Spatial Reasoning behind the Maxwell-Boltzman distribution: A Question

[corey2157]

I have been trying to figure out if, and if so, just how is the shape of the molecules in a gas is taken into account by the Maxwell-Boltzmann distribution. I know the assumption is perfect spheres, but still, a square hit and a glancing blow on a pool table yield different results. My intuition is also telling me that repeated off center collisions would be required to get those situations where atoms are traveling many-fold faster than the mean. It seems like SIN would have to be in there somewhere.

Any insights?

 

[Jano L.]

The Maxwell-Boltzmann distribution describes velocities of molecules in the situation of thermal equilibrium. It depends on molar mass and temperature of the gas, but not on the shape of the molecules.

 

[corey2157]

Well that is true, the theory has an explicit assumption that the molecules are spherical. And that is a reasonable assumption since molecules of a gas are more or less spherical. But is the spherical shape factored into the equation; that is my question.

If somehow 1 meter long carbon nanotubes were the molecules of the gas, wouldn't you need a different equation to get a reasonable model? I would think that equation would need to take into account the physical interactions between cylinders of a certain ratio.

 

[Jorriss]

Well that is true, the theory has an explicit assumption that the molecules are spherical. And that is a reasonable assumption since molecules of a gas are more or less spherical. But is the spherical shape factored into the equation; that is my question.

If somehow 1 meter long carbon nanotubes were the molecules of the gas, wouldn't you need a different equation to get a reasonable model? I would think that equation would need to take into account the physical interactions between cylinders of a certain ratio.

The Maxwell Boltzmann distribution describes an ideal gas - so no intermolecular or intramolecular potentials are taken into consideration. It is a special case of the more general Boltzmann distribution.

 

[corey2157]

I wasn't thinking of the intermolecular forces. I am thinking about the collisions between molecules, and wonder just what about those collisions give the distribution its positive skew.

The following link from MIT OCW may explain my question - I just don't understand the math well enough to know exactly what it is saying.

https://docs.google.com/viewer?a=v&...GjPLD4&sig=AHIEtbTbe-VkmUMiEIuSEXnRjnKqZk7FzA

 

[Jano L.]

... the theory has an explicit assumption that the molecules are spherical.

Not necessarily. It is important to distinguish the difference between Maxwell's and Boltzmann calculation of the distribution. They both lead to the same distribution in equilibrium, but via different, non-equivalent route.

Maxwell derived his equilibrium distribution from general statistical considerations, with no assumption as to the shape of the molecules. His result is therefore independent of the shape of the molecules. It is also independent of the intermolecular interactions, as long as they are described by weak potential energy function. Even long stick-like molecules are subject to Maxwell's derivation and the general conclusion from this theory is that they have the same velocity distribution as atom gases of the same mass and temperature.

Another way to understand Maxwell's result is to apply the general Boltzmann probability $e^{-\frac{\frac{p^2}{2m}}{k_B T}}$ to individual molecule. The only parameters this probability depends on are the temperature $T$ of the gas and the mass $m$ of the molecule.

Boltzmann's calculation is more complicated and I do not know it in detail, but if it gave different equilibrium distributions for other shapes than spheres (which is hard to calculate), I think this would be seen rather as imperfection of the Boltzmann equation.

 

[corey2157]

Thanks Jano. You've given me more food for thought.

 

[Jano L.]

No problem.