## Illustration 20.1: Maxwell-Boltzmann Distribution

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**In this animation N = nR** (i.e., k_{B} = 1). This, then, gives the ideal gas law as PV = NT. The average values shown, < >, are calculated over intervals of one time unit. Restart.

The particles that make up a gas do not all have the same speed. The temperature of the gas is related to the average speed of the particles, but there is a distribution of particle speeds called the Maxwell-Boltzmann distribution. The smooth black curve on the graph is the Maxwell-Boltzmann distribution for a given temperature. What happens to the distribution as you increase the temperature? The distribution broadens and moves to the right (higher average speed). At a specific temperature, there is a set distribution of speeds. Thus, when we talk about a characteristic speed of a gas particle at a particular temperature we use one of the following (where M is the molar mass, m is the atomic mass):

- Average speed: (8RT/πM)
^{1/2}= (8k_{B}T/πm)^{1/2} - Most probable speed: (2RT/M)
^{1/2}= (2k_{B}T/m)^{1/2} - Root-mean-square (rms) speed: (3RT/M)
^{1/2}= (3k_{B}T/m)^{1/2}

There is not simply one way to describe the speed because it is a speed distribution. This means that as long as you are clear about which one you are using, you can characterize a gas by any of them. The different characteristic speeds are marked on the graph.

Illustration authored by Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.

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