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# 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.

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