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# Illustration 20.1: Maxwell-Boltzmann Distribution

In this animation N = nR (i.e., kB = 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  =  (8kBT/πm)1/2
• Most probable speed:      (2RT/M)1/2  =  (2kBT/m)1/2
• Root-mean-square (rms) speed:     (3RT/M)1/2  =  (3kBT/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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