## Problem 15.6: Speed of particles in ideal gas

Please wait for the animation to completely load.

**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 so that the average rate change in momentum is equal to the pressure times the area,

*A*(where

*A*= 1). Restart. This animation shows the distribution of speeds in an ideal gas based on the Maxwell-Boltzmann distribution as shown by the smooth black curve on the graph for a given temperature:

*n*(v) *d*v/*N* = (2/π)^{1/2} (*m*/*k*_{B}*T*)^{3/2} v^{2}
exp(−*m*v^{2}/2*k*_{B}*T*) .

What happens to the distribution as you increase the energy (temperature)? Since there is a speed distribution, when we talk about a characteristic speed of a gas particle at a particular temperature, we use one of three characteristic speeds:

- Average speed (<v> = ∫ v
*n*(v)*d*v). - Most probable speed (find maximum of
*n*(v)). - Root-mean-square (rms) speed (<v
^{2}>^{1/2 }= [ ∫v^{2}*n*(v)*d*v ]^{1/2}).

Find an expression for each (in terms of m and *k*_{B}*T*). Identify which peak is which characteristic speed.

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