Maxwell Speed Distribution

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In the classical picture of an ideal gas, molecules bounce around at a variety of different velocities, never interacting with each other. Though this qualitative picture is obviously flawed (since molecules always do interact), it is a useful model for situations where the particle density is very low; in a more quantitative sense, this means that the particles themselves are very small when compared to the volume between them.

Accordingly, we will want to know exactly how many of these molecules are moving around at a given speed. The Maxwell Speed Distribution (MSD) is a probability distribution describing the "spread" of these molecular speeds; it is derived, and therefore only valid, assuming that we're dealing with an ideal gas. Again, no gas is truly ideal, but our own atmosphere at STP behaves enough like the ideal situation that the MSD can be used.

Note that speed is a scalar quantity, describing how fast the particles are moving, regardless of direction; velocity also describes the direction that the particles are moving.

It is elementary using statistical mechanics to find that the MSD must be proportional to the probability that a particle is moving at a given speed. Another important element is the fact that space is three dimensional, which implies that for any given speed, there are many possible velocity vectors.

The probability of a molecule having a given speed can be found by using Boltzmann factors; considering the energy to be dependent only on the kinetic energy, we find that:

$(\mbox{probability of a molecule having speed } v) \propto e^\frac{-mv^2}{2kT}$

Here, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.

In 3-dimensional velocity space, the velocity vectors corresponding to a given speed v live on the surface of a sphere with radius v. The larger v is, the bigger the sphere, and the more possible velocity vectors there are. So the number of possible velocity vectors for a given speed goes like the surface area of a sphere of radius v.

$(\mbox{number of vectors corresponding to speed } v) \propto 4 \pi v^2$

Multiplying these two functions together gives us the distribution, and normalizing this gives us the MSD in its entirety.

$D(v)dv = \left ( \frac {m}{2 \pi k T} \right) ^{3/2} 4 \pi v^2 e^\frac{-mv^2}{2kT} dv$

(Again, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.)

As this formula is a normalized probability distribution, it gives the probability of a molecule having a speed between $v$ and $v + d v$ . If you want to find the probability of a particle to be between two different velocities $v 0$ and $v 1$ , simply integrate this function with those numbers as the bounds.

[edit] Averages

There are three general methods for finding the "average" value of the speed of the Maxwell Speed Distribution.

Firstly, by differentiating the MSD and finding its maximum, we can determine the most probable speed. Calling this $v m a x$ , we find that:

$v_{max} = \left ( \frac{2 k T}{m} \right )^{1/2}$

Second, we can find the root mean square of the speed by finding the average value of $v 2$ . (Alternatively, and much simpler, we can solve it by using the equipartition theorem.) Calling this $v r m s$ :

$v_{rms} = \left ( \frac{3 k T}{m} \right )^{1/2}$

Third and finally, we can find the average value of v from the MSD. Calling this $\bar{v}$ :

$\bar{v} = \left ( \frac{8 k T}{\pi m} \right )^{1/2}$

Notice that: $v_{max} < \bar{v} < v_{rms}$

These three different ways of understanding the average velocity, though not equivalent numerically, do not each describe different physics. They are each a different way of "book-keeping," if you will. It is simply very important to be consistent in which quantity being used, and to be clear which quantity is being used.

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