To preface my problem, you should know what I'm deriving. When I try to find the most probable speed of a classical, nonrelativistic particle as described by the Maxwell speed distribution I find that it is v(adsbygoogle = window.adsbygoogle || []).push({}); _{mp}=(2*k*T/m)^(1/2). The kinetic energy associated with this particle would then be E_{K}=(m*v_{mp}^2)/2=k*T. Next, when I use the Boltzman distribution to try and find the most probable kinetic energy of a particle I find that E_{K}=(k*t)/2.

Both of these values I've verified as correct with my book and even in another topic in the forum (https://www.physicsforums.com/showthread.php?t=120947). Multiple sources I've found, including my professor, make it a point to emphasize that the most probable kinetic energy is not the same as the kinetic energy of the most probable speed. However, one would intuitively expect these values to be the same since the speed and kinetic energies of a particle are related. Obviously the math dictates that they are, but from a more physical perspective why are these values different?

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# Most probable values of the Maxwell Boltzman distribution

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