What is the measure of average kinetic energy of the particles in a sample of matter?

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Temperature!

The particles of matter (i.e. molecules) at non-extreme temperatures can be considered to have random velocity. Then the average translational kinetic energy (KE) for these molecules can be deduced by assuming that prob(KE=x) follows a Boltzmann distribution. The well known equation for kinetic energy follows:

average KE = [0.5 * m v^2] = 3kT/2,

where m = mass, v = velocity, k = Boltzmann's constant, and T = TEMPERATURE!

In other words, average KE is directly proportional to T.

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