Why gas molecules move with different speed at a given tempreture? 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 ?
 A: 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.
A: 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.
