Maxwell Speed Distribution

Previously, we described the Boltzmann factor. Multiplying a normalization constant to give an actual probability, we obtain the Boltzmann distribution:

.

Now, suppose that we have a system of tiny non-interacting particles (an ideal gas), with the energy being purely kinetic. Thus, the probability that a particle of mass *m* has a speed *v* is proportional to .

However, in 3-dimensional velocity space, the velocity vectors that give the same speed form a sphere, with radius *v*; the higher the speed *v*, the larger the number of possible velocity vectors there are. Thus, the distribution of speeds is proportional to , the surface area of the sphere in velocity space. Combining, we have

. Normalizing, we have

,

(where we’ve used the fact that )

so

.

This is the Maxwell speed distribution. Now, we can find three important speeds with this:

I. Most probable speed:

This is *v*_{p}>0 where . The derivative is

,

and this is zero for *v*_{p}>0 when

.

II. Mean speed

.

III. Root-mean-square speed:

.

The last of these gives us the average kinetic energy of ideal gas molecules:

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Tags: Boltzmann Factor, Friday Physics, Maxwell Speed Distribution, physics, Statistical Mechanics, Thermodynamics

This entry was posted on April 17, 2009 at 12:01 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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