There will be a huge range of particle velocities at any given temperature. The mean energy of the particles is actually a measure of the temperature. I suggest you google Kinetic theory of gases and then pick a page that suits your level. The "acceleration" of particles is not usually considered as the thermal energy is transferred 'to' a volume of gas by the molecules colliding with the walls and getting their extra energy (or losing it) by interaction with the surface molecules. i.e. we normally think in terms of Momentum Transfer in an unspecified duration of each collision.
But you are right, I suppose, to try to associate increasing molecular speeds with an 'acceleration'. It would be hard to make a model based on this, I think.