy states! This can be easily seen when the function is graphed.

in most cases, flow is subsonic

therefore, most of the time, distribution function is isotropic - no preferred direction - velocity distribution is symmetric

speed will also have a distribution! (c from vx,vy,vz? basically like a coordinate transformation)

When you change the average just a little bit, you change the extremes substantially!

(think about, for instance, reaction rate: dependent on the activity of the most energetic particles)

Scaling relationships are important to engineers!

"How does this scale with temperature?"

Collision Theory[edit]

Interatomic potential is not much larger than atomic diameter

Gas molecules 'feel' eachother primarily when colliding!

(an ideal gas, by definition, has no interaction and that's why ideal gas theory treats all gases the same)

You can estimate all kinds of properties from the "mean free path"

Collision cross section is defined for each type of collision!

For non-reactive, elastic, momentum-transfer reactions, a hard-sphere model yields pretty decent numbers. This is nice, because these interactions decide the diffusion coefficient.

For reactive collisions, the hard-sphere assumption fails and a different cross section must be used.

Equipartition Theorem[edit]

(1/2)mv² = (3/2)kT v = sqrt(3*kT/m)

The Mean Free Path[edit]

u = 1/(n*pi*d²)

n = molecular density d = molecular diameter


D = (1/3)uv

u = mean free path v = average speed

"The coefficient of diffusion is the product of mean velocity and mean free path, with a prefactor that can be temperature dependent. The mean velocity depends only on temperature; the mean free path is inversely proportional to the density of the gas. Thus, a thinner atmosphere has a higher diffusivity."

<more advanced models follow>


When Maxwell made these calculations in the early 1900's, it was not yet clear whether matter was composed of discrete particles.