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Is there a way to prove that $\frac{1}{2} mv^2 $$= \frac{3}{2}RT$ The first one is kinetic energy and the second is the average kinetic energy. Can this be proved by component of velocity.

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https://physics.stackexchange.com/questions/45838/how-to-deduce-e-3-2kt

http://hyperphysics.phy-astr.gsu.edu/HBASE/kinetic/molke.html#c1

http://en.wikipedia.org/wiki/Kinetic_theory#Properties

These contain proof at various level . Choose according to your mathematical level

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    $\begingroup$ Would you be willing to summarize the proof in case the links break? Additionally, none of these links do a good job of explaining the steps. $\endgroup$
    – Ben Norris
    Oct 11, 2014 at 19:44

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