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Root-mean-square speed of molecules in an ideal gas (14-15)

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Boltzmann constant

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Review

A measure of how fast gas molecules move on average is the root-mean-square speed or rms speed. To see how this is defined, first imagine finding the average value of v2, the square of the speed v, for all the molecules in a gas that have a given mass m. This is the quantity that we’ve called (v2)average. Another word for average is mean, which is why (v2)average is called the mean-square of the speed. The root-mean-square speed is the square root of the mean-square:

vrms=(v2)average

Note that (v2)average has units of m/s. From Equation 14-13, (1/2)m(v2)average=(3/2)kT, so (v2)average=3kT/m. If we substitute this into Equation 14-14 we get: