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As per my understanding we know that molecules of an ideal gas are identical in all aspects (size, shape, mass). Since collisions are elastic in nature, they don't lose their kinetic energy. That means that kinetic energy of each molecule doesn't change over time. Then how do the molecules move with different velocity regardless of possessing same mass and kinetic energy ?

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  • $\begingroup$ Related: physics.stackexchange.com/questions/417821 $\endgroup$
    – user191954
    Nov 18, 2018 at 12:04
  • $\begingroup$ Consider an elastic collision between a moving and a non-moving object. Afterwards, both will be moving (assuming same mass and off-center impact), so kinetic energy of each particle can change. $\endgroup$
    – Jasper
    Nov 18, 2018 at 12:25
  • $\begingroup$ @jasper, the OP's question assumes all particles have initially the same energy, so there won't be any stationary particles. $\endgroup$
    – S. McGrew
    Nov 18, 2018 at 12:33
  • $\begingroup$ @S.McGrew just go into the reference frame that is co-moving with one of them prior to the interaction $\endgroup$
    – Dave
    Nov 18, 2018 at 13:25
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    $\begingroup$ @Pieter not for elastic collisions, otherwise energy would not be conserved. $\endgroup$
    – Jasper
    Nov 18, 2018 at 16:11

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Here is the misunderstanding:

Since collisions are elastic in nature, they don't lose their kinetic energy

Only in the center of mass of two colliding particles the collisions have equal and opposite energy , not in the laboratory frame of the containing box. When one puts all the "identical molecules of an ideal gas" means the "molecules" not the energy momentum vector of each molecule in the laboratory frame of the box.When introduced in the box they will have an average kinetic energy according to the temperature, but there will be a distribution of possible energies and momenta. The elastic center of mass collisions of individual pairs will transform back to the lab with different energies due to the angles of scattering.

It gets worse, because of the spill over electric fields of molecules , the collisions quantum mechanically will allow for radiation, black body radiation, which will eventually lower the temperature to an equilibrium with the outside the box temperature.

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That's a good question. Elastic collisions between isolated particles will indeed conserve energy and momentum. But, consider this: Suppose the particles' momenta before the collision are uncertain: they are only known within some range. Think about it a while and you'll realize that the uncertainty grows with each collision. The Boltzmann distribution is the situation where those changes in uncertainty reach equilibrium for a large number of particles with a given total energy.

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  • $\begingroup$ The uncertainty principle is interesting but could we say that sometimes the collisions are inelastic, i.e. electrons change energy levels in the molecule and that might be the reason for spread of velocities? ( Or maybe that is what the uncertainty is?) $\endgroup$ Nov 18, 2018 at 12:49

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