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Encyclopedia > Maxwell Speed Distribution

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. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ...

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. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ... The TLA STP has several meanings: Standard Temperature and Pressure is a chemistry term for a specific controlled experimental or reaction condition. ...

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. Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t. ... The velocity of an object is simply its speed in a particular direction. ...

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. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ...

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:

Here, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

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.

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

(Again, m is the mass of the molecule, k is Boltzmann's constant, and T is the temperature.) The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

As this formula is a normalized probability distribution, it gives the probability of a molecule having a speed between v and v + dv. If you want to find the probability of a particle to be between two different velocities v0 and v1, simply integrate this function with those numbers as the bounds.

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 vmax, we find that:

Second, we can find the root mean square of the speed by finding the average value of v2. (Alternatively, and much simply, we can solve it by using the equipartition theorem.) Calling this vrms: In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ... The Equipartition Theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles will distribute itself evenly among each of the degrees of freedom allowed to the particles of the system. ...

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:

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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## References

Schroeder, Daniel V. Thermal Physics. San Francisco: Addison Wesley Longman, 2000. ISBN 0-201-38027-7

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